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This document is a thesis submitted by Nicholas J. Sharp to the University of Manchester for a Doctor of Philosophy degree in engineering and physical sciences in 2006. The thesis proposes new option-theoretic models for valuing mortgages and mortgage-backed securities. It develops partial differential equation models for valuing fixed-rate and adjustable-rate mortgages by accounting for factors like house prices, interest rates, prepayment, and default. Numerical techniques are presented for solving the PDE models. The thesis also introduces new prepayment models using concepts from occupation-time derivatives to improve mortgage valuation.
![CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 45
with the terminal (expiration) condition P(r, t = T) = 1.
The bond price takes the form
P(r, t) = A(t)e−B(t)r
, (2.7)
where
A(t) =
2γe(γ+κ)(T−t)/2
(γ + κ)(eγ(T−t) − 1) + 2γ
2κθ/σ2
r
, (2.8)
B(t) =
2(eγ(T−t)
− 1)
(γ + κ)(eγ(T−t) − 1) + 2γ
, (2.9)
γ = κ2 + 2σ2
r . (2.10)
The yield-to-maturity, R(r, t) is defined by e−(T−t)R(r,t)
= P(r, t). Therefore,
R(r, t) = [rB(t) − logA(t)]/(T − t). (2.11)
Equation (2.11) gives the pure discount bond yield. The mortgage-equivalent yield
index(r) is given by equating the yield on the the principal amount at the con-
tinuously compounded rate of R(r, t), to the yield on the principal amount at the
mortgage equivalent monthly compounded rate index(r), i.e.
P(r, t)eR(r,t)
= P(r, t) 1 +
index(r)
12
12
, (2.12)
and so
index(r) = 12[eR(r,t)/12
− 1]. (2.13)
Equation (2.13) gives the mortgage equivalent yield on a 1-year, default-free, pure
discount bond. This is used to calculate the index, given the current interest rate,
which is added to the margin when calculating the new contract rate at an adjustment
date for the ARM. Figure 2.1 shows the profile of the index as the interest rate
changes, for various different values of the long-term mean of the short-term interest
rate θ. Notice that initially when the interest rate is less than the long-term mean,
the index slightly leads the underlying interest rate, i.e. index(r) > r for r < θ. Once
the interest rate is above the long-term mean, the index lags behind the underlying
interest rate, i.e. index(r) < r for r > θ.](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-45-320.jpg)
![CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 51
The spatial derivatives for interest rate r are approximated by
∂F(H, r, τ + 1
2
∆τ)
∂r
≈
(Fk+1
i,j+1 − Fk+1
i,j−1 + Fk
i,j+1 − Fk
i,j−1)
4∆r
, (2.24)
∂2
F(H, r, τ + 1
2
∆τ)
∂r2
≈
(Fk+1
i,j+1 − 2Fk+1
i,j + Fk+1
i,j−1 + Fk
i,j+1 − 2Fk
i,j + Fk
i,j−1)
2(∆r)2
. (2.25)
The cross-spatial derivative is approximated by
∂2
F(H, r, τ + 1
2
∆τ)
∂H∂r
≈
1
8∆H∆r
(Fk+1
i+1,j+1 − Fk+1
i−1,j+1 − Fk+1
i+1,j−1 + Fk+1
i−1,j−1
+Fk
i+1,j+1 − Fk
i−1,j+1 − Fk
i+1,j−1 + Fk
i−1,j−1). (2.26)
Finally, the asset F(H, r, τ) is approximated by
F H, r, τ +
1
2
∆τ ≈
Fk+1
i,j + Fk,l
i,j
2
. (2.27)
Overall the error in the approximate solution F k
i,j is of second-order accuracy in ∆H,
∆r and ∆τ.
2.5.2 Discrete representation
Upon substituting the derivative approximations from section 2.5.1 into the governing
PDE (1.17) and rearranging, the problem of solving the PDE reduces to solving the
following set of simultaneous linear equations for F k+1
i,j
ai,jFk+1
i,j−1 + bi,jFk+1
i,j + ci,jFk+1
i,j+1 + di,jFk+1
i−1,j + ei,jFk+1
i+1,j
+fi,j[Fk+1
i+1,j+1 − Fk+1
i,j+1 − Fk+1
i+1,j−1 + Fk+1
i−1,j−1] =
−ai,jFk
i,j−1 −
2
∆τ
+ bi,j Fk
i,j − ci,jFk
i,j+1 − di,jFk
i−1,j
−ei,jFk
i+1,j − fi,j[Fk
i+1,j+1 − Fk
i,j+1 − Fk
i+1,j−1 + Fk
i−1,j−1], (2.28)
where
ai,j =
rjσ2
r
4(∆r)2
−
κ(θ − rj)
4∆r
, (2.29)
bi,j = −
H2
j σ2
H
2(∆H)2
−
rjσ2
r
2(∆r)2
−
1
∆τ
−
rj
2
, (2.30)](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-51-320.jpg)
![CHAPTER 4. A NEW PREPAYMENT MODEL 120
(analogous to the total debt for a UK FRM) is given by
FV (τm) = [1 + c(Tm − τm)]OB(i). (4.23)
The details of the numerical solution of this free boundary problem are given in
section 4.4.1.
Note the value of the mortgage, and the other mortgage components are now
functions of an additional time variable, the time until prepayment occurs ¯τ, see
section 4.3.2 for more details.
An illustration of how the consecutive occupation-time derivative (the Parisian
call option) triggers prepayment is shown in figure 4.7. The figure shows that the
value of the mortgage V must be beyond the face value of debt FV (t1) (the amount
paid if a borrower choses to prepay at time t1) consistently until the decision time ¯T
elapses for prepayment to occur. If V is still above this level at time t2 = t1 + ¯T the
borrower prepays the mortgage at the greater face value FV (t2). The value of the
mortgage to the lender, when prepayment occurs at t2, is increased by the borrower
waiting to prepay and the increase is given by d = FV (t2) − FV (t1).
4.3.2 Modified PDE
It is essential that the time the mortgage value spends beyond the prepayment barrier
(4.22) can be recorded, which then necessitates the introduction of a new time variable
- the time until the decision to prepay is made ¯τ, this is analogous to the time to knock
out for the Parisian option on a stock. Although it was mentioned previously that
all the mortgage components are now functions of this new time variable, this is not
strictly true. All the components which are affected by the borrower terminating the
contract prior to maturity now include this variable, but the value of the remaining
payments A(r, τ) does not depend on the borrower’s actions and so is not a function
of ¯τ.
When the value of the mortgage to the lender V (H, r, τm, ¯τ) is less than the
prepayment barrier (4.22), the usual mortgage valuation PDE (1.17) applies. When
the mortgage value increases, such that it is greater than or equal to this barrier, the](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-120-320.jpg)
![CHAPTER 4. A NEW PREPAYMENT MODEL 123
At maturity
At maturity the option to prepay is not relevant, therefore the payment-date condi-
tions are not affected by the new prepayment model. The payment-date conditions
for the value of the remaining payments, the value of the mortgage, and the value of
default, are the same as for the standard UK FRM model, and are given in section
3.3.1. For the US FRM the insurance is defined differently (see Kau et al., 1995),
since the issuer covers the lender’s shortfall on the lost future cash flows up to some
fraction φ of the face value of debt (at maturity this is simply the value of the final
monthly payment MP). If default does not occur at the payment date, then the
value of the insurance becomes its value in the future. The payment-date condition
at maturity is given by
I(H, r, τn = 0) = min(MP − H, φMP) (4.25)
if default occurs, and by
I(H, r, τn = 0) = 0 (4.26)
if the final monthly payment is made (default does not occur), where n is the length
of the mortgage in months.
Earlier payment dates
At earlier payment dates, there is the possibility that prepayment will occur, and
because prepayment is now determined using a different model, the payment-date
conditions change from when prepayment was modelled using the optimal call condi-
tion in chapter 3. The value of the remaining payments A(r, τ) is not affected by the
either of the borrower’s options, so the payment-date condition for this component
at earlier payment dates does not change; see equation (3.8) for the condition.
The payment-date condition for the value of the mortgage to the lender at the
end of month 1, 2, . . . , n − 2 and n − 1 is
V (H, r, τm = 0, ¯τ) =
min[V (H, r, τm+1 = Tm+1) + MP, H] if ¯τ > 0,
FV (τm = 0) if ¯τ = 0,
(4.27)](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-123-320.jpg)
![CHAPTER 4. A NEW PREPAYMENT MODEL 124
where 1 ≤ m ≤ n − 1. When the time until prepayment is zero (¯τ = 0), obviously
prepayment occurs and the mortgage is worth the face value at this time; otherwise,
the condition takes the same form as previously.
For default the payment-date conditions prior to maturity are
D(H, r, τm = 0, ¯τ) =
D(H, r, τm+1 = Tm+1) if V = H and ¯τ > 0,
A(r, τm = 0) − H if V = H and ¯τ > 0,
0 if ¯τ = 0,
(4.28)
where V = H refers to default not occurring and V = H is when default occurs,
which happens when the value of the house is so low that the borrower decides not
to make the scheduled monthly payment. The option to default is worthless when
prepayment occurs, which occurs when the value of the mortgage is greater than the
face value for the decision time ¯T, which corresponds to ¯τ = 0.
The payment-date conditions for the prepayment option prior to maturity are
C(H, r, τm = 0, ¯τ) =
C(H, r, τm+1 = Tm+1) if V = H and ¯τ > 0,
0 if V = H and ¯τ > 0,
A(r, τm = 0) − FV (τm = 0) if ¯τ = 0.
(4.29)
When prepayment occurs (¯τ = 0) the option value is calculated by rearranging the
relation V = A − D − C to determine C. The third equation in (4.29) is due to the
option to default being worthless, and as the value of the mortgage is the face value
when prepayment occurs.
Finally, the insurance component has the following conditions at the earlier pay-
ment dates
I(H, r, τm = 0, ¯τ) =
I(H, r, τm+1 = Tm+1) if V = H and ¯τ > 0,
min[FV (τm = 0) − H, φFV (τm = 0)] if V = H and ¯τ > 0,
0 if ¯τ = 0.
(4.30)
The occurrence of prepayment (¯τ = 0) renders the option to default to be worthless,
implying that the insurance value is also zero, as this only has any value if default
occurs.](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-124-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 151
1. The prepayment option held by the borrower is American, the solution of which
is readily solved using a backward valuation technique.
2. The mortgage components have known values at maturity (dependent on the
contract rate being known at this point in time) from which their unknown
values at origination can eventually be calculated.
As discussed in section 5.3, the difficulty in pricing a variable-rate contract, such as an
ARM, by using a backward valuation technique is that at the start of the procedure
vital information is unknown. The contract rate is determined at the beginning of
each year, which means its value is unknown at the initial valuation point, i.e. at
the maturity of the loan. The auxiliary-variable approach overcomes this difficulty
by effectively valuing the mortgage and its components for all possible realisations of
the adjustable contract rate. The actual values of the components can be determined
once the contract rate is specified. This scheme, as developed by Kau et al. (1985,
1990, 1993), does provide a solution, but at a cost. The procedure requires having to
value repeatedly the mortgage components for all possible contract rates, so that by
an adjustment date, when the adjusted contract rate is known, the appropriate values
of the components have been calculated and the valuation procedure can continue.
A much more efficient method to solve the problem is detailed in section 5.4, where
the auxiliary variable and its problems are overcome. The next sections detail how
the auxiliary variable is used to solve the adjustable contract-rate path-dependency
problem.
Valuing the remaining payments
At maturity of the loan, the final payment MP is made. The value of the remaining
payments at this time is
A−
[r, t(n − 1, 12); c(n − 1), OB(n − 1, 0)] = MP[n − 1; c(n − 1), OB(n − 1, 0)]. (5.6)
The path-dependency problem now becomes clear, as the current contract rate c(n−1)
is not considered until the beginning of the year is reached, and even at that point](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-151-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 152
neither this rate nor the outstanding balance OB(n − 1, 0) are known.1
Indeed, they
depend on yet earlier contract rates.
The problem with the unpaid balance OB can be eliminated by normalising the
current unpaid balance to unity. This can be done as the value of the remaining
payments is directly proportional to the outstanding balance OB. At the beginning
of the year, when the outstanding balance is determined, and the required scaling
is known, the value of A is merely changed in that same proportion. For ease of
presentation, OB is omitted whenever its value is taken to be unity.
The unknown contract rate c(n − 1) is treated by introducing it as an auxiliary
state variable, so that effectively all its possible values are considered. The remainder
of the valuation procedure for A and how the auxiliary variable is treated follows.
The valuation PDE (1.17) is solved backwards, initially using the payment-date
condition (5.6), until the beginning of the month, when another payment is due
(the numerical technique used to value all the mortgage components is covered in
section 5.3.3). At payment dates other than adjustment dates t(i, j), for j = 0 or
12, A+
[r, t(i, 0); c(i)] has been solved and the payment-date condition to begin the
succeeding month may be written as
A−
[r, t; c(i)] = A+
[r, t; c(i)] + MP[i, c(i)] for t = t(i, j), j = 0 or 12. (5.7)
The value of the remaining future payments changes by the value of the monthly
payment.
At the beginning of each year the contract rate is adjusted, at time t(i, 0) =
t(i − 1, 12), a new auxiliary variable c(i − 1) must be introduced and the subsequent
outstanding balance OB(i − 1, 0) must be set to unity. When c(i − 1) is introduced
the previous auxiliary variable c(i) can be dropped. The new auxiliary variable and
the current interest rate determines the old contract rate c(i) using the adjustment
formula (5.1). To complete the transition from one contract rate to the next, the
value of A−
[r, t(i, 0); c(i)] must be adjusted to correspond to the value at which the
outstanding balance OB(i, 0) must be reset. This later value is determined using the
1
In the payment-date conditions the variables that are determined by the auxiliary variable
appear after a semicolon.](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-152-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 153
assumed value of the new contract rate c(i − 1) and the normalised new outstand-
ing balance OB(i − 1, 0) in equation (5.4). Thus, the payment-date condition on
adjustment dates is
A−
[r, t; c(i − 1)] = A+
[r, t; c(i)]OB(i, 0) + MP[i − 1, c(i − 1)]
for t = t(i, 0) = t(i − 1, 12), (5.8)
where c(i) is determined by r and c(i−1) together, while both OB(i, 0) and MP(i−1)
are determined by c(i) and OB(i − 1, 0) = 1. Once the boundary conditions are
specified (see section 2.6.1), the valuation procedure for A is closed and the value of
the remaining future payments at origination A(r(0), t = 0), may be obtained.
Valuing other mortgage components
The borrower has two options, either to default on a monthly payment and hand the
house over to the lender, or to prepay the face value of the mortgage if it is financially
rational to do so. These options, and the insurance component, are dependent on
the house price H. House price has no direct effect on the borrower’s option to
prepay, since prepayment is assumed to occur when the value of the mortgage V is
equal to the face value of the loan FV (the borrower attempting to minimise V ).
House price does have a direct effect on the default option, if the borrower follows a
ruthless default strategy (see Foster and Van Order, 1984, 1985 for further details)
by exercising this option when the value of the mortgage is less than the value of
the house. Since exercise of one of these options renders the other worthless, these
options cannot be considered independently. As a result prepayment C, default D
and the insurance I all depend on both house price H and interest rate r.
The value of the mortgage component at maturity is
V −
[H, r, t(n − 1, 12); c(n − 1)] = min (MP[n − 1; c(n − 1)], H) , (5.9)
as the lender either receives the scheduled monthly payment or the house. The](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-153-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 154
payment-date condition for the other payment dates, excluding adjustment dates, is
V −
[H, r, t; c(i − 1)] = min V +
[H, r, t] + MP[i; c(i − 1)], H
for t = t(i, j), j = 0 or 12. (5.10)
Valuation throughout the year proceeds without complication until an adjustment
date occurs, at the beginning of a year. Here the outstanding balance for the previous
year must be adjusted. A difficulty arises as both default and prepayment are not
directly proportional to OB, since they both depend on the house price. According to
Kau et al. (1993), the value of default becomes twice as great when the loan becomes
twice the amount and the house becomes twice as valuable. As a result the value of
the mortgage at an adjustment date is
V −
[H, r, t; c(i−1)] = min V + H
OB(i, 0)
, r, t; c(i) OB(i, 0) + MP[i − 1; c(i − 1)], H
for t = t(i, 0) = t(i − 1, 12), (5.11)
where c(i) is determined by c(i − 1) and r, while both OB(i, 0) and MP(i − 1) are
determined by c(i−1) and OB(i−1, 0) = 1. Again, once the boundary conditions are
specified (see section 2.6.2 for the precise boundary conditions required), the valuation
procedure for V is closed and the value of the mortgage to the lender at origination
V [H(0), r(0), t = 0], may be obtained. As usual, the value of the mortgage involves
a free boundary due to the optimal stopping problem created by the American-type
prepayment option (see section 3.6.2 for a full exposition of the PSOR method used
to deal with the free boundary problem). As explained in section 2.6.3 the conditions
for default are fully specified by the payment-date conditions, as default is assumed
to occur only at the end of each month.
The general payment-date condition for the value of the default option, other than
at adjustment dates, is given by
D−
[H, r, t(i, j)] =
D+
[t(i, j)] if V −
[t(i, j)] = V +
[t(i, j)] + MP(i) (no default)
A−
[t(i, j)] − H if V −
[t(i, j)] = H (default)
for j = 0 or 12. (5.12)](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-154-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 155
The borrower either chooses to make the scheduled monthly payment, then the default
option is simply worth its value in the future, or the borrower exercises the right to
default and gives up the house H for the promised remaining payments A−
. The
technique for adjusting outstanding balances and carrying the auxiliary variable is
precisely the same as that discussed for the mortgage component (see earlier in this
section), and so is not repeated. Once the value of the remaining payments, the
mortgage component and the default option are determined, the value of prepayment
may be inferred using C = V − A − D.
The general payment-date condition for the insurance is
I−
[H, r, t(i, j)] =
I+
[t(i, j)] if V −
[t(i, j)] = V +
[t(i, j)] + MP(i) (no default)
max(0, min(FV −
[t(i, j)] − H, φFV −
[t(i, j)])) (default)
if V −
[t(i, j)] = H
for j = 0 or 12. (5.13)
If default occurs, the insurer covers the lender’s shortfall on the lost future cash flows
up to some fraction φ of the face value of debt. If default does not occur at the
payment date, then the value of the insurance becomes its value in the future.
Equilibrium condition
At the origination of the contract, the value of the mortgage to the lender should
equal the amount lent to the borrower; this idea is discussed in more detail in section
2.2. The avoidance of an arbitrage opportunity leads to the equilibrium condition,
analogous to equation (3.22) for FRMs; for an ARM the equilibrium condition is
V [H(0), r(0), t(0, 0) = 0; margin]+I[H(0), r(0), t(0, 0) = 0; margin] = 1−ξ, (5.14)
where ξ is the amount of points deducted from the loan (for the UK counterpart
this is the arrangement fee). Note that in this chapter the value of all assets will be
expressed to par (i.e. as a percentage of the loan amount), so that the value of the
loan is unity.2
This is why the borrower’s position is not written as (1−ξ) multiplied
2
This is in accordance with the literature on US ARM contracts.](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-155-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 158
Defining the range
No special attention was given in the work of Kau et al. (1993) to define the in-
cremental change in the auxiliary variable ∆c. This could cause a non-linearity
error (error caused when required information falls between nodes) to occur, as the
auxiliary-variable technique involves solving the valuation problem for a range of
contract rates (with the premise that at origination the values of all the mortgage
components can be calculated). If the discrete auxiliary contract-rate range (5.16)
does not include the initial contract rate then mortgage assets will not have any value
for the initial contract rate. This can be overcome by first taking the nearest integer
value of [c(0) + l]/c(0), then letting this equal the temporary variable temp, ∆c can
then be defined as
∆c =
c(0) ∗ [temp]
smax
, (5.17)
which ensures that c(0) coincides with a node in the auxiliary-variable space.
Interpolation
The asset value just after the payment date is required in all payment-date conditions.
On adjustment dates, once a year, the value of the asset at the new contract rate
is required in these payment-date conditions (see equation (5.11) for example). The
new contract rate c(i) is determined using the auxiliary variable c(i − 1). A problem
arises as the new contract rate may fall between two auxiliary-variable nodes, which
means the asset value, calculated using the new contract rate, may not exist. The
scaled house price H
OB(i,0)
causes the same type of problem, since the house price may
fall between two house price nodes.3
Linear interpolation must be used to calculate
asset values lying between grid points.
An example of repeated linear interpolation is given to illustrate how new data
points can be constructed from a discrete set of known data points to find unknown
3
There is no mention in Kau et al. (1993) of the scaled house price causing a non-lineararity
error. This matter requires as much attention as the contract-rate problem and must be dealt with
accordingly.](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-158-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 159
terms. For example, consider V +
[ H
OB(i,0)
, r, t; c(i)] in the mortgage component ad-
justment condition (5.11). Let x be the scaled house price H
OB(i,0)
and y be the new
contract rate c(i). Then let x1, x2 be the known values of x such that x1 ≤ x < x2,
and y1, y2 be the known values of y such that y1 ≤ y < y2, then Vapprox(x, y) is the
constructed value of V + H
OB(i,0)
, r, t; c(i) . This example is illustrated in figure 5.1,
which shows the unknown data point (x, y) encircled and the known data points,
used to interpolate the value of V (x, y), shown as dots. The approximate value of
the mortgage component Vapprox(x, y) is found by evaluating
u1 =
x − x1
x2 − x1
and u2 =
y − y1
y2 − y1
, (5.18)
then
g1 = (1 − u1)V (x1, y1) + u1V (x2, y1), (5.19)
g2 = (1 − u1)V (x1, y2) + u1V (x2, y2), (5.20)
and finally
Vapprox = (1 − u2)g1 + u2g2. (5.21)
This procedure is also used for the unknown terms in the adjustment conditions
for the default option and the insurance component. Simple linear interpolation is
used to calculate A+
[r, t; c(i)] in the adjustment payment-date condition, equation
(5.7). Simple linear interpolation can be used here, as there is just one possible
unknown quantity, the new contact rate c(i), which might not coincide with the
discrete auxiliary contract-rate range (5.16).
Crank-Nicolson methods vs explicit methods
Kau et al. (1993) opted to use the explicit finite-difference method for the numerical
solution of the ARM contract. Although the utilisation of this method is easier than
implicit methods, it has two main drawbacks.
1. The explicit method has a poor convergence rate of O(∆t, ∆H2
, ∆r2
).
2. The explicit method has stability constraints (see section 2.4) which require the
ratio of ∆t to ∆H and ∆r to be ‘fine tuned.’](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-159-320.jpg)
![CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 165
5.5 Results
5.5.1 Error in Kau, Keenan, Muller and Epperson (1993)
Kau et al. (1993) make comparisons of equilibrium contract rates and option values
between fixed-rate contracts and adjustable-rate contracts using the same framework.
It is sensible to do the same here. Unfortunately it can be shown that the results in
Kau et al. (1993) contain errors in components critical to the paper’s conclusions. The
main conclusion in the paper was that ARMs produce lower contract rates (compared
to equivalent FRM contracts), which lowers the value of payments. Quantifying this
conclusion relies completely on the accuracy of Kau et al.’s calculation of the value
of payments for FRMs, which can be shown to be incorrect.
The majority of a mortgage’s value comes from the value of the payments made
by the borrower (recall mortgage value is the value of payments minus the sum of
the value of the borrower’s options). It is imperative that the value of payments can
be calculated accurately. Under the framework used in Kau et al. (1993) and that
used in this thesis, the value of payments (for a FRM) can be calculated precisely, it
is ultimately the solution of the single-factor CIR PDE (2.36) which has an analytic
solution.
The value of payments A at origination can be calculated as follows
A[r = r(0), t = 0] =
12n
i=1
MPαie−βir
, (5.25)
where n is the length of the mortgage in years, MP is the value of the monthly
payments (equation (2.1)), where
αi =
2γe(γ+κ)(n− i−1
12 )/2
(γ + κ) eγ(n− i−1
12 ) − 1 + 2γ
2κθ/σ2
r
, (5.26)
βi =
2 eγ(n− i−1
12 ) − 1
(γ + κ) eγ(n− i−1
12
)
− 1 + 2γ
, (5.27)
and where
γ = κ2 + 2σ2
r . (5.28)](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-165-320.jpg)
![APPENDIX B. ANALYTIC APPROXIMATION DERIVATION 216
be the general initial condition for month m.1
Now, from equation (B.7),
dr
κ(θ − r)
= −dτm. (B.9)
Integrating equation (B.9),
⇒ −
1
κ
ln κ(θ − r) = −τm + c1 ⇒ κ(θ − r) = eκτm−κc1
∴ r =
κθ − eκτm−κc1
κ
, (B.10)
where c1 is a constant.
Note e−κc1
= κ(θ − r)e−κτm
this will be important later in the solution for F
Also from equation (B.7),
−dτm =
dF
rF
. (B.11)
Substituting equation (B.10) for r and integrating (B.11),
⇒ −dτm[κθ − eκτm−κc1
] = κ
dF
F
⇒ dτm[eκτm−κc1
− κθ] = κ
dF
F
⇒
1
κ
eκτm−κc1
− κθτm = κln(c2F)
∴ F =
1
c2
e(1/κ2)eκτm−κc1 −θτm
, (B.12)
where c2 is a constant. Finally, from equation (B.7),
dH
(r − δ)H
= −dτm. (B.13)
Again substituting equation (B.10) for r and integrating (B.13),
⇒
dH
H
= −dτm
κθ − eκτm−κc1
κ
− δ ⇒ ln(c3H) = (δ − θ)τm +
1
κ2
eκτm−κc1
∴ H =
1
c3
e(δ−θ)τm+(1/κ2)eκτm−κc1
, (B.14)
1
Due to the transformation which produced the forward PDE an initial condition is required
to solve the problem. These initial conditions were found by making the same transformation, as
shown in sections 3.3.1 and 3.3.2.](https://image.slidesharecdn.com/8f25f6c3-8adc-480c-88ab-e201f9323bd8-150620211614-lva1-app6892/85/final_version-216-320.jpg)
final_version
- 1. ADVANCES IN MORTGAGE VALUATION: AN OPTION-THEORETIC APPROACH A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2006 Nicholas J. Sharp School of Mathematics
- 2. Contents Abstract 13 Declaration 14 Copyright 15 Acknowledgements 16 Dedication 17 The Author 18 1 Introduction 19 1.1 Types of mortgage considered . . . . . . . . . . . . . . . . . . . . . . 20 1.1.1 Fixed-rate mortgage . . . . . . . . . . . . . . . . . . . . . . . 21 1.1.2 Adjustable-rate mortgage . . . . . . . . . . . . . . . . . . . . 23 1.2 Mortgages as derivative assets . . . . . . . . . . . . . . . . . . . . . . 24 1.2.1 Reduced-form models . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.2 Structural models . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.3 Mortgage-backed securities . . . . . . . . . . . . . . . . . . . . 27 1.2.4 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3 Underlying state variables . . . . . . . . . . . . . . . . . . . . . . . . 33 1.3.1 House price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.3.2 The term structure of interest rates . . . . . . . . . . . . . . . 35 1.3.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2
- 3. 1.3.4 Risk adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.4 Derivation of the asset valuation PDE . . . . . . . . . . . . . . . . . 37 2 Foundations of mortgage valuation 40 2.1 Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.1 Value of monthly payments . . . . . . . . . . . . . . . . . . . 40 2.1.2 Value of the outstanding balance . . . . . . . . . . . . . . . . 41 2.2 Equilibrium condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.1 Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3 Interest-rate index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.1 Calculation of the index . . . . . . . . . . . . . . . . . . . . . 44 2.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.1 Finite-difference methods . . . . . . . . . . . . . . . . . . . . . 48 2.5 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.1 Derivative approximations . . . . . . . . . . . . . . . . . . . . 50 2.5.2 Discrete representation . . . . . . . . . . . . . . . . . . . . . . 51 2.5.3 Solution of the difference equations . . . . . . . . . . . . . . . 52 2.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6.1 The value of the remaining payments . . . . . . . . . . . . . . 53 2.6.2 The value of the other mortgage components . . . . . . . . . . 54 2.6.3 Default boundary . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.6.4 Prepayment boundary . . . . . . . . . . . . . . . . . . . . . . 57 3 An improved fixed-rate mortgage valuation methodology 58 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Valuation framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 Mortgage contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.1 Mortgage payment-date conditions . . . . . . . . . . . . . . . 65 3.3.2 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3.3 Coinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 The equilibrium condition . . . . . . . . . . . . . . . . . . . . . . . . 68 3
- 4. 3.5 Benchmark method: the enhanced finite-difference approach . . . . . 68 3.6 The boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.1 The free boundary condition . . . . . . . . . . . . . . . . . . . 71 3.6.2 Solving the free boundary problem . . . . . . . . . . . . . . . 72 3.7 Rapid approximation method: perturbation approach . . . . . . . . . 74 3.7.1 General solution of the mth month . . . . . . . . . . . . . . . 75 3.7.2 Algorithm for value at origination . . . . . . . . . . . . . . . . 76 3.7.3 Value of the mortgage asset at origination . . . . . . . . . . . 77 3.7.4 Value of the future payments at origination . . . . . . . . . . 79 3.7.5 Value of the default option at origination . . . . . . . . . . . . 80 3.7.6 Value of insurance against default at origination . . . . . . . . 81 3.7.7 Perturbation approach pseudocode . . . . . . . . . . . . . . . 82 3.8 Satisfying the equilibrium condition . . . . . . . . . . . . . . . . . . . 83 3.8.1 Terms of the mortgage contract . . . . . . . . . . . . . . . . . 84 3.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4 A new prepayment model: an occupation-time derivative approach 99 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Introduction to occupation-time derivatives . . . . . . . . . . . . . . . 103 4.2.1 Definition of the barrier option . . . . . . . . . . . . . . . . . 103 4.2.2 Occupation-time derivatives . . . . . . . . . . . . . . . . . . . 104 4.2.3 Definition of the Parisian option . . . . . . . . . . . . . . . . . 106 4.2.4 Definition of the Parisian up-and-out option . . . . . . . . . . 106 4.2.5 Derivation of the occupation-time derivative PDE . . . . . . . 107 4.2.6 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2.8 Definition of the ParAsian option . . . . . . . . . . . . . . . . 115 4.2.9 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.10 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 116 4
- 5. 4.3 FRM valuation framework including lagged prepayment . . . . . . . . 116 4.3.1 New prepayment model . . . . . . . . . . . . . . . . . . . . . 119 4.3.2 Modified PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3.3 New payment-date conditions . . . . . . . . . . . . . . . . . . 122 4.4 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.1 Solution of the free boundary problem . . . . . . . . . . . . . 128 4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5 Advancements in adjustable-rate mortgage valuation 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.2.1 Prepayment-only ARMs . . . . . . . . . . . . . . . . . . . . . 145 5.2.2 Prepayable and defaultable ARMs . . . . . . . . . . . . . . . . 146 5.3 An improved auxiliary-variable approach . . . . . . . . . . . . . . . . 147 5.3.1 Mortgage Contract . . . . . . . . . . . . . . . . . . . . . . . . 148 5.3.2 Valuation procedure . . . . . . . . . . . . . . . . . . . . . . . 150 5.3.3 Improved numerical method . . . . . . . . . . . . . . . . . . . 156 5.4 New valuation methodology . . . . . . . . . . . . . . . . . . . . . . . 161 5.4.1 Contract rate preprocessing . . . . . . . . . . . . . . . . . . . 162 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.5.1 Error in Kau, Keenan, Muller and Epperson (1993) . . . . . . 165 5.5.2 Comparison of both methods . . . . . . . . . . . . . . . . . . 167 5.5.3 Tracking the contract rate . . . . . . . . . . . . . . . . . . . . 172 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6 Conclusions 174 6.1 Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.3 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5
- 6. 7 References 177 A Fixed-rate mortgage valuation pseudocode 186 B Analytic approximation derivation 214 B.1 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 214 B.2 Derivation of the general solution for any month . . . . . . . . . . . . 215 C Bridging solutions 218 Word count 49849 (main text only) 6
- 7. List of Tables 3.1 Contract specifications and other parameters which are fixed, all based on parameters used in the literature. . . . . . . . . . . . . . . . . . . . . . 84 3.2 Comparison of equilibrium setting contract rates for σr = 5%, σH = 5% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 88 3.3 Comparison of equilibrium setting contract rates for σr = 5%, σH = 10% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 89 3.4 Comparison of equilibrium setting contract rates for σr = 10%, σH = 5% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 90 3.5 Comparison of equilibrium setting contract rates for σr = 10%, σH = 10% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 91 3.6 Comparison of mortgage component values for σr = 5%, σH = 5%, calcu- lated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . 93 7
- 8. 3.7 Comparison of mortgage component values for σr = 5%, σH = 10%, calcu- lated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . 94 3.8 Comparison of mortgage component values for σr = 10%, σH = 5%, calcu- lated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . 95 3.9 Comparison of mortgage component values for σr = 10%, σH = 10%, cal- culated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . 96 4.1 Comparison of equilibrium contract rates and mortgage component values for σr = 5%, σH = 5%, for different prepayment assumptions. The loan is for 15 years, r(0) = spot interest rate (%). . . . . . . . . . . . . . . . . . 138 4.2 As in figure 4.1 except that σr = 5%, σH = 10%. . . . . . . . . . . . . . 138 4.3 As in figure 4.1 except that σr = 10%, σH = 5%. . . . . . . . . . . . . . 139 4.4 As in figure 4.1 except that σr = 10%, σH = 10%. . . . . . . . . . . . . . 139 5.1 Error in value of payments (for a FRM) published in Kau et al. (1993). The analytic value of payments is calculated using equation (5.25). θ = 0.1, κ = 0.25, n = 180, initial house price $100000. LTV = ratio of loan to initial value of house, r(0) = initial interest rate and contract rate are shown as percentages. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8
- 9. 5.2 Component values for the ARM calculated using the improved auxiliary- variable approach. Results without parentheses are for a 1.5% teaser; results with parentheses are without teasers. All results are to par value for a 15- year loan: spot interest rate r(0) = 8%, long-term mean θ = 10%, speed of reversion κ = 25%, correlation coefficient ρ = 0, service flow δ = 8.5%, interest-rate volatility σr = 10%, house-price volatility σH = 15%, points ξ = 1.5%, insurance coverage φ = 25%, and a 90% loan-to-value ratio. Initial margin was set at 100 basis point. Fixed-rate component values given for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.3 Component values for the ARM calculated using the new valuation method- ology. Parameter details identical to table 5.2. . . . . . . . . . . . . . . . 170 9
- 10. List of Figures 1.1 An illustration of the creation of a generic MBS showing the movement of cash. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1 A graph of index(r) against interest rate r. For each line style, the long- term mean of the short-term interest rate θ, is 0.1, 0.2 and 0.3 from the bottom to the top, with σr = 0.1 and κ = 0.25. . . . . . . . . . . . . . . 46 4.1 An illustration of the state space for a Parisian option. . . . . . . . . . . 107 4.2 Valuation of the Parisian up-and-out call option with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . 113 4.3 Parisian up-and-out call option at three different barrier times with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . 114 4.4 The delta of the Parisian up-and-out call with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5 The delta of the ParAsian up-and-out call with E = 10, ¯S = 12, ¯τ = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . 117 4.6 Comparison of the Parisian and ParAsian options with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . 117 4.7 An illustration of the effect of waiting to prepay on the value of the mort- gage, modelled using a consecutive occupation-time derivative. . . . . . . 121 4.8 An illustration of the general solution space at any time step for a FRM mortgage with the new prepayment model. . . . . . . . . . . . . . . . . 122 10
- 11. 4.9 An illustration of the finite grid in the house price H and interest rate r dimensions, the approximate location taken as the free boundary position is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.10 Mortgage value at origination V (H = 120000, r, τ1 = T1, ¯τ = ¯T) against interest rate r for eight different decision times. For each line style, the decision time ¯T is zero (this corresponds to the original prepayment as- sumption), T/8, T/4, T/2, 3T/4, T, 5T/4 and 3T/2 from the bottom to the top. For the case when κ = 0.25, θ = 0.1, δ = 0.085, σH = 0.1, σr = 0.1, ρ = 0, c = 0.111805, ratio of loan to initial value of house = 0.9, H(0) = $100000, r(0) = 0.1 and ξ = 0.015 for a 15 year loan. . . . . . . . 131 4.11 Mortgage value at origination V (H, r, τ1 = T1, ¯τ = ¯T) when the decision time is zero, ¯T = 0. The other parameters are identical to those stated in figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.12 Mortgage value at origination V (H, r, τ1 = T1, ¯τ = ¯T) when the decision time tends to infinity ¯T → ∞. The other parameters are identical to those stated in figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.13 Prepayment value at origination C(H = 120000, r, τ1 = T1, ¯τ = ¯T) against interest rate r for four different decision times. For each line style, the de- cision time ¯T is zero (this corresponds to the original prepayment assump- tion), T/2, T and 3T/2 from the top to the bottom. The other parameters are identical to those stated in figure 4.10. . . . . . . . . . . . . . . . . . 135 4.14 Mortgage ‘value’ at origination V (H = 100000, r, τ1 = T1, ¯τ) against interest rate r at selected times until prepayment ¯τ (equal intervals). The decision time is 1.5 months ¯T = 3T/2, other parameters are identical to those stated in figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1 An illustration of the unknown data point V (x, y) surrounded by its nearest grid points, at which the value of V is known. . . . . . . . . . . . . . . . 160 11
- 12. 5.2 A graph of contract rate c(i, r) against interest rate r. Shown is the initial contract rate c(0, r) (solid line), the contract rate after the first adjust- ment date c(1, r) (thick dashed line), the contract rate after the second adjustment date c(2, r) (thiner dashed line) and the contract rate after the final adjustment date c(14, r) (smallest dashed line). For the case when r(0) = 0.08, κ = 0.25, σr = 0.1, margin = 0.019, teaser = 0.015, y = 0.01, l = 0.05, 15 year loan. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.3 A graph of contract rate c(i, r) against adjustment date i. For each line style, interest rates are 0, 0.016, 0.032, 0.048, 0.064, 0.08, 0.096, 0.112 and 0.12 from the bottom to the top. For the case when r(0) = 0.08, κ = 0.25, σr = 0.1, margin = 0.019, teaser = 0.015, y = 0.01, l = 0.05, 15 year loan. 164 C.1 An illustration of the solution space for the default option in the final month n. The thick line represents the position of the required bridging solution. 219 12
- 13. Abstract This thesis improves on existing theoretical work on the pricing of mortgages as derivative assets, generally termed the option-pricing approach to mortgage valuation. In order that mortgage valuation is realistic and consequently not trivial, the future must be uncertain; therefore, the problems considered in this thesis operate within a stochastic economic environment. A highly accurate numerical scheme is presented, to tackle the partial differential equations that arise in fixed-rate mortgage valuation, and further a novel (analytic) singular perturbation approach is also developed. The analytic approximations pro- duced result in a significant increase in the efficiency of solution. A new prepayment model is also developed, which improves the modelling of the borrower’s decision pro- cess by incorporating occupation-time derivatives in the valuation framework. This simulates a delay in prepayment by the borrower, thus increasing the value of the mortgage to the lender. Empirical work supports this theory, and the new model should have positive implications for accurate mortgage-backed security pricing. For the more complex problem of adjustable-rate mortgage valuation, improvements are made to an existing approach by employing a superior numerical technique, and then a new drastically more efficient valuation methodology is developed. 13
- 14. Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institution of learning. 14
- 15. Copyright Copyright in text of this thesis rests with the Author. Copies (by any process) either in full, or of extracts, may be made only in accordance with instructions given by the Author and lodged in the John Rylands University Library of Manchester. Details may be obtained from the Librarian. This page must form part of any such copies made. Further copies (by any process) of copies made in accordance with such instructions may not be made without the permission (in writing) of the Author. The ownership of any intellectual property rights which may be described in this thesis is vested in The University of Manchester, subject to any prior agreement to the contrary, and may not be made available for use by third parties without the written permission of the University, which will prescribe the terms and conditions of any such agreement. Further information on the conditions under which disclosures and exploitation may take place is available from the Head of the School of Mathematics. 15
- 16. Acknowledgements I would like to thank my supervisors, Peter Duck and David Newton, for their advice, ideas and guidance throughout the last three years. In particular, to David for introducing me to this topic, helping me with suggestions and motivation and to Peter for his patience and encouragement with technical aspects of my work. I thank my parents for their continual encouragement and for their faith in me. I would like to thank my colleague and great friend Paul Johnson for, among other things, his insights on some numerical aspects of my work. And I acknowledge all of my close friends for keeping me sane, especially Claire, for being there when I needed her. Finally, I give thanks to the EPSRC for their generous funding, without which I would not have been able to attend conferences in Boston, Cambridge and Tokyo. 16
- 17. Dedication To my parents Margaret and Bill, and my brothers Chris and Jon. 17
- 18. The Author A Mancunian born and bred, Nicholas Sharp received his BSc in Mathematics from the University of Manchester in 2000. He continued his studies in the School of Mathematics to pursue his PhD in Mathematical Finance under the joint supervision of Prof. David P. Newton and Prof. Peter W. Duck. Nicholas Sharp started his PhD programme in September 2003. Since then, one paper (an adapted version of chapter 3 of this thesis) has been accepted at the 4th World Congress of the Bachelier Finance Society 2006 in Tokyo, Japan. This paper as also been accepted for publication in the Journal of Real Estate Finance and Economics. His principal research interests lie in the field of option-theoretic mortgage valua- tion. He is currently improving theoretical models through advancements in the nu- merical methods employed in pricing mortgage contracts and through the utilisation of exotic option pricing techniques to improve the modelling of borrower behaviour. 18
- 19. Chapter 1 Introduction A mortgage (literally meaning a dead pledge) is a type of financial contract which falls under the fixed-income product umbrella. It is a legal document by which a real estate asset is pledged as security for the repayment of a loan; the pledge is cancelled when the debt is paid in full. This type of debt instrument can be treated as a derivative security. The mortgage derives its value from the evolution of the global economy, via the underlying house price and the term structure of interest rates. A mortgage is a prime example of a financial product that can be modelled and then valued using option-pricing theory. The lender, who issues the contract, would like to know the value of the future cashflows that will be received as the result of the borrower making the scheduled monthly payments. The value of the mortgage is not simply the time value of these payments, since the borrower may terminate the contract prior to maturity, thus terminating the projected cashflows. The valuation of mortgages involves the bor- rower’s two options embedded in the contract to minimise the market value of the loan. The borrower has the option to prepay the remainder of the outstanding bal- ance owed if interest rates are financially favourable; this is an American call option which spans the whole mortgage. The borrower also has the option to default on the mortgage when a monthly payment falls due; this amounts to a series of linked monthly European options. As the mortgage is a contract between two parties, it is assumed that neither 19
- 20. CHAPTER 1. INTRODUCTION 20 would enter into an agreement unless it was fair at the onset. This means that the value of the mortgage to the lender at origination (when the contract begins) must be equal to the amount lent to the borrower. If this is in fact the case then it can be said that the contract is in equilibrium at origination. The mortgage value at origination will permit contractual arbitrage unless this is the case. To give an idea of the amount of money outstanding on residential mortgages, £1 trillion was owed by British borrowers alone by the second quarter of 2006. This figure is dwarfed by the collective worth of the unmortgaged property across the UK, which stands at £3.6 trillion (according to figures from the Bank of England). A related financial derivative is a mortgage-backed security. This product derives its value from sets of mortgages where cash flows have been combined (securitised), to form a more desirable debt instrument. If mortgage-backed securities are to be priced accurately, then it is vital that mortgage loans themselves are valued just as accurately. In 2003, the daily mortgage-backed security trading volume exceeded $200 billion in the United States of America. For price calculations to be useful, these must be timely and there remains a need for more rapid valuation. This thesis introduces some novel techniques to value several different types of mortgage loans accurately and efficiently. Also, a new termination model of prepay- ment is introduced with a view to eventually improving the modelling and pricing of mortgage-backed securities. Mortgage valuation is a complex derivative problem which involves many intricate subtleties; these make the framework required to price a mortgage very appealing from a mathematical point of view. 1.1 Types of mortgage considered If a bank agrees to grant a loan to a borrower who uses the money for the specific purpose of building or purchasing a house, the borrower typically pays a lower interest rate than for standard consumer credit, because the home offers security to the bank. If the loan cannot be repaid (this is a situation when the borrower exercises his option to default on a monthly payment), the house can be sold and the proceeds
- 21. CHAPTER 1. INTRODUCTION 21 can be used for repayment. This mechanism is formally agreed in the terms of the mortgage that the borrower receives from the bank. A mortgage is the legal claim the bank holds, allowing the bank to satisfy the debt through foreclosure and sale of the property, if necessary; the related loan is called a mortgage loan. There are various types of mortgage loans. The interest rate the borrower has to pay can be fixed or floating according to a specific index (Libor, for example); typically such loans have a maturity of up to 30 years (in the US). Various types of fixed-rate mortgage (abbreviated to FRM hereafter) are available. The constant-payment mortgage is the most common type in the United States. In the United Kingdom, variable-rate mortgages have been the historical norm, but in the past two decades especially, FRMs have been offered. For a constant-payment mortgage, the monthly payment is constant over the life of the loan. All the information, for example, terminology, formulae, conditions etc, that is necessary for the reader’s understanding of later chapters, is described in chapter 2. 1.1.1 Fixed-rate mortgage This thesis will first concentrate on the typical case of constant-payment mortgages, or fixed-repayment mortgages as they are known in the United Kingdom, with a known initial maturity and a fixed contract rate. These mortgages are the dominant collateral in the mortgage securitisation market. Also, since the mortgage market is the largest component of the outstanding US bond market debt, it is important that efficient models of the possible cash flows from these types of mortgages can be realised.1 This thesis initially attempts to simplify the highly complex problem that is the contingent claims mortgage valuation model. In chapter 3 an improved FRM valuation methodology is introduced. An approximate analytic (singular per- turbation) approach is used in a huge simplification of the valuation problem so that it is reduced to calculating a few simple equations. As a benchmark, rather than 1 According to The Bond Market Association, as of June 2006, mortgage-related bond market debt exceeded all other types of bond market debt (including municipal, treasury, corporate, federal agency securities, money markets and asset backed).
- 22. CHAPTER 1. INTRODUCTION 22 using the finite-difference techniques already available in the literature, an improved technique was first developed and this was employed as the benchmark with which to compare the singular perturbation approach, to test the latter technique for efficiency and accuracy. Although the term fixed-repayment mortgage might suggest that future cash flows are also fixed, this is not the case. In fact the real cash flows that originate from a mortgage loan are not fixed at all. The reason is that borrowers generally have the right to prepay the outstanding balance before the maturity of the loans. The prepayment feature has similarities with the callability of a more usual bond. In financial terms, the prepayment right can be viewed as an American call option. Exercise is only considered here when it makes sense from a purely financial viewpoint. No attempt is made to model any form of exogenous termination of the mortgage. Further, only one form of endogenous termination is considered, which occurs when the value of the mortgage to the bank is equal to the total debt the borrower has to pay if they decide to prepay their mortgage. This is called financially rational prepayment. The idea of optimality for a borrower can be complicated and causes much debate as to how to model correctly this idea in mortgage valuation. In reality a borrower choses to prepay based on individual circumstances. For example, the borrower may chose to prepay for any of the following reasons: • The borrower comes into money and is risk-averse; as a result the money is used to pay off the mortgage early; • The borrower moves house and pays off the mortgage with the proceeds from the sale; • The house is catastrophically lost (fire, earthquake, severe flood, etc), falls down and the insurance payment goes to the lender; • Interest rates fall and the borrower finds a better deal from another lender; this is known as refinancing.
- 23. CHAPTER 1. INTRODUCTION 23 For the first three reasons above, prepayment could be modelled in this way en- dogenously using some type of hazard process, see section 1.2.1 for details regarding this modelling approach. The final reason, which depends on the movement of the underlying interest rate, is ideally suited to be modelled using an option-theoretic approach. To model the prepayment decision within an endogenously driven frame- work the usual assumption is that the borrower pays off the mortgage when interest rates decrease sufficiently. It is clear that prepayment significantly affects the value of a mortgage. The second significant piece of work in this thesis, chapter 4, concentrates on improving the model of a borrower’s option to prepay. It has been suggested that borrowers do not actually choose to prepay when it is financially rational to do so, rather, that there is a time lag between the arrival of the information to them and the actual decision being made to prepay the mortgage (see section 4.1 for a discussion of why borrowers would wait to prepay). A new model of the borrower’s decision process regarding prepayment is offered, in which occupation-time derivatives are incorporated into the mortgage termination framework, that allows for more flexibility when modelling prepayment by the borrower, as a time lag from when it is initially financially optimal is permitted. This improvement in model flexibility will hopefully have implications in the im- provement of mortgage-backed security pricing. Empirical research suggests that conventional option-pricing mortgage valuation models do not contain the necessary features to price mortgage-backed securities accurately. It is hoped that the contribu- tions made by improving the way prepayment is modelled will allow mortgage-backed securities to be valued accurately within an option-theoretic framework. 1.1.2 Adjustable-rate mortgage The final contribution in this thesis, chapter 5, moves on to the valuation of adjustable- rate mortgages (abbreviated to ARMs hereafter). The more contractually and math- ematically complex problem of ARMs poses some interesting modelling and valuation
- 24. CHAPTER 1. INTRODUCTION 24 problems; for a review of the literature in this field, see section 5.2. In summary, the difficulty in modelling an ARM occurs as the common solution technique using back- wards valuation, of the asset valuation partial differential equation (abbreviated to PDE hereafter), is in the opposing temporal direction to the propagation of informa- tion about the varying contract rate. Innovative solution techniques must be used to overcome this difficulty. The work on this topic first utilises the auxiliary contract- rate variable approach of Kau et al. (1993) and then improves the numerical scheme employed by these authors, and further introduces a new technique to circumvent the problem with the opposing direction of the contract rate information and the solution scheme. This new technique removes the need for the auxiliary variable, simultaneously overcoming the problems with accuracy and solution efficiency that are inherent in the approach of Kau et al. (1993). 1.2 Mortgages as derivative assets In the past two decades, theoretical pricing models of mortgages as derivative assets have been accepted by the financial community as tools to improve the understanding of markets themselves. It is very rare that economic reasoning, applied to understand the workings of markets, leads to tools that have practical consequences. This thesis examines and extends the option-pricing approach to mortgage valuation. Although applying financial mathematics to price options is a relatively recent de- velopment (dating back to the early 1970’s), the foundations of option-based pricing models were laid down far earlier in economic research. At the turn of the twen- tieth century, the French mathematician Louis Bachelier (1900) was the pioneer of the random walk of financial market prices, Brownian motion and martingales. His innovations predated the famous work by Einstein (1906) on Brownian motion for physical processes. It is usually suggested that financial mathematics borrows theo- ries from leading physicists, but in this case at least, finance arrived at a theory first. In recent times the work of Merton (1973), and Black and Scholes (1973) on option- pricing theory produced closed-form solutions to the problem of valuing a European
- 25. CHAPTER 1. INTRODUCTION 25 call option on an underlying asset for short-run scenarios, in which the interest rate may be regarded as constant. Moving to mortgages, two sources of uncertainty are present: term structure risk and default risk. As house price, the source of default risk, is itself a traded asset (if not a standardised one, with frequent trading), the analogy between a mortgage on a house and an option on a stock is quite close. Default by a borrower acts in a manner similar to a put option, since by defaulting the borrower returns the asset. As interest rates are not a directly traded asset, an equilibrium model can be used to value interest-dependent contracts. This means attitudes toward interest-rate risk as well as the trend of interest movements enter into the valuation of mortgages, the corresponding elements for house prices are not a consideration. If, as typically assumed in the literature, the term structure is captured by a single variable, the spot rate, the result is a single market price of risk. The commonly assumed local expectations hypothesis concerning the term structure is nothing but the requirement that this market price of risk disappear (Cox et al., 1981). Unfortunately it is not the case, unlike for Black and Scholes (1973), that closed- form solutions exist for complex contracts such as mortgages. To value these, nu- merical solution techniques must be generally employed. Nevertheless, good analytic approximations of mortgage valuation can be found - see chapter 3 for details of an approximation method involving a novel perturbation approach to value FRMs. The approach in chapter 3 simplifies the complexities of mortgage valuation, by first appealing to the assumption of mathematically small volatilities for house-price and interest rate and then using these small parameters in an asymptotic analysis of the asset valuation PDE. This results in some especially simple analytic formulae which can be used specifically to determine equilibrium contract rates. This technique not only reduces the complexity of the problem but also vastly increases the efficiency of obtaining a solution to the valuation problem. This is abundantly evident when com- paring the computation time required to achieve a solution using the perturbation approach with that required using a full numerical solution scheme. With mortgages there are three ascending levels of assets: the underlying physical
- 26. CHAPTER 1. INTRODUCTION 26 real estate asset, the house; the contracts financing this real estate, which makes up the primary mortgage market; and mortgage-backed securities (MBS) that arise from pooling such mortgages, which make up the secondary mortgage market. Since MBS are pools of individual mortgages, analysing their performance depends on understanding the behaviour of the constituent mortgages. The main difficulties in pricing MBS are that borrowers: 1. do not all act the same, so how can this be modelled within a single framework? 2. do not exercise their option to prepay when this appears to be financially opti- mal, so how should this be modelled? The work in chapter 4 addresses the second of these difficulties by contributing a new prepayment model. By incorporating occupation-time derivatives into the option- theoretic framework for mortgages, time lagged prepayment can be included. The next section introduces MBS more formally and describes past research on pricing these derivatives. When the ultimate aim is to value a MBS, it is still necessary to value the under- lying mortgage with precision and with great care to model the subtleties that affect the security. There are two strands in the valuation literature, the split depends on whether an econometric (reduced-form) valuation model is used or whether an option-theoretic rational (structural) model is employed. 1.2.1 Reduced-form models Reduced-form models are such that borrower decisions are related to a set of predic- tors via a functional form chosen by the modeller. There are no theoretical restrictions on this functional form, speeding development time and providing a great deal of flex- ibility to match the historical data record closely. Unfortunately, there is no way to determine how the estimated parameters should change in response to changes in the economic environment, and this results in reduced-form models performing badly out-of-sample (see section 1.2.4 for more details).
- 27. CHAPTER 1. INTRODUCTION 27 1.2.2 Structural models Structural models are used exclusively throughout this thesis. The reason will be ex- plained after the following survey of the techniques used in the literature. Structural models can produce informative forecasts in economic environments unlike those seen in the past, since mortgage terminations are the result of optimising behaviour by agents in the model. The structural methodology links option exercise events to the underlying fundamentals faced by the borrower. 1.2.3 Mortgage-backed securities A mortgage-backed security is a security based on a pool of underlying mortgages. Investors then buy a piece of this pool and in return receive a fraction of the sum of all the interest and principal payments. MBS are usually based on mortgages that are guaranteed by a government agency for payment of principal and a guarantee of timely payment. The analysis of MBS concentrates on the nature of the underlying payment stream, particularly the prepayments of principal prior to maturity. By buying into this pool of mortgages, the investor gets a stake in the housing-loan market, but with less of the prepayment risk (if payments are arranged to be on equal terms to all bond holders). Since there are so many individual mortgages in each security, it would seem only necessary to model the average behaviour, but what is the behaviour of the average borrower? Recent work introduces borrower heterogeneity to model different types of borrower actions, see section 1.2.4 for details. An illustration of the creation of a generic MBS is shown in figure 1.1, which shows the flow of the mortgage loan cash flows as they are securitised by the MBS issuer, then sold to investors, as a more desirable debt instrument (where the investors are not subjected to the prepayment risk in exchange for a lower return). A more specific example of a MBS is a Collateralized Mortgage Obligations (CMO). These securities are based on a MBS but in which there has been further pooling and/or splitting so as to create securities, with different maturities for example. A typical CMO might receive interest and principal only over a certain future time frame. MBS
- 28. CHAPTER 1. INTRODUCTION 28 Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments Interest Scheduled principal repayment Prepayments $ $ $ $ Rule for distribution of cash flow Pro rata basis Pooled monthly cash flow: Interest Scheduled principal repayment Monthly cash flow Investors Pool Loans Prepayments Figure 1.1: An illustration of the creation of a generic MBS showing the movement of cash.
- 29. CHAPTER 1. INTRODUCTION 29 can be stripped into principal and interest components. Principal Only (PO) MBS receive only the principal payments and become worth more as prepayment increases. Interest Only (IO) MBS receive only the interest payments. The latter can be very risky since high levels of prepayment mean many fewer interest payments. 1.2.4 Previous work Confirmation that a contingent claims approach to pricing MBS is useful is given by Dunn and McConnell (1981a). This early work was based on a general equilibrium theory of the term structure of interest rates under uncertainty, modelled a default- free fixed-rate version of a MBS, where the interest-rate process is modelled using a mean reverting stationary Markov process; suboptimal prepayment is modelled using a Poisson-driven process. In their follow up work, Dunn and McConnell (1981b) compare the value of an amortising callable bond, with MBS specification, to three types of default-free bonds to show the impact of call, amortisation and prepayment features on pricing, returns and risks of MBS. They conclude that amortisation and prepayment features increase the price of a MBS and the callability feature decreases its price. The effect of all three features is to reduce the interest-rate risk, and consequently the expected return of a MBS relative to other securities which do not have these features. Dunn and McConnell (1981b) also express the need for an empirical study to determine if the prices generated by their model were consistent with observed market prices. In Brennan and Schwartz (1985) three arbitrage-based models for MBS are con- trasted. Both the interest-rate uncertainty and the call policy used are said to have an effect on pricing. A two-factor term structure model consisting of the short rate and the consol rate (the yield on a bond of infinite maturity) is used. Stepping away from optimal call policy models, Schwartz and Torous (1989) offer an empirically based, reduced-form model. Here, maximum-likelihood techniques are used to estimate a prepayment function from recent (at the time of publication) price information. A proportional-hazards model is then used to make prepayment
- 30. CHAPTER 1. INTRODUCTION 30 decisions. This prepayment function is integrated into the Brennan and Schwartz (1985) two-factor model for valuing default-free interest-dependent claims. Monte Carlo simulation methods are used in the solution of the problem. A comparison with an optimal, value-minimising call policy model is given. In conclusion, they claim optimal call policy models cannot explain the fact that borrowers prepay their loans when the prevailing refinancing rate exceeds their loan’s contract rate, and conversely other borrowers do not prepay even when the contract rate on their loan exceeds the prevailing refinancing rate. Also, the use of an estimated prepayment function produces mortgage prices which are consistent with traded MBS prices. Although the model results are closer to actual prices used in practice than those produced using optimal, value-minimising call conditions, these reduced-form models are notorious for performing badly out-of-sample; see Downing et al. (2005) for a discussion of this problem. Regardless of their shortcomings reduced-form models can provide an understanding of the empirical performance of existing contracts and their pricing dynamics. The later work of Schwartz and Torous (1992) is the first to introduce the pos- sibility of default in the valuation of MBS. The borrower’s conditional probability of prepayment is given by a prepayment function, while the borrower’s conditional probability of default is given by a default function. Although a MBS is default free, as it is guaranteed by the government, the cash flows to the security are not. There- fore, the value of the security is affected by default. In the work of Schwartz and Torous (1993), still with the inclusion of a default possibility, a Poisson regression technique is developed, instead of a likelihood method, to estimate the parameters of a proportional-hazards model for prepayment and default decisions. This is simply an alternative way to find the parameters for the termination functions from real data. Stanton (1995) comments on the shortcomings of reduced-form models and their inability to perform in different economic environments; instead, a rational mortgage prepayment model is proposed. Akin to the work of Dunn and McConnell (1981a,b), Stanton’s model estimates heterogeneity in the transaction costs faced by mortgage
- 31. CHAPTER 1. INTRODUCTION 31 holders, and borrowers make prepayment decisions at discrete intervals. Burnout dependence is produced by letting expected prepayment rates depend on cumulative historical prepayment levels. This allows prices to exceed par by more than the transaction costs. The model gives a simple rational representation for prepayment, and MBS value is given as a weighted sum of the market values of the underlying mortgages. So far, the discussion of the literature for MBS valuation has focused on what type of model to use, be it structural or reduced-form, but Chen and Yang (1995) explore deeper and discuss which actual interest-rate process should be used in pricing MBS. Four processes are used to compare MBS prices, and they conclude that mortgage pricing models are less sensitive to the underlying interest-rate process than a simple coupon bond, and that this is due to the prepayment feature with mortgages. In the work of Kariya et al. (2002), borrowers are allowed to act differently within mortgage pools. A framework is provided whereby a short term interest rate is used for discounting and a mortgage rate is used as an incentive factor for refinancing; a second prepayment incentive factor based upon rising property values is also used. This falls into the non-option methodology for pricing. The more recent work of Kau and Slawson (2002) incorporates frictions into a theoretical options-pricing model for mortgages. Here the model is still a rational model of mortgage valuation, where prepayment and default are financial decisions but the effect of borrower characteristics is introduced without destroying the options theoretic framework. Three categories of friction are allowed for, including fixed and variable transaction costs, sub-optimal termination and sub-optimal non-termination. The adaptability and flexibility of an option-theoretic model is illustrated. The ability to include borrower heterogeneity is shown not to require the loss of optimality. A full spectrum of refinancing behaviour is modelled using a notion of refinancing efficiency by Kalotay et al. (2004). They focus on understanding the market value of the mortgage, rather than trying to predict future cashflows. Two separate yield curves are used, one for discounting mortgage cashflows and the other for MBS cash- flows. They give the following reasons why option-theoretic models, at present, tend
- 32. CHAPTER 1. INTRODUCTION 32 not to be used for prepayment modelling: most homeowners do not exercise their options optimally; and option-based models are not able to explain observed MBS prices. The authors show that in fact, a ‘rigorously constructed’ option-pricing model does explain MBS prices well. MBS are said to be priced well with the assumption that most homeowners exercise their refinancing option near-optimally. As mortgages are not always refinanced using an optimal strategy (sold at par by the borrower to the lender), the authors account for borrower heterogeneity by breaking the mort- gage pool into buckets and assume that each bucket represents different refinancing behaviour to price MBS well. The empirical test of Downing et al. (2005) as to the importance of a second factor confirmed that including the house price as a factor in mortgage valuation and MBS pricing (to capture the effect of default by the borrower) is necessary. This adds weight to the research of Schwartz and Torous (1992) and that of Kau and Slawson (2002), which both include the possibility that the borrower will default. In the recent research of Longstaff (2005), a multi-factor term structure approach is used to incorporate borrower credit into the analysis. Results show that optimal refinancing strategy can delay prepayment relative to conventional models, and that mortgage values can exceed par by much more than the cost of refinancing. The notion that a borrower’s financial situation affects the rate at which he can refinance, including credit worthiness, is introduced. The borrower’s optimal refinancing strat- egy involves considering the life of loan affects of refinancing. If the borrower’s credit is poor he will have to refinance at a premium rate; this is modelled by adding a credit spread to the prepayment cost. A Poisson process is used to add in the chance of exogenous refinancing reasons. Borrowers then find it optimal to delay prepay- ment beyond the point at which conventional models imply the mortgage should be prepaid. Other recent models include Dierker et al. (2005) and Dunn and Spatt (2005). As in the two previously discussed articles, option exercise is modelled by endogenous decisions made by borrowers to minimise the present value of their current mortgage position.
- 33. CHAPTER 1. INTRODUCTION 33 Both reduced-form and structural models have the same goal, which is to account realistically for all the embedded options in mortgage contracts. Kalotay et al. (2004) comment that there is evidence for the use of both types of model in practice. As mentioned previously, reduced-form models contain a great deal of flexibility to match historical data closely. However, there is no guarantee that a functional form which works well in-sample will perform as well out-of-sample. Also, there is no way to determine how the estimated parameters should change in response to a change in the economic environment. Although with basic structural models it is true that it is difficult to give prepayment predictions that match observed prepayment behaviour, and impossible to allow prices to exceed par by more than the transaction costs, Kalotay et al. (2004) and Longstaff (2005) show that they are flexible enough that mortgage features such as friction, borrower heterogeneity and many other subtleties can be included. These make it possible to achieve realistic mortgage values within a rational framework (Kalotay et al., 2004 and Longstaff, 2005). Evidence has also been given for the importance of house price as a necessary factor. It is a crucial factor in capturing the information about the default behaviour of the borrower. Although MBS are generally guaranteed against default, default affects the cashflows from the underlying mortgages themselves, as changes in house value affect the borrower’s decision to default, the cashflows to the MBS will also be affected indirectly by house price changes. The next section details the two state variables (house price and interest rate) which are used in this thesis as the sources of uncertainty in the economic environment for which all problems will be set. The modelling will be of the rational option- theoretic variety for the reasons discussed above and the processes chosen for the house price and interest rate variables are discussed next. 1.3 Underlying state variables It is possible to treat a mortgage as a derivative asset which exists within a stochastic economic environment. The uncertainty which comes with this modelling setup could
- 34. CHAPTER 1. INTRODUCTION 34 affect risk preferences. Option theory can be used to show that the role of prefer- ences is actually quite limited when applied to derivative assets. Valuation can be performed as if the world were risk neutral (with some risk adjustments), so that the value of a derivative asset is simply the expected present value of its future payoffs; see Cox et al. (1985a) for a discussion of risk-neutralised pricing. A mortgage derives its value from two state variables. Possibly the most obvious (from the borrower’s point of view) is the price of the underlying real estate asset, the house. The term structure of interest rates is the other state variable. This could be considered the most relevant to the lender, as it will ultimately determine the value of the payments made by the borrower, but it is the interaction between these two factors that must be considered simultaneously to determine the value of the mortgage. The assumption that underpins the whole option-theoretic approach to mortgages is that even though mortgages depend on the real economy through the house price and term structure, mortgages themselves are not necessary to determine this un- derlying economy. As a derivative asset is one that is not necessary to describe the underlying real economy, it is a redundant asset and its value depends entirely on the variables that do determine the underlying economy. The choice of the processes in this thesis for the two state variables that model the economic environment is consistent with recent literature (Kau et al. 1995; Kau and Slawson 2002; Azevedo-Pereira et al. 2002, 2003). 1.3.1 House price Merton (1973) lognormal diffusion process The house-price process, equation (1.1) below, models house price behaviour as a lognormal diffusion process; see Merton (1973) for more details. In the contingent claim framework, let the true process describing the underlying estate asset, the house price H, be dH = (µ − δ)Hdt + σHHdXH, (1.1)
- 35. CHAPTER 1. INTRODUCTION 35 where: µ is the instantaneous average rate of house-price appreciation, δ is the ‘dividend-type’ per unit service flow provided by the house, σH is the house-price volatility, XH is the standardised Wiener process for house price. The house-price appreciation µ, is analogous to the drift term for the more stan- dard stock-price model. The service flow δ is analogous to a dividend on a stock as the borrower benefits from the underlying asset (the borrower is allowed to live in the real estate asset during the life of the mortgage contract). The borrower benefits from the asset, therefore the price must drop by this amount otherwise arbitrage would occur. 1.3.2 The term structure of interest rates CIR (1985) mean reverting square root process The term structure of interest rates is modelled using the single factor Cox et al. (1985b) mean-reverting square root process. The single factor r is taken to be the spot rate of interest. Within the contingent claim framework let the true process describing the term structure of interest rates, the spot rate r, be dr = κ(θ − r)dt + σr √ rdXr, (1.2) where: κ is the speed of adjustment in the mean reverting process, θ is the long-term mean of the short-term interest rate r, σr is the interest-rate volatility, Xr is the standardised Wiener process for interest rate.
- 36. CHAPTER 1. INTRODUCTION 36 1.3.3 Correlation The stochastic elements of the house-price process (1.1) and the spot interest-rate process (1.2) which involve the standardised Wiener processes, XH for house price and Xr for interest rate respectively, are correlated according to dXHdXr = ρdt, (1.3) where ρ is the instantaneous correlation coefficient between the two Wiener processes. 1.3.4 Risk adjustment For house price and term structure to have any value, preferences, technology and supply and demand considerations are incorporated into the price. The only other factor that influences a derivative asset’s price is its market price of risk, which is included in the risk adjustment of that variable. As the real estate asset underly- ing a mortgage is itself a traded asset (Kau et al., 1993, 1995; Kau and Slawson, 2002; Azevedo-Pereira et al. 2002, 2003), the risk adjustment involves no external parameters whatsoever. As the term structure follows the Cox et al. (1985b) process, the market price of risk can be regarded as having been absorbed into the estima- tion of reversion κ and long-term average θ parameters of the interest rate-process. As a result, the local expectations hypothesis requires that this market price of risk also disappear (Cox et al. 1981). This means that with risk adjustments taken care of, it is possible to proceed with the expected present value calculation. Using risk neutrality arguments the instantaneous average rate of house-price appreciation (the drift term) can be taken as the interest rate (as the market price of risk for the house price is taken as zero, which is explained above), Therefore µ = r in the process for the house price, equation (1.1). These risk adjustment reasons are standard in the literature.
- 37. CHAPTER 1. INTRODUCTION 37 1.4 Derivation of the asset valuation PDE This section demonstrates a derivation of the asset valuation PDE using standard hedged portfolio arguments. The PDE for the valuation of any asset F = F(H, r, t) whose value is a function only of house price H, interest rate r, and time t, can be found as follows. House price is described by the stochastic differential equation (1.1) and stochastic interest rate follows equation (1.2). Using Itˆo’s lemma (see Itˆo, 1951, for the details) on the function F(H, r, t), it can be shown that, dF = ∂F ∂t dt + ∂F ∂H dH + ∂F ∂r dr + 1 2 ∂2 F ∂H2 dH2 + 2 ∂2 F ∂H∂r dHdr + ∂2 F ∂r2 dr2 + · · · (1.4) From stochastic calculus; dt2 → 0, dX2 → dt, and dXdt = o(dt), as dt → 0; then from equation (1.1) dH2 → σ2 HH2 dX2 H → σ2 HH2 dt; (1.5) and from equation (1.2) note that, dr2 → σ2 r rdX2 r → σ2 r rdt; (1.6) and finally from equation (1.1), (1.2) and (1.3) dHdr → σHσrH √ rdXHdXr = ρσHσrH √ rdt. (1.7) Thus, Itˆo’s lemma for the two stochastic variables governed by (1.1) and (1.2) is, dF = ∂F ∂t dt + ∂F ∂H dH + ∂F ∂r dr + 1 2 σ2 HH2 ∂2 F ∂H2 + 2ρσHσrH √ r ∂2 F ∂H∂r + σ2 r r ∂2 F ∂r2 dt. (1.8) Now construct a portfolio Π, long one asset F1(H, r, t) with maturity T1, short ∆2 of an asset F2(H, r, t) with maturity T2, and short ∆1 of the underlying asset H. Thus, Π = F1 − ∆2F2 − ∆1H. (1.9) The change in this portfolio over a time dt is, dΠ = dF1 − ∆2dF2 − ∆1dH, (1.10)
- 38. CHAPTER 1. INTRODUCTION 38 where ∆1 and ∆2 are constant during this time. The effect of the service flow δ is to cause the price of the underlying asset H to drop in value by δH over a time dt. Therefore, the portfolio must change by an amount −δH∆1dt during this time. Thus, the correct change in the value of the portfolio over a time dt is dΠ = dF1 − ∆2dF2 − ∆1(dH + δHdt). (1.11) With a careful choice of ∆2 = ∂F1/∂r ∂F2/∂r (1.12) and ∆1 = ∂F1 ∂H − ∆2 ∂F2 ∂H (1.13) the risk from the portfolio can be eliminated, i.e. the random components of the dH and dr terms vanish, and dΠ becomes dΠ = ∂F1 ∂t dt + 1 2 σ2 HH2 ∂2 F1 ∂H2 + 2ρσHσrH √ r ∂2 F1 ∂H∂r + σ2 r r ∂2 F1 ∂r2 dt − δH ∂F1 dH dt − ∂F1/∂r ∂F2/∂r ∂F2 ∂t dt + 1 2 σ2 HH2 ∂2 F2 ∂H2 + 2ρσHσrH √ r ∂2 F2 ∂H∂r + σ2 r r ∂2 F2 ∂r2 dt − δH ∂F2 dH dt = r F1 − ∂F1/∂r ∂F2/∂r F2 − ∂F1 ∂H H + ∂F1/∂r ∂F2/∂r ∂F2 ∂H H dt. (1.14) Here arbitrage arguments have been used to set the return on the portfolio equal to rΠdt, since the growth of the portfolio in a time step dt is equal to the risk-free growth rate of the portfolio, as the portfolio is now completely deterministic. Dividing by dt and separating the F1 and F2 terms leads to, 1 ∂F1/∂r ∂F1 ∂t + 1 2 σ2 HH2 ∂2 F1 ∂H2 + ρσHσrH √ r ∂2 F1 ∂H∂r 1 2 σ2 r r ∂2 F1 ∂r2 + (r − δ)H ∂F1 ∂H − rF1 = 1 ∂F2/∂r ∂F2 ∂t + 1 2 σ2 HH2 ∂2 F2 ∂H2 +ρσHσrH √ r ∂2 F2 ∂H∂r 1 2 σ2 r r ∂2 F2 ∂r2 +(r−δ)H ∂F2 ∂H −rF2 . (1.15) Although this is one equation in two unknowns, the left-hand side is a function of T1 but not of T2 and the right-hand side is a function of T2 but not of T1. The only way for this to be possible is for both sides to be independent of maturity date. Thus,
- 39. CHAPTER 1. INTRODUCTION 39 removing the subscript from F, 1 ∂F/∂r ∂F ∂t + 1 2 σ2 HH2 ∂2 F ∂H2 + ρσHσrH √ r ∂2 F ∂H∂r + 1 2 σ2 r r ∂2 F ∂r2 + (r − δ)H ∂F ∂H − rF = a(H, r, t), (1.16) is obtained for some function a(H, r, t). It is convenient to write a(H, r, t) = −κ(θ−r) (this is a standard procedure in the literature, see Kau et al. 1993, 1995; Azevedo- Pereira et al. 2002, 2003), which leads to the asset valuation PDE for F(H, r, t), 1 2 H2 σ2 H ∂2 F ∂H2 + ρH √ rσHσr ∂2 F ∂H∂r + 1 2 rσ2 r ∂2 F ∂r2 +κ(θ − r) ∂F ∂r + (r − δ)H ∂F ∂H + ∂F ∂t − rF = 0. (1.17) This PDE will be used extensively in this thesis to value fixed-rate (chapter 3) and adjustable-rate mortgages (chapter 5), and a modified version will be used to value a fixed-rate mortgage containing a new prepayment assumption (chapter 4).
- 40. Chapter 2 Foundations of mortgage valuation It is first necessary to describe some concepts and ideas that underlie the complex problem of mortgage valuation. This chapter acts as a reference source and will be referred to where necessary later in this thesis, which will remove excess formulation in later discussions of the improvements to be implemented. 2.1 Formulae Initially, the problem of valuing a FRM is explored. This type of loan is repaid by a series of equal monthly payments, made on pre-determined, equally-spaced dates. The monthly payment MP and the outstanding balance following each payment OB(i) are calculated using standard annuity formulae, which will be given in the next section. 2.1.1 Value of monthly payments The asset valuation PDE (1.17) is solved using a backward valuation procedure (see section 2.5 for details). It is necessary to start the process from the known information at maturity, referring to these known cashflows at the final moment of the contract, rather than using the more common actuarial procedure of referring all the cashflows at the origination of the loan. To define the value of each monthly payment it is necessary to recognise that the 40
- 41. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 41 future value of the outstanding debt in the terminal period of the contract must be equal to the future value of all the payments, when this value is also referred to the terminal moment of the contract. Consequently OB(0) 1 + c 12 n = MP 1 + c 12 n 1 − 1 + c 12 −n c 12 , which upon slightly simplifying yields OB(0) 1 + c 12 n = MP 1 + c 12 n − 1 c 12 , and then making MP the subject of this equation MP = OB(0) 1 + c 12 n c 12 1 + c 12 n − 1 , (2.1) gives the formula for the value of the monthly payments, where OB(0) is the amount initially loaned to the borrower, c is the fixed yearly contract rate, and n is the life of the mortgage in months. 2.1.2 Value of the outstanding balance Immediately after the ith monthly payment has been made, the outstanding balance OB(i) the borrower still has to repay can be expressed in the following way OB(i) = OB(0) − MP 1 − 1 + c 12 −i c 12 1 + c 12 i . Making the substitution for MP from equation (2.1) yields OB(i) = OB(0) − OB(0) 1 + c 12 n c 12 1 + c 12 n − 1 1 − 1 + c 12 −i c 12 1 + c 12 i , and then simplifying gives OB(i) = OB(0) 1 + c 12 n − 1 + c 12 i 1 + c 12 n − 1 , (2.2) which is the formula for the value of the outstanding balance OB(i) after the ith monthly payment has been made. For an ARM the formulae to calculate the monthly payment and the outstanding balance are given by equation (5.3) and (5.4), respectively. As the contract rate can
- 42. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 42 change each year the two formulae are also functions of the current year as well as the current month. The only change in the formulae is that a year index is introduced so the specific contract rate for the present year can be used in the calculation. Therefore the derivation of the formulae for an ARM is omitted. 2.2 Equilibrium condition As mentioned in chapter 1, the mortgage contract would not be agreed originally by the two counter parties unless it was fair. This means that at origination the contract must be in financial equilibrium, which is the case if the value of the mortgage to the bank is equal to the amount lent to the borrower. A generalised equilibrium condition for a generic mortgage loan (where the type of mortgage is irrelevant) is as follows V (t = 0; c) + I(t = 0; c) = (1 − fee)loan. (2.3) The bank’s position in the contract is V = A − D − C, i.e. the scheduled payments minus the sum of the value of the borrower’s options to terminate the mortgage (D is the value of the default option and C is the value of the prepayment option), plus any insurance I the bank may have against the borrower defaulting on a payment. The borrower’s position is the amount lent by the bank, which will be some percentage of the initial house value, minus an arrangement fee (for a UK contract) or the points (for a US loan) charged as a percentage of the loan amount. The equilibrium constraint (2.3) is to avoid contractual arbitrage. The specific equilibrium condition for a UK and US FRM is given by equation (3.22), and by equation (5.14) for a US ARM. The difference between the conditions for a UK contract and a US contract is only in the terminology used for the fee paid by the borrower when the contract is set up. The fixed-rate and the adjustable-rate conditions vary according to the free parameter c which is used to balance the equilibrium condition. This is discussed in the next section, as well as the method used to calculate the free parameter c that will provide a contract in equilibrium at origination.
- 43. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 43 2.2.1 Newton method All mortgage contracts discussed in this thesis require an equilibrium condition to be set prior to the contract commencing, and so it is necessary to find the value of the free parameter c (which is the contract rate for FRM valuation and the margin for ARM valuation) which balances the relevant equilibrium condition. The free variable can be found easily using an iterative process following Newton’s method. Let f(c) be a function of c only, where c is the free variable used to balance the equilibrium condition and f(c) is given by rearranging equation (2.3) to form f(c) = V (t = 0; c) + I(t = 0; c) − (1 − fee)loan, (2.4) which must be zero to satisfy the equilibrium condition. An initial estimate for the value of c is made, let this estimate be c0. Then the values of the mortgage components involved in the equilibrium condition are calculated with the initial estimate c0 used as the value of the free parameter. Next, a tolerance to which the absolute value of f(c) must be less than is specified; once f(c) is less than this tolerance the iterative process is terminated. For FRMs, c at this point is the equilibrium setting contract rate, and for ARMs, c at this point is the equilibrium setting margin. An estimate is required for the initial increment change in c0; call this increment ∆0 (which is specified). The next potential equilibrium setting free parameter c is given by c1 = c0 + ∆0. Given this information it is then possible to calculate f(c1) and check if its absolute value is less than the tolerance. If the absolute value of f(c1) and any further f(ci) is greater than the tolerance the new increment for the change in c is calculated as follows, ∆i+1 = − ∆if(ci) f(ci) − f(ci−1) ; where i ≥ 1. (2.5) 2.3 Interest-rate index This section discusses interest-rate indices, how they drive the contract rate for ARMs, and how the particular index used in chapter 5, on ARM valuation, is calcu- lated.
- 44. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 44 Traditionally, in the UK, the variable rate has been adjustable at the discretion of the lender, but in recent years mortgages that charge a rate tied to a specific interest rate (such as LIBOR or the Bank of England repurchase rate) have become popular. In the USA the adjustable rate can be based on any rate. Currently, the Federal National Mortgage Association (FNMA), or Fannie Mae for short, offers a two-step purchase programme which specifies that the new rate be calculated by adding 250 basis points to a weekly average of the 10-year constant maturity Treasury yield. Fannie Mae also limit any increase in the mortgage rate to no more than 600 basis points over the initial mortgage rate. When valuing a theoretical ARM, it is necessary to decide how the contract rate will change during the life of the mortgage. Usually a contract rate will change according to an index. The precise details of how the index is derived may vary, but it will depend on a specific interest rate. The model used in chapter 5 for the valuation of an ARM uses an index which depends on the current interest rate plus a margin - this is just one way to model an index. Another example is illustrated in Stanton and Wallace (1995), where an index is used which lags behind shifts in the term structure; this is discussed in section 5.2.1. 2.3.1 Calculation of the index The index that is used in section 5.3 as part of the adjustment rule (see equation (5.1)), is the mortgage-equivalent rate or yield, index(r), for a 1-year, default-free pure discount bond (as used by Kau et al., 1993). Given the assumption of the Local Expectations Hypothesis (see Cox et al., 1981) and that the interest-rate process is the single-factor spot interest rate, there exists a closed-form solution for the pure discount bond yield. Cox et al. (1985b) give full details, but the solution is sum- marised next. The mortgage equivalent conversion takes into account the monthly compounding. From Cox et al. (1985b), the PDE for the price of a discount bond P(r, t), in the absence of the market price of risk, is 1 2 σ2 r r ∂2 P ∂r2 + κ(θ − r) ∂P ∂r + ∂P ∂t − rP = 0, (2.6)
- 45. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 45 with the terminal (expiration) condition P(r, t = T) = 1. The bond price takes the form P(r, t) = A(t)e−B(t)r , (2.7) where A(t) = 2γe(γ+κ)(T−t)/2 (γ + κ)(eγ(T−t) − 1) + 2γ 2κθ/σ2 r , (2.8) B(t) = 2(eγ(T−t) − 1) (γ + κ)(eγ(T−t) − 1) + 2γ , (2.9) γ = κ2 + 2σ2 r . (2.10) The yield-to-maturity, R(r, t) is defined by e−(T−t)R(r,t) = P(r, t). Therefore, R(r, t) = [rB(t) − logA(t)]/(T − t). (2.11) Equation (2.11) gives the pure discount bond yield. The mortgage-equivalent yield index(r) is given by equating the yield on the the principal amount at the con- tinuously compounded rate of R(r, t), to the yield on the principal amount at the mortgage equivalent monthly compounded rate index(r), i.e. P(r, t)eR(r,t) = P(r, t) 1 + index(r) 12 12 , (2.12) and so index(r) = 12[eR(r,t)/12 − 1]. (2.13) Equation (2.13) gives the mortgage equivalent yield on a 1-year, default-free, pure discount bond. This is used to calculate the index, given the current interest rate, which is added to the margin when calculating the new contract rate at an adjustment date for the ARM. Figure 2.1 shows the profile of the index as the interest rate changes, for various different values of the long-term mean of the short-term interest rate θ. Notice that initially when the interest rate is less than the long-term mean, the index slightly leads the underlying interest rate, i.e. index(r) > r for r < θ. Once the interest rate is above the long-term mean, the index lags behind the underlying interest rate, i.e. index(r) < r for r > θ.
- 46. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 46 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 PSfragreplacements Interest Rate r index(r) Figure 2.1: A graph of index(r) against interest rate r. For each line style, the long-term mean of the short-term interest rate θ, is 0.1, 0.2 and 0.3 from the bottom to the top, with σr = 0.1 and κ = 0.25. 2.4 Numerical methods Most problems that arise in financial mathematics cannot be solved analytically. Instead, numerical methods must be employed to obtain their solution. There are several numerical methods that can be used to approximate the value of such deriva- tive securities. Popular methods include the Monte Carlo method, lattice methods (binomial or trinomial trees), quadrature and finite-difference methods. This study will solely involve the implementation of the latter, although a few brief details, of each method, is given below. The Monte Carlo approach is a forward method, in that the solution starts from the initiation of the option at time t = 0. Random sample paths are generated according to which stochastic process is used to model the underlying asset. The sample paths are then discounted at a specified interest rate to find the implied option value. Boyle (1977) was first to develop a Monte Carlo simulation method for solving option-valuation problems. The main drawback with this numerical method is that it is a complex matter to value options which have early exercise features. As simulated paths are generated forward in time, it is difficult to decide when it is
- 47. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 47 optimal to exercise the option. As mortgage valuation involves finding the optimal time for the borrower to prepay, this effectively rules out the use of the Monte Carlo method. Moving to backward methods, the lattice approach and the finite-difference ap- proach can both readily handle early exercise features. Lattice, or tree methods were developed independently by Cox et al. (1979) and Rendlemann and Bartter (1979). The theory behind the method is that at each discrete moment in time, the asset price can either move up to a new level, down to a new level, or, in the case of the trinomial lattice, move to a third level. As the value of an option is known at expiry (the payoff) this value can be used to evaluate the option price at T − δt. This is performed recursively so that ultimately the value of the option at t = 0 can be cal- culated. Early exercise features are no problem, as valuation is performed backwards in time the option value can be compared to the value of the option price if early exercise is taken. The lattice approach suffers as the tree itself is not very flexible, it is difficult to align nodes with important asset prices, such as a barrier or the exercise price. Also, computations, are rather inefficient, as only a single option price is found from each calculation, unlike the finite-difference approach which produces a range of option values for each calculation. The first author to employ numerical integration or quadrature techniques to option pricing was Parkinson (1977). The more recent work of Andricopoulos et al. (2003, 2004), for a single underlying, and Andricopoulos et al. (2006) for multi- asset and complex path-dependent options, contain considerable improvements in convergence and accuracy over previous quadrature benchmark methods. The main difficulty with this numerical method is that the mathematics required to formulate the integrand itself is often very difficult, even if it is actually possible to then perform the integration. As such, the exact integrand required for assets following processes other than the lognormal diffusion processes is still in the early stages of development. The PDEs that arise in Mathematical Finance can often be best solved directly using finite-difference methods. This type of numerical method is ideally suited to optimal-stopping problems and it was for this type of problem that Brennan and
- 48. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 48 Schwartz (1977) first recognised the utility of this method to price American put options. Solution via a backward method makes it simple to track the free bound- ary, which determines when it is optimal to exercise the option. Finite-difference methods are extremely flexible and allow the inclusion of complex path-dependent option features, such as occupation-time derivatives, to cause very little problem to the financial engineer attempting to value a problem of this nature. See section 4.2 for a detailed description of the method for pricing these types of derivatives. 2.4.1 Finite-difference methods The framework for pricing options is built around the Black-Scholes (1973) equation. Although this backward parabolic equation can be solved analytically (in simple cases), it can be very efficiently solved by use of finite-difference methods. These can be adapted to handle with ease many problems based on the Black-Scholes equation, with mortgages treated as derivatives being no exception. Finite-difference methods provide the user with an intuitive feel for the problem and how the solution is produced. The underlying problem is converted from one which exists over a continuous domain to a problem that can be described on a finite domain. The derivatives in the partial differential equation are discretised to form linear difference equations and the state space for the problem is replaced with a mesh on which the problem is defined. Brennan and Schwartz (1977), used an explicit method to obtain the price for an American put option. There are other (improved) variations of finite-difference methods that exist and are used today in more complicated financial problems. Other methods can give better accuracy and can be used in a wider variety of situations. Next the derivation of the most basic difference equations is shown. From standard calculus, the following approximations are valid for the derivative of a function u(x, t). A forward derivative approximation is ∂u(x, t) ∂x = u(x + ∆x, t) − u(x, t) ∆x + O(∆x), (2.14)
- 49. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 49 a backward derivative approximation is ∂u(x, t) ∂x = u(x, t) − u(x − ∆x, t) ∆x + O(∆x), (2.15) and a central derivative approximation is ∂u(x, t) ∂x = u(x + ∆x, t) − u(x − ∆x, t) 2∆x + O(∆x2 ). (2.16) The difference equations are obtained by removing the error terms indicated by the ‘O’ notation. The order of the error for each of the equations is easily seen by considering the Taylor series expansion of u about x, u(x + ∆x, t) = u(x, t) + ∆x ∂u ∂x + ∆x2 2 ∂2 u ∂x2 + . . . = ∞ n=0 ∆xn n! ∂n u ∂xn , (2.17) and u(x − ∆x, t) = u(x, t) − ∆x ∂u ∂x + ∆x2 2 ∂2 u ∂x2 + . . . = ∞ n=0 (−1)n ∆xn n! ∂n u ∂xn , (2.18) where ∂nu ∂xn denotes the n-th order derivative of u with respect to x. Equation (2.17) leads to the forward derivative approximation (2.14), whilst (2.18) leads to the back- ward derivative equation (2.15), and both approximations have an error O(∆x). The central derivative equation (2.16) is obtained by subtracting equation (2.17) from equation (2.18), and has error O(∆x2 ). The terms O(∆x) and O(∆x2 ) indicate the truncation error of the difference equations. If a better approximation is required, the computational mesh (on which the approximated solution is calculated), can be made finer (by making ∆x smaller) or information can be added by including higher-order neighbouring terms, which will involve additional mesh points. In order to simplify the notation, when convenient, the discretisation points will be labelled with appropriate indices. With uk i ≡ u(xi, tk) where xi = i∆x, tk = k∆t, for example equation (2.16) becomes ∂u ∂x ≈ uk i+1 − uk i−1 2∆x . (2.19)
- 50. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 50 2.5 Numerical solution Throughout the following study of various mortgage pricing techniques, the Crank- Nicolson finite-difference method will be employed. Finite-difference algorithms re- place derivatives with difference equations and approximate the solution of the PDE by a set of algebraic equations. For convenience, the analysis will involve the following transformation: τ = T − t. (2.20) This transforms the governing PDE (1.17) to a forward parabolic equation in τ. In the physical world, parabolic equations are generally solved forward in time, starting from an initial condition. The Crank-Nicolson finite-difference method will be used as convergence for this method is superior to the more basic explicit and implicit methods (which of both have convergence at the rate O(∆τ, ∆H2 , ∆r2 )). The Crank- Nicolson method converges at the rate O(∆τ2 , ∆H2 , ∆r2 ) and unlike the explicit method, there is no stability constraint. Section 3.5 provides a full exposition of how to space the finite-difference grid, and how the mortgage valuation PDE (1.17) is discretised according to the Crank- Nicolson finite-difference scheme follows. 2.5.1 Derivative approximations The valuation PDE (1.17) is discretised following a Crank-Nicolson finite-difference scheme to ensure second-order accuracy in underlying house price, interest rate and time. The time derivative is approximated as ∂F(H, r, τ + 1 2 ∆τ) ∂τ ≈ Fk+1 i,j − Fk i,j ∆τ . (2.21) The spatial derivatives for house price H are approximated by ∂F(H, r, τ + 1 2 ∆τ) ∂H ≈ (Fk+1 i+1,j − Fk+1 i−1,j + Fk i+1,j − Fk i−1,j) 4∆H , (2.22) ∂2 F(H, r, τ + 1 2 ∆τ) ∂H2 ≈ (Fk+1 i+1,j − 2Fk+1 i,j + Fk+1 i−1,j + Fk i+1,j − 2Fk i,j + Fk i−1,j) 2(∆H)2 . (2.23)
- 51. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 51 The spatial derivatives for interest rate r are approximated by ∂F(H, r, τ + 1 2 ∆τ) ∂r ≈ (Fk+1 i,j+1 − Fk+1 i,j−1 + Fk i,j+1 − Fk i,j−1) 4∆r , (2.24) ∂2 F(H, r, τ + 1 2 ∆τ) ∂r2 ≈ (Fk+1 i,j+1 − 2Fk+1 i,j + Fk+1 i,j−1 + Fk i,j+1 − 2Fk i,j + Fk i,j−1) 2(∆r)2 . (2.25) The cross-spatial derivative is approximated by ∂2 F(H, r, τ + 1 2 ∆τ) ∂H∂r ≈ 1 8∆H∆r (Fk+1 i+1,j+1 − Fk+1 i−1,j+1 − Fk+1 i+1,j−1 + Fk+1 i−1,j−1 +Fk i+1,j+1 − Fk i−1,j+1 − Fk i+1,j−1 + Fk i−1,j−1). (2.26) Finally, the asset F(H, r, τ) is approximated by F H, r, τ + 1 2 ∆τ ≈ Fk+1 i,j + Fk,l i,j 2 . (2.27) Overall the error in the approximate solution F k i,j is of second-order accuracy in ∆H, ∆r and ∆τ. 2.5.2 Discrete representation Upon substituting the derivative approximations from section 2.5.1 into the governing PDE (1.17) and rearranging, the problem of solving the PDE reduces to solving the following set of simultaneous linear equations for F k+1 i,j ai,jFk+1 i,j−1 + bi,jFk+1 i,j + ci,jFk+1 i,j+1 + di,jFk+1 i−1,j + ei,jFk+1 i+1,j +fi,j[Fk+1 i+1,j+1 − Fk+1 i,j+1 − Fk+1 i+1,j−1 + Fk+1 i−1,j−1] = −ai,jFk i,j−1 − 2 ∆τ + bi,j Fk i,j − ci,jFk i,j+1 − di,jFk i−1,j −ei,jFk i+1,j − fi,j[Fk i+1,j+1 − Fk i,j+1 − Fk i+1,j−1 + Fk i−1,j−1], (2.28) where ai,j = rjσ2 r 4(∆r)2 − κ(θ − rj) 4∆r , (2.29) bi,j = − H2 j σ2 H 2(∆H)2 − rjσ2 r 2(∆r)2 − 1 ∆τ − rj 2 , (2.30)
- 52. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 52 ci,j = rjσ2 r 4(∆r)2 + κ(θ − rj) 4∆r , (2.31) di,j = H2 i σ2 H 4(∆H)2 − (rj − δ)Hi 4∆H , (2.32) ei,j = H2 i σ2 H 4(∆H)2 + (rj − δ)Hi 4∆H , (2.33) fi,j = ρσHσrHi √ rj 8∆H∆r . (2.34) Next, the solution of this set of algebraic equations is considered. 2.5.3 Solution of the difference equations At any time step during the valuation of the mortgage, the value of the asset F(H, r, τ) must be calculated for all house-price and interest-rate values. Moving to the discrete representation of the problem, this means that F k+1 i,j must be found all for i and j. It is not easy to directly solve the system of equations (2.28), since the two-dimensional matrix problem produced is particularly complicated. For the results produced in chapter 3, for the case of the solution of a two-factor UK FRM, a general LU solver standard library package is employed (see Wilmott et al., 1993, for more on LU decomposition). The default D, insurance I and coinsurance CI components are calculated using the general LU solver standard library package. The coefficients ai,j, bi,j, etc, are the input for the package; the output is the value of the particular component at the present time step, for further details of the actual implementation and for full details of the solution for the two-factor UK FRM valuation (using the Crank-Nicolson finite-difference method), see the pseudocode given in Appendix A. The value of the remaining payments is dependent only on the interest rate and time, and this is discussed in section 2.6.1. This implies that the matrix problem produced to calculate Ak+1 j using the Crank-Nicolson method is tridiagonal at each time step, which can readily be solved using Gaussian elimination (see Smith, 1978). The value of the mortgage to the lender V has the added complication of a free boundary, which
- 53. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 53 determines when it is optimal for the borrower to prepay. This component is valued by considering the problem in its linear complementarity form (see Wilmott et al., 1993) and then solving the resulting constrained-matrix problem using the projected successive over-relaxation method (PSOR); see section 3.6.1 for the details. An alternative method to the general LU solver library package is used in chapters 4 and 5 to solve for the default and insurance components. The successive over- relaxation method (SOR) is used. The only difference between this method and the PSOR method (Wilmott et al. 1993), as described generally in section 3.6.1, is that equation (3.39) is simplified to xk+1 i = xk i + ω(yk+1 i − xk i ), (2.35) as the test to ensure that xk+1 i ≥ ci is not required since there is no free boundary constraint for these components. It could be thought that this alternative iterative technique would be less computationally efficient than the direct solution technique, using the library package, but this is not the case, as discussed in section 4.4. The solution for the value of the mortgage component with the new prepayment model is discussed in chapter 4. The modelling details require that a more sophisti- cated solver be used, as described in section 4.4.1. Finally throughout this thesis the prepayment component is valued by rearranging the relation V = A−D −C, so that once V , A and D are calculated the prepayment component C can be inferred. 2.6 Boundary conditions The valuation of the different mortgage models, discussed in chapters 3, 4 and 5, require the following boundary conditions to close each problem. 2.6.1 The value of the remaining payments The value of the remaining payments A(r, t) is dependent only on the term structure of interest rates and time. Since A is independent of house price the valuation PDE
- 54. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 54 (1.17) for A reduces to 1 2 rσ2 r ∂2 A ∂r2 + κ(θ − r) ∂A ∂r + ∂A ∂t − rA = 0. (2.36) Condition at r = 0 Setting the interest rate to zero, r = 0, directly into equation (2.36) leads to, κθ ∂A ∂r + ∂A ∂t = 0, (2.37) which serves as a boundary condition for A(r = 0, t). Condition as r → ∞ In the limit of large interest rates any expected future payment is worthless. In accordance with Azevedo-Pereira et al. (2002), then lim r→∞ A(r, t) → 0. (2.38) However, it is more computationally convenient to impose the corresponding Neu- mann boundary condition, namely lim r→∞ ∂A(r, t) ∂r = 0. (2.39) This is a ‘softer’ condition than equation (2.38), and enables a smaller domain trun- cation rmax to be used. The differing modelling details, between the contracts considered in this thesis, make it more appropriate to discuss the payment-date conditions, which complete the boundary conditions for the value of the remaining payments A(r, t), at the relevant points in chapters 3, 4 and 5. 2.6.2 The value of the other mortgage components Here, unless stated otherwise, F(H, r, t), represents any of the components: V , D, C, I and CI (the coinsurance is only relevant to the UK FRM considered in chapter 3). For these remaining mortgage components, consider the boundary conditions to be imposed at the extremes of the grid. Condition at H = 0, r = 0
- 55. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 55 An intuitive condition can be derived if H and r are simply set equal to zero in equation (1.17); the asset pricing PDE becomes, κθ ∂F(0, 0, t) ∂r + ∂F(0, 0, t) ∂t = 0. (2.40) Condition as H → ∞, r = 0 If H tends to infinity and r = 0 in equation (1.17); the asset pricing PDE reduces to, lim H→∞ ∂F(H, 0, t) ∂H → 0. (2.41) Condition as H → ∞, r → ∞ and at H = 0, r → ∞ In the limit of large interest rate any asset is worthless, therefore, lim r→∞ F(H, r, t) → 0. (2.42) For the above condition the following equivalent Neumann boundary condition can be used if it is numerically expedient to do so, namely lim r→∞ ∂F(H, r, t) ∂r → 0. (2.43) Condition along H = 0 If the house price becomes zero, the borrower will default and the mortgage is now worth the same as the house, and so V (0, r, t) = 0. (2.44) Prepayment at this point is worthless, thus C(0, r, t) = 0. (2.45) The option to default is now equal to the value of the remaining payments. Since D = A − C − V , then D(0, r, t) = A(r, t). (2.46)
- 56. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 56 The value of I and CI is given by equation (2.47) with F(0, r, t) replaced by either I or CI, 1 2 rσ2 r ∂2 F ∂r2 + κ(θ − r) ∂F ∂r + ∂F ∂t − rF = 0, (2.47) which is a degenerate form of equation (1.17) with H = 0. Condition when r = 0 Substituting r = 0 directly into equation (1.17) gives 1 2 H2 σ2 H ∂2 F ∂H2 + κθ ∂F ∂r − δH ∂F ∂H + ∂F ∂t = 0. (2.48) Condition along H → ∞ As H → ∞ the value of the default option tends to zero. Since there is no value in default, the insurance and the coinsurance have no value, and therefore: lim H→∞ D(H, r, t) = 0, (2.49) lim H→∞ I(H, r, t) = 0, (2.50) lim H→∞ CI(H, r, t) = 0. (2.51) The value of the mortgage V is constant as H tends to infinity, implying ∂V/∂H → 0. V is then determined by a degenerate form of equation (1.17) with ∂V/∂H and ∂2 V/∂H2 both set to zero, namely 1 2 rσ2 r ∂2 V ∂r2 + κ(θ − r) ∂V ∂r + ∂V ∂t − rV = 0. (2.52) Since the value of the mortgage is the difference between the value of the remaining future payments and the borrower’s joint option to terminate the mortgage, the prepayment option value at this extreme is given by lim H→∞ C(H, r, t) = A(r, t) − lim H→∞ V (H, r, t). (2.53)
- 57. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 57 Condition as r → ∞ Since, in the limit of infinite interest rate, any asset is worthless, the following is taken as the boundary condition for F as r tends to infinity, lim r→∞ F(H, r, t) = 0. (2.54) Alternatively, if necessary, it is possible to use the corresponding Neumann condition (similar to the limit when r → ∞) for the corners of the grid (see equation (2.43)). Again, due to the differing modelling details between the contracts considered, it is more appropriate to discuss the payment-date conditions, which complete the boundary conditions for the mortgage components V , D, C, I and CI, at the relevant points in chapters 3, 4 and 5. 2.6.3 Default boundary The option to default is serial-European in nature, since it can only be exercised (if the borrower chooses to do so) on the payment date in any particular month. Also, default cannot occur if the option to prepay is exercised. Therefore, default is only rational outside the prepayment region and the default boundary is described fully by the payment-date conditions. 2.6.4 Prepayment boundary The prepayment boundary is discussed at the relevant point in each chapter. Section 3.6.1 explains the prepayment boundary condition when prepayment occurs ratio- nally, as an effort by borrower’s to minimise the cost of the mortgage to themselves. This assumption is used again in chapter 5 on valuing ARMs. Chapter 4 considers an alternative prepayment assumption, as an attempt to improve FRM valuation, as actual borrowers tend to wait for a time after theory says it is optimal to prepay, a time lag before the borrowers prepay is introduced. This is explained in detail in section 4.3.1.
- 58. Chapter 3 An improved fixed-rate mortgage valuation methodology with interacting prepayment and default options The work in this chapter draws extensively on that presented in Sharp et al. (2006). 3.1 Introduction This chapter considers in detail a realistic mortgage valuation model (including the potential for early prepayment and the risk of default), based on stochastic house- price and interest-rate models. As well as the development of a highly accurate nu- merical scheme to tackle the resulting partial differential equations, this chapter also exploits singular perturbation theory (a mathematically rigorous procedure, based on the idea of the smallness of the volatilities), whereby mortgage valuation can be quite accurately approximated by very simple closed-form solutions. Determination of equilibrium contract rates, previously requiring many computational hours (using the highly accurate numerical scheme) is reduced to just a few seconds, rendering this a highly useful portfolio management tool; these approximations compare favourably 58
- 59. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 59 with the full numerical solutions. The method is of wide applicability in US or other mortgage markets and is demonstrated for UK FRMs, including insurance and coin- surance. Contingent claims analysis leads to the modelling of many derivative securities as PDEs. The need to model increasingly sophisticated products realistically has resulted in the development of extremely complex valuation frameworks, whose com- putation often prove excessively time-consuming. Collin-Dufresne and Harding (1999) note the utility of reduced calculation times for mortgage values as a useful tool for portfolio management and develop a closed-form formula for the value of a fixed- rate residential mortgage dependent only on a single state variable. However, until now, for models using two state variables including both default and prepayment, no closed-form solutions of any kind have been available. Several numerical procedures based on the explicit finite-difference method have been published for the solution of a contingent claims valuation model aimed at valuing mortgage-related products, including the work of Kau et al. (1992, 1995), on US mortgages and Azevedo-Pereira et al. (2000, 2002, 2003), on UK mortgages, which use two state variables. Brunson et al. (2001), describe a three-state variable model with a two factor term structure and a one factor property process. They argue that two-state variable models lack flexibility, failing to reflect the evolution of the whole term structure by using just one factor to represent the term structure and therefore these misprice the mortgage value. Conversely, empirical work by Chatterjee et al. (1998) indicates that a two variable model (short rate and building value) is the most efficient, in terms of pricing accuracy, of all the alternative mortgage valuation models that are available. The problem addressed in this chapter will follow a two-state variable model, and will also establish a new technique for extremely rapid valuation, using UK FRMs as the practical example. However, the technique is more broadly applicable to US and foreign mortgages; for example, it would be directly applicable to the US mortgage- backed securities model of Downing et al. (2005). The model falls into the structural category, where default and prepayment are treated as the exercise of options held
- 60. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 60 by the borrower. The common alternative is to use a reduced-form model where termination is modelled as a function of a set of externally based explanatory vari- ables, as in the early work by Schwartz and Torous (1989, 1992) in which a two-factor model including a hazard rate was used to derive a PDE for the mortgage contract value. For a helpful survey, including recent applications of credit risk techniques to mortgage valuation, see Pliska (2005). For a practitioner’s viewpoint, focusing on refinancing behaviour in a model based on an optimal exercise strategy, see Kalotay et al. (2004), and for a multi-factor term structure approach incorporating premium rate refinancing, see Longstaff (2005). However, the thrust of this chapter is not to favour a particular model category but to show how simpler (and very much quicker) solutions may be had for models where the solution of a PDE is required. This chapter presents a singular perturbation approach to the valuation at origi- nation of FRMs with default and prepayment; no previous literature appears to have employed this approach in mortgage valuation. In this model, mortgages are treated as derivative assets whose prices depend on the evolution of the global economy via both house prices and the term structure of interest rates, which are determined us- ing appropriate models. Following on from these determinations, the value of the mortgage is set through a process of arbitrage inference. All other factors that might exert some influence are taken into consideration through the market price of risk associated with each state variable. The first state variable, house price, is taken as a traded asset, and so risk adjustment becomes unnecessary; it is modelled as a lognor- mal diffusion. This is a standard assumption, but could be modified with trivial effect on the computational method. The second state variable, interest rate, is represented by the instantaneous spot interest rate modelled as a mean-reverting, square root pro- cess. Any interest-rate risk premium from this state variable is assumed either to be embedded in the reversion and long-term average parameters (see equation (1.2)) or, alternatively, its absence can be explained via the Local Expectation Hypothesis (see Cox et al., 1979, 1981). The valuation procedure considers two forms of endogenous termination prior to maturity of the mortgage, whereby the borrower minimises the market value of
- 61. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 61 the loan, namely prepayment and default by the mortgage borrower. Financially rational termination by the borrower is used and the effect of exogenous factors or the borrower’s individual characteristics on the termination of the mortgage are neglected (such as a sudden increase in wealth from an inheritance). Just prior to each monthly mortgage payment, the borrower decides whether to default or to make the scheduled monthly payment and it is assumed that the borrower shows a financially ruthless default behaviour (first employed by Foster and Van Order, 1984, 1985; see also Vandell, 1995, for a review). This implies that the borrower defaults when the value of the property drops below the value of the mortgage. It is assumed that default results in the loss of the house in exchange for forgiveness of the debt. The borrower can also choose to prepay the mortgage in full at any time prior to maturity if interest-rate changes make this financially favourable. As the value of the mortgage is affected by the options to prepay and default, it is necessary to use a procedure allowing for valuation backwards in time. The aim was to follow the valuation structure laid down by Kau et al. (1995) and by Azevedo-Pereira et al. (2000), who used finite-difference techniques, and then to establish a practically simpler technique through the application of singular perturbation theory. The UK mortgage problem of Azevedo-Pereira et al. (2000) is used for demonstration in which, for a 25 year mortgage, there is a series of 300 monthly European-style options to default and an overarching (and interacting) American-style option to prepay at any time. In this early work on UK mortgages, the most basic type of finite-difference approach, the explicit method, was employed, focusing on the finance in what remains a fairly complex mortgage options problem to solve numerically. However, in this chapter, since a new, simpler and much faster technique is presented, it is more appropriate to set up comparisons with a much superior finite-difference technique. The approach employed in this chapter is a significant improvement and involves a semi-implicit Crank-Nicolson finite-difference scheme to discretise the PDE. The treatment of the free boundary associated with the prepayment aspect of the problem, using the linear complementarity method coupled with the projected successive over- relaxation (PSOR) method (details are in section 3.6.1) completes the upgrading
- 62. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 62 of the numerical technique, against which the new analytic (singular perturbation) method will be compared. It is notable that even with the improvements in raw computing power since the earlier work of Kau et al. (1995) and Azevedo-Pereira et al. (2000), and using improved finite-difference schemes, fully numerical calculations remain ponderously slow. The chapter proceeds as follows. In section 3.2, the valuation framework is spec- ified. This establishes the governing PDE for the valuation of the mortgage compo- nents. The structure of the mortgage contract and the payment-date conditions for all the components are given in section 3.3. Two solution methods are then described. Section 3.5 details the improved finite-difference (fully numerical) approach (which is used as the ‘exact’ solution), and subsequently section 3.7 outlines how a perturba- tion approach produces a simple analytic approximation. The crucial aspect to both approaches is how they produce an equilibrium setting contract rate, as described in section 3.8, along with the base parameters. In section 3.9 the contract rates found using the two procedures are discussed and compared. The results show that the analytic approximation is good and computationally trivial. The chapter concludes in section 3.10 with some comments on extensions to increase the accuracy of the analytic approximations and directions for future research. 3.2 Valuation framework In the contingent claim framework, house price H(t) is modelled as a lognormal diffusion process (Merton 1973), see equation (1.1), and interest rate r(t) is modelled as a CIR mean-reverting square root process (Cox et al., 1985b), see equation (1.2). The stochastic elements of H(t) and r(t) involve the standardised Wiener processes, XH(t) and Xr(t) respectively, which are correlated according to equation (1.3); the parameters involved are discussed in section 1.3. Further discussion of parameters and their relevance in the market and valuation under the risk-neutral measure may be found in Kau et al. (1995), Azevedo-Pereira (1997) and Azevedo-Pereira et al. (2002).
- 63. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 63 Using standard arguments, the PDE for the valuation of any asset F(H, r, t) whose value is a function only of house price H, interest rate r, and time t, takes the form of equation (1.17) (Cox et al., 1985a, 1985b; Epperson et al., 1985; Kau et al., 1992; Azevedo-Pereira et al., 2000, 2002, 2003). The solution of equation (1.17) must include the value of the remaining payments to the lender and the borrower’s options to terminate the contract prior to maturity by either prepayment or default. These components of the mortgage cannot be valued independently. The resulting mortgage valuation problem is made up of a series of European options to default (one for each month of the mortgage) with an American option to prepay overarching the entire contract; no analytic solution is available to this ‘full’ problem. Next, basic, specific details of the mortgage valuation problem will be laid out in preparation for solution by a finite-difference technique and for further consideration, and ultimate simplification, via singular perturbation theory. 3.3 Mortgage contract For a fixed-rate repayment mortgage the loan is repaid by a series of equal monthly payments on pre-determined, equally-spaced dates. The formulae for the monthly payment MP and the outstanding balance OB(i) after the ith monthly payment has been made, are given by equation (2.1) and equation (2.2), respectively. There are two embedded option types within the mortgage contract under consid- eration. First, the option to default on the mortgage, which occurs only at payment dates when the borrower makes the decision whether to pay the required monthly payment MP or default on the mortgage, and secondly the option to prepay the mortgage, which can occur at any time during the life-time of the mortgage. If the decision to prepay the mortgage is made, the borrower is liable to pay the lender an amount referred to as the total debt payment TD(τm), which will include a penalty for the early termination of the contract; this payment is not standardised across mort- gages. Here a general payment is modelled, which is a proportion of the outstanding
- 64. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 64 balance plus accrued interest. Also, it is convenient to define τm = Tm − tm (3.1) as the time until the payment date in month m where Tm is the total time in one month and tm is the time that has elapsed in month m.1 The formula to calculate the total debt payment in the event of prepayment occurring at time τm until the payment date in month m is, TD(τm) = (1 + ψ) 1 + c(Tm − τm) OB(i), (3.2) where ψ is the prepayment penalty. The problem consists of valuing several different assets simultaneously. Each asset has its own payment-date conditions for each month of the mortgage. To find the value of each of these assets at the start of the mortgage, the valuation PDE for F(H, r, τm) must be solved in 0 ≤ τm ≤ Tm, 0 ≤ H < ∞, 0 ≤ r < ∞. Each month contains a separate option for the borrower to default. At the payment time, occurring at the end of a given month, the borrower must make the decision whether to pay the required monthly payment or default and hand over the house to the lender. The option to prepay is an American option and gives the borrower the right to exercise the prepayment option at any time during the lifetime of the mortgage. The prepayment option spans the length of the mortgage. The mortgage value V (H, r, τm) is the difference between the value of remaining future payments promised to the lender A(r, τm) and the value of the borrower’s op- tions. The borrower’s joint option value is the sum of the option to prepay C(H, r, τm), eliminating the debt early, and the option to default D(H, r, τm), reneging on the debt and turning over the house to the lender. The equation which describes the value of the mortgage is as follows, V (H, r, τm) = A(r, τm) − D(H, r, τm) − C(H, r, τm). (3.3) The mortgage value is discussed in terms of its value to the lender (excluding the insurance the lender has on the loan). 1 This transforms (1.17) to a forward PDE which is solved starting from an initial condition.
- 65. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 65 Although it may seem from equation (3.3) that the three components of the mort- gage can be considered in isolation, this is not the case. At the extremes, if default occurs, prepayment cannot then happen, and alternatively, if the borrower chooses to prepay, then the default option becomes worthless. The values then interact across all circumstances. 3.3.1 Mortgage payment-date conditions At maturity, the value of the scheduled monthly payment is, A(r, τn = 0) = MP. (3.4) The borrower may either pay the required monthly amount MP or default. The value of the mortgage to the lender immediately before the payment at maturity is the minimum of MP and the house value, V (H, r, τn = 0) = min(MP, H). (3.5) The default option will be worthless if the value of the house is greater than the value of the final monthly payment. Otherwise, the option will be equal to the difference between the two, D(H, r, τn = 0) = max(0, MP − H). (3.6) However, the option to prepay at maturity, by definition, has no meaning; therefore the value of this option is zero, C(H, r, τn = 0) = 0. (3.7) Given next are the value of all the mortgage components for the other payment dates, namely at the end of month 1, 2, . . . , n − 2 and n − 1. The value of the remaining future payments promised to the lender at these times is, A(r, τm = 0) = A(r, τm+1 = Tm+1) + MP, (3.8) where 1 ≤ m ≤ n − 1. The value of the mortgage to the lender is the lesser of MP plus its value after the payment, and the house price itself, V (H, r, τm = 0) = min V (H, r, τm+1 = Tm+1) + MP, H . (3.9)
- 66. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 66 Default occurs if the value of the house is less than the sum of the monthly payment plus the value of the mortgage immediately after the payment is made. In this situation the option components take the following values: D(H, r, τm = 0) = A(r, τm = 0) − H (3.10) and C(H, r, τm = 0) = 0, (3.11) since if default occurs, prepayment cannot happen. If default does not occur the borrower’s options have the values: D(H, r, τm = 0) = D(H, r, τm+1 = Tm+1), (3.12) C(H, r, τm = 0) = C(H, r, τm+1 = Tm+1). (3.13) 3.3.2 Insurance If the borrower chooses to exercise the option to default, the lender will lose future payments that would have been received had default not occurred. The lender may take advantage of an insurance policy which would cover a fraction of this loss. In the UK this policy is called a Mortgage Indemnity Guarantee (abbreviated to MIG hereafter). The MIG = I(H, r, τm) only adds to the lender’s position in the contract, as a financially rational borrower would not need to take this into account. The model for the MIG to be used is as follows: the insurer agrees to pay a fraction γ of the total loss TD(τm) − H suffered by the lender but only up to a maximum indemnity, or cap, of Γ. It is assumed this cap is 0.2 times the original value of the house, and that γ = 0.8. These assumptions are based on values a lender would normally expect, as utilised in the work of Azevedo-Pereira et al. (2000, 2002, 2003). The conditions at maturity are either that the MIG has some value because it is worthwhile for the borrower to default, or that the MIG has no value since the value of default to the lender is zero. Therefore at maturity, if default occurs I(H, r, τn = 0) = min γ(MP − H), Γ , (3.14)
- 67. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 67 and if default does not occur I(H, r, τn = 0) = 0. (3.15) At earlier payment dates, the value of the MIG to the lender if default occurs is I(H, r, τm = 0) = min γ TD(τm = 0) − H , Γ , (3.16) and if default does not occur I(H, r, τm = 0) = I(H, r, τm+1 = Tm+1), (3.17) where 1 ≤ m ≤ n − 1. 3.3.3 Coinsurance Coinsurance is the fraction of the potential loss not covered by the MIG and includes any loss above the cap. Since the coinsurance provides information about the extent of the coverage the MIG supplies, its value not only imparts useful information to the insurer, but also for the lender and for any third party insurers which may be interested in selling coverage for this source of risk. At each payment date the value of the coinsurance CI(H, r, τm) is equal to the difference between the potential loss and the insurance coverage provided by the MIG. Consequently, at the final payment date at maturity, the value of the coinsurance if default occurs is CI(H, r, τn = 0) = max (1 − γ)(MP − H), (MP − H) − Γ , (3.18) and if default does not occur, CI(H, r, τn = 0) = 0. (3.19) If default occurs at earlier payment dates, the value of the coinsurance is CI(H, r, τm = 0) = max (1−γ) TD(τm = 0)−H , TD(τm = 0)−H −Γ , (3.20) and if default does not occur at earlier payment dates, CI(H, r, τm = 0) = CI(H, r, τm+1 = Tm+1), (3.21) where 1 ≤ m ≤ n − 1.
- 68. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 68 3.4 The equilibrium condition At the origination of the contract, it is assumed that neither the borrower nor the lender would enter into an agreement unless both parties agreed that the mortgage was fair. From a financial point of view, this means that the terms of the mortgage have to be set such that arbitrage is avoided, i.e. if the value of the contract to the lender (including the MIG which the lender may hold) is equal to the amount lent to the borrower. The equilibrium or no-arbitrage condition for the mortgage contract is V Hinitial, rinitial, τ1 = T1; ψ, c + I Hinitial, rinitial, τ1 = T1; ψ, c = (1 − ξ)OB(0). (3.22) A feature which affects the borrower’s position is the arrangement fee ξ which is modelled as a proportion of the amount lent to the borrower. The arrangement fee, the prepayment penalty ψ and whether or not the lender holds a MIG are known details specified in the contract. A contract rate c which will satisfy the equilibrium condition, is as yet unknown. Since this condition must be set prior to the contract commencing, it is necessary to find an appropriate contract rate capable of balancing equation (3.22). For details of this, see section 3.8. 3.5 Benchmark method: the enhanced finite-difference approach This section details how the mortgage valuation problem can be solved numerically using an improved finite-difference method. Explicit methods have been the tech- niques of choice in previous research on this problem; see Kau et al. (1995) and Azevedo-Pereira et al. (2002). Here, a Crank-Nicolson finite-difference method is employed. Although the explicit method is adequate for the purpose, the present method is far superior to the more basic explicit and implicit methods, which both have convergence at the rate O(∆τm, ∆H2 , ∆r2 ) (where the ‘∆’ quantities refer to the grid sizes in the appropriate dimensions, as detailed below), whilst for Crank- Nicolson methods, convergence is at the faster rate O(∆τ2 m, ∆H2 , ∆r2 ). Further, in
- 69. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 69 contrast with explicit methods, Crank-Nicolson methods, have no intrinsic stability constraints. Also, a different approach is taken to the free boundary aspect of the problem (compared to previous work in this area), the choice being to use the linear comple- mentarity method and then to solve the resulting non-linear problem iteratively using PSOR (see section 3.6.1), as suggested by Wilmott et al. (1993) to value American options. As described in section 3.2, the mortgage valuation problem to be considered is a serial European option problem with an overarching American feature, where the values from options to be calculated are dependent on previously calculated values. For the numerical solution it is necessary to solve equation (1.17) in terms of each of the assets in the contract, for each month of the mortgage. When defining a (finite, truncated) equally spaced grid suppose, 0 ≤ H ≤ Hmax, 0 ≤ r ≤ rmax and 0 ≤ τm = (Tm − tm) ≤ Tm. Then the function F(H, r, τm) is represented by its values on the discrete set of points: H = Hi = i∆H where 0 ≤ i ≤ imax, (3.23) r = rj = j∆r where 0 ≤ j ≤ jmax, (3.24) τm = τmk = k∆τ where 0 ≤ k ≤ kmax. (3.25) ∆H, ∆r and ∆τ are the grid spacings in the H, r and τm dimensions respectively. imax and jmax are the number of nodes along the spatial H and spatial r axes respectively and kmax is the number of time steps dividing each month of the contract. Writing Fk i,j ≡ F(Hi, rj, τmk ), (3.26) for each (i, j, k) triple. The Crank-Nicolson method reduces equation (1.17) to the form, MFk+1 = Dk , (3.27) where M is a block-banded matrix, Fk+1 is a vector whose elements are the Fk+1 i,j , and Dk is a vector whose elements are functions of F k i,j. This system of equations relates the values of the asset for different values of H and r at time τmk to possible values
- 70. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 70 at time τmk+1 . To perform the mortgage valuation: start at the final month when time until the payment date is zero; here the value of the mortgage V , its embedded assets (A, D, C) and the value of the insurance products related to it (I, CI) are all known; equation (3.27) is solved repeatedly, working backwards one step at a time until the values of all the assets are known at the beginning of the final month; these values provide information necessary for the payment-date conditions in the previous month. Then, these known conditions enable a time stepping procedure towards the solution at the beginning of the penultimate month. This process continues until the values of all the mortgage components at the origination of the contract are obtained. In particular, it is then possible to evaluate whether the no-arbitrage condition is satisfied by using these values. For a problem, such as mortgage valuation, consisting of two state variables, solution using the Crank-Nicolson finite-difference method is not straightforward. The value of the remaining future payments A(r, τm) is dependent only on the interest rate and time, so the matrix (tridiagonal) problem produced for this component can be solved easily using Gaussian elimination. However, the other components are dependent on both state variables and finding their value is more difficult. The Crank-Nicolson method produces a matrix equation of the form (3.27) for each of the other components, solution of which was found using a standard library package. The value of the mortgage to the lender V (H, r, τm) is discussed further in the next section, which also considers the boundary conditions which constrain the underlying valuation equation (1.17). 3.6 The boundary conditions The boundary conditions used are similar to those in the work of Azevedo-Pereira et al. (2000, 2002, 2003) and Kau et al. (1995). A full discussion of these conditions is presented in section 2.6. However, the prepayment boundary will be considered further here.
- 71. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 71 3.6.1 The free boundary condition As noted earlier, the option to prepay is American in type, in the sense that prepay- ment could occur at any time during the lifetime of the contract (as usual, allowance for the inability to exercise the option outside working hours is ignored as unnec- essary). This produces a free boundary which must be applied in the appropriate position. On one side of the free boundary it is financially optimal for the borrower to prepay and on the other it is not. The prepayment boundary condition is obtained by observing that at each moment in time the value of the mortgage to the lender (not including insurance the lender has on the loan) can be no greater than the value of the total debt TD, V (H, r, τm) ≤ TD(τm), (3.28) otherwise the lender would choose to prepay the mortgage. This occurs when the value of the mortgage to the lender is equal to the total debt required to be paid by the borrower if the mortgage was chosen to be prepaid at that time, V (H, r, τm) = TD(τm). (3.29) Clearly, it is important to position the free boundary accurately. In this chapter the problem is treated using the linear complementarity method, as discussed by Wilmott et al. (1993). The free boundary is not calculated per se, but can be found if necessary. The method involves writing the finite-difference equation (3.27) for the mortgage component V (H, r, τm) and the discretised equation for the prepayment boundary condition (3.28) as a constrained matrix problem. This problem can then be solved using the PSOR method which involves iterating on the equations produced from the matrix problem until the difference in successive iterates is small enough to be regarded as negligible. For a full exposition of the solution scheme including the free boundary, see section 3.6.2 below. Linear complementarity coupled with the PSOR method ensure an internally con- sistent solution where the solution is unique (see Crank, 1984 for more information on linear complementarity problems and free boundary problems). Rather than constant tracking, the linear complementarity formulation (for the valuation of the mortgage
- 72. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 72 component) eliminates the explicit dependence on the free boundary. This process leads to an efficient, straightforward and accurate numerical solution scheme. 3.6.2 Solving the free boundary problem It can be shown that the free boundary problem associated with valuing the mortgage (discussed in section 3.6.1), can be reduced to a linear complementarity problem. A linear complementarity problem is such that in general A ≥ 0, B ≥ 0, AB = 0 (3.30) (i.e. either A = 0 or B = 0). The valuation PDE for the mortgage component V (H, r, τm) is given by the linear operator, L{V } ≡ 1 2 H2 σ2 H ∂2 V ∂H2 + ρH √ rσHσr ∂2 V ∂H∂r + 1 2 rσ2 r ∂2 V ∂r2 +κ(θ − r) ∂V ∂r + (r − δ)H ∂V ∂H − ∂V ∂τm − rV = 0. (3.31) Then L{V } ≥ 0. (3.32) The prepayment constraint, equation (3.28), can be rearranged as TD(τm) − V (H, r, τm) ≥ 0. (3.33) Thus, (3.32) and (3.33) can be written in the linear complementarity form L{V } ≥ 0, TD(τm) − V (H, r, τm) ≥ 0, L{V } TD(τm) − V (H, r, τm) = 0. (3.34) The two possibilities in the formulation correspond to situations in which it is optimal to prepay (V = TD) and those in which it is not (L{V } = 0). The advantage of the linear complementarity formulation is that there is no ex- plicit mention of the free boundary. The optimal exercise boundary can be found by the condition that defines it, namely that it divides the continuation region where
- 73. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 73 V < TD from the prepayment region where V = TD; see Wilmott et al. (1993) for a discussion of the existence and uniqueness of the linear complementarity formulation to solve free boundary problems. The method of choice for solving the resulting constrained linear problem (3.34) is the PSOR solution scheme. To illustrate the method consider a more general version of (3.30), which applies to any linear complementarity problem. Let A = ax−b and B = x − c. Hence, (3.30) can now be written as ax ≥ b, x ≥ c (x − c) · (ax − b) = 0. (3.35) It can be shown that there is one and only one solution vector x for this problem. The algorithm for finding the solution is iterative. Start with an initial guess x0 ≥ c (for the mortgage problem the general linear solver is used, which would find the required solution if the free boundary were not present, to provide the initial guess). During each iteration a new vector is formed xk+1 = (xk+1 1 , xk+1 2 , . . . , xk+1 n ), (3.36) from the current vector xk , xk = (xk 1, xk 2, . . . , xk n), (3.37) by the following two-step process. For each i = 1, 2, . . . , n sequentially form the intermediate quantity yk+1 i , given by yk+1 i = 1 aii bi − i−1 j=1 aiixk+1 j − n j=i+1 aiixk i (3.38) and then define the new xk+1 i to be xk+1 i = max ci, xk i + ω(yk+1 i − xk i ) . (3.39) The constant ω is the relaxation parameter, and provided that x0 ≥ c and 0 < ω < 2, the method converges. At each iteration this defines a new vector xk+1 ≥ c, as k → ∞ xk → x, the solution of the problem. Iteration is terminated once the following condition is satisfied xk+1 − xk < (3.40)
- 74. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 74 where > 0 is some pre-chosen small tolerance (for this problem 10−12 was found to be sufficient); then xk+1 is taken as the solution. The PSOR method is discussed in some detail in Crank (1984), where details of the convergence of the method are also given. For a pseudocode of the improved finite-difference method described in this chapter to value a FRM, the associated embedded options and the related insurance products, see Appendix A. Before going on to consider numerical results obtained using the techniques de- scribed in this section, an alternative, analytic approximation to the mortgage valu- ation equations is considered. 3.7 Rapid approximation method: perturbation approach Having established a strong benchmark technique with the ‘enhanced’ finite-difference approach, consider now the development of an alternative, simple, very much faster but approximate technique via a perturbation approach. Asymptotic expansions have recently been shown to simplify considerably the mathematical effort required to solve problems involving the Black-Scholes PDE (Widdicks et al., 2005). Equation (1.17), with its boundary conditions for real estate finance valuation problems, has been solved via finite-difference techniques in section 3.5, but it is an attractive proposition to simplify to make solution easier and faster, albeit approximate. The key initial observation is that σ2 H and σ2 r multiply the high- est order derivatives (see Widdicks et al., 2005; for financial applications and for a general treatment of singular perturbation theory see van Dyke, 1975; Holmes, 1995; Kevorkian and Cole, 1996; Nayfeh, 2000) and so conditions are ripe for application of (singular) perturbation theory to (1.17), since these two volatilities are, in practice, small, in a numerical sense. The typical value (see Buser and Hendershott, 1984; Kau et al., 1992, 1995 for examples) of the two volatilities σH and σr is only around 0.05
- 75. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 75 (per (annum) 1 2 ). The basic technique, therefore is to assume the following expansion F(H, r, τm, σH, σr) = ∞ n=0 σn Fn(H, r, τm), (3.41) where σ is a measure of both volatilities, which we take to be of comparable size, i.e. we may write σH = σˆσH, σr = σˆσr, where ˆσH, ˆσr = O(1). (3.42) In its basic form, simplification to give a leading-order approximation can be per- formed on sight, by setting the volatilities in (1.17) to zero, and so the valuation of any asset F(H, r, τm) assumes the much reduced form (to leading-order) κ(θ − r) ∂F ∂r + (r − δ)H ∂F ∂H − ∂F ∂τm − rF = 0. (3.43) At this point, the reader should note the simplicity of equation (3.43) compared with the equation whose solution we ultimately seek, equation (1.17). Equation (3.43) is a first-order, three-dimensional PDE which has an especially simple analytic solution. The following subsection describes in detail how this equation can be solved with regard to each of the mortgage components and related insurance products. A single algorithm is presented to determine the value of any of the mortgage components at origination. The most important step in the algorithm is calculation of the general solution of the valuation PDE, equation (3.43). 3.7.1 General solution of the mth month In contrast with the brief presentation of the finite-difference technique, here more details of the new technique will be given in order that it can be easily replicated. This is the first step in the algorithm to value any of the mortgage components at origination, and hinges on the solution of equation (3.43). Using the Method of Characteristics (see Appendix B.1 for an explanation and Garabedian, 1998, for further details) equation (3.43) can be reduced to the following series of equations: dr κ(θ − r) = −dτm = dH (r − δ)H = dF rF , (3.44)
- 76. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 76 which can be solved by simple integration. To determine F(H, r, τm) from equation (3.44), for a particular month m, the only requirement is the appropriate initial condition. Let, F(H, r, τm = 0) = F0m (H0, r0) (3.45) be the general initial condition for month m. The general solution for any month is then F(H, r, τm) = F0m (H0, r0) exp 1 κ (θ − r)(1 − e−κτm ) − θτm , (3.46) where, H0 = Hexp (θ − δ)τm + 1 κ (θ − r)(e−κτm − 1) (3.47) r0 = θ − (θ − r)e−κτm . (3.48) For a full exposition of obtaining the general solution for any month see Appendix B.2. The algorithm begins by solving equation (3.43) for month n, the final month of the problem. It is shown next how the solution for month n feeds directly into the final condition needed to solve the problem in month n − 1. 3.7.2 Algorithm for value at origination (i) Find the general solution of the mth month for F(H, r, τm). (ii) Use (i) along with the final payment-date condition for a particular component to calculate the solution in the nth month, i.e. F(H, r, τn). (iii) Substitute τn = Tn into the solution from (ii) to find the solution at the start of month n. (iv) Find the payment-date condition of month n − 1 using (iii) to calculate the solution of month n − 1, i.e. F(H, r, τn−1). (v) Repeat steps (iii) and (iv) working progressively back one month at a time until the first month of the contract is reached.
- 77. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 77 (vi) Use the solution of first month, F(H, r, τ1), to find F(H, r, τ1 = T1), which is the value of the component at origination. 3.7.3 Value of the mortgage asset at origination The algorithm given in section 3.7.2 is used in conjunction with the general solu- tion given in section 3.7.1, to find the value of the mortgage asset at origination, V (H, r, τ1 = T1). (i) The first step in the asset valuation algorithm is to find the general solution for the mth month. This is demonstrated in section 3.7.1. (ii) Use the final payment-date condition, V (H, r, τn = 0) = min(MP, H), along with the general solution for month n to calculate the actual solution of month n for V (H, r, τn), V (H, r, τn) = min MPe 1 κ (θ−r)(1−e−κτn )−θτn , He−δτn . (3.49) (iii) Use (ii) to simply find V (H, r, τn = Tn). (iv) Using (iii) find the payment-date condition of month n − 1, V (H, r, τn−1 = 0) = min V (H, r, τn = Tn) + MP, H , (3.50) to then calculate the solution for month n − 1. From equation (3.50) it is seen necessary to substitute in the value of the mortgage asset at τn = Tn, which is the value at the start of month n (when the time until the payment is Tn). Thus, the solution for the penultimate month is, V (H, r, τn−1) = min min MPe 1 κ (θ−r) 1−e−κ(Tn+τn−1) −θ(Tn+τn−1) , He−δ(Tn+τn−1) + MPe 1 κ (θ−r)(1−e−κτn−1 )−θτn−1 , He−δτn−1 . (3.51) (v) Repeat steps (iii) and (iv) working progressively back one month at a time until the solution for the first month of mortgage is obtained.
- 78. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 78 (vi) From (v) it is straightforward to find the value of mortgage asset at origination, V (H, r, τ1 = T1) = n-terms min min · · · min · · · min MPe 1 κ (θ−r) 1−e−κMn −θMn , He−δMn + MPe 1 κ (θ−r) 1−e−κMn−1 −θMn−1 , He−δMn−1 · · · + MPe 1 κ (θ−r) 1−e−κMm −θMm , He−δMm · · · + MPe 1 κ (θ−r)(1−e−κM1 )−θM1 , He−δM1 , (3.52) where Mm = Tm + Tm−1 + · · · + T2 + T1. Equation (3.52) represents the method of characteristics solution for the value at the origination of the mortgage asset, in the limit of small volatilities. Early termination To calculate the value of the mortgage component, the possibility that the mortgage will not reach maturity must be considered carefully. The model allows two forms of early termination, by either default or prepayment. It makes financial sense to default only on a scheduled payment-date, when a monthly mortgage payment falls due. For the rest of the time, the house occupier chooses to enjoy the benefits of the house even in the knowledge of likely default in the next few weeks. This results in a series of European options to default, with an overarching single American option to prepay (remortgage). Default enters the approximations through the payment-date conditions and the leading-order equation is heavily dependent on these conditions. This is why default is not considered as a free boundary condition and why house price cannot be dropped as a state variable. Financially, as well as mathematically, it is important to bear in mind that the recurring monthly European option to default and the overarching American early exercise option interact (Azevedo-Pereira et al., 2000, 2002, 2003). The free boundary does not appear at the level of the approximation of the leading-order equation in
- 79. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 79 the asymptotic expansion of the governing PDE (3.43) (the free boundary is in fact a second-order effect), but the bridging solutions do include the free boundary (see Appendix C for a detailed discussion of bridging solutions and extensions to the perturbation approach). Although bridging solutions could be included, this would necessarily lead to more complex solutions, whilst the goal here is to approximate mortgage valuation by very simple closed-form solutions. Applying a full asymptotic analysis at this point would detract from the very purpose here, but the implications (and some limitations) of treating the problem to a fuller analysis are discussed in Appendix C. The equilibrium contract rate (the crux of the problem) obtained this way is, in reality, quite accurate, as will be seen in section 3.9. An effective route to solutions which do involve the free boundary, is to feed the approximate equilibrium contract rate, described above, into the finite-difference program, to yield an improved solution (without iteration) or if the ‘exact’ value is required, this can be used as a first, good approximation for the contract rate, which can then be iterated upon (using the finite-difference program). It is actually the calculation of the contract rate, iteratively, which makes finite-difference approaches so very slow in mortgage valuation calculations such as this. The algorithm outlined in section 3.7.2 can also be used to determine the value at origination of: the value of the remaining future payments; the value of the borrower’s option to default; and the value of the mortgage-related insurance component. The value, at origination, of all these mortgage components needs be found to determine the equilibrium contract rate (this is the purpose of the simple closed-form solutions). It is a simple matter to follow the steps in the algorithm using the necessary payment- date conditions (given in sections 3.3.1 and 3.3.2). 3.7.4 Value of the future payments at origination In section 3.7.3, the algorithm used to value any mortgage asset was implemented to calculate the value of the mortgage component at origination. Next, it is shown how the value of the remaining future payments at origination can also be found using
- 80. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 80 the perturbation method. Following the algorithm (shown in section 3.7.2) using the final payment-date condition (see equation (3.4)) for A(r, τn = 0), the solution in month n for the remaining future payments is, A(r, τn) = MPe 1 κ (θ−r)(1−e−κτn )−θτn . (3.53) Working through the algorithm, it is found that the solution at the beginning of the first month is, A(r, τ1 = T1) = MPe 1 κ (θ−r) 1−e−κMn −θMn + MPe 1 κ (θ−r) 1−e−κMn−1 −θMn−1 + · · · + MPe 1 κ (θ−r) 1−e−κMm −θMm + · · · + MPe 1 κ (θ−r) 1−e−κM1 −θM1 , (3.54) where Mm = Tm + Tm−1 + · · · + T2 + T1. Equation (3.54) represents the method of characteristics solution for the value at origination of remaining payments, in the limit of small volatilities. 3.7.5 Value of the default option at origination Following the algorithm again but considering the payment-date conditions for the default option, the solution at the beginning of month n is, D(H, r, τn = Tn) = max 0, MPe 1 κ (θ−r)(1−e−κTn )−θTn − He−δTn , (3.55) where the option is equal to zero if the decision to default is worthless. If the default option value is zero at the start of month n then it is zero at the end of month n − 1, i.e. at the payment-date in month n − 1. Considering the previous result and the algorithm, it can be shown that at the beginning of the contract D(H, r, τ1 = T1) = max 0, A(r, τ1 = T1) − He−δT1 , (3.56) where A(r, τ1 = T1) is given by equation (3.54). Equation (3.56) represents the method of characteristics solution for the value at origination of the default option, in the limit of small volatilities.
- 81. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 81 3.7.6 Value of insurance against default at origination The value of the insurance against default at origination can be calculated in a similar manner to that for the mortgage component in section 3.7.3, following the algorithm from section 3.7.2 and using the payment-date conditions given for the insurance asset in section 3.3.2. This leads to the following solution for the value of the insurance asset at the beginning of the contract: For the house-price and interest-rate values (H, r) which result in the default component having some worth, I(H, r, τ1 = T1) = min γ TD(τ1 = 0)e 1 κ (θ−r)(1−e−κT1 )−θT1 − He−δT1 , Γe 1 κ (θ−r)(1−e−κT1 )−θT1 , (3.57) where TD(τ1 = 0) = (1 + ψ)(1 + cT1)OB(0). (3.58) When the option to default is worthless, insurance is not necessary. Therefore, for these (H, r) values, the insurance component is worthless, I(H, r, τ1 = T1) = 0. (3.59) Equations (3.57) and (3.59) represent the approximate solution for the value at origination of insurance against default, in the limit of small volatilities. Computation times required to calculate the analytic solutions produced, using this asymptotic approach to the mortgage valuation problem are insignificant in com- parison with the time required to solve the problem using the numerical method as in described in section 3.5. The perturbation approach used only requires the value of V , A, D and I to calculate the equilibrium contract rate. The prepayment option does not feature in the solution to leading order and the coinsurance does not affect the equilibrium con- dition. Hence, these components are not discussed in this section. The perturbation approach is used to find an equilibrium contract rate. This rate can be substituted as a base parameter in the benchmark mark model to produce values for all the
- 82. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 82 mortgage assets. The ability of the perturbation approach to determine equilibrium contract rates and how well these contract rates determine the value of individual mortgage components is described by the results in section 3.9. The analytic approximations are the solution of the reduced form of the govern- ing PDE, (3.43), which is the leading-order equation in the asymptotic expansion of the governing PDE (1.17). Further, in the expansion for F(H, r, τm, σH, σr), tak- ing the leading-order approximation alone gives acceptable results, thus keeping the procedure highly tractable (though it is important to note that for the simple exam- ples, accuracy can be increased by including more terms, although this would also necessitate the inclusion of bridging solutions, see Appendix C). In the following section, a pseudocode is given, showing how to program the perturbation approach. 3.7.7 Perturbation approach pseudocode DO i = 1,n h(i) = (i-1)*dh END DO DO j = 1,m r(j) = (j-1)*dr END DO contract_rate = initial_estimate DO newton_iteration = 0,max_iteration CALL MonthlyPayment(MP,contract_rate,monthmax,loan) CALL FindA(A(j),MP,kappa,theta,r(j)) CALL FindV(MP,kappa,theta,delta,r(j),h(i)) CALL FindD(D(i,j),A(j),delta,r(j),h(i)) TD = (1._wp+pen)*(1._wp+crate*(1._wp/12._wp))*loan IF(D(i,j).ne.0)THEN !.. Default occurs, Ins has value .. CALL FindINS(TD,kappa,theta,delta,r(j),h(i),cap,fracloss)
- 83. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 83 ELSE !.. Default does not occur .. INS(i,j) = 0 END IF IF(V_orig+INS_orig-(1-arrange_fee)*loan < newton_tol)THEN EXIT !.. exit if equilibrium condition holds .. END IF CALL NewtonMethod(V(i,j),I(i,j),loan,arrange_fee,contract_rate) END DO In the above pseudocode, the subroutine ‘MonthlyPayment’ calculates the required monthly payment using the latest value of the contract rate; the subroutine ‘FindA’ calculates the value of the remaining monthly payments at origination, using equa- tion (3.54); the subroutine ‘FindV’ calculates the value of the mortgage asset at origination, using equation (3.52); the subroutine ‘FindD’ calculates the value of the borrower’s option to default at origination, using equation (3.56); the subroutine ‘FindINS’ calculates the value of the insurance component at origination, using equa- tion (3.57); finally the subroutine ‘NewtonMethod’ checks the equilibrium condition (3.22), and updates the contract rate if a further iteration is required. In the following sections, the equilibrium contract rate will be considered. A heavy computational load is faced to calculate this when using the finite-difference method, but it will be shown that with the new technique, calculations requiring hours on a PC are reduced to mere seconds using the analytical approximation approach described in this section. 3.8 Satisfying the equilibrium condition When the mortgage contract is arranged, it must be set up to avoid arbitrage, such that neither the borrower nor the lender can make an immediate profit. For both parties to agree on the contract, it is necessary that the value of the mortgage to the lender be equal to the amount lent.
- 84. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 84 Base parameters. ECONOMIC ENVIRONMENT Steady state spot rate, θ 0.1 Speed of reversion, κ 0.25 House service flow, δ 0.075 Correlation coefficient, ρ 0 CONTRACT Value of the house at origination £100000 Ratio of loan to initial value of house 0.95 Initial estimate for contract rate, c0 0.1 Prepayment penalty, ψ 0.05 INSURANCE Guaranteed fraction of total loss, γ 0.8 Cap, Γ 0.2Hinitial Table 3.1: Contract specifications and other parameters which are fixed, all based on parameters used in the literature. The contract rate is the vital parameter in the mortgage valuation problem, yet it is unknown at the time when the contract details are first decided; if it is set correctly, the possibility of an arbitrage situation occurring is eliminated. Next, details are given of the contract parameters, which are fixed. 3.8.1 Terms of the mortgage contract So that results from the finite-difference method could be compared with those from the perturbation approach, a basic set of parameters were chosen. The choice was made in accordance with parameters reported in the literature (see Titman and Torous 1989; Kau et al. 1995; Azevedo-Pereira et al. 2002); table 3.1 details these values. Unless noted otherwise, this set of economic parameters has been applied in all calculations described below. The perturbation approach was first considered because the state space volatilities associated with mortgage valuation are typically very low (in a numerical sense). To test the analytic approximations thoroughly, both the volatilities and the other mort- gage model parameters are flexed to show the ability of the perturbation approach to calculate equilibrium setting contract rates.
- 85. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 85 The equilibrium condition for setting up the mortgage contract is given in equation (3.22). Considering (3.22) and table 3.1, it is clear that the initial value of the house and the initial value of the interest rate are both known values. The arrangement fee ξ and the prepayment penalty ψ are specified in the terms of the contract. Thus, from (3.22), the only unknown parameter in the equilibrium condition is the contract rate c. The contract rate can be found easily using an iterative process following Newton’s method, see section 2.2.1 for details of this method. The next section describes results which illustrate how accurately (compared to the finite-difference approach) the perturbation approach approximates the equilib- rium setting contract rate. 3.9 Results The solution of the mortgage valuation problem is investigated, comparing results from the new singular perturbation approach and the high quality finite-difference technique (which may be regarded as giving ‘exact’ solutions). The results for the finite-difference method are based on Hmax = 2Hinitial with imax = 200; rmax = 5rinitial with jmax = 50; and kmax = 30; these choices were deemed satisfactory through extensive computational experimentation. Tables 3.2, 3.3, 3.4 and 3.5, show the contract rates produced using both methods, for several combinations of loan time, initial spot rate r(0) and arrangement fee ξ, for different interest-rate and house-price volatilities, respectively. Both approaches incorporate Newton’s method in order to determine the contract rates, as described in the section above. The relatively insignificant computation times using the perturbation approach contrast with those for the finite-difference approach as seen in these four tables.2 The smallest reduction in computation time seen is from 5.47 hours, for the finite- difference approach, to 5.78 seconds, for the perturbation approach (15 year loan); the largest from 13.52 hours to 11.78 seconds (25 year loan). 2 The results were obtained using a 2412 MHz AMD Athlon computer.
- 86. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 86 Table 3.2 shows a comparison of the contract rates produced using both meth- ods for the smallest state space volatilities. As expected, the analytic approximation to the problem (Pert) produces very acceptable substitutes for the ‘exact’ contract rates calculated using the finite-difference approach (FD), for all parameter combi- nations. As discussed in section 3.7 the perturbation approach lends itself well as an approximation when the volatilities are low. The percentage error in the contract rate produced using FD and Pert can be seen generally to increase in tables 3.3, 3.4, and 3.5, as the two volatilities are increased. Although this is expected, it is interesting to note that increasing interest-rate volatil- ity produces larger errors than increasing house-price volatility. For example, consider the change in percentage error for a loan time of 25 years, initial interest rate r(0) of 10% and arrangement fee ξ of 1.5%. When interest-rate volatility and house-price volatility are both 5% the error between the contract rates produced using the two methods is 0.08%. As house-price volatility increases to 10% the error increases to 0.37%, whereas when interest-rate volatility increases to 10% the error increases by a larger amount, to 3.74%. Within each of these four tables, as the initial interest rate r(0) increases, so too does the percentage error. This is consistent with the observation that interest-rate volatility has a stronger affect on the accuracy of the perturbation approach. The percentage error in the Pert contract rate, compared to the ‘exact’ FD con- tract rate, is less than 4% for all parameters tested in tables 3.2 and 3.3. This is for the lowest state-space volatilities and when there is an increase in house-price volatility. Considering all possibilities of the volatilities that were tested, only 11 out of 144 parameter setups had a percentage error above 6%. As the arrangement fee decreases, for a fixed loan time and initial interest rate r(0), the error in the contract rate found using the perturbation method increases. To understand the reason for this, consider the following. Mortgages with contracts which, in regions critical to the equilibrium condition, move the values of components closer to the region where the approximate solutions meet, reduce the accuracy of the contract rate produced. It is here, at the union between two approximate solutions, that they are least powerful.
- 87. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 87 This is a small/thin region where a fuller and more sophisticated asymptotic anal- ysis would require an investigation of bridging solutions in order to join the simple approximate solutions. The inclusion of these regions would increase the accuracy of the perturbation approach further in these critical regions, but would also increase the complexity of the solutions, which would also result in an increase in the compu- tation time required to calculate these solutions. The insignificant computation time as well as the level of accuracy are the major payoffs from the current approach. A more detailed discussion of these bridging solutions is given in Appendix C.
- 88. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 88 Equilibrium setting contract rates and computation times. σr = 5% σH = 5% Loan Contract Rate (%) Time (sec) (years) r(0) ξ FD Pert % error FD Pert 15 8 0 9.086 9.152 0.73 19700 5.78 0.5 8.990 9.066 0.85 19700 5.78 1 8.898 8.980 0.92 19700 5.78 1.5 8.810 8.894 0.95 24600 7.23 10 0 10.146 10.042 1.03 24600 7.23 0.5 9.973 9.954 0.19 24600 7.23 1 9.859 9.866 0.07 24600 7.23 1.5 9.754 9.777 0.24 24600 7.23 12 0 11.197 10.947 2.23 19700 5.78 0.5 11.069 10.857 1.92 19700 5.78 1 10.930 10.766 1.50 24600 5.78 1.5 10.795 10.675 1.11 24600 5.78 20 8 0 9.197 9.264 0.73 33700 7.55 0.5 9.114 9.194 0.88 33700 7.55 1 9.034 9.123 0.99 33700 7.55 1.5 8.957 9.052 1.06 27000 7.55 10 0 10.124 10.042 0.81 40400 9.44 0.5 10.020 9.969 0.51 33700 9.44 1 9.916 9.896 0.20 33700 9.44 1.5 9.816 9.822 0.06 40400 9.44 12 0 11.200 10.836 3.25 33700 7.55 0.5 11.069 10.761 2.78 33700 7.55 1 10.944 10.685 2.37 27000 7.55 1.5 10.809 10.610 1.84 33700 7.55 25 8 0 9.266 9.336 0.76 48700 11.78 0.5 9.191 9.274 0.90 32500 9.42 1 9.118 9.211 1.02 32500 9.42 1.5 9.047 9.149 1.13 40600 9.42 10 0 10.146 10.042 1.03 32500 11.78 0.5 10.048 9.977 0.71 40600 11.78 1 9.951 9.912 1.04 40600 11.78 1.5 9.856 9.848 0.08 48700 11.78 12 0 11.197 10.767 3.84 32500 9.42 0.5 11.071 10.700 3.35 32500 9.42 1 10.944 10.632 2.85 32500 9.42 1.5 10.819 10.565 2.35 40600 9.42 Table 3.2: Comparison of equilibrium setting contract rates for σr = 5%, σH = 5% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 89. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 89 Equilibrium setting contract rates and computation times. σr = 5% σH = 10% Loan Contract Rate (%) Time (sec) (years) r(0) ξ FD Pert % error FD Pert 15 8 0 9.014 9.152 1.53 24600 5.78 0.5 8.913 9.066 1.72 19700 5.78 1 8.816 8.980 1.86 19700 5.78 1.5 8.721 8.894 1.98 19700 5.78 10 0 10.028 10.042 0.14 29600 7.23 0.5 9.905 9.954 0.49 29600 7.23 1 9.791 9.866 0.77 29600 7.23 1.5 9.685 9.777 0.95 29600 7.23 12 0 11.152 10.947 1.84 19700 5.78 0.5 11.004 10.857 1.34 19700 5.78 1 10.861 10.766 0.87 19700 5.78 1.5 10.730 10.675 0.51 24600 5.78 20 8 0 9.142 9.264 1.33 33700 7.55 0.5 9.053 9.194 1.56 33700 7.55 1 8.967 9.123 1.74 33700 7.55 1.5 8.884 9.052 1.89 33700 7.55 10 0 10.082 10.042 0.40 40400 9.44 0.5 9.971 9.969 0.02 33700 9.44 1 9.863 9.896 0.33 47200 9.44 1.5 9.764 9.822 0.59 47200 9.44 12 0 11.163 10.836 2.93 40400 7.55 0.5 11.022 10.761 2.37 40400 7.55 1 10.886 10.685 1.85 26900 7.55 1.5 10.756 10.610 1.36 33700 7.55 25 8 0 9.220 9.336 1.26 40600 11.78 0.5 9.139 9.274 1.48 40600 9.42 1 9.061 9.211 1.66 40600 9.42 1.5 8.985 9.149 1.83 40600 9.42 10 0 10.113 10.042 0.70 32500 11.78 0.5 10.009 9.977 0.32 40600 11.78 1 9.908 9.912 0.04 40600 11.78 1.5 9.812 9.848 0.37 48700 11.78 12 0 11.171 10.767 3.62 40600 9.42 0.5 11.033 10.700 3.02 32500 9.42 1 10.903 10.632 2.49 40600 9.42 1.5 10.775 10.565 1.95 40600 9.42 Table 3.3: Comparison of equilibrium setting contract rates for σr = 5%, σH = 10% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 90. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 90 Equilibrium setting contract rates and computation times. σr = 10% σH = 5% Loan Contract Rate (%) Time (sec) (years) r(0) ξ FD Pert % error FD Pert 15 8 0 9.317 9.152 1.77 19700 5.78 0.5 9.175 9.066 1.19 19700 5.78 1 9.046 8.980 0.73 19700 5.78 1.5 8.921 8.894 0.30 19700 5.78 10 0 10.534 10.042 4.67 24600 7.23 0.5 10.368 9.954 3.99 24600 7.23 1 10.204 9.866 3.31 24600 7.23 1.5 10.048 9.777 2.70 24600 7.23 12 0 11.835 10.947 7.50 24600 5.78 0.5 11.653 10.857 6.83 2460 5.78 1 11.469 10.766 6.13 19700 5.78 1.5 11.289 10.675 5.44 24600 5.78 20 8 0 9.463 9.264 2.10 27000 7.55 0.5 9.332 9.194 1.48 27000 7.55 1 9.211 9.123 0.96 27000 7.55 1.5 9.097 9.052 0.49 27000 7.55 10 0 10.625 10.042 5.49 33700 9.44 0.5 10.469 9.969 4.78 33700 9.44 1 10.315 9.896 4.06 33700 9.44 1.5 10.167 9.822 3.39 33700 9.44 12 0 11.888 10.836 8.85 33700 7.55 0.5 11.712 10.761 8.12 33700 7.55 1 11.539 10.685 7.40 33700 7.55 1.5 11.368 10.610 6.67 33700 7.55 25 8 0 9.542 9.336 2.16 40600 11.78 0.5 9.419 9.274 1.54 40600 9.42 1 9.304 9.211 1.00 40600 9.42 1.5 9.196 9.149 0.51 40600 9.42 10 0 10.673 10.042 5.91 40600 11.78 0.5 10.521 9.977 5.17 40600 11.78 1 10.374 9.912 4.45 40600 11.78 1.5 10.231 9.848 3.74 40600 11.78 12 0 11.913 10.767 9.76 40600 9.42 0.5 11.743 10.700 8.88 40600 9.42 1 11.573 10.632 8.13 40600 9.42 1.5 11.407 10.565 7.38 40600 9.42 Table 3.4: Comparison of equilibrium setting contract rates for σr = 10%, σH = 5% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 91. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 91 Equilibrium setting contract rates and computation times. σr = 10% σH = 10% Loan Contract Rate (%) Time (sec) (years) r(0) ξ FD Pert % error FD Pert 15 8 0 9.270 9.152 1.27 19700 5.78 0.5 9.128 9.066 0.68 19700 5.78 1 9.046 8.980 0.73 24600 5.78 1.5 8.921 8.894 0.30 24600 5.78 10 0 10.491 10.042 4.28 29600 7.23 0.5 10.322 9.954 3.57 24600 7.23 1 10.157 9.866 2.87 24600 7.23 1.5 10.004 9.777 2.27 19700 7.23 12 0 11.798 10.947 7.21 29600 5.78 0.5 11.608 10.857 6.47 24600 5.78 1 11.420 10.766 5.73 24600 5.78 1.5 11.239 10.675 5.02 24600 5.78 20 8 0 9.420 9.264 1.66 33700 7.55 0.5 9.290 9.194 1.03 33700 7.55 1 9.167 9.123 0.48 33700 7.55 1.5 9.050 9.052 0.02 33700 7.55 10 0 10.590 10.042 5.17 33700 9.44 0.5 10.430 9.969 4.42 33700 9.44 1 10.276 9.896 3.70 33700 9.44 1.5 10.131 9.822 3.05 33700 9.44 12 0 11.859 10.836 8.63 33700 7.55 0.5 11.679 10.761 7.86 33700 7.55 1 11.499 10.685 7.08 33700 7.55 1.5 11.326 10.610 6.32 33700 7.55 25 8 0 9.498 9.336 1.39 40600 11.78 0.5 9.375 9.274 1.08 40600 9.42 1 9.260 9.211 0.53 40600 9.42 1.5 9.150 9.149 0.01 40600 9.42 10 0 10.637 10.042 5.59 40600 11.78 0.5 10.482 9.977 4.82 48700 11.78 1 10.335 9.912 4.09 40600 11.78 1.5 10.197 9.848 3.42 40600 11.78 12 0 11.891 10.767 9.45 48700 9.42 0.5 11.710 10.700 8.63 40600 9.42 1 11.538 10.632 7.85 40600 9.42 1.5 11.369 10.565 7.07 40600 9.42 Table 3.5: Comparison of equilibrium setting contract rates for σr = 10%, σH = 10% calculated using the finite-difference approach (FD) and the perturbation approach (Pert). The computation times for the two methods are also shown. r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 92. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 92 The final four tables of this section (3.6, 3.7, 3.8 and 3.9), show the ability of the contract rates, as produced using the perturbation approach (Pert), to value accurately the individual mortgage components for different state space volatilities. The values of the components are calculated using the finite-difference approach, with the Pert contract rate as a base parameter. Rather than show the component values for every parameter combination tested, only the results for a loan time of 25 years are illustrated. This presentation was chosen because the Pert approach showed the largest range of accuracy when calculating the equilibrium setting contract rates for this contract length (from tables 3.2, 3.3, 3.4 and 3.5). Thus, component values, calculated using the Pert contract rates, and the accuracy with which the Pert contract rates reproduce the ‘exact’ values, are shown for contract rates which exhibited a broad range of accuracy. The percentage error, between the value of the components calculated using the benchmark contract rate (found using FD) and the value of the components calculated using the simply approximated contract rate (found using Pert) is given. From table 3.6, it is seen that the contract rates calculated using the perturbation approach can be used to value accurately the mortgage components (typically to within 5%) for all contract specifications shown, when the lower state space volatilities are tested. As mentioned previously, as the volatilities increase it is expected that the accuracy of the component values will decrease, and this is indeed the case; again, more so when the interest rate volatility is increased. Invariably, for corresponding parameter choices of initial interest rate r(0) and arrangement fee ξ, the increase in percentage error is greater when moving from both interest-rate volatility σr and house-price volatility σH being low (both 5%, see table 3.6) to an increase in σr (see table 3.8), rather than the smaller increase in error when σH is increased (see table 3.7).
- 93. CHAPTER3.IMPROVEDFIXED-RATEMORTGAGEVALUATION93 Component values calculated using the FD contract rate and the Pert contract rate (in £). σr = 5% σH = 5% Value of Remaining Payments, A Default, D Prepayment, C Insurance, I Coinsurance, CI r(0) ξ FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err 8 0 95155 95692 0.56 936 949 1.39 228 232 1.75 1010 1017 0.69 252 254 0.79 0.5 94574 95216 0.68 846 860 1.65 172 175 1.74 969 976 0.72 242 244 0.83 1 94020 94734 0.76 765 779 1.83 127 139 2.36 921 929 0.87 230 232 0.87 1.5 93482 94260 0.83 690 705 2.17 96 99 3.13 879 887 0.91 220 222 0.91 10 0 96509 95736 0.80 981 960 2.14 1199 1163 3.00 670 664 0.90 168 166 1.79 0.5 95779 95255 0.55 867 855 1.38 1023 1003 1.96 636 631 0.79 159 157 1.26 1 95059 94774 0.30 773 765 1.03 842 828 1.66 605 601 0.66 151 149 1.32 1.5 94356 94302 0.06 691 686 0.72 670 659 1.64 580 578 0.34 145 144 0.69 12 0 98871 95779 3.13 1080 1027 4.91 3234 3082 4.70 439 427 2.73 110 107 2.73 0.5 97961 95301 2.72 923 881 4.55 2919 2782 4.69 406 398 1.97 102 100 1.96 1 97048 94816 2.30 806 774 3.97 2575 2467 4.19 382 375 1.83 95 93 2.11 1.5 96154 94339 1.89 705 680 3.55 2232 2156 3.41 358 352 1.68 90 89 1.11 Table 3.6: Comparison of mortgage component values for σr = 5%, σH = 5%, calculated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 94. CHAPTER3.IMPROVEDFIXED-RATEMORTGAGEVALUATION94 Component values calculated using the FD contract rate and the Pert contract rate (in £). σr = 5% σH = 10% Value of Remaining Payments, A Default, D Prepayment, C Insurance, I Coinsurance, CI r(0) ξ FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err 8 0 94805 95692 0.94 3266 3341 2.30 126 130 3.17 3587 3652 1.81 897 913 1.78 0.5 94185 95216 1.09 3098 3179 2.61 91 94 3.30 3530 3608 2.21 883 902 2.15 1 93589 94734 1.22 2941 3031 3.06 67 64 4.48 3469 3559 2.59 868 890 2.53 1.5 93012 94260 1.34 2794 2892 3.51 49 47 4.08 3406 3508 2.99 852 877 2.93 10 0 96264 95736 0.55 3356 3306 1.49 804 788 1.99 2897 2863 1.17 724 716 1.10 0.5 95492 95255 0.25 3157 3119 1.20 663 654 1.36 2853 2844 1.32 713 711 0.28 1 94743 94774 0.03 2973 2981 0.27 553 547 1.08 2813 2816 0.11 703 704 0.14 1.5 94034 94302 0.29 2805 2777 1.01 426 420 1.41 2772 2786 0.51 693 697 0.58 12 0 98680 95779 2.94 3602 3422 5.00 2375 2235 5.89 2297 2194 4.48 574 548 4.53 0.5 97689 95301 2.44 3344 3211 3.98 2088 1991 4.65 2268 2173 4.19 567 543 4.23 1 96754 94816 2.00 3110 3009 3.25 1821 1748 4.01 2228 2152 3.41 557 538 3.41 1.5 95837 94339 1.56 2904 2832 2.48 1556 1503 3.41 2197 2119 3.55 549 530 3.46 Table 3.7: Comparison of mortgage component values for σr = 5%, σH = 10%, calculated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 95. CHAPTER3.IMPROVEDFIXED-RATEMORTGAGEVALUATION95 Component values calculated using the FD contract rate and the Pert contract rate (in £). σr = 10% σH = 5% Value of Remaining Payments, A Default, D Prepayment, C Insurance, I Coinsurance, CI r(0) ξ FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err 8 0 99291 97670 1.63 2918 2821 3.32 2487 2381 4.26 1115 1093 1.97 279 273 2.15 0.5 98318 97185 1.15 2696 2635 2.26 2202 2135 3.04 1106 1087 1.72 277 272 1.81 1 97417 96692 0.74 2492 2455 1.48 1965 1925 2.04 1090 1077 1.19 273 269 1.47 1.5 96574 96209 0.38 2307 2290 0.74 1766 1746 1.13 1074 1069 0.47 269 267 0.74 10 0 102663 97840 4.70 3097 2936 5.20 5286 5036 4.73 725 700 3.45 181 175 3.31 0.5 101490 97348 4.08 2825 2694 4.64 4854 4645 4.31 712 689 3.23 178 172 3.37 1 100362 96857 3.49 2588 2481 4.13 4424 4254 3.84 698 678 2.87 175 170 2.86 1.5 99276 98374 0.91 2392 2305 3.64 3997 3866 3.28 689 670 2.76 172 167 2.91 12 0 106491 98003 7.97 3207 3033 5.43 8779 8215 6.42 493 464 5.88 123 116 5.69 0.5 105222 97513 7.33 2863 2719 5.03 8305 7788 6.23 468 444 5.13 117 111 5.13 1 103964 97017 6.68 2581 2460 4.69 7778 7316 5.94 454 432 4.85 113 108 4.42 1.5 102727 96529 6.03 2343 2103 4.28 7237 6832 5.60 435 417 4.14 109 105 3.67 Table 3.8: Comparison of mortgage component values for σr = 10%, σH = 5%, calculated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 96. CHAPTER3.IMPROVEDFIXED-RATEMORTGAGEVALUATION96 Component values calculated using the FD contract rate and the Pert contract rate (in £). σr = 10% σH = 10% Value of Remaining Payments, A Default, D Prepayment, C Insurance, I Coinsurance, CI r(0) ξ FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err FD c Pert c % err 8 0 98941 97670 1.28 5531 5367 2.97 1858 1780 4.20 3449 3431 0.52 864 860 0.46 0.5 97977 97185 0.81 5250 5149 1.92 1638 1593 2.75 3435 3420 0.44 861 857 0.46 1 97075 96692 0.39 4992 4945 0.94 1450 1429 1.45 3418 3409 0.26 857 855 0.23 1.5 96216 96209 0.01 4755 4743 0.25 1282 1281 0.08 3396 3390 0.18 852 851 0.12 10 0 102384 97840 4.44 6004 5710 4.90 4072 3848 5.50 2691 2633 2.16 673 659 2.08 0.5 101196 97348 3.80 5657 5415 4.28 3696 3526 4.60 2685 2622 2.35 672 656 2.38 1 100071 96857 3.21 5345 5147 3.70 3358 3221 4.08 2672 2615 2.13 668 655 1.95 1.5 99013 96374 2.67 5056 4898 3.13 3036 2930 3.49 2655 2601 2.03 664 651 1.96 12 0 106332 98003 7.83 6505 6134 5.70 6949 6474 6.84 2126 1995 6.16 532 499 6.20 0.5 104986 97513 7.12 6094 5772 5.28 6472 6051 6.50 2111 1983 6.06 528 496 6.06 1 103694 97017 6.44 5711 5434 4.85 6020 5650 6.15 2090 1973 5.60 523 493 5.74 1.5 102440 96529 5.77 5368 5131 4.42 5569 5248 5.76 2072 1965 5.16 518 492 5.02 Table 3.9: Comparison of mortgage component values for σr = 10%, σH = 10%, calculated using the ‘exact’ contract rate and the contract rate found using the perturbation method, for different contract specifications. The loan is for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%).
- 97. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 97 When interest-rate volatility is increased (see table 3.8) and when both volatili- ties are increased (see table 3.9) it is expected that the perturbation approach will be less accurate. Even so, the error in the component values calculated using the perturbation approach, compared to the ‘exact’ finite-difference approach, is small. The percentage error in the value of all the components is slightly more than 6% for only 2 parameter senerios and this occurs when the initial interest rate is very high (12%) and the arrangement fee is either zero or 0.5%. The error in the con- tract rate is a good indication of the error in the component values. Overall, even as the method is extended away from parametric regions of practical importance, it performs well. Most of all, although the technique might at first appear unfamiliar, it is actually easier to implement than the corresponding finite-difference program, and many orders of magnitude faster. 3.10 Conclusions This chapter has considered a mortgage valuation model, which includes the potential for early prepayment and for default. An improved finite-difference procedure has been presented, together with a perturbation analysis (based on the assumption of numerically small volatility of house price and interest rate), which leads to closed- form solutions. Using this analytic approximation, calculation of the equilibrium contract rates (one of the crucial unknowns in mortgage valuation) can be achieved in a tiny fraction of the time required by fully numerical techniques (for example, 13.5 hours reduced to 11.8 seconds). The contract rates can then be used to value the mortgage components to within a few percent of the ‘exact’ value (except in an easily identified region, close to zero arrangement fee when the initial interest rate is high and only when both house-price volatility and interest-rate volatility are high). This chapter has shown that perturbation theory is a very efficient and effective tool in the solution of a contingent claims mortgage valuation model. The algorithm given in section 3.7.2 is easy to implement and could be applied to any FRM model.
- 98. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 98 Closed-form analytic approximations of mortgage components are then trivially sim- ple to calculate. The major benefits of the technique, as developed and presented here, are speed and simplicity. It would be expected that the improvement in accuracy over a narrow range of parameters would display the need for the bridging solutions (see Appendix C) in thin regions of the state space. At present, the approximation has been shown to be accurate with both volatilities as high as 10% in the original model. It is antic- ipated that even as the volatilities increase, a full asymptotic analysis would provide a more accurate approximation (Widdicks et al., 2005, obtained extremely accurate American option values for volatilities as large as 100% using these techniques). It might also be possible to explore adjustable-rate (variable-rate) mortgages, where the simplification provided by this technique offers obvious benefits.
- 99. Chapter 4 A new prepayment model: an occupation-time derivative approach 4.1 Introduction Relaxing the assumption of ruthless prepayment, used in all previous research on purely option-theoretic valuation of mortgages (and in chapter 3 on efficient FRM valuation), a new prepayment policy is described in this chapter. It is well known that basic mortgage option-pricing models cannot replicate mortgage values which are greater than par, i.e. greater than the initial loan amount (see Downing et al., 2005).1 Also, it has been shown that in practice borrowers prepay their mortgages later than when standard option-pricing models indicate they should. According to Stanton (1995) some mortgages are not prepaid even when their contract rate is above current mortgage rates. Also, Longstaff (2005) expresses that borrowers who believe mortgage rates may decrease further in the near future may choose to delay prepayment. Kalotay et al. (2004) mention that one failure of past option-based approaches has been in their inability to model borrowers who should prepay but do 1 The ruthless, optimal call condition results in the face value of debt (analogous to the total debt payment for UK loans) being the maximum value that the mortgage can achieve. At origination the face value is the initial loan amount (in the absence of transaction costs). 99
- 100. CHAPTER 4. A NEW PREPAYMENT MODEL 100 not. Thus, utilising an American call option to model prepayment produces mortgage values that are lower than those observed in reality. Classic prepayment or rational ruthless prepayment (Kau et al., 1995; Azevedo-Pereira et al. 2000, 2002, 2003), as a result of borrowers minimising their mortgage costs, occurs if interest rates decrease sufficiently, so that it is financially favourable for borrowers to prepay (the instant the value of the mortgage to the lender is greater than the cost of prepaying) and take out a new mortgage. Actual prepayment appears significantly suboptimal, relative to the optimal behaviour implied by standard call policy. By using occupation-time derivatives, a lag in prepayment being exercised from when it is initially financially optimal can be simulated, and it is then possible to achieve mortgage values greater than more basic models, within a rational structural framework. Rather than prepayment being modelled as an American call option, exercised when it is financially optimal for borrowers to do so, prepayment is modelled as a Parisian call option (a consecutive occupation-time derivative - see section 4.2 for full details of this type of option). A lag in prepayment is created by including a borrower waiting or decision time, during which the value of the mortgage must be greater than the cost of prepayment. The decision time could be regarded as a measure of the difficulty and time involved in deciding whether to prepay is optimal at any time. This results in the borrower incurring a cost when making the decision (which results in a reduction in value of the prepayment option). The endogenously produced time lag results in the the prepayment decision being the result of optimising behaviour by the borrower and still only depends on the interest rate and time. Although the American optimal-stopping free boundary is no longer present in the valuation, the introduction of the Parisian prepayment call option creates its own free boundary problem; see section 4.3.1 for the description of this new problem. In reality there could be a great deal of inertia preventing borrowers from prepay- ing, or even knowing that the possibility exists of the availability of a better contract, prepayment penalties, transaction costs, etc; see Boudoukh, Whitelaw, Richardson and Stanton (1997) for more details. By creating a lag endogenously before the pre- payment option is exercised, the value of the mortgage to the lender is increased,
- 101. CHAPTER 4. A NEW PREPAYMENT MODEL 101 since the lender benefits from the borrower not terminating the loan prematurely. This enables mortgage values to be produced that are higher than those implied by less sophisticated models even without explicit transaction costs. A failing of past structural models was an inability to produce mortgage values that exceeded par; under the framework detailed in this chapter, this difficulty can be overcome. The main contribution from this chapter is the improved manner in which pre- payment is modelled, which can provide more realistic FRM valuation by predicting increased mortgage values and allowing for different time lags in prepayment by the borrower (by varying the decision times by the borrower). The next step would be to apply this framework to price more accurately a MBS, a closely related financial derivative (for the problems with MBS valuation, see section 1.2.3). Borrowers have been shown to follow a near-optimal call policy, see Kalotay et al. (2004). One possibility is that rather than exercising immediately when it is apparently financially optimal, they wait to see if interest rates stay low, or they may be waiting to see if rates may drop even further before they decide to prepay. Borrowers following this strategy are attempting to minimise the lifetime cost of their mortgage. The following is a summary of reasons why borrowers might wait to prepay: 1. To see if interest rates increase and return to their previous value, so they avoid having to prepay and continue with their current mortgage. 2. They wait to see if interest rates will decrease even further. 3. They attempt to minimise the life-of-loan cost of their mortgage (by not pre- paying an unnecessary number of times). There have been several previous attempts at improving termination modelling and, in particular, prepayment modelling. Termination within a structural model arises from a borrower’s optimising behaviour, which means that constraints are imposed on the relation between terminations and the underlying state variables. According to Downing et al. (2005), this results in basic structural models (Dunn
- 102. CHAPTER 4. A NEW PREPAYMENT MODEL 102 and McConnell, 1981a,b) which produce mortgage prices and termination behaviour that differs in important ways from what is observed in practice. Specifically these models predict mortgages (or MBS) can never trade above par, since borrowers will exercise their prepayment option the instant the mortgage value exceeds par (ruthless option exercise), when actually mortgages are often traded above par. These models also assume that all borrowers are identical, whereas it has been observed that not all borrowers prepay simultaneously. Kelly and Slawson (2001) show prepayment decisions of commercial property owners which appear to be slow, may in fact be quite rational when time-varying prepayment penalties are considered in an option- theoretic mortgage model. Longstaff (2005) incorporated borrower credit into the framework to value FRMs and shows that an optimal refinancing strategy (rational prepayment) can delay pre- payment relative to conventional models, and that mortgage values can exceed par by more than the cost of refinancing (the usual structural limit). Choosing how borrowers prepay is said to have inhibited the formulation of a fundamental theory of mortgage pricing. Dunn and McConnell (1981a,b) and Brennan and Schwartz (1985) applied contingent claims techniques (structural modelling) to the problem by modelling the prepayment decision as a result of the borrower minimising life- time mortgage costs. Schwartz and Torous (1989, 1992, 1993) comment that actual prepayment behaviour appears considerably suboptimal, relative to the optimal, be- haviour implied by these early models. Stanton (1995) and Boudoukh et al. (1997) demonstrate this upper bound is nearly always violated in practice. The model de- veloped by Stanton (1995) bases prepayment by borrowers on rational decisions, and the results indicate that borrowers act as though they face transaction costs that far exceed the explicit costs usually incurred on refinancing, i.e. they wait, even when it is optimal to prepay. Longstaff (2005) concludes that borrowers find it optimal to de- lay prepayment far beyond the point at which simple rational models imply that the mortgage should be prepaid. The use of an occupation-time derivative can model this theory within a structural framework. The features of this type of derivative allow for the representation of a lag between investment information and its implementation,
- 103. CHAPTER 4. A NEW PREPAYMENT MODEL 103 where optimality is maintained (purely financial decisions remain the foundation of this theory). A new prepayment model with lagged prepayment by the borrower, rather than the usual ‘optimal’ call policy is now proposed to overcome the existing problem with the borrower’s call policy, by incorporating an occupation-time derivative in the valuation framework. Borrowers are still only motivated financially and the behaviour is determined endogenously. A future empirical study could confirm whether the Parisian feature is a more accurate model of termination. Before giving the precise details of the new model, occupation-time derivatives are first introduced and the details of their valuation are given. 4.2 Introduction to occupation-time derivatives This type of derivative has a path-dependency which depends on how much time an underlying spends beyond a given barrier level. There are many varieties of this basic concept, and this section discusses two variations: the Parisian option and the ParAsian option, in preparation for the more complex matter of incorporating occupation-time derivatives into mortgage valuation. It is first necessary to clarify the meaning of a vanilla barrier option before describing the more complex path- dependent barrier options. 4.2.1 Definition of the barrier option The term barrier can be attached to many different options. Specified in the contract is an agreed value which causes the right to exercise to be forfeited if the underlying asset value crosses this given value (an out barrier), or the option comes into existence only if the asset value crosses this given value (an in barrier). A barrier is described as either a knock in or a knock out and both have the extra specification of being either up barriers or down barriers, depending on whether the barrier feature is triggered by the underlying asset crossing above or below the barrier level.
- 104. CHAPTER 4. A NEW PREPAYMENT MODEL 104 Uses Barrier options are attractive to buyers who would rather not pay a premium for scenarios they think are unlikely to occur. Buyers of barrier options have to choose whether they would rather lower the risk of being knocked out, which increases the option price, or pay less for the option, whilst reducing the chance of being knocked in. 4.2.2 Occupation-time derivatives In essence, occupation-time derivatives are exotic barrier options, where the action executed at the barrier is not as straightforward as a simple knock in or knock out. Compared to their vanilla counterparts, barrier options allow a more flexible deriva- tive for investors by letting them make certain decisions regarding possible future changes in the direction of the market. Unlike a straightforward barrier option, an occupation-time derivative is not affected by single outliers in the value of the under- lying. This makes them resilient to manipulation of the underlying asset by market makers. The extra specification required to classify an occupation-time derivative is whether it is a consecutive option (this will be referred to as a Parisian option hereafter) or cumulative (hereafter referred to as a ParAsian option, see section 4.2.8 for further details about the ParAsian option). Consecutive refers to the manner in which the underlying must be beyond the barrier level for a consecutive number of prescribed time steps for the knock in/out feature to be activated. If the underlying moves back across the given barrier level, the barrier clock is reset to zero, whereas for the ParAsian the time steps in all excursions across the barrier are summed together. The initial framework was proposed by Chesney et al. (1997), who derived an upper and lower bound for the value of the Parisian option. The calculation involved evaluation of Laplace transforms, followed by an inversion via the Euler method. Cornwall and Kentwell (1995) extended the approach of Chesney et al. (1997) to a quasi-analytical model and also incorporated discrete time monitoring (often the
- 105. CHAPTER 4. A NEW PREPAYMENT MODEL 105 case in actual markets for Parisian options). Their framework has also been extended by Hugonnier (1999) to price and hedge any type of occupation-time derivative. Moraux (2002) stresses that while Hugonnier (1999) uses proper numerical techniques to price the ParAsian option, his results do not match the results given by the quasi- analytical pricing formula given in his proposition 14 (page 166). Although Moraux claims to have provided closed-form solutions which perform quasi-analytical pricing of ParAsian options, no numerical results are given so it is difficult to fully confirm these claims. Initial attempts at pricing Parisian options using lattice based methods have been performed by Avellaneda and Wu (1999). They develop a method which uses a modified trinomial scheme and involves the density function of the first-passage time at which the asset price first reaches the barrier. Kwok and Lau (2001) use the forward shooting grid method, a variant of the lattice-based method. Their method can be used to price Parisian options, options with a reset feature and alpha-quantile options (where the barrier level is a stochastic variable that defines the terminal payoff). The square-root rate of convergence that the forward shooting grid approach achieves for these path-dependent options is only improved by the use of a non-linear extrapolation technique. This has to be used in conjunction with an adjustment method to avoid oscillatory convergence behaviour; these features further add to the complexity of this method. Effective numerical methods using a finite-difference scheme by direct discretisa- tion of the governing PDE have been presented by Vetzal and Forsyth (1999); also Haber et al. (1999) have developed a finite-difference scheme to price both consecu- tive and cumulative continuously monitored Parisian options. For an application of occupation-time derivatives in corporate debt valuation see Yu et al. (2006). Here the firm value is modelled as the underlying asset of a ParAsian option, with the intention of properly modelling the endogenous recovery rate for firms in distress.
- 106. CHAPTER 4. A NEW PREPAYMENT MODEL 106 The method set out by Haber et al. (1999) is reproduced here, except the (supe- rior) Crank-Nicolson finite-difference scheme is used, rather than an explicit finite- difference scheme. The new prepayment model, see section 4.3.1, utilises the basic occupation-time derivative model outlined by Haber et al. (1999). The underlying, which must be monitored to check whether the barrier clock is activated, is not the underlying stochastic process, as in Haber et al. (1999) (where the underlying was the stock value), but is in fact the value of the mortgage which depends on two stochastic processes. This results in the barrier level generating a free boundary, which must be set so that the appropriate scheme can be applied in the necessary region of the state space (beyond the barrier when the barrier clock is activated, a separate PDE must be solved, see section 4.2.5). As an introduction to this type of option, the derivation for the governing PDE for the Parisian option is demonstrated, along with the numerical solution for a Parisian up-and-out call option. This is then compared to a ParAsian up-and-out call option. 4.2.3 Definition of the Parisian option Parisian options are barrier options for which the barrier feature is activated only after the price process has spent a certain prescribed, consecutive time beyond the barrier. As proposed by Haber et al. (1999), Parisian options serve several purposes. They are not affected by single outliers in underlying asset price in the way that barrier options are. Also the problem caused when hedging a standard barrier option close to the barrier, due to the gamma becoming very large, is easier to manage with this type of option. 4.2.4 Definition of the Parisian up-and-out option The barrier feature is only activated if the underlying stays above the barrier level ¯S for a certain prescribed, consecutive time. If this happens during the lifetime of the contract then the option immediately expires worthless. If the barrier feature is not activated, then the option has the standard payoff at expiry.
- 107. CHAPTER 4. A NEW PREPAYMENT MODEL 107 PSfrag replacements S 0 ¯S ¯T ¯t T t Figure 4.1: An illustration of the state space for a Parisian option. 4.2.5 Derivation of the occupation-time derivative PDE This section uses the Parisian up option to demonstrate the derivation of the occupation- time derivative PDE. This can be easily generalised for other Parisian and ParAsian options. The Parisian option is path-dependent since the payoff depends on the value of the underlying at expiration and on the path taken to get there. In the PDE framework, the value of the barrier-time variable ¯t is required, which is the length of time the underlying has spent above the barrier during its current excursion. The value of the Parisian option can be written as V (S, t, ¯t) (for this section only, V will be used to refer to the value of the option in question and not the value of the mortgage to the lender). This means the option value is a function of three independent variables: the current asset price S, time t and the barrier time ¯t. The valuation problem is split into two regions. The first is below the barrier, where the barrier clock is always zero and the second above the barrier, where the barrier clock increases by d¯t at each time step; these regions are shown in figure 4.1. Continuing with the Parisian up-and-out option for explanatory purposes, when the underlying asset is below the barrier S < ¯S, the barrier clock ¯t remains unchanged and the basic Black-Scholes
- 108. CHAPTER 4. A NEW PREPAYMENT MODEL 108 equation determines V : ∂V ∂t + 1 2 σ2 S2 ∂2 V ∂S2 + rS ∂V ∂S − rV = 0. (4.1) When the underlying rises above the barrier S > ¯S, the barrier time ¯t increases at the same rate as real time. Now consider a function of the random variable S, of time t and time ¯t, V (S, t, ¯t). If V (S + dS, t + dt, ¯t + d¯t) is expanded as follows about (S, t, ¯t) to obtain dV = ∂V ∂S dS + ∂V ∂t dt + ∂V ∂¯t d¯t + 1 2 ∂2 V ∂S2 dS2 + 1 2 ∂2 V ∂t2 dt2 + 1 2 ∂2 V ∂¯t2 d¯t2 + . . . (4.2) Above the barrier, by definition d¯t = dt, see Haber et al. (1999), and by applying the standard rules from stochastic calculus that as, dt → 0, dX2 → dt (where X is the standardised Wiener process for the underlying asset S), equation (4.2) becomes dV = σS ∂V ∂S dX + µS ∂V ∂S + 1 2 σ2 S2 ∂2 V ∂S2 + ∂V ∂t + ∂V ∂¯t dt; (4.3) this prescribes the random walk followed by V .2 Now construct the usual portfolio Π = V − ∆S, (4.4) where ∆ is as yet an unknown parameter which is constant across a time period dt. The jump in the value of this portfolio over this time step is dΠ = dV − ∆dS. (4.5) By substituting (4.3) into (4.5) and choosing ∆ = ∂V ∂S , (4.6) the result is a completely deterministic portfolio: dΠ = 1 2 σ2 S2 ∂2 V ∂S2 + ∂V ∂t + ∂V ∂¯t dt. (4.7) 2 Haber et al. 1999 define the dynamics of barrier time ¯t so that it increases at the same rate as the real time t, therefore d¯t = dt, if the underlying S is beyond the barrier. The barrier time ¯t is reset to zero if S hits the barrier S = ¯S, and does not change if S < ¯S.
- 109. CHAPTER 4. A NEW PREPAYMENT MODEL 109 Appealing to the no arbitrage assumption associated with the Black-Scholes PDE framework, the change in value of this portfolio over a time dt is equal to the growth on an amount Π at the riskless interest rate r, rΠdt = 1 2 σ2 S2 ∂2 V ∂S2 + ∂V ∂t + ∂V ∂¯t dt. (4.8) Substituting (4.4) and (4.6) into (4.8) and dividing throughout by dt results in ∂V ∂t + 1 2 σ2 S2 ∂2 V ∂S2 + rS ∂V ∂S − rV + ∂V ∂¯t = 0, (4.9) which is the modified form of the Black-Scholes PDE for the Parisian option when the underlying is beyond the barrier level. 4.2.6 Numerical solution The Crank-Nicolson finite-difference method is used to approximate the governing PDE for the Parisian option; the following algorithm outlined is for the Parisian up-and-out call option. The problem separates into two distinct PDEs that must be solved. Below the barrier level ¯S, the standard Black-Scholes PDE (4.1) is solved numerically. Above the barrier, the barrier time increases and the modified Black- Scholes PDE (4.9) for the Parisian option is solved. Before discretising the problem, the usual transformation in the time variable t is made, i.e. τ = T − t, where τ is the time to expiry. The corresponding transformation is also made in the barrier time variable ¯t, i.e. ¯τ = ¯T − ¯t, where ¯τ is the time to knock out and ¯T is the activation time (when ¯t = ¯T knock out comes into affect and the option expires worthless). The transformed (forward) PDE which models the price of the option above the barrier is − ∂V ∂τ − ∂V ∂¯τ + 1 2 σ2 S2 ∂2 V ∂S2 + rS ∂V ∂S − rV = 0. (4.10) The discrete approximation for this PDE requires differencing in both the τ and ¯τ time variables. This problem can be regarded as being bound inside a cube, see figure 4.1 for an illustration, with one spatial and two temporal axes. When defining an equally spaced finite cube (for the numerical solution the S domain is truncated for
- 110. CHAPTER 4. A NEW PREPAYMENT MODEL 110 convenience) let, 0 ≤ S ≤ Smax, 0 ≤ τ = (T − t) ≤ T and 0 ≤ ¯τ = ( ¯T − ¯t) ≤ ¯T so for the cube: S = Si = i∆S where 0 ≤ i ≤ imax, τ = τk = k∆τ where 0 ≤ k ≤ kmax, ¯τ = ¯τl = l∆¯τ where 0 ≤ l ≤ lmax. Here imax is the number of nodes along the S axis, kmax is number of time steps the contract is divided into and lmax = ¯T/∆¯τ is the number of time steps the activation time is divided into. By definition, as noted above, d¯t = dt, and therefore the step size in the barrier time is set equal to the step size in time, ∆¯τ = ∆τ, so that lmax = ¯T/∆τ. When the contract starts, the underlying stock is assumed to be below the barrier, and the time until knock out is ¯τ = ¯T, which corresponds to l = lmax. The barrier level ¯S is defined by ¯i = ¯S/∆S and V k,l i is denoted as the numerical approximation to the option value V (S, τ, ¯τ). The PDE describing the option value below the barrier is discretised in a manner similar to that employed in section (2.5.1), except that the ‘j’ index can be omitted which is only relevant if the derivative is dependent on two stochastic variables. Derivative approximations for S above the barrier When the underlying is above the barrier level, S > ¯S, the necessary derivative approximations are: ∂V (S, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂τ ≈ V k+1,l+1 i − V k,l+1 i + V k+1,l i − V k,l i 2∆τ , (4.11) ∂V (S, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂¯τ ≈ V k+1,l+1 i − V l+1,k i + V k,l+1 i − V k,l i 2∆¯τ , (4.12) ∂V (S, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂S ≈ 1 8∆S (V k+1,l+1 i+1 − V k+1,l+1 i−1 + V k+1,l i+1 − V k+1,l i−1 +V k,l+1 i+1 − V k,l+1 i−1 + V k,l i+1 − V k,l i−1), (4.13) ∂2 V (S, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂S2 ≈ 1 4(∆S)2 (V k+1,l+1 i+1 − 2V k+1,l+1 i + V k+1,l+1 i−1 +V k,l+1 i+1 − 2V k,l+1 i + V k,l+1 i−1 +V k,l i+1 − 2V k,l i + V k,l i−1). (4.14)
- 111. CHAPTER 4. A NEW PREPAYMENT MODEL 111 The option value V (S, τ, ¯τ) is approximated as V S, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ ≈ V k+1,l+1 i + V k+1,l i + V k,l+1 i + V k,l i 4 . (4.15) Overall the error in the approximate solution V k,l i is of second-order accuracy in ∆S and ∆τ. As the step size in the barrier time is equal to the step size in time, ∆¯τ = ∆τ, this produces a simplification in the matrix problem, described below, as several terms cancel. Upon substitution of the two time derivatives approximations, equations (4.11) and (4.12), into the pricing PDE (4.10) (for the case when the underlying S is above the barrier) the resulting simplification occurs − ∂V ∂τ − ∂V ∂¯τ ≈ − V k+1,l+1 i − V k,l i ∆τ , (4.16) as ∆¯τ = ∆τ. The matrix problem is similar to valuing the remaining future payments (see section 2.5.3), where solving the basic Black-Scholes PDE (4.1) below the barrier and solving the modified form of the Black-Scholes PDE (4.9) above the barrier reduces to solving a set of linear equations for V k+1,l+1 i , except that in addition to solving the tridiagonal matrix problem produced at each time step (which can be readily solved using Gaussian elimination, see Smith, 1978), the valuation must be carried out for all barrier times. This is a simple matter of looping the valuation over all possible ¯τ, from ¯τ = 0 (knock out, i.e. l = 0), until the ¯τ = ¯T (when the barrier time is zero, i.e. l = lmax). The solution gives the price of the Parisian up-and-out call option, when time and barrier time are both zero, V kmax,lmax i . Initial condition and boundary conditions All that remains to close the valuation is to specify the initial/boundary conditions. For a Parisian up-and-out call option, the initial condition at the time until expiry is zero, i.e. τ = 0, is the usual call option condition (see Wilmott et al., 1993) as shown in equation (4.17), where E is the exercise price, unless the time until knock out is also zero, and then the option expires worthless, since then the barrier is activated
- 112. CHAPTER 4. A NEW PREPAYMENT MODEL 112 and knock out occurs (this region is shown by the shaded plane in figure 4.1), V (S, τ = 0, ¯τ) = max(S − E, O) if ¯τ > 0, 0 if ¯τ = 0. (4.17) This condition is implemented in the algorithm as follows V 0,l i = max(Si − E, 0) for l > 0, and V 0,0 i = 0 for l = 0. (4.18) The usual call option condition applies when the underlying asset is zero, the option is worthless for all time and barrier time V (S, τ, ¯τ) = 0 at S = 0. (4.19) As the underlying increases in value it is obvious that knock out will occur and the option will be worthless V (S, τ, ¯τ) → 0 as S → ∞. (4.20) These conditions can be implemented in the algorithm as follows V k,l 0 = 0 at i = 0 and V k,l imax = 0 at i = imax. (4.21) Finally, the option is worthless when it has been knocked out, and so V k,0 i = 0 for all k. A feature specific to the Parisian option is that if the underlying is below the barrier the barrier clock is set to zero, i.e. for i < ¯i set l = lmax (here time to knock out ¯τ = ¯T). 4.2.7 Numerical results Figures 4.2, 4.3 and 4.4 are obtained using the numerical algorithm described in section 4.2.6. The results in sections 4.2.7 and 4.2.10 are based on Smax = 3E with imax = 300; kmax = 1000 and lmax = kmax ¯T (as ∆¯τ = ∆τ), choices which were deemed satisfactory through extensive computational experimentation. Figure 4.2 shows the value of a Parisian up-and-out call option for varying times until knock out ¯τ, and for varying values of the underlying S, at τ = T. Notice below
- 113. CHAPTER 4. A NEW PREPAYMENT MODEL 113 7 8 9 10 11 12 13 140 0.02 0.04 0.06 0.08 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 PSfragreplacements ¯τ S VV Figure 4.2: Valuation of the Parisian up-and-out call option with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. the strike price, E = 10, well away from the impending barrier, the option value behaves like a vanilla call option. As the underlying increases and approaches the barrier level, ¯S = 12, the value of the option decreases. The option value tends to zero once the underlying is not much greater than the barrier level. Also in figure 4.2, as the time until knock out approaches zero, i.e. ¯τ → 0, the option value decreases at a greater rate (for S > ¯S), which is to be expected since V = 0 for ¯τ = 0. This point is illustrated further in figure 4.3, where the Parisian up-and-out call option value is shown for three barrier times. The bold curve, for barrier time ¯t = 0, is the price of the option for all values of the underlying. To contrast this, the two other curves show the option value as the barrier time tends towards the activation time. As expected, the gradient of the option value is much steeper around the barrier level, as knock out draws closer. Figure 4.4 shows how the gradient of the option value (the option delta) varies for different times to knock out. The rapid change in the gradient in figure 4.3, as knock out looms closer, corresponds to the spike seen in figure 4.4.
- 114. CHAPTER 4. A NEW PREPAYMENT MODEL 114 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 6 8 10 12 14 barrier time = 0.000 barrier time = 0.050 barrier time = 0.095 PSfragreplacements ¯t S V Figure 4.3: Parisian up-and-out call option at three different barrier times with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. 7 8 9 10 11 12 13 140 0.02 0.04 0.06 0.08 0.1 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 PSfragreplacements ¯τ S ∂V ∂S ∂V ∂S Figure 4.4: The delta of the Parisian up-and-out call with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05.
- 115. CHAPTER 4. A NEW PREPAYMENT MODEL 115 The delta, ∂V ∂S , is the rate of change of the option value with respect to the value of the underlying asset. The associated delta of the option is shown in figure 4.4. The delta is smooth away from knock out, where 0.06 < ¯τ ≤ 0.1 but decreases for S ∼ ¯S as knock out approaches. The gamma of the option, ∂2V ∂S2 would be large in the region close to S = ¯S and close to ¯τ = 0. In practice this makes hedging Parisian options difficult, whereas a ParAsian option (described in section 4.2.8) would reduce some of this difficulty in hedging, since the gamma for a ParAsian option does not vary as rapidly as the equivalent Parisian option. Remarks The algorithm to value the Parisian up-and-out call option, section 4.2.6, can be easily modified for all possible combinations of a Parisian option. The changes required in the original algorithm are minimal and involve only slight variations in the initial and boundary conditions. There are a total of eight different variants of Parisian options. This total doubles if the possibility of an American early exercise feature is included in the contract. 4.2.8 Definition of the ParAsian option A cumulative Parisian option, which is more commonly referred to as a ParAsian option, is a further type of barrier option. The barrier is triggered if the underlying asset spends a prescribed time across a given barrier level. The activation time does not have to be reached in a single excursion for the barrier to be triggered, unlike the case for a consecutive Parisian option. Each time the underlying asset crosses the barrier the excursion time is recorded, and all the excursion times are summed together over the length of the contract. If the barrier clock, which keeps a record of the total excursion time, reaches the barrier activation time, the barrier is triggered. The contract specifies whether the barrier is an up or a down barrier and also whether activation causes the option to be knocked in or knocked out.
- 116. CHAPTER 4. A NEW PREPAYMENT MODEL 116 4.2.9 Numerical solution The algorithm to value a ParAsian up-and-out option is almost identical to that described in section 4.2.6. The only difference that arises is the barrier clock is cumulative rather than consecutive. The time to knock out ¯τ is not set at ¯T when the underlying is below the barrier level and the value of the option is only set to zero in the initial condition V 0,0 i = 0. 4.2.10 Numerical results Figure 4.5 shows the delta of the up-and-out ParAsian call option with the same parameters as the Parisian option discussed in section 4.2.6. The effect on the option delta due to the barrier clock being cumulative rather than consecutive can be clearly seen by comparing figure 4.4 with figure 4.5. Since the barrier time ¯t is not reset to zero when the underlying asset drops below ¯S, we do not see the drastically decreasing delta characteristic of a Parisian option, instead the delta of the ParAsian option remains within a much smaller range. For all values of the underlying S, the Parisian option has a greater value than the ParAsian option. Intuitively this is easy to understand, since with ¯t being cumulative, the ParAsian version increases the likelihood of knock out, making the option value lower. This is shown in figure 4.6, where the upper surface is the Parisian option value and the lower surface corresponds to the ParAsian option value. Now the definitions of occupation-time derivatives and their numerical solution are established, the details of how these are incorporated into FRM valuation can be given. 4.3 FRM valuation framework including lagged pre- payment The general valuation framework is the same as for the standard FRM model, as detailed in section 3.2, the main difference being that in this section the prepayment
- 117. CHAPTER 4. A NEW PREPAYMENT MODEL 117 6 8 10 12 14 0 0.02 0.04 0.06 0.08 0.1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 PSfragreplacements ¯τ S ∂V ∂S ∂V ∂S Figure 4.5: The delta of the ParAsian up-and-out call with E = 10, ¯S = 12, ¯τ = 0.1, T = 1, σ = 0.2 and r = 0.05. 6 8 10 12 14 0 0.02 0.04 0.06 0.08 0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 PSfragreplacements ¯τ S VV Figure 4.6: Comparison of the Parisian and ParAsian options with E = 10, ¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05.
- 118. CHAPTER 4. A NEW PREPAYMENT MODEL 118 option is governed by a new model regarding the borrower’s behaviour, see section 4.3.1 for details. The contract details are slightly different, as the contract under discussion is that of a US FRM contract, rather than the UK FRM contract considered in chapter 3. The reason for this is that future work, leading from this study, will be to improve MBS pricing, a security predominantly traded in the US, which is usually based on pools of US FRM loans. The difference with a US contract is mainly in terminology, the arrangement fee is referred to as percentage points charged or more simply the points on the loan, and the total debt payment required to be paid when the borrower chooses to prepay is referred to as the face value of the loan. As the focus in this section is on the new prepayment assumption, only the components which affect the equilibrium of the mortgage are considered, and so coinsurance is not considered. The equilibrium condition, which must be satisfied at the origination of the con- tract (explained in section 3.4), takes the same form as for a UK FRM, equation (3.22). As mentioned above, the arrangement fee ξ is now the called the points on the loan. As in chapter 3, the contract consists of the value of the remaining payments to the lender A(r, τm) and the borrower’s options. Default D(H, r, τm) is again modelled as a monthly European put option, where the borrower’s decision is modelled using a financially ruthless default behaviour, see Foster and Van Order (1984, 1985). The borrower’s decision when to prepay, C(H, r, τm), has a new assumption dictating when this option is exercised, see section 4.3.1. The insurance I(H, r, τm), only pays if the borrower defaults, and covers only the outstanding balance of the loan plus accrued interest up to some predetermined maximum coverage percentage, and not the forgone payments. Again it is assumed that insurance is paid for as part of the contract rate that determines monthly payments, and the coverage limit is known.
- 119. CHAPTER 4. A NEW PREPAYMENT MODEL 119 4.3.1 New prepayment model A new termination model for prepayment is proposed, rather than prepaying as soon as the mortgage value is equal to the face value of debt, the borrower waits until the mortgage value is equal to or exceeds the face value of debt for the decision time ¯T and then prepays the mortgage. The effect of different borrower decisions or waiting times on the value of the mortgage is analysed in section 4.5. By incorporating an occupation-time derivative, a lag in the investment informa- tion and its implementation can be modelled within an endogenous model. Prepay- ment is now modelled as a Parisian call option rather than an American call option. The American free boundary, on one side of which (in house price and interest rate space) prepayment occurs and on the other it does not, is associated with standard prepayment, and so the new termination model has a different free boundary. As discussed in section 4.2, for simple occupation-time derivatives, a particular value of the underlying is specified as the barrier level, beyond which a counter is triggered, upon reaching the barrier time (the decision time for this mortgage application of occupation-time derivatives) the derivative will either knock in or knock out. The prepayment Parisian call option is slightly different; instead of the underlying asset being monitored across a barrier, the value of the mortgage V (a derivative depen- dent on two underlying factors) is monitored above a time dependent barrier - the face value of debt FV (τm). The barrier position in V corresponds to the free bound- ary location in (H, r) space, i.e. at each instant in time the barrier FV (τm) for V is known but the free boundary location, the value of house price and interest rate which correspond to the mortgage value being equal to the barrier, are unknown. The values of these two state variables are required to value V , so that the modified PDE can be valued in the appropriate region of the state space. The free boundary must be positioned such that V (H, rb, τm, ¯τ) = FV (τm), (4.22) where, rb is a function of H and is the interest-rate value which separates the con- tinuation region from the prepayment region, and the face value of debt FV (τm)
- 120. CHAPTER 4. A NEW PREPAYMENT MODEL 120 (analogous to the total debt for a UK FRM) is given by FV (τm) = [1 + c(Tm − τm)]OB(i). (4.23) The details of the numerical solution of this free boundary problem are given in section 4.4.1. Note the value of the mortgage, and the other mortgage components are now functions of an additional time variable, the time until prepayment occurs ¯τ, see section 4.3.2 for more details. An illustration of how the consecutive occupation-time derivative (the Parisian call option) triggers prepayment is shown in figure 4.7. The figure shows that the value of the mortgage V must be beyond the face value of debt FV (t1) (the amount paid if a borrower choses to prepay at time t1) consistently until the decision time ¯T elapses for prepayment to occur. If V is still above this level at time t2 = t1 + ¯T the borrower prepays the mortgage at the greater face value FV (t2). The value of the mortgage to the lender, when prepayment occurs at t2, is increased by the borrower waiting to prepay and the increase is given by d = FV (t2) − FV (t1). 4.3.2 Modified PDE It is essential that the time the mortgage value spends beyond the prepayment barrier (4.22) can be recorded, which then necessitates the introduction of a new time variable - the time until the decision to prepay is made ¯τ, this is analogous to the time to knock out for the Parisian option on a stock. Although it was mentioned previously that all the mortgage components are now functions of this new time variable, this is not strictly true. All the components which are affected by the borrower terminating the contract prior to maturity now include this variable, but the value of the remaining payments A(r, τ) does not depend on the borrower’s actions and so is not a function of ¯τ. When the value of the mortgage to the lender V (H, r, τm, ¯τ) is less than the prepayment barrier (4.22), the usual mortgage valuation PDE (1.17) applies. When the mortgage value increases, such that it is greater than or equal to this barrier, the
- 121. CHAPTER 4. A NEW PREPAYMENT MODEL 121 t Tt V(t) FV(t ) FV(t ) t1 1 2 2 d T Figure 4.7: An illustration of the effect of waiting to prepay on the value of the mortgage, modelled using a consecutive occupation-time derivative. time until knock out ¯τ begins to decrease, which produces an extra time derivative in the valuation PDE to track the time until the prepayment decision, while V is beyond the barrier (4.22) in a single excursion (see section 4.2.5 for full explanation of how the extra time derivative arises for a straightforward Parisian option). The PDE to value a general mortgage component F(H, r, τ, ¯τ) (which is affected by the borrower’s decision time), is − ∂F ∂¯τ + 1 2 H2 σ2 H ∂2 F ∂H2 + ρH √ rσHσr ∂2 F ∂H∂r + 1 2 rσ2 r ∂2 F ∂r2 +κ(θ − r) ∂F ∂r + (r − δ)H ∂F ∂H − ∂F ∂τ − rF = 0. (4.24) By making a straightforward comparison of the derivation (see section 1.4) for the standard mortgage valuation PDE (1.17) and the derivation (see section 4.2.5) for the simple Parisian option PDE (4.9) (except that the underlying asset is now the house price H and the interest rate r is no longer constant), the derivation of the PDE above is obvious. This PDE will be used in this chapter to value all the components which are affected by the borrower’s decision time.
- 122. CHAPTER 4. A NEW PREPAYMENT MODEL 122 Free boundary position Solve modified PDE beyond free boundary V(H,r,t) < FV(t) H r 0 0 00 00 (H,r) space Figure 4.8: An illustration of the general solution space at any time step for a FRM mortgage with the new prepayment model. An illustration of the solution space for all the mortgage components, other than the value of the remaining payments, at any time step, with prepayment triggered by an occupation-time derivative, is shown in figure 4.8, including the position of a general free boundary position implied by the prepayment barrier. On one side of this boundary, when the value of the mortgage is less than the face value, the standard governing PDE (1.17) must be solved; beyond the free boundary position the modified PDE (4.24) must be solved. 4.3.3 New payment-date conditions Now that the new time variable ¯τ has been defined (as the time until the decision to prepay is made) it is appropriate to state the new payment-date conditions for the mortgage components as they are affected by this new variable.
- 123. CHAPTER 4. A NEW PREPAYMENT MODEL 123 At maturity At maturity the option to prepay is not relevant, therefore the payment-date condi- tions are not affected by the new prepayment model. The payment-date conditions for the value of the remaining payments, the value of the mortgage, and the value of default, are the same as for the standard UK FRM model, and are given in section 3.3.1. For the US FRM the insurance is defined differently (see Kau et al., 1995), since the issuer covers the lender’s shortfall on the lost future cash flows up to some fraction φ of the face value of debt (at maturity this is simply the value of the final monthly payment MP). If default does not occur at the payment date, then the value of the insurance becomes its value in the future. The payment-date condition at maturity is given by I(H, r, τn = 0) = min(MP − H, φMP) (4.25) if default occurs, and by I(H, r, τn = 0) = 0 (4.26) if the final monthly payment is made (default does not occur), where n is the length of the mortgage in months. Earlier payment dates At earlier payment dates, there is the possibility that prepayment will occur, and because prepayment is now determined using a different model, the payment-date conditions change from when prepayment was modelled using the optimal call condi- tion in chapter 3. The value of the remaining payments A(r, τ) is not affected by the either of the borrower’s options, so the payment-date condition for this component at earlier payment dates does not change; see equation (3.8) for the condition. The payment-date condition for the value of the mortgage to the lender at the end of month 1, 2, . . . , n − 2 and n − 1 is V (H, r, τm = 0, ¯τ) = min[V (H, r, τm+1 = Tm+1) + MP, H] if ¯τ > 0, FV (τm = 0) if ¯τ = 0, (4.27)
- 124. CHAPTER 4. A NEW PREPAYMENT MODEL 124 where 1 ≤ m ≤ n − 1. When the time until prepayment is zero (¯τ = 0), obviously prepayment occurs and the mortgage is worth the face value at this time; otherwise, the condition takes the same form as previously. For default the payment-date conditions prior to maturity are D(H, r, τm = 0, ¯τ) = D(H, r, τm+1 = Tm+1) if V = H and ¯τ > 0, A(r, τm = 0) − H if V = H and ¯τ > 0, 0 if ¯τ = 0, (4.28) where V = H refers to default not occurring and V = H is when default occurs, which happens when the value of the house is so low that the borrower decides not to make the scheduled monthly payment. The option to default is worthless when prepayment occurs, which occurs when the value of the mortgage is greater than the face value for the decision time ¯T, which corresponds to ¯τ = 0. The payment-date conditions for the prepayment option prior to maturity are C(H, r, τm = 0, ¯τ) = C(H, r, τm+1 = Tm+1) if V = H and ¯τ > 0, 0 if V = H and ¯τ > 0, A(r, τm = 0) − FV (τm = 0) if ¯τ = 0. (4.29) When prepayment occurs (¯τ = 0) the option value is calculated by rearranging the relation V = A − D − C to determine C. The third equation in (4.29) is due to the option to default being worthless, and as the value of the mortgage is the face value when prepayment occurs. Finally, the insurance component has the following conditions at the earlier pay- ment dates I(H, r, τm = 0, ¯τ) = I(H, r, τm+1 = Tm+1) if V = H and ¯τ > 0, min[FV (τm = 0) − H, φFV (τm = 0)] if V = H and ¯τ > 0, 0 if ¯τ = 0. (4.30) The occurrence of prepayment (¯τ = 0) renders the option to default to be worthless, implying that the insurance value is also zero, as this only has any value if default occurs.
- 125. CHAPTER 4. A NEW PREPAYMENT MODEL 125 While some of the payment-date conditions change, for those which are affected by the new prepayment model, the other boundary conditions do not and are as stated in section 2.6. 4.4 Numerical solution As in section 3.5, on the valuation of the more straightforward FRM under the more simplistic prepayment assumption, the Crank-Nicolson finite-difference scheme is used to discretise the valuation PDEs. The addition of the new prepayment model, driven by the Parisian occupation-time derivative, requires the solution of two distinct PDEs (as noted already). On one side of the free boundary (4.22), discussed in section 4.3.1, the standard governing PDE (1.17) must be solved numerically; on the other side, the borrower decision time increases and the modified valuation PDE (4.24) must be solved numerically. The usual transformation in the temporal direction is made for both time variables as the valuation begins at maturity of the loan, so that τm = Tm − tm, where τm is the time until the payment date in month m and the corresponding transformation is made in the borrower decision time variable ¯t.3 ¯τ = ¯T − ¯t, (4.31) where ¯τ, is the time until the call to prepay the mortgage is made by the borrower. When defining a (finite, truncated) equally spaced grid (for the numerical solution the H domain and r domain are truncated for convenience) suppose, 0 ≤ H ≤ Hmax, 0 ≤ r ≤ rmax, 0 ≤ τm ≤ Tmax and 0 ≤ ¯τ ≤ ¯Tmax. Then the function F(H, r, τm, ¯τ) is 3 The real time is separated into months for clarity as each month has its own payment-date conditions.
- 126. CHAPTER 4. A NEW PREPAYMENT MODEL 126 represented by values on a discrete set of points: H = Hi = i∆H where 0 ≤ i ≤ imax, r = rj = j∆r where 0 ≤ j ≤ jmax, τ = τk = k∆τ where 0 ≤ k ≤ kmax, ¯τ = ¯τl = l∆¯τ where 0 ≤ l ≤ lmax. As in chapter 3, ∆H, ∆r and ∆τ are the grid spacings in the H, r and τm dimensions respectively. Further, when the valuation problem is above the barrier and the PDE includes the extra time derivative, ∆¯τ is the grid spacing in the ¯τ dimension. As in the numerical solution for the straightforward Parisian option (section 4.2.6), again ∆¯τ = ∆τ, by definition. imax and jmax are the number of nodes along the spatial H and spatial r axes respectively, kmax and lmax are the number of time steps dividing each month of the contract and the borrower decision time respectively. Writing F k i,j ≡ F(Hi, rj, τmk ) for each (i, j, k) triple below the barrier and F k,l i,j ≡ F(Hi, rj, τmk , ¯τl) for each (i, j, k, l) quartet above the barrier. Below the barrier the valuation PDE (1.17) is discretised as shown in section 2.5.1. Above the barrier the modified PDE (4.24) is discretised in the following manner to account for the second time dimension ¯τ. Following a Crank-Nicolson finite-difference scheme, to retain second-order accuracy in house price and interest rate, the time derivatives are approximated as ∂F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂τ ≈ Fk+1,l+1 i,j − Fk,l+1 i,j + Fk+1,l i,j − Fk,l i,j 2∆τ , (4.32) ∂F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂¯τ ≈ Fk+1,l+1 i,j − Fk+1,l i,j + Fk,l+1 i,j − Fk,l i,j 2∆¯τ . (4.33) The spatial derivatives for house price H are approximated by ∂F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂H ≈ 1 8∆H (Fk+1,l+1 i+1,j − Fk+1,l+1 i−1,j + Fk+1,l i+1,j − Fk+1,l i−1,j +Fk,l+1 i+1,j − Fk,l+1 i−1,j + Fk,l i+1,j − Fk,l i−1,j), (4.34)
- 127. CHAPTER 4. A NEW PREPAYMENT MODEL 127 ∂2 F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂H2 ≈ 1 4(∆H)2 (Fk+1,l+1 i+1,j − 2Fk+1,l+1 i,j + Fk+1,l+1 i−1,j +Fk,l+1 i+1,j − 2Fk,l+1 i,j + Fk,l+1 i−1,j +Fk+1,l i+1,j − 2Fk+1,l i,j + Fk+1,l i−1,j +Fk,l i+1,j − 2Fk,l i,j + Fk,l i−1,j). (4.35) The spatial derivatives for interest rate r are approximated by ∂F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂r ≈ 1 8∆r (Fk+1,l+1 i,j+1 − Fk+1,l+1 i,j−1 + Fk+1,l i,j+1 − Fk+1,l i,j−1 +Fk,l+1 i,j+1 − Fk,l+1 i,j−1 + Fk,l i,j+1 − Fk,l i,j−1), (4.36) ∂2 F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂r2 ≈ 1 4(∆r)2 (Fk+1,l+1 i,j+1 − 2Fk+1,l+1 i,j + Fk+1,l+1 i,j−1 +Fk,l+1 i,j+1 − 2Fk,l+1 i,j + Fk,l+1 i,j−1 +Fk+1,l i,j+1 − 2Fk+1,l i,j + Fk+1,l i,j−1 +Fk,l i,j+1 − 2Fk,l i,j + Fk,l i,j−1). (4.37) The cross-spatial derivative is approximated by ∂2 F(H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ) ∂H∂r ≈ 1 16∆H∆r (Fk+1,l+1 i+1,j+1 − Fk+1,l+1 i−1,j+1 − Fk+1,l+1 i+1,j−1 +Fk+1,l+1 i−1,j−1 + Fk,l+1 i+1,j+1 − Fk,l+1 i−1,j+1 − Fk,l+1 i+1,j−1 +Fk,l+1 i−1,j−1 + Fk+1,l i+1,j+1 − Fk+1,l i−1,j+1 − Fk+1,l i+1,j−1 +Fk+1,l i−1,j−1 + Fk,l i+1,j+1 − Fk,l i−1,j+1 − Fk,l i+1,j−1 +Fk,l i−1,j−1). (4.38) Finally, the asset F(H, r, τ, ¯τ) is approximated by F H, r, τ + 1 2 ∆τ, ¯τ + 1 2 ∆¯τ ≈ Fk+1,l+1 i,j + Fk+1,l i,j + Fk,l+1 i,j + Fk,l i,j 4 . (4.39) Overall the error in the approximate solution F k,l i,j is of second-order accuracy in ∆H, ∆r and ∆τ. The above approximations are substituted into the modified PDE (4.24), and then the matrix problem produced for each component is similar to that described in section 2.5.2, except that the equations are rearranged for F k+1,l+1 i,j (so that the new
- 128. CHAPTER 4. A NEW PREPAYMENT MODEL 128 value of F is found at each time step). The solution of the set of linear equations produced is described in section 2.5.3, and the valuation must be carried out for all barrier times, as explained for the Parisian option in section 4.2.6; the solution gives the value of a general mortgage component at origination. As described for the straightforward Parisian option valuation, a simplification in the matrix problem can also be made (see section 4.2.6 for the details) for the mortgage valuation matrix problem (containing the Parisian prepayment feature), as the step size in the barrier time is set equal to the step size in time, ∆¯τ = ∆τ. As mentioned in section 2.5.3, regarding the solution of the linear equations that arise from the finite-difference methods used in this thesis, this chapter and chapter 5 both use iterative solvers for the default and insurance components, whereas in chap- ter 3 a general LU library package was used (see section 2.5.3 for specific details about both solution techniques). The improvement in computational efficiency is approxi- mately 30 times. The development of the iterative solvers was due to the recognition that the valuation of a FRM using the new prepayment model and valuation of the ARM (chapter 5) would both require a huge increase in computational effort from the straightforward FRM in chapter 3. The solution of the new prepayment model mortgage effectively requires the straightforward FRM model solution to be looped over the decision time, and the ARM model requires the solution to be looped over all the possible contract rates. Without the speed up, by switching to iterative solvers, the time required to compute a single valuation would have made collecting results very computationally expensive. 4.4.1 Solution of the free boundary problem As mentioned in section 4.3.1 the barrier level, associated with the Parisian prepay- ment feature, introduces a free boundary problem to divide the regions of state space in which the modified PDE (4.24) must be solved and where the usual PDE (1.17) must be solved (the general solution space is illustrated in figure 4.7). When the mortgage value is above the prepayment barrier level (4.22) (here the
- 129. CHAPTER 4. A NEW PREPAYMENT MODEL 129 H r i j Actual positionApproximate location r H Rj(i) Figure 4.9: An illustration of the finite grid in the house price H and interest rate r dimensions, the approximate location taken as the free boundary position is shown. mortgage value is calculated using the modified PDE (4.24)), borrowers are assumed to wait until the decision time ¯T elapses until they prepay. The free boundary problem occurs, at every time step, as the valuation of the mortgage requires the region of state space in which the mortgage value is greater than the prepayment barrier level to be known, but this region is unknown until the after the valuation has taken place. The boundary position is denoted by the variable rb in equation (4.22) and a typical location is shown by the solid curve in figure 4.9. Figure 4.9 shows a sketch of the finite-difference grid in H and r space, the solid curve is a typical position of the free boundary and the dashed line is the approximate location of the boundary that is used to separate the state space. The discrete approximation Rj(i) to the free boundary position rb (as shown in figure 4.9) separates the zones where the modified PDE (4.24) and the usual mortgage valuation PDE (1.17) must be implemented. It is shown next how the discrete approximation to the free boundary is found.
- 130. CHAPTER 4. A NEW PREPAYMENT MODEL 130 Algorithm to locate the discrete approximation Rj(i) At each time step: (i) Take an initial guess to the free boundary location, let this be Rold j (i). (ii) Value V k,l i,j for all i and for j = 0 to j = Rold j (i) using the modified PDE (4.24) as explained in section 4.4.1, and for j = Rold j (i) + 1 to j = jmax using the standard governing PDE (1.17). (iii) Find the new location of the discrete approximation Rnew j (i) by checking where V ≥ FV , i.e. where the mortgage value is above the prepayment barrier. (iv) If Rnew j (i) = Rold j (i) then stop, as Rnew j (i) is the approximate location of the free boundary position at the present time step, otherwise let Rold j (i) = Rnew j (i) and goto (ii). Step (ii) sweeps across the house price dimension Hi and a maximum value of interest rate rj (for each Hi value) is imposed so that the modified PDE (4.24) is valued for interest rates less than the approximate free boundary, i.e. for rj < Rj(i), and for house prices greater than the approximate free boundary, i.e. for Hi > Rj(i), and the usual PDE (1.17) is valued everywhere else. Steps (ii) and (iii) are repeated until the boundary used in the valuation is the same as that which is found by checking the mortgage value afterwards. The procedure must be carried out at every time step in the valuation of the mort- gage component V k,l i,j . Other than at maturity (the initial point in the valuation), it is convenient to take the initial guess to the free boundary location as the approximate location from the previous time step. 4.5 Results The results in figures 4.10, 4.11, 4.12, 4.13 and 4.14 are based on Hmax = 2Hinitial with imax = 200; rmax = 5rinitial with jmax = 200; kmax = 32; and lmax = kmax ¯T (as
- 131. CHAPTER 4. A NEW PREPAYMENT MODEL 131 89000 89500 90000 90500 91000 91500 0 0.02 0.04 0.06 0.08 0.1 PSfragreplacements r V(H=120000,r,τ1=T1,¯τ=¯T) Figure 4.10: Mortgage value at origination V (H = 120000, r, τ1 = T1, ¯τ = ¯T) against interest rate r for eight different decision times. For each line style, the decision time ¯T is zero (this corresponds to the original prepayment assumption), T/8, T/4, T/2, 3T/4, T, 5T/4 and 3T/2 from the bottom to the top. For the case when κ = 0.25, θ = 0.1, δ = 0.085, σH = 0.1, σr = 0.1, ρ = 0, c = 0.111805, ratio of loan to initial value of house = 0.9, H(0) = $100000, r(0) = 0.1 and ξ = 0.015 for a 15 year loan. ∆¯τ = ∆τ); these choices were deemed satisfactory through extensive computational experimentation. This section investigates the effect of modelling the borrower decision process, regarding when to prepay, as a consecutive occupation-time derivative. Borrowers who are very apprehensive about the future cost of their mortgages can be modelled using a short decision time, i.e. they prepay soon after the mortgage value is equal to the face value of debt. Alternatively, borrowers who would rather wait before they prepay (possible reasons for this are given in section 4.1) are modelled using a longer decision time. The affect of using different decision times, in the prepayment model, for the value of a US FRM can be seen in figure 4.10, which shows the mortgage value at origination at a constant house price of H = 120000, for interest-rate values of zero to 10% p.a. The results show how the value of the mortgage changes inside the prepayment region for various decision times; the prepayment region exists for low interest rates r < r(0), a house price of H = 120000 was inside the prepayment region for all the decision times shown. The new model of prepayment only affects
- 132. CHAPTER 4. A NEW PREPAYMENT MODEL 132 the results inside and close to the prepayment region. For a decision time of zero, the bottom curve, the result corresponds to the simplistic prepayment assumption of ruthless prepayment, which is to exercise as soon as the mortgage value is equal to the face value, i.e. FV (τ1 = T1) = 90000; this is the maximum mortgage value possible for the simplistic assumption. As the decision time is increased, the possible mortgage values also increase. When the decision time is 1.5 months ( ¯T = 3T/2), the top curve, the maximum possible mortgage value, as r → 0, is approximately 91250. By creating a lag in the prepayment decision, mortgage values greater than the par value can be achieved within a structural model; this was not possible under the more basic prepayment model. Note that the contract rate used for the results in this figure is the equilibrium contract rate value for the US FRM using the simplistic prepayment assumption. This is then used as a base parameter for comparative purposes when testing the various decision times. The comment above, that a decision time of zero (which causes excitation of prepayment the instant the mortgage value is equal to the face value) replicates the simplistic prepayment assumption, is expanded here. Figure 4.11 shows the mortgage value at origination for various house price and interest rate values around the prepayment region for a decision time of zero. This result is indistinguishable from the equivalent US FRM result using the simplistic prepayment assumption. The conventional prepayment assumption can be included within this more general framework by specifying the borrower decision time to be zero. It is possible to model different borrower behaviour types under the framework introduced in this chapter, and this has favourable implications for MBS pricing (the ability to include several borrower behaviour types within a model of a mortgage pool is advantageous, as it may lead to more accurate MBS pricing) as well as providing a more complete model of single FRM loans. Note that the prepayment region for the mortgage value is clearly visible as a plateau in (H, r) space, for increasing house price and low interest rates. For greater decision times, this region expands and the mortgage value increases as interest rates decrease. The default region is located at low house prices and for all interest rates, and this is visible as the curved surface towards the bottom
- 133. CHAPTER 4. A NEW PREPAYMENT MODEL 133 0 0.02 0.04 0.06 0.08 0.1 0.12 80000 85000 90000 95000 100000 105000 110000 115000 120000 78000 80000 82000 84000 86000 88000 90000 PSfragreplacements r H V (H, r, τ1 = T1, ¯τ = ¯T)V (H, r, τ1 = T1, ¯τ = ¯T) Figure 4.11: Mortgage value at origination V (H, r, τ1 = T1, ¯τ = ¯T) when the decision time is zero, ¯T = 0. The other parameters are identical to those stated in figure 4.10. left of the figure. Finally, the value of the mortgage essential in determining a contract in equilibrium is located at H(0) = 100000, r(0) = 0.1, this point in state space is located (as required for a contract in equilibrium at origination) in the continuation region, the curved surface towards the back right of the figure. This must be true, otherwise the borrower would terminate the contract immediately. Figure 4.12 shows the mortgage value at origination when the borrower is assumed to have an infinite decision time, i.e. ¯T → ∞.4 The purpose of this figure is to illustrate that a borrower who adopts this prepayment strategy would never prepay. The region of the state space shown is identical to that shown in figure 4.11, in that case the borrower’s decision time is zero, i.e. ¯T = 0, which enables an easy comparison of the affects of the two extreme prepayment assumptions to be made by comparing these two figures. The expanding nature of the mortgage value in figure 4.12, as house price increases and as interest rate decreases, is as a result of prepayment not being possible. By removing the possibility of prepayment, the prepayment region, which 4 An infinite decision time can be simulated by letting the decision time be greater than the length of the contract - prepayment will never occur.
- 134. CHAPTER 4. A NEW PREPAYMENT MODEL 134 0 0.02 0.04 0.06 0.08 0.1 0.12 80000 85000 90000 95000 100000 105000 110000 115000 120000 75000 80000 85000 90000 95000 100000 105000 110000 115000 120000 PSfragreplacements r H V (H, r, τ1 = T1, ¯τ = ¯T)V (H, r, τ1 = T1, ¯τ = ¯T) Figure 4.12: Mortgage value at origination V (H, r, τ1 = T1, ¯τ = ¯T) when the decision time tends to infinity ¯T → ∞. The other parameters are identical to those stated in figure 4.10. is visible as a plateau in figure 4.11, vanishes as the decision time tends to infinity. In reality a borrower would always prepay no matter how apprehensive they were about prepaying, since if interest rates continued to decline there would eventually be a time when the cost of borrowing becomes zero. If interest rates were zero, a borrower could prepay and then take out an interest-free loan with another lender. Figure 4.13 shows the prepayment value at origination at a constant house price of H = 120000, for interest rate values of zero to 10% p.a., for four decision times. The same region is chosen as that which was used to illustrate the affect of increasing decision time on the value of the mortgage (figure 4.10), so that a direct comparison of the affect on the value of prepayment, as the decision time increases, can be seen. The top curve is for a decision time of zero ¯T = 0, which corresponds exactly to the prepayment value for the simplistic prepayment assumption (the borrower exercises the option as soon as the mortgage value is equal to the face value of debt). As the decision time increases the value of prepayment, within the prepayment region, decreases. The bottom curve is for the greatest borrower decision time of 1.5 months,
- 135. CHAPTER 4. A NEW PREPAYMENT MODEL 135 10000 15000 20000 25000 30000 35000 0 0.02 0.04 0.06 0.08 0.1 PSfragreplacements r C(H=120000,r,τ1=T1,¯τ=¯T) Figure 4.13: Prepayment value at origination C(H = 120000, r, τ1 = T1, ¯τ = ¯T) against interest rate r for four different decision times. For each line style, the decision time ¯T is zero (this corresponds to the original prepayment assumption), T/2, T and 3T/2 from the top to the bottom. The other parameters are identical to those stated in figure 4.10. ¯T = 3T/2. By not prepaying immediately when the mortgage value is equal to the face value, the borrower reduces the value of the right to minimise the market value of the loan via prepayment. As a result of this, the value of the mortgage to the lender increases as the decision time increases, as already shown in figure 4.10. Figure 4.14 shows the mortgage ‘values’ at origination for several times until prepayment occurs (the only true value of the mortgage at origination is for a time until prepayment which is equal to the decision time ¯τ = ¯T).5 This figure illustrates a typical profile of the mortgage ‘value’ as prepayment becomes more imminent. The solid curve at the top is the actual value of the mortgage at origination, moving down from this curve, the curves below this represent the ‘value’ of the mortgage as the time until prepayment decreases. As expected the ‘value’ of the mortgage decreases as prepayment becomes more likely, i.e. as ¯τ → 0. When prepayment occurs (the bottom curve, ¯τ = 0) the mortgage ‘value’ is equal to the face value at that time, V = FV (τm = T1) (at origination the face value is equal to the loan amount). As prepayment is now modelled by a Parisian call feature, the ‘value’ of the mortgage 5 References to the mortgage value other than the true value are in inverted commas.
- 136. CHAPTER 4. A NEW PREPAYMENT MODEL 136 89600 89800 90000 90200 90400 90600 90800 91000 91200 0 0.02 0.04 0.06 0.08 0.1 PSfragreplacements r V(H=100000,r,τ1=T1,¯τ) ¯τ = 3T/2 ¯τ = 0 Figure 4.14: Mortgage ‘value’ at origination V (H = 100000, r, τ1 = T1, ¯τ) against interest rate r at selected times until prepayment ¯τ (equal intervals). The decision time is 1.5 months ¯T = 3T/2, other parameters are identical to those stated in figure 4.10. when prepayment occurs is constant for all house price and interest rate values, due to nature of the barrier feature, inherent in occupation-time derivatives. The results in the previous five figures illustrated the direct affect of the decision time on FRM valuation, for a contract rate held fixed; the results in the following four tables illustrate the affect that the decision time has on the equilibrium contract rate and the on the mortgage component values. The results in tables 4.1, 4.2, 4.3 and 4.4 are obtained using the same truncated underlying dimension sizes and the same grid sizes (as noted earlier) as for the figures 4.10, 4.11, 4.12, 4.13 and 4.14; except that now jmax = 50. Note that for all the earlier figures ∆r was much smaller as jmax = 200, this improved the resolution in the figures only, and did not affect the results. The tables show the equilibrium contract rate and the associated mortgage component values for four prepayment assumptions. ‘simp’ is the simple prepayment assumption ( ¯T = 0) that was the basis for all the results in section 3.9 on UK FRM valuation using ruthless prepayment. Results are also shown using the new prepayment model for decision times ¯T of 0.5 months ( ¯T/2), one month ( ¯T) and 1.5 months ( ¯3T/2). The parameters which are fixed are κ = 0.25, θ = 0.1, δ = 0.085, ρ = 0, the initial guess to the contract rate cinitial = 0.1, the ratio of the
- 137. CHAPTER 4. A NEW PREPAYMENT MODEL 137 loan to the initial value of the house = 0.9, the initial house value H(0) = 100000 and the points on the loan ξ = 0.015 for a 15 year loan.6 The choice of fixed parameters, for the results in this chapter, is made in accordance with parameters reported in the literature (see Titman and Torous 1989; Kau et al. 1995 Azevedo-Pereira et al. 2002). The parameters which were allowed to vary are the initial interest rate r(0), the house-price volatility σH and the interest-rate volatility σr. To understand the results in the tables, consider the equilibrium condition (3.22), and recall that the value of the mortgage is the value of the remaining payments minus the borrowers options to terminate the contract V = A−D−C. As the decision time ¯T increases it is expected that the value of prepayment will decrease, as borrowers are less inclined to prepay, and as the decision time does not affect the default decision, the only way that (3.22) will balance is if the value of payments decreases; this occurs if the contract rate decreases. These features are evident in all the tables for the different initial interest rates and volatilities tested, in particular as the decision time increases the equilibrium contract rate decreases (and as a result the value of payments decreases), and the value of prepayment decreases. The mortgage values in tables 4.1, 4.2, 4.3 and 4.4 do not vary significantly, for fixed initial interest rate r(0) and fixed volatilities, as the decision time increases. This is because the majority of the effect of the decision time is focused in the prepayment region of the state space (as shown in figure 4.10), which is away from the mortgage value vital to the equilibrium condition (at the initial house price and the initial interest rate). The following three observations are independent of the decision time analysis and are general features of mortgage valuation. (i) As the initial interest rate r(0) increases, the possible prepayment region ex- pands, as the prepayment region is for r < r(0) (otherwise a contract in equilib- rium would not exist), which means there is a greater possibility of prepayment. As such, the value of prepayment increases as r(0) increases, and the required equilibrium contract rate increases as a result. 6 Results were taken for longer maturities but were not qualitatively different.
- 138. CHAPTER 4. A NEW PREPAYMENT MODEL 138 Equilibrium contract rates and component values (in $). σr = 5% σH = 5% Contract Mortgage Payments Default Prepayment Insurance r(0) ¯T Rate (%) V A D C I 8 simp 9.0858 88630 90013 26 1357 19 T/2 9.0658 88640 89908 12 1256 10 T 9.0391 88641 89768 9 1118 9 3T/2 9.0243 88627 89690 21 1042 9 10 simp 10.3345 88641 91904 23 3240 9 T/2 10.3187 88642 91822 23 3159 8 T 10.3054 88642 91753 23 3089 9 3T/2 10.2918 88641 91683 23 3018 9 12 simp 11.7037 88648 94247 13 5586 3 T/2 11.6824 88646 94138 12 5479 3 T 11.6614 88647 94031 13 5372 3 3T/2 11.6415 88647 93930 13 5270 3 Table 4.1: Comparison of equilibrium contract rates and mortgage component values for σr = 5%, σH = 5%, for different prepayment assumptions. The loan is for 15 years, r(0) = spot interest rate (%). Equilibrium contract rates and component values (in $). σr = 5% σH = 10% Contract Mortgage Payments Default Prepayment Insurance r(0) ¯T Rate (%) V A D C I 8 simp 9.0712 87915 89937 801 1221 798 T/2 9.0596 87860 89876 818 1198 816 T 9.0481 87806 89815 847 1162 844 3T/2 9.0377 87752 97669 879 1128 897 10 simp 10.2950 88073 91699 712 2914 521 T/2 10.2830 88095 91637 752 2791 556 T 10.2703 88069 91571 810 2693 581 3T/2 10.2540 88032 91487 819 2637 618 12 simp 11.6731 88230 94090 702 5158 356 T/2 11.6563 88270 94005 728 5006 380 T 11.6373 88253 93909 788 4868 379 3T/2 11.6156 88227 93798 781 4790 423 Table 4.2: As in figure 4.1 except that σr = 5%, σH = 10%.
- 139. CHAPTER 4. A NEW PREPAYMENT MODEL 139 Equilibrium contract rates and component values (in $). σr = 10% σH = 5% Contract Mortgage Payments Default Prepayment Insurance r(0) ¯T Rate (%) V A D C I 8 simp 9.8993 88639 95156 47 6471 12 T/2 9.8160 88643 94975 25 6307 7 T 9.7976 88640 94875 36 6198 9 3T/2 9.7726 88639 94739 43 6057 11 10 simp 11.3063 88647 98214 21 9547 4 T/2 11.2677 88647 98007 15 9344 3 T 11.2333 88647 97822 22 9153 3 3T/2 11.1949 88646 97616 25 8945 4 12 simp 12.8384 88649 101460 10 12801 1 T/2 12.7902 88650 101204 8 12546 1 T 12.7428 88649 100953 12 12292 1 3T/2 12.6936 88649 100693 14 12030 1 Table 4.3: As in figure 4.1 except that σr = 10%, σH = 5%. Equilibrium contract rates and component values (in $). σr = 10% σH = 10% Contract Mortgage Payments Default Prepayment Insurance r(0) ¯T Rate (%) V A D C I 8 simp 9.7871 88319 94818 515 5984 301 T/2 9.7767 88292 94761 665 5804 358 T 9.7609 88228 94679 831 5617 422 3T/2 9.7248 88118 94480 926 5436 532 10 simp 11.2564 88484 97946 640 8822 231 T/2 11.2327 88390 97819 682 8747 261 T 11.2027 88348 97658 845 8465 303 3T/2 11.1580 88277 97418 898 8243 373 12 simp 12.8123 88358 101321 630 12333 172 T/2 12.7614 88454 101052 690 11904 192 T 12.7184 88430 100824 844 11550 220 3T/2 12.6649 88386 100541 872 11284 265 Table 4.4: As in figure 4.1 except that σr = 10%, σH = 10%.
- 140. CHAPTER 4. A NEW PREPAYMENT MODEL 140 (ii) Consider tables 4.1 and 4.2, for corresponding parameters between the two tables, as house-price volatility σH increases the contract rate decreases. Al- though the value of default increases as σH increases (this is in close analogy to the result that an increase in stock price volatility raises the value of a stock option), it may be expected that this would increase the contract rate but as insurance covers default, and since if default occurs prepayment cannot (mean- ing that prepayment decreases in value as σH increases), means the required contract rate actually falls as a result of an increase in house price volatility. (iii) Considering again tables 4.1 and 4.3, for corresponding parameters between the two tables, as interest-rate volatility σr increases the contract rate increases. This is more straightforward, as σr increases the value of prepayment increases, this would potentially increase the borrower’s position, unless the equilibrium contract rate also increased. A note on computation time is in order; the Parisian prepayment feature requires the valuation to be looped over the decision time, which implies that the greater the decision time, the greater the computation time. The simple prepayment assumption required on average 102 seconds per valuation, whereas for a decision time of 0.5 months, one month and 1.5 months took an average of 2085 seconds, 4662 seconds and 7060 seconds, respectively.7 For the results in the tables the equilibrium contract rate took between 4 and 9 valuation iterations. 4.6 Conclusions The new model of prepayment presented in this chapter provides a parsimonious structural means of modelling a borrower’s termination behaviour that appears ‘irra- tional’ according to the results of a basic optimal exercise model, i.e. results can be obtained outside the scope of simple rational models. By incorporating an occupation- time derivative into the valuation framework a more advanced (compared with the 7 The results were obtained using a 2412 MHz AMD Athlon computer.
- 141. CHAPTER 4. A NEW PREPAYMENT MODEL 141 simple ruthless approach to prepayment modelling) borrower decision process is de- veloped, where a rational exercise structure is retained in a modified form. For straightforward Parisian options the barrier level is a fixed value of the un- derlying but when they are used in the context of prepayment modelling (to simulate a lag in prepayment being exercised), the barrier level is time dependent and also introduces a free boundary problem. The barrier level depends on the value of the mortgage, which is a function of the underlying house price and interest rate, and creates a free boundary problem to determine the region of the state space in which the modified PDE must be solved. This chapter has shown that it is possible to achieve mortgage values above par within a structural model by including a decision time by the borrower before pre- payment is made, and the results show that the direct effect of increasing the decision time is to increase the value of the mortgage above par inside the prepayment region (under simple option-theoretic models this is not possible); correspondingly the value of prepayment decreases. The limiting case of allowing the decision time to tend to zero results in the simple ruthless exercise assumption. The alternative limiting case of allowing the decision time to tend to infinity causes the prepayment region to vanish, which illustrates that under this scenario prepayment would never occur. The contract rate required to achieve equilibrium was calculated for increasing de- cision times, and the results show that the contract rate increases as the value of prepayment decreases. The approach of creating a lag in prepayment is one method of modelling borrow- ers who do not exercise their option to prepay when it appears financially optimal; borrowers who do not act the same can also be modelled under this framework by varying the decision time. A possibility for future research is to use these advance- ments in prepayment modelling in an option-theoretic MBS pricing model, and it is hoped that the flexibility of the occupation-time derivative driven framework will improve pricing. The approach used could be applied to other securities as well as MBS, for example any security with embedded options whose value is determined by
- 142. CHAPTER 4. A NEW PREPAYMENT MODEL 142 the behaviour of a large group of individuals who cannot be counted on to act ac- cording to a simple rational model, including determining the optimal call policy for corporate bonds, and modelling the conversion behaviour of the holders of convertible debt.
- 143. Chapter 5 Advancements in adjustable-rate mortgage valuation 5.1 Introduction The work of Sharp et al. (2006) and chapter 3 demonstrated how singular pertur- bation theory can be used to value a FRM contract accurately and efficiently. The analysis in this chapter concentrates on the more mathematically complex case of the ARM. Initially, this chapter describes the auxiliary variable technique of Kau et al. (1993), to value the ARM contract. This chapter improves on their approach by employing a superior numerical procedure, and then focuses on a new improved methodology to value the ARM contract. The motivation for an ARM is to adjust the monthly payments to the prevailing market and so to insulate the value of the contract from interest rate variations. The option-theoretic simulation of variable-rate (US: adjustable-rate) mortgages is one of the most complex derivative products to value. A full contract, without too many simplifying assumptions, must include a series of European put options (that the borrower holds to default on a scheduled monthly payment), an overarching American call option (whereby the borrower can prepay the mortgage at any time) and automatically exercised options (the payment caps/floors). These options are interlinked and must be considered simultaneously. 143
- 144. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 144 The optimal stopping problem produced, as a result of the American-style option to prepay, can easily be tackled by a backward method to solve the governing PDE. Although this type of path-dependence can be readily handled, more subtle path- dependency that occurs when valuing ARMs causes problems when using backward methods to value the contract. The contract rate, which determines the value of the monthly payments by the borrower, is a floating rate depending on past interest rates. For backward pricing methods, such historical information is not available since the past is unknown at the present step in the solution. This chapter details how previous researchers have overcome this problem, whilst still using backward pricing methods, and then details new approaches for modelling and valuing ARMs. This motivates the question: why not instead use a forward method such as the Monte Carlo procedure? This would remove the path-dependency problem arising due to the adjustable contract rate, however, it would introduce a greater problem caused by forward techniques having extreme difficulty in treating early exercise features. Although considerable research into developing forward schemes that can deal with early exercise features, has been carried out (see, Duck et al, 2005, for an improved recent effort), using a forward method to solve a problem that is basically an American option is generally inefficient. The previous research on ARMs can be split into two distinct categories, broadly depending on the options deemed significant that the mortgage holder could exercise. The first strand allows the borrower to prepay the full outstanding debt, prior to maturity of the loan, whilst in the second type of model the borrower is given two options: as well as being able to prepay, the borrower is also able to default on a scheduled monthly payment. Further details are described next.
- 145. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 145 5.2 Related Literature 5.2.1 Prepayment-only ARMs A prepayment-only model, sometimes referred to as default-free ARM, gives the borrower a single option to terminate the contract by prepaying the mortgage. This type of model requires the inclusion of one stochastic underlying variable: the interest rate. Buser et al. (1985) model a default-free ARM, using a one-factor Cox et al. (1985b) model for the spot interest rate; they analyse only a simple version of an ARM. The first attempt by Kau et al. (1985) to value a full ARM specifically considered default-free, rate-capped ARMs. The model includes a single factor (the spot rate) plus an auxiliary variable which overcomes the problem of path-dependency with the adjustable contract rate when following a backward-pricing procedure. The formula that determines the new contract rate depends on the present contract rate and an index, which was described in section 2.3.1, and is the mortgage equivalent of a one-year, default-free pure discount bond (Cox et al. (1985b) give a closed-form solution for the one-year, default-free pure discount bond), which yields additional information about the most recently set contract rate. The Kau et al. (1985) paper uses a given initial contract rate and iteratively solves for the required margin added to the variable rate, subject to a no-arbitrage condition being satisfied; the mortgage components can be determined once this condition is satisfied. The authors’ later work, Kau et al. (1990), uses the same approach as their previous attempt, but includes a prepayment penalty in the form of points charged as a percentage of the outstanding balance. Also, a teaser rate, a reduction on the contract rate, offered before the first adjustment is made is included in their analysis. Stanton and Wallace (1995) use similar techniques to Kau et al. (1985, 1990) in their analysis of ARMs, and employ an index supposedly more advanced than the theoretical indices used in other ARM valuations. The index used lags behind shifts in the term structure. From an empirical examination of the Eleventh District Cost
- 146. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 146 of Funds Index (EDCOFI) a deterministic model of this index is produced. This is embedded into the contingent claims ARM valuation. The authors claim the lag in EDCOFI contributes significantly to the value of the borrower’s prepayment option. In the most recent work on default-free ARMs, Stanton and Wallace (1999) ex- amine the affects of four different, commonly used indices in the US, including the EDCOFI used in their earlier work, the one-year constant maturity Treasury yield, the one-year LIBOR and the Federal Housing Finance Board national average con- tract interest rate, on the ARM value. The indices are all modelled in the same way as previously discussed. The value of the underlying term structure (usually the spot rate) drives an index which determines the value of the contract rate (the index also depends on the index value at the previous adjustment). In both studies by Stanton and Wallace, the index and the adjustable contract rate are functions of the path of interest rates. An extended Crank-Nicolson algorithm was used to value the ARM and prepayment option for each different index. The research carried out by Skinner (1999), for the UK Office of Fair Trading, examines the affects of redemption fees (charges upon prepayment) and incentives (initial discount on mortgage rate) on mortgages in the UK mortgage market. A binomial interest-rate process calibrated to the existing term structure is used. The author considers this work the only research on the UK mortgage market and therefore unique in mortgage valuation literature. 5.2.2 Prepayable and defaultable ARMs Giving the borrower the option to terminate the contract by either prepaying or defaulting on the loan requires a much more complex financial model. As well as a model of the term structure of interest rates, a model for the value of the property with which the mortgage loan is secured must be included. The work of Kau et al. (1993) is the only research, as far as the author is aware, to include the house-price process as a second stochastic variable so that default can be included in the valuation of an ARM. The same auxiliary variable that carries the
- 147. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 147 relevant past information about the contract rate, that was used in the work of Kau et al. (1985, 1990) previously, is utilised in this two-factor model for an ARM. One possible reason that no other research has been carried out on ARM valuation is that the numerical analysis of a single loan requires such a huge computational effort. From table 5.2 (the results are from the numerical analysis in this study) the quickest valuation of a single loan took over 8 hours. Also the auxiliary variable approach is complex and difficult to programme. The contract rate for an ARM, which determines the value of the monthly pay- ments required to be made by the borrower, is variable and is linked to an index. The floating nature of the contract rate causes difficulties in the valuation of the various mortgage components, and is determined by the past value of the underlying interest rate, although the valuation procedure works backwards (due to the American nature of the prepayment option) from maturity towards the origination of the contract. The difficulty of the opposing temporal direction of the valuation process and the propa- gation of the contract-rate information can be overcome by employing the auxiliary contract-rate variable method, developed by Kau et al. (1993). While solving the path-dependency problem for the solution direction, this method produces problems of its own. This chapter builds on the two-state variable model to value ARMs of Kau et al. (1993) by offering some technical improvements to their numerical method. Later a new methodology to overcome the contract-rate path-dependency problem is introduced, and it is shown that some of the pricing efficiency problems inherent in the model of Kau et al. (1993) are removed. 5.3 An improved auxiliary-variable approach The work of Kau et al. (1993), which will be referred to as the auxiliary-variable approach on the valuation of ARMs, is the most realistic model currently in the literature. The possibilities of both default and prepayment by the borrower are included. Also, the floating contract rate is determined by including an underlying index, which is deterministically driven by the term structure of interest rates. An
- 148. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 148 inclusive model such as that of Kau et al. (1993), brings a high level of complexity to the valuation procedure, which as a result contains many interesting subtleties. The basic framework for the ARM is very similar to that described in the chapter 3 treatment of FRMs. As before, the two state variables used to model the economic environment in which the contract is set, are house price H, again modelled as a lognormal diffusion process (Merton 1973), and interest rate r modelled as a CIR mean-reverting square root process (Cox et al. 1985b). The PDE for the valuation of any asset whose value is a function of house price H, interest rate r, and time t, is again equation (1.17). Thus far, the valuation framework described is identical to that of the FRM. Again, the general valuation procedure starts at the maturity of the contract when the value of all the mortgage components are known from the contractual specifica- tions. The value of the components at any time prior to maturity can be calculated by solving the governing PDE (1.17), back through time to the origination of the con- tract. The difference in the ARM valuation to that of the FRM valuation is due to the path-dependency introduced by the floating contract rate. For the ARM discussed in this chapter, the contract rate is adjusted at the beginning of each year. This causes difficulties with the backward valuation procedure, since the necessary information about the contract rate is not known at the valuation point. Kau et al. (1993) sug- gest a method to circumvent this problem. The outline of this method, together with several improvements, will follow the specific details of the ARM contract. 5.3.1 Mortgage Contract The contract considered in this chapter will be of US mortgage specification (the setup of the ARM model borrows heavily from Kau et al., 1993, unless specified); the thrust of this chapter is in improved numerical method, the contract specification merely serves as the example in question. The methodology could easily be applied in the same way to a UK contract. To be consistent with Kau et al. (1993), the contract rate for an ARM is adjusted at yearly intervals.
- 149. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 149 Some notation is given which is used to describe the contract features that follow: n is the life of the mortgage in years, t(i, j) is the jth monthly payment date after the ith yearly adjustment date, where 0 ≤ i ≤ n − 1, 0 ≤ j ≤ 12 (note that t(i, 12) = t(i + 1, 0)), index(r) is the mortgage equivalent rate for a 1-year, default-free pure discount bond set on the ith adjustment date (see section 2.3.1 for a derivation of the index used in this chapter), margin is the amount added to the index at an adjustment date, y is the yearly cap and floor, l is the life-of-loan cap, and c(i) is the contract rate set on the ith adjustment date. At the start of a new year, at time t(i + 1, 0), the contract rate is adjusted. The new contract rate c(i + 1) is determined using the following contract-rate adjustment formula c(i + 1) = max min index(r) + margin, c(i) + y, c(0) + l , c(i) − y . (5.1) The new contract rate c(i+1), equation (5.1), can be interpreted as the current value of the interest-rate dependent index plus the margin, as long as this value does not increase beyond the initial contract rate c(0) by more than the life-of-loan cap l, or deviate from the previous contract rate c(i) by more than the yearly cap y. The initial contract rate is calculated according to c(0) = index(r(0)) + margin − teaser, (5.2) where the teaser rate teaser is only relevant in the first year of the loan and r(0) is the spot interest rate at the origination of the contract, at time t(0, 0). Once the contract rate is set for the current year, the monthly payments MP(i), made by the borrower are determined by the current outstanding balance OB(i, 0) and the current contract rate c(i). The monthly payments for the forthcoming year are calculated as though the mortgage were a fixed-rate contract that would com- pletely amortise the current outstanding balance OB(i, 0) at the present contract rate c(i) over the remaining life of the loan. Thus, MP(i) = OB(i, 0) 1 + c(i) 12 12(n−i) c(i) 12 1 + c(i) 12 12(n−i) − 1 , (5.3)
- 150. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 150 where the outstanding balance at the jth monthly payment date after the ith yearly adjustment date is OB(i, j) = OB(i, 0) 1 + c(i) 12 12(n−i) − 1 + c(i) 12 j 1 + c(i) 12 12(n−i) − 1 . (5.4) As in chapters 3 and 4, for the FRM the model of the ARM has two embedded options, default D(H, r, t) and prepayment C(H, r, t). As before, the mortgage also consists of a further component, the value of the remaining future payments promised to the lender A(r, t); see section 3.3 for further explanation regarding the details of these components. At payment dates a distinction is made between the value of an asset, immediately before (superscript −) and after (superscript +) a payment is made. For the US mortgage contract there is no penalty charged to the borrower for choosing to prepay the loan. If the option to prepay is exercised, the borrower must prepay the current outstanding balance plus any accrued interest since the last monthly payment. The amount charged, the face value, is FV (t) = 1 + c(i)(τ − t(i, j)) OB(i, j), for t(i, j) ≤ τ ≤ t(i, j + 1). (5.5) The insurance I(H, r, t) is not part of the mortgage but depends on it. This means that the insurance payoffs occur as the result of the borrower acting to minimise the cost of the mortgage V (H, r, t), without regard to the presence of insurance (where V = A−D−C as described in chapter 3). The insurance adds to the lender’s position in the contract. 5.3.2 Valuation procedure This section details the mortgage payment-date conditions for each component and how the auxiliary-variable technique of Kau et al. (1985, 1990, 1993) is incorporated into the ARM valuation. A backward valuation procedure is followed to solve the governing PDE (1.17). The problem is solved in this temporal direction for two reasons:
- 151. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 151 1. The prepayment option held by the borrower is American, the solution of which is readily solved using a backward valuation technique. 2. The mortgage components have known values at maturity (dependent on the contract rate being known at this point in time) from which their unknown values at origination can eventually be calculated. As discussed in section 5.3, the difficulty in pricing a variable-rate contract, such as an ARM, by using a backward valuation technique is that at the start of the procedure vital information is unknown. The contract rate is determined at the beginning of each year, which means its value is unknown at the initial valuation point, i.e. at the maturity of the loan. The auxiliary-variable approach overcomes this difficulty by effectively valuing the mortgage and its components for all possible realisations of the adjustable contract rate. The actual values of the components can be determined once the contract rate is specified. This scheme, as developed by Kau et al. (1985, 1990, 1993), does provide a solution, but at a cost. The procedure requires having to value repeatedly the mortgage components for all possible contract rates, so that by an adjustment date, when the adjusted contract rate is known, the appropriate values of the components have been calculated and the valuation procedure can continue. A much more efficient method to solve the problem is detailed in section 5.4, where the auxiliary variable and its problems are overcome. The next sections detail how the auxiliary variable is used to solve the adjustable contract-rate path-dependency problem. Valuing the remaining payments At maturity of the loan, the final payment MP is made. The value of the remaining payments at this time is A− [r, t(n − 1, 12); c(n − 1), OB(n − 1, 0)] = MP[n − 1; c(n − 1), OB(n − 1, 0)]. (5.6) The path-dependency problem now becomes clear, as the current contract rate c(n−1) is not considered until the beginning of the year is reached, and even at that point
- 152. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 152 neither this rate nor the outstanding balance OB(n − 1, 0) are known.1 Indeed, they depend on yet earlier contract rates. The problem with the unpaid balance OB can be eliminated by normalising the current unpaid balance to unity. This can be done as the value of the remaining payments is directly proportional to the outstanding balance OB. At the beginning of the year, when the outstanding balance is determined, and the required scaling is known, the value of A is merely changed in that same proportion. For ease of presentation, OB is omitted whenever its value is taken to be unity. The unknown contract rate c(n − 1) is treated by introducing it as an auxiliary state variable, so that effectively all its possible values are considered. The remainder of the valuation procedure for A and how the auxiliary variable is treated follows. The valuation PDE (1.17) is solved backwards, initially using the payment-date condition (5.6), until the beginning of the month, when another payment is due (the numerical technique used to value all the mortgage components is covered in section 5.3.3). At payment dates other than adjustment dates t(i, j), for j = 0 or 12, A+ [r, t(i, 0); c(i)] has been solved and the payment-date condition to begin the succeeding month may be written as A− [r, t; c(i)] = A+ [r, t; c(i)] + MP[i, c(i)] for t = t(i, j), j = 0 or 12. (5.7) The value of the remaining future payments changes by the value of the monthly payment. At the beginning of each year the contract rate is adjusted, at time t(i, 0) = t(i − 1, 12), a new auxiliary variable c(i − 1) must be introduced and the subsequent outstanding balance OB(i − 1, 0) must be set to unity. When c(i − 1) is introduced the previous auxiliary variable c(i) can be dropped. The new auxiliary variable and the current interest rate determines the old contract rate c(i) using the adjustment formula (5.1). To complete the transition from one contract rate to the next, the value of A− [r, t(i, 0); c(i)] must be adjusted to correspond to the value at which the outstanding balance OB(i, 0) must be reset. This later value is determined using the 1 In the payment-date conditions the variables that are determined by the auxiliary variable appear after a semicolon.
- 153. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 153 assumed value of the new contract rate c(i − 1) and the normalised new outstand- ing balance OB(i − 1, 0) in equation (5.4). Thus, the payment-date condition on adjustment dates is A− [r, t; c(i − 1)] = A+ [r, t; c(i)]OB(i, 0) + MP[i − 1, c(i − 1)] for t = t(i, 0) = t(i − 1, 12), (5.8) where c(i) is determined by r and c(i−1) together, while both OB(i, 0) and MP(i−1) are determined by c(i) and OB(i − 1, 0) = 1. Once the boundary conditions are specified (see section 2.6.1), the valuation procedure for A is closed and the value of the remaining future payments at origination A(r(0), t = 0), may be obtained. Valuing other mortgage components The borrower has two options, either to default on a monthly payment and hand the house over to the lender, or to prepay the face value of the mortgage if it is financially rational to do so. These options, and the insurance component, are dependent on the house price H. House price has no direct effect on the borrower’s option to prepay, since prepayment is assumed to occur when the value of the mortgage V is equal to the face value of the loan FV (the borrower attempting to minimise V ). House price does have a direct effect on the default option, if the borrower follows a ruthless default strategy (see Foster and Van Order, 1984, 1985 for further details) by exercising this option when the value of the mortgage is less than the value of the house. Since exercise of one of these options renders the other worthless, these options cannot be considered independently. As a result prepayment C, default D and the insurance I all depend on both house price H and interest rate r. The value of the mortgage component at maturity is V − [H, r, t(n − 1, 12); c(n − 1)] = min (MP[n − 1; c(n − 1)], H) , (5.9) as the lender either receives the scheduled monthly payment or the house. The
- 154. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 154 payment-date condition for the other payment dates, excluding adjustment dates, is V − [H, r, t; c(i − 1)] = min V + [H, r, t] + MP[i; c(i − 1)], H for t = t(i, j), j = 0 or 12. (5.10) Valuation throughout the year proceeds without complication until an adjustment date occurs, at the beginning of a year. Here the outstanding balance for the previous year must be adjusted. A difficulty arises as both default and prepayment are not directly proportional to OB, since they both depend on the house price. According to Kau et al. (1993), the value of default becomes twice as great when the loan becomes twice the amount and the house becomes twice as valuable. As a result the value of the mortgage at an adjustment date is V − [H, r, t; c(i−1)] = min V + H OB(i, 0) , r, t; c(i) OB(i, 0) + MP[i − 1; c(i − 1)], H for t = t(i, 0) = t(i − 1, 12), (5.11) where c(i) is determined by c(i − 1) and r, while both OB(i, 0) and MP(i − 1) are determined by c(i−1) and OB(i−1, 0) = 1. Again, once the boundary conditions are specified (see section 2.6.2 for the precise boundary conditions required), the valuation procedure for V is closed and the value of the mortgage to the lender at origination V [H(0), r(0), t = 0], may be obtained. As usual, the value of the mortgage involves a free boundary due to the optimal stopping problem created by the American-type prepayment option (see section 3.6.2 for a full exposition of the PSOR method used to deal with the free boundary problem). As explained in section 2.6.3 the conditions for default are fully specified by the payment-date conditions, as default is assumed to occur only at the end of each month. The general payment-date condition for the value of the default option, other than at adjustment dates, is given by D− [H, r, t(i, j)] = D+ [t(i, j)] if V − [t(i, j)] = V + [t(i, j)] + MP(i) (no default) A− [t(i, j)] − H if V − [t(i, j)] = H (default) for j = 0 or 12. (5.12)
- 155. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 155 The borrower either chooses to make the scheduled monthly payment, then the default option is simply worth its value in the future, or the borrower exercises the right to default and gives up the house H for the promised remaining payments A− . The technique for adjusting outstanding balances and carrying the auxiliary variable is precisely the same as that discussed for the mortgage component (see earlier in this section), and so is not repeated. Once the value of the remaining payments, the mortgage component and the default option are determined, the value of prepayment may be inferred using C = V − A − D. The general payment-date condition for the insurance is I− [H, r, t(i, j)] = I+ [t(i, j)] if V − [t(i, j)] = V + [t(i, j)] + MP(i) (no default) max(0, min(FV − [t(i, j)] − H, φFV − [t(i, j)])) (default) if V − [t(i, j)] = H for j = 0 or 12. (5.13) If default occurs, the insurer covers the lender’s shortfall on the lost future cash flows up to some fraction φ of the face value of debt. If default does not occur at the payment date, then the value of the insurance becomes its value in the future. Equilibrium condition At the origination of the contract, the value of the mortgage to the lender should equal the amount lent to the borrower; this idea is discussed in more detail in section 2.2. The avoidance of an arbitrage opportunity leads to the equilibrium condition, analogous to equation (3.22) for FRMs; for an ARM the equilibrium condition is V [H(0), r(0), t(0, 0) = 0; margin]+I[H(0), r(0), t(0, 0) = 0; margin] = 1−ξ, (5.14) where ξ is the amount of points deducted from the loan (for the UK counterpart this is the arrangement fee). Note that in this chapter the value of all assets will be expressed to par (i.e. as a percentage of the loan amount), so that the value of the loan is unity.2 This is why the borrower’s position is not written as (1−ξ) multiplied 2 This is in accordance with the literature on US ARM contracts.
- 156. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 156 by the loan amount. It is assumed, following Kau et al. (1993) that the insurance is purchased up front by the borrower. Although the insurance is not strictly part of the mortgage, it does affect the margin being added to the contract rate. The next section explains how the valuation procedure is used iteratively to determine the margin value that will produce a contract in equilibrium. Recall that in the case of the FRM, see chapters 3 and 4, it was the contract rate that was the free variable in the equilibrium condition. The margin is the parameter that must be found in the ARM case, and is the free parameter which is used to provide the necessary balance between the worth of the contract to both parties. 5.3.3 Improved numerical method The mortgage valuation PDE, (1.17), is solved using the Crank-Nicolson finite- difference scheme (the discretisation of this PDE is discussed in chapter 2), subject to the valuation procedure described in section 5.3.2. It is important to make absolutely clear that the valuation procedure is not novel (it is that employed by Kau et al., 1993 for the valuation of an ARM contract), but it is the techniques used to perform this procedure that are improved. This chapter significantly improves on Kau et al.’s auxiliary method by employing a significantly superior numerical procedure, and in- troduces a new methodology which substantially reduces the complexity of the ARM valuation. The numerical method is similar to that used for the valuation of a FRM contract, as described in sections 2.5 and 3.5. Again, the state space is defined by a (finite, truncated) equally spaced grid (see section 3.5). The three dimensions of the grid are the house-price H and interest-rate r state variables and time, respectively. Section 2.5 contains further details of the solution of the resulting system of equations, which arise due to the chosen (numerically superior) finite-difference method. The boundary conditions at extreme house values and interest rates are the same as for the FRM, given in section 2.6. The prepayment free boundary is also dealt with in the same manner as for the FRM, and is discussed in detail in section 3.6.1. Another feature
- 157. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 157 similar to the numerical method already described for the FRM is the iteration of the whole valuation procedure, until a contract in financial equilibrium is produced. Newton’s method is used again (see section 2.2.1 for details) but for the ARM the iterations continue until the margin is found, rather than the contract rate (which was the parameter used to balance the FRM contract), which produces a contract in equilibrium. The auxiliary variable The inclusion of the auxiliary contract-rate variable successfully deals with the path- dependency problem (described in section 5.3) associated with using the necessary, backward valuation technique to value an ARM. The auxiliary variable must span the whole range of possible contract rates, which is given by 0 < c(i) ≤ c(0) + l, (5.15) where after the ith adjustment the maximum contract rate is the initial contract rate c(0) (given by equation (5.2)) plus the life-of-loan cap l. To include the auxiliary vari- able in the valuation procedure (see section 5.3.2 for details) c(i) must be discretised. Thus, c(i) is represented by its value on the discrete set of points: c(i) = cs = s∆c where 0 ≤ s ≤ smax, (5.16) s is the counter used to reference a specific value of the auxiliary contract rate and ∆c is the incremental change in the auxiliary variable c(i). Through computational testing a sensible choice of smax was found to be 30. The auxiliary-variable method requires performing the valuation procedure over the entire state space s number of times. Although the auxiliary-variable technique provides a solution to the problem, it is by no means ideal. Section 5.4 provides a valuation procedure that completely removes the necessity for an auxiliary variable; the improvement in the efficiency of the ARM valuation procedure is substantial.
- 158. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 158 Defining the range No special attention was given in the work of Kau et al. (1993) to define the in- cremental change in the auxiliary variable ∆c. This could cause a non-linearity error (error caused when required information falls between nodes) to occur, as the auxiliary-variable technique involves solving the valuation problem for a range of contract rates (with the premise that at origination the values of all the mortgage components can be calculated). If the discrete auxiliary contract-rate range (5.16) does not include the initial contract rate then mortgage assets will not have any value for the initial contract rate. This can be overcome by first taking the nearest integer value of [c(0) + l]/c(0), then letting this equal the temporary variable temp, ∆c can then be defined as ∆c = c(0) ∗ [temp] smax , (5.17) which ensures that c(0) coincides with a node in the auxiliary-variable space. Interpolation The asset value just after the payment date is required in all payment-date conditions. On adjustment dates, once a year, the value of the asset at the new contract rate is required in these payment-date conditions (see equation (5.11) for example). The new contract rate c(i) is determined using the auxiliary variable c(i − 1). A problem arises as the new contract rate may fall between two auxiliary-variable nodes, which means the asset value, calculated using the new contract rate, may not exist. The scaled house price H OB(i,0) causes the same type of problem, since the house price may fall between two house price nodes.3 Linear interpolation must be used to calculate asset values lying between grid points. An example of repeated linear interpolation is given to illustrate how new data points can be constructed from a discrete set of known data points to find unknown 3 There is no mention in Kau et al. (1993) of the scaled house price causing a non-lineararity error. This matter requires as much attention as the contract-rate problem and must be dealt with accordingly.
- 159. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 159 terms. For example, consider V + [ H OB(i,0) , r, t; c(i)] in the mortgage component ad- justment condition (5.11). Let x be the scaled house price H OB(i,0) and y be the new contract rate c(i). Then let x1, x2 be the known values of x such that x1 ≤ x < x2, and y1, y2 be the known values of y such that y1 ≤ y < y2, then Vapprox(x, y) is the constructed value of V + H OB(i,0) , r, t; c(i) . This example is illustrated in figure 5.1, which shows the unknown data point (x, y) encircled and the known data points, used to interpolate the value of V (x, y), shown as dots. The approximate value of the mortgage component Vapprox(x, y) is found by evaluating u1 = x − x1 x2 − x1 and u2 = y − y1 y2 − y1 , (5.18) then g1 = (1 − u1)V (x1, y1) + u1V (x2, y1), (5.19) g2 = (1 − u1)V (x1, y2) + u1V (x2, y2), (5.20) and finally Vapprox = (1 − u2)g1 + u2g2. (5.21) This procedure is also used for the unknown terms in the adjustment conditions for the default option and the insurance component. Simple linear interpolation is used to calculate A+ [r, t; c(i)] in the adjustment payment-date condition, equation (5.7). Simple linear interpolation can be used here, as there is just one possible unknown quantity, the new contact rate c(i), which might not coincide with the discrete auxiliary contract-rate range (5.16). Crank-Nicolson methods vs explicit methods Kau et al. (1993) opted to use the explicit finite-difference method for the numerical solution of the ARM contract. Although the utilisation of this method is easier than implicit methods, it has two main drawbacks. 1. The explicit method has a poor convergence rate of O(∆t, ∆H2 , ∆r2 ). 2. The explicit method has stability constraints (see section 2.4) which require the ratio of ∆t to ∆H and ∆r to be ‘fine tuned.’
- 160. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 160 . .. . c(i) H . c H x y x ,y x ,y x ,y x ,y 1 2 2 2 11 1 2 Figure 5.1: An illustration of the unknown data point V (x, y) surrounded by its nearest grid points, at which the value of V is known. Instead, by incorporating the slightly more complicated Crank-Nicolson method (see section 2.5 and Smith, 1978; for further details), the numerical approach improves on the two drawbacks stated above, the new method is unconditionally stable and has the improved convergence rate of O(∆t2 , ∆H2 , ∆r2 ). Form of state variables The state variables, house price H and interest rate r, are kept in their original form for the implementation of the numerical method, rather than using unit-square transforms, as in the work of Kau et al. (1993). The transformations used by Kau et al. (1993) were x = 1 1 + Λr , (5.22) y = 1 1 + ΩH , (5.23) where Λ and Ω are scaling factors. The transformations convert the state space from a doubly-infinite domain into a unit square. Since the problem is solved by using a
- 161. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 161 discrete approximation for the underlying PDE, which suggests the problem should be solved inside a finite domain, hence, the conversion to the unit square. However these transformations produce a more complicated PDE, it becomes necessary to carefully select the scaling parameters for each set of initial house price H(0) and initial interest rate r(0), so that sufficient grid points are located close to the region of interest. An infinite domain in (H, r) space is not a difficulty, computationally, as Kau et al. (1993) suggest. It is simple to keep the original form of the state variables and then truncate the domain. It was found through extensive computational testing that 5r(0) and 2H(0) were satisfactory for the infinity values of interest rate and house price, respectively, on the finite state space. As described in section 2.6, Neumann boundary conditions are used, where appropriate, which enable a relatively small, truncated domain to be used. Programming the problem is more straightforward as the valuation PDE is kept in its original form. Most of the inaccuracy with the auxiliary approach, as stated in Kau et al. (1993), is caused by the possibility of error due to the interpolation on the auxiliary contract- rate variable and the scaled house price. Section 5.4 presents a new methodology to value ARMs where the auxiliary variable is eliminated completely, thus, removing this problem. 5.4 New valuation methodology This section proposes a new methodology to value ARMs, which eliminates the path- dependency problem, which occurs due to the opposing direction of the valuation procedure and the available required information about the floating rate. This results in significant improvements in the efficiency of the procedure, specifically, that it is not necessary to value the mortgage components for each possible contract rate at every time step, as is the case with the auxiliary method of Kau et al. (1993). By removing the need to loop the valuation procedure over all possible contract rates, this improvement has the added advantage that possible errors produced as a result
- 162. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 162 of the necessary interpolation are eliminated. 5.4.1 Contract rate preprocessing The key observation is that the variable contract rate determining the monthly cash flows the bank receives can be calculated for all levels of interest rate r at each of the adjustment dates, up front, prior to the actual valuation, using equation (5.1). With the contract rates known, the valuation takes a much simplified form from the methodology developed by Kau et al. (1993) and described in section 5.3. There is no need for an auxiliary variable, all adjusted contract rates can be stored a priori of the valuation procedure. Given the margin (which is iterated on to achieve equilibrium) the initial contract rate c(0) is known (determined by equation, 5.2). The contract rate after the first adjustment date is as follows, c(1, r) = max (min (index(r) + margin, c(0) + y, c(0) + l) , c(0) − y) . (5.24) The new contract rate c(1, r) is a function of the current level of the interest rate r only, as at this point the value of the previous contract rate c(0) is known. Equation (5.1) can be used recursively to determine the contract rate after each adjustment date for the entire mortgage. The contract rate after the first adjustment date c(1, r), calculated for any interest rate r, using equation (5.24), can be substituted into the right-hand side of the adjustment formula (5.1), which will then determine the con- tract rate after the second adjustment date c(2, r). This procedure can be continued so that all contract rates, 0 ≤ c(i, r) ≤ n−1, can be calculated prior to the valuation of the contract. These contract rates can be used to determine the value of all the monthly payments MP(i, r) and the value of all the outstanding balances OB(i, j, r) for each payment date, using equations (5.3) and (5.4), respectively.4 The new valu- ation methodology calculates the adjustable contract rate exactly and then uses this to value the mortgage components once, whereas the auxiliary contract-rate variable 4 Note that the monthly payment MP(i, r) and outstanding balance OB(i, j, r) are now functions of the current interest rate r as they are determined by the calculated contract rate c(i, r) which is now also a function of the interest rate r.
- 163. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 163 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 PSfrag replacements r c(i,r) Figure 5.2: A graph of contract rate c(i, r) against interest rate r. Shown is the initial contract rate c(0, r) (solid line), the contract rate after the first adjustment date c(1, r) (thick dashed line), the contract rate after the second adjustment date c(2, r) (thiner dashed line) and the contract rate after the final adjustment date c(14, r) (smallest dashed line). For the case when r(0) = 0.08, κ = 0.25, σr = 0.1, margin = 0.019, teaser = 0.015, y = 0.01, l = 0.05, 15 year loan. method, Kau et al. (1993), has to perform the valuation of the mortgage components repeatedly for the full range of the discrete steps used for the contract-rate range. Figure 5.2 shows typical results for the calculated contract rate against interest rate, for four adjustment dates. The initial contract rate c(0, r), determined by equation (5.2), is 8.65%. The final adjusted contract rate c(14, r) is used to start the valuation procedure at maturity. The distribution of the adjusted contract rate c(i, r) against the interest rate r has a step-like nature. This can be explained by appealing to the adjustment rule, equation (5.1). For example, consider the contract rate after the first adjustment c(1, r) (thick dashed line) in figure 5.2. The contract rate is bounded above/below by the initial contract rate c(0, r) plus/minus the yearly cap y and increases according to index(r) + margin between these levels. Figure 5.3 shows typical results for the calculated contract rate variation with the adjustment dates on which they are set, for various interest rates. Notice that the contract rate for any adjustment date has a maximum value of the initial contract rate plus the life cap (c(0, r) + l), which for the parameters chosen is 13.65%.
- 164. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 164 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 12 14 PSfrag replacements Adjustment Date i c(i,r) Figure 5.3: A graph of contract rate c(i, r) against adjustment date i. For each line style, interest rates are 0, 0.016, 0.032, 0.048, 0.064, 0.08, 0.096, 0.112 and 0.12 from the bottom to the top. For the case when r(0) = 0.08, κ = 0.25, σr = 0.1, margin = 0.019, teaser = 0.015, y = 0.01, l = 0.05, 15 year loan. This small amount of preprocessing to calculate the adjustable contract rate c(i, r), the monthly payments MP(i, r) and the outstanding balance OB(i, j, r), re- quires a negligible amount of computational effort. This new methodology removes the previous necessity of the auxiliary contract-rate variable approach to value ARMs. The remaining valuation procedure is now only as complex as the FRM valuation pro- cedure, described in detail in sections 2.5 and 3.5. The new valuation methodology does not exhibit the problems associated with the valuation procedure of Kau et al. (1993). It is no longer necessary to use inter- polation, which is potentially a source of error, and the valuation computation time is significantly reduced. The next section presents some comparative results between the (improved) auxiliary-variable approach and the new valuation methodology.
- 165. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 165 5.5 Results 5.5.1 Error in Kau, Keenan, Muller and Epperson (1993) Kau et al. (1993) make comparisons of equilibrium contract rates and option values between fixed-rate contracts and adjustable-rate contracts using the same framework. It is sensible to do the same here. Unfortunately it can be shown that the results in Kau et al. (1993) contain errors in components critical to the paper’s conclusions. The main conclusion in the paper was that ARMs produce lower contract rates (compared to equivalent FRM contracts), which lowers the value of payments. Quantifying this conclusion relies completely on the accuracy of Kau et al.’s calculation of the value of payments for FRMs, which can be shown to be incorrect. The majority of a mortgage’s value comes from the value of the payments made by the borrower (recall mortgage value is the value of payments minus the sum of the value of the borrower’s options). It is imperative that the value of payments can be calculated accurately. Under the framework used in Kau et al. (1993) and that used in this thesis, the value of payments (for a FRM) can be calculated precisely, it is ultimately the solution of the single-factor CIR PDE (2.36) which has an analytic solution. The value of payments A at origination can be calculated as follows A[r = r(0), t = 0] = 12n i=1 MPαie−βir , (5.25) where n is the length of the mortgage in years, MP is the value of the monthly payments (equation (2.1)), where αi = 2γe(γ+κ)(n− i−1 12 )/2 (γ + κ) eγ(n− i−1 12 ) − 1 + 2γ 2κθ/σ2 r , (5.26) βi = 2 eγ(n− i−1 12 ) − 1 (γ + κ) eγ(n− i−1 12 ) − 1 + 2γ , (5.27) and where γ = κ2 + 2σ2 r . (5.28)
- 166. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 166 The analytic solution for the value of payments is an adaptation of the solution for a risk free coupon-paying bond, see Cox et al. (1985b). Here the lender is equivalent to the bond holder and each monthly payment made by the borrower can be regarded of as a coupon paid by the writer, except that where the principal is returned at maturity with a bond, there is no analogous situation for a mortgage, as the borrower retains the house instead. Table 5.1 shows the value of payments calculated using the analytic equation (5.25) and the error in the results given in the work of Kau et al. (1993), for the value of payments for a FRM. The error shown in Kau et al.’s results for the value of payments for the FRM is somewhat disconcerting since an analytic solution is available. Once the contract rate is determined, using the auxiliary-variable approach for the ARM, calculating the value of the remaining payments follows the same method as for the FRM. As this more straightforward valuation has been shown to be calculated incorrectly in Kau et al. (1993), it makes the results in their work invalid and inappropriate for comparative purposes for this chapter. Analytic solution for A(r(0), t = 0) and error in Kau et al. (1993). LTV r(0) Contract Rate σr Analytic A (to par) Kau Error ($) 80 8 10.17 5 106.46 176 10 107.67 600 15 109.55 1416 90 8 10.00 5 105.43 189 10 106.64 666 15 108.50 1575 Table 5.1: Error in value of payments (for a FRM) published in Kau et al. (1993). The analytic value of payments is calculated using equation (5.25). θ = 0.1, κ = 0.25, n = 180, initial house price $100000. LTV = ratio of loan to initial value of house, r(0) = initial interest rate and contract rate are shown as percentages. The error in the Kau et al. (1993) results for all cases shown in table 5.1 is signif- icant. The error increases in Kau et al.’s calculation for the value of the remaining payments as the interest-rate volatility increases (for both contract rates shown). As a consequence, the results obtained using the improved technique are presented and then compared with FRM results calculated using the numerical method described
- 167. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 167 in sections 2.5 and 3.5.5 These are then compared to results obtained using the new valuation methodology, described in section 5.4. Conclusions are made about these results only, and are not compared with the published erroneous results of Kau et al. (1993). 5.5.2 Comparison of both methods The improved auxiliary-variable approach, section 5.3.3, and the new valuation method- ology, section 5.4, can both be used to value the ARM. The following section presents some results using both methods for a range of adjustable-rate contracts by examining several combinations of annual and life caps on the adjustable rate. Table 5.2 shows the component values for an ARM calculated using the improved auxiliary-variable approach (section 5.3). The parameter values used were chosen in line with those used in the literature (Kau et al., 1993), which were originally estimated by Titman and Torous (1989). The numerical solution involves an initial guess for the margin; Newton’s method is then used to iterate on this parameter until the equilibrium condition, equation (5.14), is satisfied. The margin sets the initial contract rate according to equation (5.2). The value of the components given in table 5.2 are given to par, as a percentage of the initial loan amount. Also shown are the results for an equivalent FRM contract, where all relevant parameters are those given in the table for the ARM. It can seen that for all parameters taken, the ARM balances at a lower initial contract rate than the equivalent FRM contract. This is as expected and results in significantly lower values of payments than for the FRM. This is true for all combinations of caps on the adjustable rate. The tighter (lower in value) the caps are, the more the contract will resemble a FRM, whereas, the wider the caps, the more the contract will resemble a full adjustable contract, as witnessed in the table. The uncapped ARM (where both the yearly cap and the monthly cap are set to infinity6 ) has the lowest initial contract rate. Lenders can offer lower 5 Note that the numerical results for a FRM used in chapters 3, 4 and 5 were tested by comparing the value of the payments A(r, t) to the analytic solution (5.25). The error was always less than $1 based on $100000 initial house value. 6 For numerical purposes a cap of infinity was modelled by setting the cap to rmax, the finite,
- 168. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 168 contract rates, safe in the knowledge that the ARM payments will adjust according to the market interest rate. The lower initial contract rate results in prepayment value being much lower for an ARM. Initial teaser rates serve to entice borrowers into agreeing to a loan, and lenders are compensated for this initial generosity by requiring higher contract rates than otherwise after the first year. It can be seen that without teasers the margin for an ARM can become negative. This is not surprising, as a teaser rate such as 1.5% results in a considerably greater initial contract rate compared to the case without the presence of a teaser rate, the negative margin will cause the contract rate to change at a reduced rate as the underlying interest rate moves, thus, insulating the borrower from interest-rate variation but at the cost of greater initial payments. Table 5.3 shows the results using the new methodology for the same parameter combinations as table 5.2. According to Kau et al. (1993) the main source of error in the valuation was due to the auxiliary variable itself. As the new methodology does not require this technique, the results should be more reliable. truncated value of interest rate as r → ∞.
- 169. CHAPTER5.ADVANCEMENTSINARMVALUATION169 Component values calculated using the improved numerical method (in %). Contract in equilibrium. Life of Initial Loan Contract Margin Value of Computation Annual Cap Cap Rate in Basis Points Payments Default Prepayment Insurance Time y c c(0) margin A D C I (sec) 1 5 7.988 124.07 101.56 7.97 1.83 6.74 29396 (8.439) (19.14) (98.42) (6.55) (0.47) (7.10) (44094) 2 5 7.529 78.20 100.23 7.47 1.10 6.85 29396 (8.231) (-1.66) (97.90) (6.38) (0.16) (7.14) (51443) ∞ 5 7.196 44.90 99.06 7.01 0.50 6.95 39745 (8.076) (-17.11) (97.46) (6.23) (0.00) (7.17) (51443) 1 ∞ 8.028 128.20 102.17 8.17 2.18 6.67 29396 (8.416) (16.82) (98.64) (6.61) (0.60) (7.07) (44094) 2 ∞ 7.437 68.95 101.12 7.85 1.59 6.82 29396 (8.057) (-19.02) (97.98) (6.42) (0.42) (7.03) (51443) ∞ ∞ 6.395 10.43 98.83 7.01 0.70 6.82 44094 (7.940) (-30.71) (98.76) (6.80) (0.46) (7.10) (51443) Fixed-Rate mortgage 10.895 . . . 106.68 4.47 6.28 2.56 1152 Table 5.2: Component values for the ARM calculated using the improved auxiliary-variable approach. Results without parentheses are for a 1.5% teaser; results with parentheses are without teasers. All results are to par value for a 15-year loan: spot interest rate r(0) = 8%, long-term mean θ = 10%, speed of reversion κ = 25%, correlation coefficient ρ = 0, service flow δ = 8.5%, interest-rate volatility σr = 10%, house-price volatility σH = 15%, points ξ = 1.5%, insurance coverage φ = 25%, and a 90% loan-to-value ratio. Initial margin was set at 100 basis point. Fixed-rate component values given for comparison.
- 170. CHAPTER5.ADVANCEMENTSINARMVALUATION170 Component values calculated using the new methodology (in %). Contract in equilibrium. Life of Initial Loan Contract Margin Value of Computation Annual Cap Cap Rate in Basis Points Payments Default Prepayment Insurance Time y c c(0) margin A D C I (sec) 1 5 8.181 143.40 103.30 4.70 3.50 3.40 855 (8.596) (34.90) (101.01) (4.14) (1.99) (3.62) (971) 2 5 7.756 100.90 102.24 4.54 2.74 3.55 575 (8.397) (15.00) (100.43) (4.08) (1.62) (3.76) (1250) ∞ 5 7.475 72.72 101.48 4.39 2.26 3.66 855 (8.305) (5.75) (101.26) (4.04) (1.53) (3.81) (1459) 1 ∞ 8.113 136.54 103.71 4.83 3.78 3.39 855 (8.538) (29.02) (101.17) (4.20) (2.08) (3.61) (1250) 2 ∞ 7.493 74.54 102.62 4.71 2.94 3.53 855 (8.215) (-3.24) (100.52) (4.16) (1.62) (3.77) (1459) ∞ ∞ 6.914 16.70 101.23 4.46 2.01 3.73 1250 (7.992) (-25.49) (100.17) (4.08) (1.43) (3.84) (1459) Fixed-Rate mortgage 10.895 . . . 106.68 4.47 6.28 2.56 1152 Table 5.3: Component values for the ARM calculated using the new valuation methodology. Parameter details identical to table 5.2.
- 171. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 171 It would be interesting to analyse both methods in such a way that would indicate how good the underlying model is, and how much error is a result of the auxiliary variable itself. This could be done by comparing results from these two methods to actual data. This is a possibility for feature research. Comparing theoretical results with real market data is the best way to test the capability of the theoretical models discussed in this thesis to produce practically useful results. The new methodology produces equilibrium setting margins slightly higher than the improved auxiliary-variable method. This leads to slight increases in the initial contract rates for all cap scenarios considered. The maximum contract rate for the ARM, in table 5.3, is still much lower than the equivalent FRM. The same general trend can be seen, namely tighter caps produce results closest to the FRM and the widest caps, when both caps are set to infinity (here the contract is fully adjustable), produce results most resembling those expected for an ARM, specifically a low initial contract rate, a low value of payments, and a low value of prepayment. The difference between the results for the two methods should be due to the error accompanying the auxiliary variable. Since the new methodology does not require the complex numerical procedure the auxiliary-variable method necessitates, the numerical results produced using this valuation scheme should be the more reliable. The improvement in efficiency in valuing the ARM between the two methods is dramatic. Consider the final column in table 5.2 and 5.3 which displays the com- putation time required to produce a contract in equilibrium.7 The time required to compute the results depends heavily on the initial guess used for the margin. The initial margin was set at 100 basis points for all tests. This means contracts that balanced with a final margin close to 100 would have taken fewer iterations of the valuation procedure to converge to the equilibrium setting margin. It is better to con- sider the computation time for a single run of the valuation process. The improved auxiliary-variable method took approximately 7350 seconds per iteration, whereas the new methodology took only 210 seconds per iteration. These figures are in pro- portion to that expected, with the new methodology taking approximately 30 times 7 The results were obtained using a 2412 MHz AMD Athlon computer.
- 172. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 172 less time to compute than the improved auxiliary-variable method, this factor being the number of finite steps taken in the auxiliary-variable range. 5.5.3 Tracking the contract rate Two approaches have been given in this chapter to value an ARM, the first is an im- proved auxiliary-variable approach (section 5.3) to that given by Kau et al. (1993), the second approach is a new valuation methodology (section 5.4). The goal of both methods is to eliminate the path-dependency problem that occurs at the val- uation point where the necessary information about the floating contract rate is unknown. For the new methodology, the contract rate for all interest-rate levels and all adjustment dates can be easily calculated and recorded even before the valuation begins. The auxiliary-variable approach incorporates an auxiliary contract rate, re- quired throughout the valuation process, which is used as a guess to the previously set contract rate. To make a direct comparison between the two methods the contract rate would need to be tracked throughout the valuation for the auxiliary-variable approach and then compared to the contract rates calculated using the new methodology. Unfortu- nately it is not possible to do this for the auxiliary-variable approach. The contract rate cannot be tracked at each adjustment date, as the new contract rate is deter- mined for the forthcoming year (the year previously valued, due to the backward valuation) by the new auxiliary variable. Consequently each value of the new aux- iliary variable determines a different value for the new contract rate; this makes it intractable for the contract rate to be tracked for the auxiliary-variable method. 5.6 Conclusions This chapter has presented and significantly improved the framework for the most ad- vanced ARM model available in the literature, established by Kau et al. (1993). The necessity of using a backward valuation procedure and the resulting path-dependency
- 173. CHAPTER 5. ADVANCEMENTS IN ARM VALUATION 173 problem is discussed. The auxiliary-variable approach of Kau et al. (1993) to over- come this problem is described, as well as the interpolation problems necessary in using this procedure. The first contribution of this chapter is to address several of the shortcomings in the work of Kau et al. (1993). The improved auxiliary-variable ap- proach presented in this chapter uses the numerically superior Crank-Nicolson scheme rather than the explicit method used in Kau et al.’s initial work on this topic. Much more care is taken in calculating the adjustment date payment-date conditions by using repeated linear interpolation for both the new contract rate and scaled house price. A new methodology is presented to value the ARM which removes the need for the auxiliary-variable procedure, as well as all the complications that go with it. By carrying out a small amount of preprocessing, prior to the valuation procedure, the contract rate for all interest rates and at all adjustment dates can be calculated. This results in a dramatically more efficient valuation of the ARM contract. No longer does the valuation have to be performed for all possible contract rates, as is the case with the auxiliary-variable method. Instead, only a single valuation is necessary. This produces (typically) a 30 factor decrease in the computation time required for a single loan valuation. As discussed in Kau et al. (1993), the main source of error with the auxiliary-variable approach is due to the interpolation necessary on adjustment dates. The new methodology does not require any interpolation, there is no need for the auxiliary variable, as the contract rates are calculated directly a priori of the valuation procedure. Therefore the possible error associated with the auxiliary variable is eliminated with this new methodology.
- 174. Chapter 6 Conclusions Treating mortgages as derivative securities and then using option-pricing models for their valuation is an established technique. In this thesis, current theoretical models have have been improved through advancements in the necessary methods employed in pricing various mortgage contracts. As well as making increases in both accuracy and efficiency in the valuation of the complex problems involved, a sophisticated modelling adjustment to the assumed way borrowers behave overcomes a major failing of existing option-theoretic mortgage models. 6.1 Summaries Throughout this thesis a realistic mortgage valuation model (including the potential for prepayment and default by the borrower) is considered. For the case of a FRM, a perturbation approach was used to develop analytic approximations, which provided rapid solutions to value FRMs. An ‘enhanced’ finite-difference approach was devel- oped to test the ability of the approximate solutions to calculate equilibrium contract rates accurately. As well as being the first to apply singular perturbation theory to the valuation of mortgages, the closed-form solutions demonstrated are the first of any kind for models using two state variables including both default and prepayment. The relaxation of the assumption of ruthless prepayment, used in all previous re- search on option-theoretic valuation of mortgages, produced a new prepayment policy. 174
- 175. CHAPTER 6. CONCLUSIONS 175 This improves the modelling of the borrower’s decision process. An occupation-time derivative was incorporated, which delays the prepayment call option, thus increasing the value of the mortgage to the lender. For the more complex valuation problem, when the mortgage contract rate is adjustable, a more efficient and more accurate pricing methodology was developed to value ARMs. An existing valuation technique has been improved by employing an advanced numerical procedure as well as introducing several other techniques to improve accuracy. A new valuation methodology has also been presented which is drastically more efficient than the existing technique. 6.2 Implications Perturbation theory has been shown to be a very efficient tool in the solution of a contingent claims mortgage valuation model. Determination of equilibrium contract rates, previously requiring many hours can be reduced to just a few seconds, rendering this a highly useful portfolio management tool. Although the example shown is of a UK contract, the method is applicable to US or other mortgage markets. The new prepayment model developed should have implications for more accurate MBS pricing. Using a modified rational model mortgage values above par can now be replicated, which previously was not possible. It is also possible to alter the borrower prepayment call policy to allow different borrower decision times. These features are necessary to accurately price a MBS using a structural endogenous approach. 6.3 Future research Using the ‘enhanced’ finite-difference approach developed in this thesis, for valuing FRMs, and including the new prepayment model, the obvious future work would be to create an advanced rational MBS pricing model. The vital improvements made to rational FRM valuation would enable a shift back towards structural methods for MBS pricing, and away from the current reduced-form pricing methods (calibrated
- 176. CHAPTER 6. CONCLUSIONS 176 to a single data set with no way to determine how the estimated parameters should change in response to changes in the economic environment). A vital area for future research would be to introduce an empirical test for the new prepayment model - comparing the model predictions on mortgage valuations and the implied borrower’s option values to those of real cases. Ultimately, to improve MBS pricing, the underlying mortgages models themselves must be able to produce values consistent with those observed in the market.
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- 185. CHAPTER 7. REFERENCES 185 van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic Press, Palo Alto, California. Vandell, K.D. 1995. How Ruthless is Mortgage Default? A Review and Synthe- sis of the Evidence. Journal of Housing Research. 6 245-264. Vetzal, K.R., P.A. Forsyth. 1999. Discrete Parisian and Delayed Barrier Options: A General Numerical Approach. Advances in Futures and Options Research. 10 1-15. Widdicks, M., P.W. Duck, A.D. Andricopoulos, D.P. Newton. 2005. The Black- Scholes Equation Revisited: Asymptotic Expansions and Singular Perturbations. Mathematical Finance. 15 373-391. Wilmott, P., J.N. Dewynne, S. Howison. 1993. Option Pricing: Mathematical Mod- els and Computation. Oxford Financial Press, Oxford. Yu, L., D.P. Newton, P.V. Johnson, P.W. Duck. 2006. Pricing Credit Risk in a Two-Class Debt Structure. Working Paper: Manchester Business School, The Uni- versity of Manchester.
- 186. Appendix A Fixed-rate mortgage valuation pseudocode PROGRAM uk_fixed_rate_mortgage IMPLICIT NONE INTEGER :: i,j,k,n,m,l ! .. state space indices INTEGER :: month,monthmax REAL(dp) :: kappa,theta,delta,sigma_h,sigma_r,rho,dh,dr,dt,crate REAL(dp) :: mp,rltv,horig,rspot,arrange,tol,loan,pen,nltv,fracloss,cap REAL(dp), ALLOCATABLE :: a(:),v(:,:),c(:,:),d(:,:),ins(:,:),coins(:,:) REAL(dp), ALLOCATABLE :: h(:),r(:),Z(:,:),td(:,:),ob(:) REAL(dp), ALLOCATABLE :: alp(:,:),bet(:,:),gam(:,:),del(:,:),eps(:,:) REAL(dp), ALLOCATABLE :: mu(:,:),x(:),x_inc(:),f(:) REAL(dp) :: newton_tol ! .. dp denotes double precision INTEGER :: iterat,i_horig,j_rspot INTEGER, PARAMETER :: in=9,out=10,iter=10000 ! .. Mortgage Components .. ! .. a is the value of the remaining future payments .. ! .. v is the value of the mortgage to the lender .. ! .. c is the value of the option to prepay .. ! .. d is the value of the option to default .. 186
- 187. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 187 ! .. ins is the value of the insurance against default .. ! .. coins is the value of the coinsurance .. ! .. Read in parameters from Input file .. OPEN(UNIT=9,file=’Input’) CALL Input(in,out,n,m,l,monthmax,kappa,theta,delta,sigma_h,sigma_r,rho,& crate,rltv,horig,rspot,arrange,dh,dr,dt,tol,loan,nltv,fracloss,cap,pen,& newton_tol) ! .. Allocate size of arrays .. ALLOCATE(h(n),r(m),v(n,m),a(m),c(n,m),d(n,m),ins(n,m),coins(n,m)) ALLOCATE(ob(0:monthmax-1),td(l,monthmax)) ALLOCATE(alp(n,m),bet(n,m),gam(n,m),del(n,m),eps(n,m),mu(n,m),Z(n,m)) ALLOCATE(x(0:20),x_inc(0:20),f(0:20)) DO i = 1,n ! .. Define house price dimension h(i) = (i-1)*dh END DO DO j = 1,m ! .. Define spot rate dimension r(j) = (j-1)*dr END DO ! .. Newton method .. ! ! .. To satisfy no-arbitrage condition find contract rate .. ! ! .. that satisfies the equilibrium condition .. ! ! .. V(H(0),r(0),t=0,c,pen)+I(H(0),r(0),t=0,c,pen)=(1-arrange)Loan .. ! i_horig = NINT(horig/dh) + 1 ! .. + 1 due to how h(i) is calculated j_rspot = NINT(rspot/dr) + 1 x(0) = crate ! .. initial guess for contract rate x_inc(0) = 0.01 ! .. initial increment change DO iterat = 0,100 IF(iterat==0)THEN x(iterat) = x(iterat) ELSE
- 188. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 188 x(iterat) = x(iterat-1)+x_inc(iterat-1) ENDIF crate = x(iterat) mp = ((crate/12._dp)*((1.0_dp+(crate/12._dp))**monthmax)*(loan))& /(((1._dp+(crate/12._dp))**monthmax)-1._dp) ! .. Solution for month 300 .. ! month = monthmax ! .. Terminal conditions for month 300 .. k = 1 ! .. Payment date .. DO j = 1,m a(j) = mp END DO DO i = 1,n DO j = 1,m v(i,j) = MIN( h(i) , mp ) d(i,j) = MAX( 0._dp , (mp-h(i)) ) c(i,j) = 0._dp IF (v(i,j) == mp) THEN ! .. Condition: default doesn’t occur .. ins(i,j) = 0._dp coins(i,j) = 0._dp ELSE ! .. Condition: default occurs .. ins(i,j) = MIN( (fracloss*(mp-h(i))) , cap ) coins(i,j) = MAX( (1._dp-fracloss)*(mp-h(i)) , (mp-h(i))-cap ) END IF END DO END DO ! .. The outstanding balance after the previous monthly payment .. CALL Outstanding_balance(month,monthmax,ob,crate,loan)
- 189. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 189 ! .. TD at payment date = 0 since prepayment is not possible here .. td(1,monthmax) = 0._dp ! .. V Coeff’s independent of time, call outside time loop .. CALL Coefficient_matrix_v(alp,bet,gam,del,eps,mu,n,m,r,& sigma_r,h,sigma_h,delta,dr,dt,dh,kappa,theta,rho) ! .. Time loop for month 300 .. DO k = 2,l ! .. Calculate the total debt payment .. CALL Total_debt(k,l,crate,td,dt,ob,month,monthmax,pen) ! .. Calculate a(r) the value of the remaining future payments .. CALL Calculate_a_value(m,r,sigma_r,dr,dt,kappa,theta,a) ! .. Due to Lin Comp method coeff’s & Z for v need to be set .. CALL Z_matrix_v(alp,bet,gam,del,eps,mu,Z,v,dt,n,m) ! .. V using Lin Comp & PSOR, others use gen Lin solver .. CALL Find_v_d_c_ins_coins(v,Z,td,n,m,alp,bet,gam,del,eps,mu,& tol,iter,OUT,l,monthmax,k,month,a,d,c,ins,coins,& r,h,sigma_r,sigma_h,dr,dt,dh,kappa,theta,delta,rho) END DO ! .. Solution for month 299 to month 1 of the contract .. ! ! .. Time loop for months .. DO month = monthmax-1,1,-1 ! .. Terminal conditions .. k = 1 ! .. Payment date .. ! .. The outstanding balance after the monthly payment .. CALL Outstanding_balance(month,monthmax,ob,crate,loan) ! .. Calculate the total debt payment .. CALL Total_debt(k,l,crate,td,dt,ob,month,monthmax,pen) DO j = 1,m a(j) = a(j) + mp END DO
- 190. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 190 DO i = 1,n DO j = 1,m v(i,j) = MIN( h(i) , v(i,j) + mp ) v(i,j) = MIN( td(k,month) , v(i,j) ) ! .. We know v on a payment date but we must compare it .. ! .. with the td to see if prepayment should occur .. IF (v(i,j) == td(k,month)) THEN ! .. if v = TD we are inside the prepayment region .. d(i,j) = 0._dp c(i,j) = max( (a(j)-v(i,j)-d(i,j)) , 0._dp ) ins(i,j) = 0._dp; coins(i,j) = 0._dp ELSE IF (v(i,j) == h(i)) THEN ! .. Condition: default occurs .. d(i,j) = a(j) - h(i); c(i,j) = 0._dp ins(i,j) = MIN( (fracloss*(td(1,month)-h(i))) , cap ) coins(i,j) = MAX( (1._dp-fracloss)*(td(1,month)-h(i)) , & (td(1,month)-h(i))-cap ) ELSE ! .. Condition: default doesn’t occur .. d(i,j) = d(i,j); c(i,j) = c(i,j) ins(i,j) = ins(i,j); coins(i,j) = coins(i,j) END IF END DO END DO ! .. V coeff’s are independent of time, call outside time loop .. CALL Coefficient_matrix_v(alp,bet,gam,del,eps,mu,n,m,r,& sigma_r,h,sigma_h,delta,dr,dt,dh,kappa,theta,rho) ! .. Time loop for month .. DO k = 2,l ! .. Calculate the total debt payment ..
- 191. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 191 CALL Total_debt(k,l,crate,td,dt,ob,month,monthmax,pen) ! .. Calculate a(r) the value of the remaining future payments .. CALL Calculate_a_value(m,r,sigma_r,dr,dt,kappa,theta,a) ! .. Due to Lin Comp method coeff’s & Z for v need to be stated CALL Z_matrix_v(alp,bet,gam,del,eps,mu,Z,v,dt,n,m) ! .. V using Lin Comp & PSOR, others use gen Lin solver .. CALL Find_v_d_c_ins_coins(v,Z,td,n,m,alp,bet,gam,del,eps,mu,& tol,iter,OUT,l,monthmax,k,month,a,d,c,ins,coins,& r,h,sigma_r,sigma_h,dr,dt,dh,kappa,theta,delta,rho) END DO END DO ! .. Now check the equilibrium condition. If not satisfied the ! .. Newton method is used to find the correct contract rate .. f(iterat)=v(i_horig,j_rspot)+ins(i_horig,j_rspot)-(1.0_dp-arrange)*loan ! .. first uses initial guess for crate .. IF(ABS(f(iterat)) < newton_tol)THEN WRITE TO SCREEN ’converged after’,iterat,’iterations’ EXIT ENDIF IF(iterat==100)THEN WRITE TO SCREEN ’Failed to converge’ STOP ENDIF IF(iterat==0)THEN x_inc(iterat)=x_inc(iterat) ELSE x_inc(iterat)=(-x_inc(iterat-1)*f(iterat))/(f(iterat)-f(iterat-1)) ENDIF ENDDO END PROGRAM uk_fixed_rate_mortgage
- 192. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 192 SUBROUTINE Input(in,out,n,m,l,monthmax,kappa,theta,delta,& sigma_h,sigma_r,rho,crate,rltv,horig,rspot,arrange,dh,dr,dt,tol,loan,& nltv,fracloss,cap,pen,newton_tol) IMPLICIT NONE INTEGER, PARAMETER :: dp = KIND(1.0D0) INTEGER,INTENT(OUT):: n,m,l,monthmax INTEGER,intent(in) :: in,out REAL(dp),INTENT(OUT) :: kappa,theta,delta,sigma_h,sigma_r,rho,loan REAL(dp),INTENT(OUT) :: crate,rltv,horig,rspot,arrange,dh,dr,dt,tol REAL(dp),INTENT(OUT) :: nltv,fracloss,cap,pen,newton_tol n = 201; m = 51; l = 31; monthmax = 25*12 kappa = 0.25_dp; theta = 0.1_dp; delta = 0.075_dp; sigma_h = 0.05_dp sigma_r = 0.05_dp; rho = 0._dp; crate = 0.1_dp; rltv = 0.95_dp horig = 1._dp; rspot = 0.08_dp; arrange = 0.015_wp; tol = 1.e-12 dH = 2._dp*horig/FLOAT(n-1); dr = 5._dp*rspot/FLOAT(m-1) dt = 1._dp/12._dp/FLOAT(l-1) nltv = 0.75_dp; fracloss = 0.8_dp; pen = 0.05_dp; newton_tol = 1.e-5 loan = horig*rltv; cap = (rltv-nltv)*horig END SUBROUTINE Input SUBROUTINE Outstanding_balance(month,monthmax,ob,crate,loan) IMPLICIT NONE INTEGER :: month,monthmax REAL(dp) :: crate,loan,ob(0:monthmax-1) ! .. In month m ob(m-1) is the outstanding balance after the recent MP .. ob(month-1) = ( (1._dp+(crate/12._dp))**monthmax - & (1._dp+(crate/12._dp))**(month-1) )* loan / & ( (1._dp+(crate/12._dp))**monthmax - 1._dp ) ENDSUBROUTINE Outstanding_balance
- 193. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 193 SUBROUTINE Total_debt(k,l,crate,td,dt,ob,month,monthmax,pen) IMPLICIT NONE INTEGER :: k,l,month,monthmax REAL(dp) :: crate,dt,pen,ob(0:monthmax-1),td(l,monthmax) ! .. Total debt = outstanding balance plus accrued interest .. ! .. k = counter for time until the payment date in the present month .. ! .. therefore (l-k)dtau is the time since the last payment .. td(k,month) = (1._dp+pen)*( 1._dp+crate*( (l-k)*dt ) )*ob(month-1) ENDSUBROUTINE Total_debt ! .. Solver for value of remaining future payments .. ! SUBROUTINE Calculate_a_value(m,r,sigma_r,dr,dt,kappa,theta,a) IMPLICIT NONE INTEGER :: m,j REAL(d),DIMENSION(m) :: alp,bet,gam,Z,a,r REAL(dp) :: dr,dt,theta,kappa,sigma_r ! .. Note: a is a function of r only therefore the pde is like CIR .. ! .. Single factor means tridag solver can be used .. ! .. Note Neumann b.c. at r=rmax to smooth solution .. ! .. Boundary r=0 .. alp(1) = -1/dt - 0.75_dp*kappa*theta/dr bet(1) = kappa*theta/dr; gam(1) = -0.25_dp*kappa*theta/dr Z(1) = (-2._dp/dt - alp(1))*a(1) - bet(1)*a(2) - gam(1)*a(3) ! .. Intermediate points .. DO j = 2,m-1 alp(j) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))) bet(j) = -1._dp/dt - 0.5_dp/dr/dr*sigma_r*sigma_r*r(j) - 0.5_dp*r(j) gam(j) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) &
- 194. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 194 + 0.25_dp/dr*(kappa*(theta-r(j))) Z(j) = - alp(j)*a(j-1) + (-2._dp/dt - bet(j))*a(j) - gam(j)*a(j+1) END DO ! .. Boundary at r=infty .. alp(m) = 0._dp; bet(m) = 1._dp; gam(m) = 0._dp; Z(m) = a(m-1) ! .. Use Tridiag solver to find a at present time step .. CALL tridag(alp,bet,gam,Z,a,m) END SUBROUTINE Calculate_a_value SUBROUTINE tridag(alp,bet,gam,Z,a,m) IMPLICIT NONE INTEGER :: m,j REAL(dp),DIMENSION(m) :: alp,bet,gam,Z,a ! .. Tridiagonal solver uses Gaussian elimination .. ! .. System had 3 points in the first row and 1 point in the last row .. ! .. Gaussian elimination .. bet(2) = bet(2) - bet(1)*alp(2)/alp(1) gam(2) = gam(2) - gam(1)*alp(2)/alp(1) Z(2) = Z(2) - Z(1)*alp(2)/alp(1) DO j = 3,m-1 bet(j) = bet(j)-gam(j-1)*alp(j)/bet(j-1) Z(j) = Z(j)-Z(j-1)*alp(j)/bet(j-1) END DO ! .. Calculate solution using back substitution .. a(m) = Z(m)/bet(m) DO j = m-1,2,-1 a(j) = (Z(j) - gam(j)*a(j+1))/bet(j) END DO a(1) = (Z(1) - bet(1)*a(2) - gam(1)*a(3))/alp(1) END SUBROUTINE tridag
- 195. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 195 SUBROUTINE Coefficient_matrix_v(alp,bet,gam,del,eps,mu,n,m,r,& sigma_r,h,sigma_h,delta,dr,dt,dh,kappa,theta,rho) IMPLICIT NONE INTEGER :: n,m,i,j REAL(dp) :: h(n),r(m),sigma_r,sigma_h REAL(dp) :: dr,dt,dh,theta,kappa,rho,delta REAL(dp),DIMENSION(n,m) :: alp,bet,gam,del,eps,mu ! .. Set the coefficients which are indep of time .. ! .. All coefficients are zero unless specified otherwise .. alp = 0._dp; bet = 0._dp; gam = 0._dp del = 0._dp; eps = 0._dp; mu = 0._dp ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. new improved b.c. ! .. h=r=0 substituted into PDE .. alp(1,1) = -1._dp/dt - 0.75_dp*kappa*theta/dr !v(1,1) bet(1,1) = kappa*theta/dr !v(1,2) gam(1,1) = -0.25_dp*kappa*theta/dr !v(1,3) ! bet(1,1) = 1._dp ! .. Boundary at h=infty, r=0 .. bet(n,1) = 1._dp ! V(n,1)=V(n,2) ! .. Boundary at h=infty, r=infty .. ! .. Impose Neumann b.c. V(H=hmax,r=rmax)=V(H,r=rmax-dr) .. bet(n,m) = 1._dp ! .. Boundary at h=0, r=infty .. bet(1,m) = 1._dp ! .. Edges of the grid .. ! .. Boundary at h=0, r varies .. DO j = 2,m-1 bet(1,j) = 1._dp
- 196. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 196 END DO ! .. Boundary as h varies, r=0 .. DO i = 2,n-1 alp(i,1) = 0.25_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) + & 0.25_dp/dh*delta*h(i) ! v(i-1,1) bet(i,1) = - 0.5_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) - & 0.75/dr*kappa*theta - 1._dp/dt ! v(i,1) gam(i,1) = 0.25_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) - & 0.25_dp/dh*delta*h(i) ! v(i+1,1) del(i,1) = kappa/dr*theta ! v(i,2) eps(i,1) = - 0.25_dp/dr*kappa*theta ! v(i,3) END DO ! .. Boundary at h=infty, r varies .. DO j = 2,m-1 alp(n,j) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))) ! v(n,j-1) bet(n,j) = -1/dt -.5_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.5_dp*r(j) ! v(n,j) gam(n,j) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & + 0.25_dp/dr*(kappa*(theta-r(j))) ! v(n,j+1) END DO ! .. Boundary as h varies, r=infty .. ! .. Impose Neumann b.c. V(H,r=rmax)=V(H,r=rmax-dr) .. DO i = 2,n-1 bet(i,m) = 1._dp END DO ! .. Intermediate points .. DO i = 2,n-1 DO j = 2,m-1 alp(i,j) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) &
- 197. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 197 - 0.25_dp/dr*(kappa*(theta-r(j))) bet(i,j) = -1._dp/dt - 0.5_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) & - 0.5_dp/dr/dr*sigma_r*sigma_r*r(j) - 0.5_dp*r(j) gam(i,j) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & + 0.25_dp/dr*(kappa*(theta-r(j))) del(i,j) = 0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & - 0.25_dp/dH*(r(j)-delta)*H(i) eps(i,j) = 0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & + 0.25_dp/dH*(r(j)-delta)*H(i) mu(i,j) = 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)*SQRT(r(j))*rho END DO END DO END SUBROUTINE Coefficient_matrix_v SUBROUTINE Z_matrix_v(alp,bet,gam,del,eps,mu,Z,v,dt,n,m) IMPLICIT NONE INTEGER :: n,m,i,j REAL(dp) :: dt REAL(dp),DIMENSION(n,m)::alp,bet,gam,del,eps,mu,Z,v ! .. Sets Z coefficients in C-N scheme (RHS). Change as v is updated .. ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. ! Z(1,1) = v(1,2) Z(1,1) = (-2._dp/dt - alp(1,1))*v(1,1) & - bet(1,1)*v(1,2) - gam(1,1)*v(1,3) ! .. Boundary at h=infty, r=0 .. Z(n,1) = v(n,2) ! .. Boundary at h=infty, r=infty ! .. Impose Neumann b.c. V(H=hmax,r=rmax)=V(H,r=rmax-dr) .. Z(n,m) = v(n,m-1)
- 198. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 198 ! .. Boundary at h=0, r=infty .. Z(1,m) = 0._dp ! ! .. Edges of the grid .. ! .. Boundary at h=0, r varies .. DO j = 2,m-1 Z(1,j) = 0._dp END DO ! .. Boundary as h varies, r=0 .. DO i = 2,n-1 Z(i,1) = - alp(i,1)*v(i-1,1) + (-2._dp/dt - bet(i,1))*v(i,1)& - gam(i,1)*v(i+1,1) - del(i,1)*v(i,2) - eps(i,1)*v(i,3) END DO ! .. Boundary at h=infty, r varies .. DO j = 2,m-1 Z(n,j) = - alp(n,j)* v(n,j-1) + (-2._dp/dt - bet(n,j))*v(n,j) & - gam(n,j)*v(n,j+1) END DO ! .. Boundary as h varies, r=infty .. ! .. Neumann b.c. to smooth solution .. DO i = 2,n-1 Z(i,m) = v(i,m-1) END DO ! .. Intermediate points .. DO i = 2,n-1 DO j = 2,m-1 Z(i,j) = (-2/dt-bet(i,j))*v(i,j) - alp(i,j)*v(i,j-1) & - gam(i,j)*v(i,j+1) - del(i,j)*v(i-1,j) & - eps(i,j)*v(i+1,j) - mu(i,j)*(v(i+1,j+1)-v(i-1,j+1) & - v(i+1,j-1)+v(i-1,j-1)) END DO
- 199. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 199 END DO END SUBROUTINE Z_matrix_v ! .. PSOR solver for mortgage asset .. ! SUBROUTINE Find_v_d_c_ins_coins(v,Z,td,n,m,alp,bet,gam,del,eps,mu,& tol,iter,OUT,l,monthmax,k,month,a,d,c,ins,coins,& r,h,sigma_r,sigma_h,dr,dt,dh,kappa,theta,delta,rho) IMPLICIT NONE INTRINSIC MIN INTEGER :: n,m,iter,i,j,counter,out,l,monthmax,k,month REAL(dp) :: r(m),h(n),sigma_r,sigma_h,dr,dt,dh,kappa,theta,delta,rho REAL(dp),DIMENSION(n,m) :: alp,bet,gam,eps,del,mu REAL(dp),DIMENSION(n,m) :: v,Z,d,c,ins,coins REAL(dp) :: omega=1._dp,tol,error,td(l,monthmax),y,a(m) ! .. Uses Linear Complementarity method to value mortgage component .. ! .. Iteration counter for PSOR method .. DO counter = 1,iter error = 0._dp ! .. Linear Complementarity Method using PSOR technique .. ! .. This converts free boundary prob to a fixed boundary problem .. ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. y = 1._dp/alp(1,1)*(Z(1,1) - bet(1,1)*v(1,2) - gam(1,1)*v(1,3)) y = MIN( td(k,month) , v(1,1)+omega*(y-v(1,1)) ) error = error+(v(1,1)-y)*(v(1,1)-y) v(1,1) = y ! .. Boundary at h=infty, r=0 .. y = 1._dp/bet(n,1)*Z(n,1) y = MIN( td(k,month) , omega*y ) v(n,1) = y
- 200. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 200 ! .. Boundary at h=infty, r=infty .. y = 1._dp/bet(n,m)*Z(n,m) y = MIN( td(k,month) , omega*y ) v(n,m) = y ! .. Boundary at h=0, r=infty .. v(1,m) = 0._dp ! .. Edges of the grid .. ! .. Boundary at h=0, r varies .. DO j = 2,m-1 v(1,j) = 0._dp END DO ! .. Boundary as h varies, r=0 .. DO i = 2,n-1 y = 1._dp/bet(i,1)*(Z(i,1) - alp(i,1)*v(i-1,1)- gam(i,1)*v(i+1,1) & - del(i,1)*v(i,2)- eps(i,1)*v(i,3)) y = MIN( td(k,month) , v(i,1)+omega*(y-v(i,1)) ) error = error+(v(i,1)-y)*(v(i,1)-y) v(i,1)=y END DO ! .. Boundary at h=infty, r varies .. DO j = 2,m-1 y = 1._dp/bet(n,j)*(Z(n,j) - alp(n,j)*v(n,j-1)- gam(n,j)*v(n,j+1)) y = MIN( td(k,month) , v(n,j)+omega*(y-v(n,j)) ) error = error+(v(n,j)-y)*(v(n,j)-y) v(n,j)=y END DO ! .. Boundary as h varies, r=infty .. DO i = 2,n-1 y = 1._dp/bet(i,m)*Z(i,m) y = MIN( td(k,month) , omega*y )
- 201. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 201 v(i,m) = y END DO ! .. Intermediate points DO i = 2,n-1 DO j = 2,m-1 y = 1._dp/bet(i,j)*(Z(i,j) - alp(i,j)*v(i,j-1)- & gam(i,j)*v(i,j+1) - del(i,j)*v(i-1,j)- & eps(i,j)*v(i+1,j) - mu(i,j)* & (v(i+1,j+1) - v(i-1,j+1) - v(i+1,j-1) + v(i-1,j-1))) ! .. Constraint to determine if prepayment occurs .. y = MIN( td(k,month) , v(i,j)+omega*(y-v(i,j)) ) ! .. Calculate L1 norm, sum over i,j of error_{i,j} .. error = error+(v(i,j)-y)*(v(i,j)-y) v(i,j) = y END DO END DO IF(error < tol)THEN WRITE TO FILE ’converged after ’,counter,’ iterations’ ! .. If true set v, otherwise iterate again .. EXIT END IF IF(counter==iter)THEN WRITE TO SCREEN ’Failed to converge’ STOP END IF END DO ! .. Once v has been set d,c,ins and coins can be calculated .. DO i = 1,n DO j = 1,m
- 202. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 202 IF (v(i,j) == td(k,month)) THEN ! .. Prepayment has occurred .. d(i,j) = 0._dp; ins(i,j) = 0._dp; coins(i,j) = 0._dp ELSE ! .. otherwise .. d(i,j) = d(i,j); ins(i,j) = ins(i,j); coins(i,j) = coins(i,j) END IF END DO END DO ! .. Calculate d, ins, coins with new information from above .. CALL Calculate_default_value(n,m,dh,dr,dt,sigma_h,sigma_r,& h,r,kappa,theta,delta,rho,d,a) CALL Calculate_insurance_value(r,sigma_r,h,sigma_h,dr,dt,dh,kappa& ,theta,delta,rho,ins,n,m) ! .. solve for coinsurance component .. CALL Calculate_insurance_value(r,sigma_r,h,sigma_h,dr,dt,dh,kappa& ,theta,delta,rho,coins,n,m) DO i = 1,n DO j = 1,m IF (v(i,j) == td(k,month)) THEN ! .. Inside the prepayment region .. c(i,j) = max( (a(j)-v(i,j)-d(i,j)) , 0._dp ) ELSE ! .. Inside the continuation region .. c(i,j) = max( 0._dp , (a(j)-v(i,j)-d(i,j)) ) END IF END DO END DO END SUBROUTINE Find_v_d_c_ins_coins
- 203. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 203 ! .. Solver for default option .. ! .. Value is found using a standard library package - ! .. general linear solver gives solution of real band system of linear ! .. equations, matrix already (LU) factorised .. SUBROUTINE Calculate_default_value(n,m,dh,dr,dt,sigma_h,sigma_r,& h,r,kappa,theta,delta,rho,d,a) IMPLICIT NONE INTEGER :: n,m,i,j REAL(dp):: r(m),h(n),a(m),d(n,m) REAL(dp):: sigma_r,sigma_h,dr,dt,dh,kappa,theta,delta,rho INTEGER :: LDAB,KU, KL,IPIV(1:m*N) REAL(dp):: AB(1:3*m+4,1:n*m),BB(1:m*n) ! .. Store coefficients as required by library package LDAB = 3*m+4; KU = m+1; KL = m+1 ! .. All coefficients are zero unless specified otherwise .. AB = 0._dp ! .. AB is LHS matrix .. BB = 0._dp ! .. BB is RHS matrix .. ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. AB(2*m+3,1) = -1/dt - 0.75_dp*kappa*theta/dr AB(2*m+2,2) = kappa*theta/dr AB(2*m+1,3) = -0.25_dp*kappa*theta/dr ! .. Boundary at h=infty, r=0 .. AB(2*m+3,(n-1)*m+1) = 1._dp ! d_bet d(n,1) not sure about this yet ! .. Boundary at h=infty, r=infty .. AB(2*m+3,(n-1)*m+m) = 1._dp ! d_bet d(n,m) ! .. Boundary at h=0, r=infty .. ! .. Neumann b.c. d(1,m)=d(1,m-1) .. AB(2*m+3,m) = 1._dp ! d_bet d(1,m) ! .. Edges of the grid ..
- 204. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 204 ! .. Boundary at h=0, r varies .. DO j = 2,m-1 AB(2*m+3,j) = 1._dp ! d_bet d(1,j) END DO ! .. Boundary at h varies, r=0 .. DO i = 2,n-1 AB(3*m+3,(i-2)*m+1) = 0.25_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) & + 0.25_dp/dh*delta*h(i) ! d(i-1,1) AB(2*m+3,(i-1)*m+1) = - 0.5_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) & - 0.75/dr*kappa*theta - 1._dp/dt ! d(i,1) AB(m+3,i*m+1) = 0.25_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) & - 0.25_dp/dh*delta*h(i) ! d(i+1,1) AB(2*m+2,(i-1)*m+2) = kappa/dr*theta ! d(i,2) AB(2*m+1,(i-1)*m+3) = - 0.25_dp/dr*kappa*theta ! d(i,3) END DO ! .. Boundary at h=infty, r varies .. DO j = 2,m-1 AB(2*m+3,(n-1)*m+j) = 1._dp END DO ! .. Boundary as h varies, r=infty .. ! .. Neumann b.c. d(i,m)=d(i,m-1) .. DO i=2,n-1 AB(2*m+3,(i-1)*m+m) = 1._dp ! d_bet d(i,m) END DO ! .. Intermediate points .. DO i = 2,n-1 DO j = 2,m-1 ! .. Here AB(ku+kl+1+i-j,j) = A(i,j) .. ! ..B((i-1)*m+j) = b(i,j) .. AB(2*m+4,(i-1)*m+j-1) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) &
- 205. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 205 - 0.25_dp/dr*(kappa*(theta-r(j))) ! d_alp d(i,j-1) AB(2*m+3,(i-1)*m+j) = -1._dp/dt-0.5_dp/dh/dh*sigma_h*sigma_h& *h(i)*h(i)-0.5_dp/dr/dr*sigma_r*sigma_r*r(j)-0.5_dp*r(j)!bet(i,j) AB(2*m+2,(i-1)*m+j+1) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & + 0.25_dp/dr*(kappa*(theta-r(j))) ! d_gam d(i,j+1) AB(m+4,i*m+j-1) = - 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho !-d_mu d(i+1,j-1) AB(m+3,i*m+j) = 0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & + 0.25_dp/dH*(r(j)-delta)*H(i) ! d_eps d(i+1,j) AB(m+2,i*m+j+1) = 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho !-d_mu d(i+1,j-1) ! d_mu d(i+1,j+1) AB(3*m+4,(i-2)*m+j-1) = 0.125_dp/dr/dH*sigma_H*sigma_H& *H(i)*SQRT(r(j))*rho !-d_mu d(i+1,j-1) ! d_mu d(i-1,j-1) AB(3*m+3,(i-2)*m+j) = 0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & - 0.25_dp/dH*(r(j)-delta)*H(i) ! d_del d(i-1,j) AB(3*m+2,(i-2)*m+j+1) = - 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho !-d_mu d(i+1,j-1) !-d_mu d(i-1,j+1) END DO END DO ! .. Define right hand side .. ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. BB(1) = (-2._dp/dt - (-1/dt - 0.75_dp*kappa*theta/dr))*d(1,1) & - (kappa*theta/dr)*d(1,2) - (-0.25_dp*kappa*theta/dr)*d(1,3) ! d_Z(1,1) ! .. Boundary at h=infty, r=0 .. BB((n-1)*m+1) = d(n,2) ! d_Z(n,1) ! .. Boundary at h=infty, r=infty .. BB((n-1)*m+m) = 0._dp ! d_Z(n,m) ! .. Boundary at h=0, r=infty .. ! .. Neumann b.c. d(1,m)=d(1,m-1) ..
- 206. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 206 BB(m) = d(1,m-1) ! d_Z(1,m) ! .. Edges of the grid .. ! .. Boundary at h=0, r varies .. DO j = 2,m-1 BB(j) = a(j) ! d_Z(1,j) END DO ! .. Boundary as h varies, r=0 .. DO i = 2,n-1 BB((i-1)*m+1) = - (0.25_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) & + 0.25_dp/dh*delta*h(i))*d(i-1,1) + (-2/dt & -(- 0.5_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i)- 0.75/dr*kappa*theta & - 1._dp/dt) )*d(i,1) - (0.25_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) & - 0.25_dp/dh*delta*h(i))*d(i+1,1) & - (kappa/dr*theta)*d(i,2) & - (-0.25_dp/dr*kappa*theta)*d(i,3) ! d_Z(i,1) END DO ! .. Boundary at h=infty, r varies .. DO j=2,m-1 BB((n-1)*m+j) = 0._dp ! d_Z(n,j) END DO ! .. Boundary as h varies, r=infty .. ! .. Neumann b.c. d(i,m)=d(i,m-1) .. DO i=2,n-1 BB((i-1)*m+m) = d(i,m-1) ! d_Z(i,m) END DO ! .. Intermediate points .. DO i=2,n-1 DO j=2,m-1 BB((i-1)*m+j) = (-2/dt - (-1._dp/dt& -0.5_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i) &
- 207. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 207 - 0.5_dp/dr/dr*sigma_r*sigma_r*r(j)-0.5_dp*r(j)))*d(i,j)& - (0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))))*d(i,j-1) & - (0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & + 0.25_dp/dr*(kappa*(theta-r(j))))*d(i,j+1) & - (0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & - 0.25_dp/dH*(r(j)-delta)*H(i))*d(i-1,j) & - (0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & + 0.25_dp/dH*(r(j)-delta)*H(i))*d(i+1,j) & - (0.125_dp/dr/dH*sigma_H*sigma_H*H(i)*SQRT(r(j))*rho)& *(d(i+1,j+1)-d(i-1,j+1)-d(i+1,j-1)+d(i-1,j-1)) END DO END DO ! .. solve for d .. CALL LUCOMP(m*n,m*n,KL,KU,AB,ldab,IPIV) CALL SOLVEAB(n*M,ku,kl,1,ab,ldab,ipiv,BB,n*m) DO i = 1,n DO j = 1,m d(i,j) = BB((i-1)*m+j) END DO END DO END SUBROUTINE Calculate_default_value ! .. Solver for insurance asset .. ! ! .. Value is found using a standard library package - ! .. general linear solver gives solution of real band system of linear ! .. equations, matrix already (LU) factorised .. SUBROUTINE Calculate_insurance_value(r,sigma_r,h,sigma_h,dr,dt,dh,kappa,& theta,delta,rho,ins,n,m) IMPLICIT NONE
- 208. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 208 INTEGER :: n,m,i,j REAL(dp):: r(m),h(n),ins(n,m) REAL(dp):: sigma_r,sigma_h,dr,dt,dh,kappa,theta,delta,rho INTEGER :: LDAB,KU, KL,IPIV(1:m*N) REAL(dp):: AB(1:3*m+4,1:n*m),BB(1:m*n) ! .. Store coefficients as required by library package dab = 3*m+4; ku = m+1; kl = m+1 ! .. All coefficients are zero unless specified otherwise .. AB = 0._dp; BB = 0._dp ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. AB(2*m+3,1) = -1/dt - 0.75_dp*kappa*theta/dr AB(2*m+2,2) = kappa*theta/dr AB(2*m+1,3) = -0.25_dp*kappa*theta/dr ! .. Boundary at h=infty, r=0 .. AB(2*m+3,(n-1)*m+1) = 1._dp ! .. Boundary at h=infty, r=infty .. AB(2*m+3,(n-1)*m+m) = 1._dp ! ins_bet d(n,m) ! .. Boundary at h=0, r=infty .. ! .. Neumann b.c. ins(1,m)=ins(1,m-1) .. AB(2*m+3,m) = 1._dp ! ins_bet ins(1,m) ! .. Edges of the grid .. ! .. Boundary at h=0, r varies .. DO j = 2,m-1 AB(2*m+4,j-1) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))) ! ins(1,j-1) AB(2*m+3,j) = -1._dp/dt -.5_dp/dr/dr*sigma_r*sigma_r*r(j) & -.5_dp*r(j) ! ins(1,j) AB(2*m+2,j+1) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & + 0.25_dp/dr*(kappa*(theta-r(j))) ! ins(1,j+1)
- 209. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 209 END DO ! .. Boundary at h varies, r=0 .. DO i =2,n-1 AB(2*m+3,(i-1)*m+1) = 1._dp END DO ! .. Boundary at h=infty, r varies .. DO j = 2,m-1 AB(2*m+3,(n-1)*m+j) = 1._dp END DO ! .. Boundary as h varies, r=infty .. ! .. Neumann b.c. ins(i,m)=ins(i,m-1) .. DO i = 2,n-1 AB(2*m+3,(i-1)*m+m) = 1._dp ! ins_bet ins(i,m) END DO ! .. Intermediate points .. DO i=2,n-1 DO j=2,m-1 ! .. Here AB(ku+kl+1+i-j,j) = A(i,j) .. ! ..B((i-1)*m+j) = b(i,j) .. AB(2*m+4,(i-1)*m+j-1) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))) ! ins_alp ins(i,j-1) AB(2*m+3,(i-1)*m+j) = -1._dp/dt - & 0.5_dp/dh/dh*sigma_h*sigma_h*h(i)*h(i)& -0.5_dp/dr/dr*sigma_r*sigma_r*r(j)-0.5_dp*r(j)!bet(i,j) AB(2*m+2,(i-1)*m+j+1) = 0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & + 0.25_dp/dr*(kappa*(theta-r(j))) ! ins_gam ins(i,j+1) AB(m+4,i*m+j-1) = - 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho !-ins_mu ins(i+1,j-1) AB(m+3,i*m+j) = 0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & + 0.25_dp/dH*(r(j)-delta)*H(i) ! ins_eps ins(i+1,j)
- 210. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 210 AB(m+2,i*m+j+1) = 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho ! ins_mu ins(i+1,j+1) AB(3*m+4,(i-2)*m+j-1) = 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho ! ins_mu ins(i-1,j-1) AB(3*m+3,(i-2)*m+j) = 0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & - 0.25_dp/dH*(r(j)-delta)*H(i) ! ins_del ins(i-1,j) AB(3*m+2,(i-2)*m+j+1) = - 0.125_dp/dr/dH*sigma_H*sigma_H*H(i)& *SQRT(r(j))*rho !-ins_mu ins(i-1,j+1) END DO END DO ! .. Define right hand side .. ! .. Corners of the grid .. ! .. Boundary at h=0, r=0 .. BB(1) = (-2._dp/dt-(-1/dt-0.75_dp*kappa*theta/dr))*ins(1,1)& -(kappa*theta/dr)*ins(1,2)-(-0.25_dp*kappa*theta/dr)*ins(1,3) ! .. Boundary at h=infty, r=0 .. BB((n-1)*m+1) = ins(n,2) ! ins_Z(n,1) -(kappa*theta/dr)*ins(n,2)-(-0.25_dp*kappa*theta/dr)*ins(n,3) ! .. Boundary at h=infty, r=infty .. BB((n-1)*m+m) = 0._dp ! ins_Z(n,m) ! .. Boundary at h=0, r=infty .. ! .. Neumann b.c. ins(1,m)=ins(1,m-1) .. BB(m) = ins(1,m-1) ! ins_Z(1,m) ! .. Edges of the grid .. ! .. Boundary at h=0, r varies .. DO j = 2,m-1 BB(j) = - (0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))))*ins(1,j-1)& + (-2_dp/dt-(-1_dp/dt -.5_dp/dr/dr*sigma_r*sigma_r*r(j) & -0.5_dp*r(j)))*ins(1,j) - (0.25_dp/dr/dr*sigma_r*sigma_r*r(j) &
- 211. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 211 + 0.25_dp/dr*(kappa*(theta-r(j))))*ins(1,j+1) ! ins_Z(n,j) END DO ! .. Boundary as h varies, r=0 .. DO i = 2,n-1 BB((i-1)*m+1) = ins(i,2) END DO ! .. Boundary at h=infty, r varies .. DO j = 2,m-1 BB((n-1)*m+j) = 0._dp ! ins_Z(n,j) END DO ! .. Boundary as h varies, r=infty .. ! .. Neumann b.c. to smooth solution .. DO i = 2,n-1 BB((i-1)*m+m) = ins(i,m-1) ! ins_Z(i,m) END DO ! .. Intermediate points .. DO i = 2,n-1 DO j = 2,m-1 BB((i-1)*m+j) =(-2/dt - (-1._dp/dt-0.5_dp/dh/dh*sigma_h*sigma_h*h(i)& *h(i) -0.5_dp/dr/dr*sigma_r*sigma_r*r(j)-0.5_dp*r(j)))*ins(i,j)& - (0.25_dp/dr/dr*sigma_r*sigma_r*r(j) & - 0.25_dp/dr*(kappa*(theta-r(j))))*ins(i,j-1) & - (0.25_dp/dr/dr*sigma_r*sigma_r*r(j) + 0.25_dp/dr*& (kappa*(theta-r(j))))*ins(i,j+1) & ! ins_Z(i,j) - (0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & - 0.25_dp/dH*(r(j)-delta)*H(i))*ins(i-1,j) & - (0.25_dp/dH/dH*sigma_H*sigma_H*H(i)*H(i) & + 0.25_dp/dH*(r(j)-delta)*H(i))*ins(i+1,j) & - (0.125_dp/dr/dH*sigma_H*sigma_H*H(i)*SQRT(r(j))*rho)& *(ins(i+1,j+1)-ins(i-1,j+1)-ins(i+1,j-1)+ins(i-1,j-1))
- 212. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 212 END DO END DO ! .. solve for ins .. CALL LUCOMP(m*n,m*n,KL,KU,AB,ldab,IPIV) CALL SOLVEAB(n*M,ku,kl,1,ab,ldab,ipiv,BB,n*m) DO i = 1,n DO j = 1,m ins(i,j) = BB((i-1)*m+j) END DO END DO END SUBROUTINE Calculate_insurance_value ! .. The b.c.’s for coins are the same as for ins asset so use ! .. Calculate_insurance_value with ins variable changed to coins .. ! SUBROUTINE LUCOMP(M,N,KL,KU,AB,ldab,IPIV)! LU factorisation IMPLICIT NONE INTEGER :: LDAB,KU, KL, M, N INTEGER :: IFAIL, INFO REAL(dp) :: ab(1:ldab,1:n) INTEGER :: ipiv(1:n) EXTERNAL :: dgbtrf CALL dgbtrf(M,N,KL,KU,AB,LDAB,IPIV,INFO) ifail = 0 END SUBROUTINE LUCOMP SUBROUTINE SOLVEAB(n,ku,kl,nrhs,ab,ldab,ipiv,b,ldb)! Solves equations IMPLICIT NONE INTEGER :: LDAB,KU, KL, N INTEGER :: IFAIL, INFO
- 213. APPENDIX A. FIXED-RATE MORTGAGE VALUATION PSEUDOCODE 213 INTEGER :: LDB,nrhs REAL(dp) :: ab(1:ldab,1:n),b(1:LDB,1:NRHS) INTEGER :: ipiv(1:n) CHARACTER (len=1) :: TRANS EXTERNAL :: dgbtrs,X04CAF TRANS = ’N’ CALL dgbtrs(TRANS,n,kL,kU,nrhs,ab,ldab,ipiv,b,ldb,INFO) IFAIL = 0 END SUBROUTINE SOLVEAB
- 214. Appendix B Analytic approximation derivation B.1 Method of Characteristics What follows is the working to calculate the general solution of the mth month for the value of an asset F(H, r, τm), in the limit of small volatilities. This is the first step in the algorithm to value any of the assets at origination. This step hinges on the successful solution of equation (3.43) which, in section 3.7, was said to have a simple analytic solution. The Method of Characteristics is used to solve the PDE in equation (3.43) by finding curves in the H − r − τm surface that reduce this equation to an ordinary differential equation (see for example Garabedian, 1998). In general, any curve on the H − r − τm surface can be expressed in parametric form by, H = H(ζ), r = r(ζ), τm = τm(ζ), (B.1) where the parameter, ζ, gives a measure of the distance along the curve. The curve starts at the initial point, H = H0, r = r0, τm = 0, when ζ = 0. Assuming that the resulting ordinary differential equation can be solved means that F is known everywhere along this curve, i.e. along the curve picked out by the value of (H0, r0). Another choice for (H0, r0) gives a different curve and the value of F can be deter- mined along this curve. In this manner, F can be determined at any point in the H − r − τm plane by choosing the curve, defined by (H0, r0), that passes through 214
- 215. APPENDIX B. ANALYTIC APPROXIMATION DERIVATION 215 this point and taking the correct value of ζ, the distance along the curve. Hence, F(H, r, τm) can be evaluated. Consider F(H, r, τm) = F H(ζ), r(ζ), τm(ζ) and so F is a function of ζ. Hence, the derivative of F with respect to ζ is, dF dζ = dH dζ ∂F ∂H + dr dζ ∂F ∂r + dτm dζ ∂F ∂τm . (B.2) By comparing equation (3.43) with (B.2) equation (3.43) can be converted into an ordinary derivative of F with respect to ζ, i.e. dF dζ = rF, (B.3) provided the parametric representation of the curve satisfies: dH dζ = (r − δ)H, (B.4) dr dζ = κ(θ − r), (B.5) dτm dζ = −1. (B.6) Equations (B.4), (B.5) and (B.6) give the characteristic curves. If these four equations are rearranged it is possible to eliminate dζ. Then equate all four equations which leads to the following series of ODE’s: dr κ(θ − r) = −dτm = dH (r − δ)H = dF rF , (B.7) which can be solved by simple integration (as shown in the next section). B.2 Derivation of the general solution for any month From section 3.7.1 it is known that for a particular month m all that is required is the appropriate initial condition to determine F(H, r, τm) from equation (B.7). Let, F(H, r, τm = 0) = F0m (H0, r0) (B.8)
- 216. APPENDIX B. ANALYTIC APPROXIMATION DERIVATION 216 be the general initial condition for month m.1 Now, from equation (B.7), dr κ(θ − r) = −dτm. (B.9) Integrating equation (B.9), ⇒ − 1 κ ln κ(θ − r) = −τm + c1 ⇒ κ(θ − r) = eκτm−κc1 ∴ r = κθ − eκτm−κc1 κ , (B.10) where c1 is a constant. Note e−κc1 = κ(θ − r)e−κτm this will be important later in the solution for F Also from equation (B.7), −dτm = dF rF . (B.11) Substituting equation (B.10) for r and integrating (B.11), ⇒ −dτm[κθ − eκτm−κc1 ] = κ dF F ⇒ dτm[eκτm−κc1 − κθ] = κ dF F ⇒ 1 κ eκτm−κc1 − κθτm = κln(c2F) ∴ F = 1 c2 e(1/κ2)eκτm−κc1 −θτm , (B.12) where c2 is a constant. Finally, from equation (B.7), dH (r − δ)H = −dτm. (B.13) Again substituting equation (B.10) for r and integrating (B.13), ⇒ dH H = −dτm κθ − eκτm−κc1 κ − δ ⇒ ln(c3H) = (δ − θ)τm + 1 κ2 eκτm−κc1 ∴ H = 1 c3 e(δ−θ)τm+(1/κ2)eκτm−κc1 , (B.14) 1 Due to the transformation which produced the forward PDE an initial condition is required to solve the problem. These initial conditions were found by making the same transformation, as shown in sections 3.3.1 and 3.3.2.
- 217. APPENDIX B. ANALYTIC APPROXIMATION DERIVATION 217 where c3 is a constant. Once the constants c1, c2 and c3 are determined using the initial condition (B.8), H0 and r0 can be fixed. From equation (B.14), the value of H0 at τm = 0 is, H0 = 1 c3 e(1/κ2)e−κc1 = 1 c3 e(1/κ)(θ−r)e−κτm and rearranging equation (B.14) gives, 1 c3 = He(θ−δ)τm−(1/κ)(θ−r) (B.15) ⇒ H0 = He(θ−δ)τm+(1/κ)(θ−r)(e−κτm −1) . (B.16) The value of r0 at τm = 0 is, r0 = κθ − e−κc1 κ , (B.17) and using the result given in the note for e−κc1 , ⇒ r0 = θ − (θ − r)e−κτm . (B.18) Using the initial condition (B.8) and equation (B.12) at τm = 0, 1 c2 = F0m (H0, r0)e−(1/κ2)e−κc1 = F0m e−(1/κ)(θ−r)e−κτm . (B.19) Finally the equation for c2 is substituted into equation (B.12). This gives the general solution for any month: F(H, r, τm) = F0m (H0, r0)exp 1 κ (θ − r)(1 − e−κτm ) − θτm , (B.20) where, H0 = Hexp (θ − δ)τm + 1 κ (θ − r)(e−κτm − 1) (B.21) r0 = θ − (θ − r)e−κτm . (B.22)
- 218. Appendix C Bridging solutions As mentioned in section 3.9, the perturbation approach could be improved by includ- ing bridging solutions in the asymptotic analysis for each of the mortgage components. To understand the role of these bridging solutions, consider equation (3.55). This is the analytic approximation for the value of the default option at the beginning of month n. This solution has a discontinuity in the delta of the option along the line H = MPe(1/κ)(θ−r)(1−e−κTn )−(θ−δ)Tn ; a bridging solution would smooth this disconti- nuity. Since ∂D ∂H and ∂D ∂r are both discontinuous along this line, this suggests that ∂2D ∂H2 and ∂2D ∂r2 become large along this same line, implying that the approximation equation (3.55), which neglects these terms if σH and σr are both small, must become invalid. Therefore, the solution consists of three regions, the two from equation (3.55) sep- arated by a thin bridging solution.1 A schematic representation is shown in figure C.1. The crucial zone is the thin region in the vicinity of the following line H = MPe(1/κ)(θ−r)(1−e−κTn )−(θ−δ)Tn . This line has variation in both H and r and examina- tion of the two dimensional bridging solution in detail would require interest focused around this zone by introducing scaled variables. With this type of approach, an appropriate asymptotic expansion of D, with inclusion of the bridging solution in the asymptotic analysis would be possible. 1 Where consideration of the effect of the free boundary associated with prepayment (when the prepayment option has some value the default option is worthless) has been neglected. 218
- 219. APPENDIX C. BRIDGING SOLUTIONS 219 PSfrag replacements r H0 0 D = MPe(1/κ)(θ−r)(1−e−κTn )−θTn − He−δTn H = MPe(1/κ)(θ−r)(1−e−κTn )−(θ−δ)Tn D = 0 Figure C.1: An illustration of the solution space for the default option in the final month n. The thick line represents the position of the required bridging solution. This type of analysis becomes important when the critical point in state space (H, r at origination) for the equilibrium condition coincides with a bridging solution, as it is here that the approximation is required to be extremely accurate (otherwise, the contract rate may be set at a value which produces arbitrage opportunities). However, the full inclusion of these bridging solutions would defeat the point of the work, which was to provide quick simple solutions, since second-order derivative terms would be introduced, resulting in the necessity of finite-difference techniques in addition to the perturbation analysis.