Stochastic processes for_finance_dictaat2 (1)

STOCHASTIC PROCESSES FOR FINANCE
RISK MANAGEMENT TOOLS
Notes for the Course by
F. Boshuizen, A.W. van der Vaart, H. van Zanten, K. Banachewicz and P. Zareba
CORRECTED 15 October 2006

ii
CONTENTS
1. Pricing derivatives . . . . . . . . . . . . . . . . . . . . . 1
1.1. Hedging a Forward . . . . . . . . . . . . . . . . . . . 1
1.2. Note on Continuous Compounding . . . . . . . . . . . . 3
2. Binomial Tree Model . . . . . . . . . . . . . . . . . . . . 5
2.1. One Period Model . . . . . . . . . . . . . . . . . . . 5
2.2. Two Period Model . . . . . . . . . . . . . . . . . . . 7
2.3. N Period Model . . . . . . . . . . . . . . . . . . . . 8
3. Discrete Time Stochastic Processes . . . . . . . . . . . . . 10
3.1. Stochastic Processes . . . . . . . . . . . . . . . . . . 10
3.2. Conditional Expectation . . . . . . . . . . . . . . . . 11
3.3. Filtration . . . . . . . . . . . . . . . . . . . . . . 12
3.4. Martingales . . . . . . . . . . . . . . . . . . . . . 13
3.5. Change of Measure . . . . . . . . . . . . . . . . . . 14
3.6. Martingale Representation . . . . . . . . . . . . . . . 15
4. Binomial Tree Model Revisited . . . . . . . . . . . . . . . 17
4.1. Towards Continuous Time . . . . . . . . . . . . . . . 20
5. Continuous Time Stochastic Processes . . . . . . . . . . . . 23
5.1. Stochastic Processes . . . . . . . . . . . . . . . . . . 23
5.2. Brownian Motion . . . . . . . . . . . . . . . . . . . 23
5.3. Filtrations . . . . . . . . . . . . . . . . . . . . . . 25
5.4. Martingales . . . . . . . . . . . . . . . . . . . . . 25
5.5. Generalized Brownian Motion . . . . . . . . . . . . . 25
5.6. Variation . . . . . . . . . . . . . . . . . . . . . . 26
5.7. Stochastic Integrals . . . . . . . . . . . . . . . . . . 28
5.8. Geometric Brownian Motion . . . . . . . . . . . . . . 29
5.9. Stochastic Diﬀerential Equations . . . . . . . . . . . . 29
5.10. Markov Processes . . . . . . . . . . . . . . . . . . . 30
5.11. Quadratic variation - revisited . . . . . . . . . . . . . 30
5.12. Itˆo Formula . . . . . . . . . . . . . . . . . . . . . 32
5.13. Girsanov’s Theorem . . . . . . . . . . . . . . . . . . 34
5.14. Brownian Representation . . . . . . . . . . . . . . . 35
5.15. Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . 36
5.16. Stopping . . . . . . . . . . . . . . . . . . . . . . . 38
5.17. Extended Stochastic Integrals . . . . . . . . . . . . . 39
6. Black-Scholes Model . . . . . . . . . . . . . . . . . . . . 41
6.1. Portfolios . . . . . . . . . . . . . . . . . . . . . . 42
6.2. The Fair Price of a Derivative . . . . . . . . . . . . . 42
6.3. European Options . . . . . . . . . . . . . . . . . . . 44
6.4. The Black-Scholes PDE and Hedging . . . . . . . . . . 45
6.5. The Greeks . . . . . . . . . . . . . . . . . . . . . 47
6.6. General Claims . . . . . . . . . . . . . . . . . . . . 48
6.7. Exchange Rate Derivatives . . . . . . . . . . . . . . . 48
7. Extended Black-Scholes Models . . . . . . . . . . . . . . . 50

iii
7.1. Market Price of Risk . . . . . . . . . . . . . . . . . 51
7.2. Fair Prices . . . . . . . . . . . . . . . . . . . . . . 51
7.3. Arbitrage . . . . . . . . . . . . . . . . . . . . . . 53
7.4. PDEs . . . . . . . . . . . . . . . . . . . . . . . . 54
8. Interest Rate Models . . . . . . . . . . . . . . . . . . . 57
8.1. The Term Structure of Interest Rates . . . . . . . . . . 57
8.2. Short Rate Models . . . . . . . . . . . . . . . . . . 59
8.3. The Hull-White Model . . . . . . . . . . . . . . . . . 63
8.4. Pricing Interest Rate Derivatives . . . . . . . . . . . . 66
8.5. Examples of Interest Rate Derivatives . . . . . . . . . . 68
9. Risk Measurement . . . . . . . . . . . . . . . . . . . . 72
9.1. Value-At-Risk . . . . . . . . . . . . . . . . . . . . 72
9.2. Normal Returns . . . . . . . . . . . . . . . . . . . 74
9.3. Equity Portfolios . . . . . . . . . . . . . . . . . . . 75
9.4. Portfolios with Stock Options . . . . . . . . . . . . . 77
9.5. Bond Portfolios . . . . . . . . . . . . . . . . . . . . 79
9.6. Portfolios of Bonds and Swaptions . . . . . . . . . . . 80
9.7. Diversiﬁed Portfolios . . . . . . . . . . . . . . . . . 82

iv
LITERATURE
[1] Baxter, M. and Rennie, A., (1996). Financial calculus. Cambridge
University Press, Cambridge.
[2] Chung, K.L. and Williams, R.J., (1990). Introduction to stochastic
integration, second edition. Birkhäuser, London.
[3] Etheridge, A., (2002). A Course in Financial Calculus. Cambridge
University Press.
[4] Campbell, J.Y., Lo, A.W. and MacKinlay, A.C., (1997). The Econo-
metrics of Financial Markets. Princeton University Press.
[5] Hunt, P.J. and Kennedy, J.E., (1998). Financial Engineering. Wiley.
[6] Jorion, P., (2001). Value at Risk the New Benchmark for Managing
Financial Risk. McGraw-Hill, New York.
[7] Musiela, M. and Rutkowski, M., (1997). Martingale Methods in Fi-
nancial Modelling. Springer-Verlag, Berlin.
[8] Smithson, C.W., Smith, C.W. and Wilford, D.S., (1995). Managing
Financial Risk. Irwin, Burr Ridge, Illinois.
Baxter and Rennie is a book written for an audience of people in practice,
but using the correct mathematical concepts and results. The result is a
text with much intuition, and with mathematical theorems that are stated
with some precision, but never proved or rigorously interpreted. Etheridge
is a more mathematical version of Baxter and Rennie. Together these books
are close to the content of the course. We recommend that you read these
books, next to the following notes, which are really very brief.
The other books on the list are for further reading.
Musiela and Rutkowski is a step up in mathematical level. Hunt and
Kennedy and Chung and Williams are mathematically completely rigorous.
Chung and Williams has very little on finance. This is just a tiny selection
of the books on mathematical finance that have appeared in the past ten
years.
Campbell, Lo and MacKinLay gives a much wider view on finance,
including some historical analysis and economic theories of price form-
ing through utility, next to subjects close this course, all from a low-level
mathematics point of view. The authors of “Managing Financial Risk” are
bankers, not mathematicians. In their preface they write:
This stuff is not as hard as some people make it sound.
The financial markets have some complicated features, but good com-
mon sense goes a lot further than mathematical flash and dash.
Keep that mind when following this course. We do not entirely disagree,
but do believe that some flash (and dash) will certainly help to clarify this
complicated area.

1
Pricing derivatives
Financial instruments can be divided in two basic classes: underlying and
derivative ones. The former can be stocks, bonds or various trade goods,
while the latter are financial contracts that promise some payment to the
owner, depending on the behavior of the “underlying” (e.g. a price at a
given time T or an average price over a certain time period). Derivatives
are extremely useful for risk management (apart from obvious investment
properties) — we can reduce our financial vulnerability by fixing a price for
a future transaction now.
In this chapter we introduce some basic concepts through examples;
formal definitions and theory follow in later chapters.
1.1 Hedging a Forward
A forward is a contract that pays the owner an amount ST − K at a fixed
time T in the future, the expiry time, where St is the price of an asset at
time t and K is a fixed number, the strike price. Both T and K are written
in the contract, but ST will be known only at the expiry time. For which
number K is the value of this contract equal to 0 at time 0?
Suppose that we have, besides buying the contract, two other options
to invest our money:
(i) We can put our money in a savings account against a fixed, predeter-
mined interest rate r. One unit of money placed in the account grows
to ert
units during a time interval [0, t] and is freely available. A neg-
ative balance in our account is permitted, thus allowing us to borrow
money at the same interest rate r. If we borrow one unit at time 0,
than we owe ert
units at time t, which is equivalent to having a capital
of −ert
units.

2 1: Pricing derivatives
(ii) We can invest in the asset. The asset price St at time t is a stochastic
variable, dependent on t. It may be assumed that we know the proba-
bility distributions of these variables. For instance, a popular model is
that the variable ST /S0 is log normally distributed.
We discuss two answers to the pricing question.
A naive (wrong) answer is to argue as follows. A payment of ST − K
at time T is worth e−rT
(ST − K) at time 0. This is an unknown stochastic
quantity from the perspective of time 0. The reasonable price is the expec-
tation E e−rT
(ST − K) of this variable. The strike price that gives the
value 0 is therefore the solution of the equation
0 = E e−rT
(ST − K) = e−rT
(EST − K).
In other words, K = EST . For instance, using a log normal distribution with
parameters µT and σ2
T for ST /S0, i.e. log(ST /S0) is normally distributed
with these parameters, we find that
K = EST = S0eµT +
1
2 σ2
T
.
Somewhat surprisingly, this does not depend on the interest rate, meaning
that somehow no trade-off has been made between investing in the asset or
putting money in the savings account.
An accepted solution to the pricing question compares buying the con-
tract to a “hedging strategy”, as follows:
(i) at time 0 borrow an amount S0 at interest rate r, and buy the asset
at price S0,
(ii) sit still until expiry time T.
At time T we own the asset, worth ST , and we owe S0erT
to the bank. If
K = S0erT
, then together this is exactly the amount ST − K paid by the
forward contract at time T. Thus in that case we ought to be indifferent
towards buying the forward or carrying out the strategy (i)-(ii). The strat-
egy costs us nothing at time 0 and hence K = S0erT
is the correct strike
price for a forward with value 0 at time 0; this is also a maximal price, that
the buyer of the contract can accept. Since the seller of the contract can
perform a symmetric reasoning with S0erT
as a minimal accepted price,
K = S0erT
is the unique fair price acceptable to both parties.
The correct solution K = S0erT
does not depend on the distribu-
tion of the asset price ST , and in fact depends on the asset price process
only through its (observable) value at time 0. This is remarkable if you
are used to evaluate future gains through expectations on the random vari-
ables involved. Is there no role for probability theory in evaluating financial
contracts?
There is. First we note that the expected gain of owning a contract
is equal to E(ST − K), which does depend on the distribution of the asset
price. It evaluates to
S0(eµT +
1
2 σ2
T
− erT
)

1.2: Note on Continuous Compounding 3
if the asset price is log normally distributed and K = S0erT
.
Second, it turns out that the correct solution can in fact be found from
computing an expectation, but the expectation should be computed under
a special probability measure, called a “martingale measure”. To evaluate
the price of a forward, this route would be overdone, as the preceding hedg-
ing strategy is explicit and simple. However, the prices of other contracts
may not be so easy to evaluate, and in general probabilities and expec-
tations turn out to be very useful. For instance, consider a European call
option, which is a contract that pays an amount (ST − K)+
at time T, i.e.
ST − K if ST > K and nothing otherwise. There is no simple universal
hedging strategy to price this contract, but it turns out that given rea-
sonable probabilistic models for the asset price process St, there are more
complicated trading strategies that allow to parallel the preceding reason-
ing. These strategies require continuous trading during the term [0, T] of
the contract, and a big mathematical apparatus for their evaluation.
1.2 Note on Continuous Compounding
Our model for a savings account is that a capital of size R0 placed in such an
account at some time t increases to the amount R0er∆T
at time t+∆t. The
capital in the account remains ours without restriction: we can withdraw
it at no cost at any time. The constant r is the continuously compounded
interest rate and is not quite an ordinary rate for a savings account, which
is more often a yearly or monthly rate. If interest is added to the account
at the end of a time period of one unit, then this would increase the capital
from R0 to R1 = (1 + r1)R0, the interest being r1R0, and r1 being the rate
per time unit. If instead we would obtain the interest in two installments,
the first after half a time unit, and the second after one time unit, then
the initial capital would first increase to R1/2 = (1 + r2)R0 and next to
R1 = (1 + r2)R1/2 = (1 + r2)2
R0. The second time we receive “interest on
interest”. The rate r2 would be the rate per half time unit, and hence 2r2
should be compared to r1. However, the comparison would not be exact,
because (1+r1/2)2
> 1+r1. It would be logical that r1 and r2 relate through
the equation (1 + r2)2
= 1 + r1, apart from possibly a correction for the
benefit of early payment in the second scheme. The interest on interest is
making the difference.
We could continue this thought experiment and break the time unit in
n equal parts. The reasonable rate rn per (1/n)th time unit would satisfy
(1 + rn)n
= 1 + r1, or
1 +
nrn
n
n
= 1 + r1.
Here nrn is the rate per time unit. Taking the limit as n → ∞, and assuming

4 1: Pricing derivatives
that nrn tends to a limit r we obtain the equation er
= 1 + r1. This r is
the rate of interest per time unit earned in a savings account in which the
interest is added “continuously in time”. Using the continuous rate r is
convenient, as exponentials multiply.

2
Binomial Tree Model
A financial derivative is a contract that is based on the price of an underly-
ing asset, such as a stock price or bond price. An “option”, of which there
are many different types, is an important example. A main objective of “fi-
nancial engineering” is to find a “fair price” of such a derivative, where by
fair we mean a price acceptable both for the buyer and the seller. Following
the work by Black and Scholes in the 1970s the prices of derivatives are
found through the principle of “no arbitrage” introduced in the previous
chapter.
In this chapter we discuss, as an introduction, the pricing of a European
call option using a discrete time framework.
2.1 One Period Model
Suppose that at time 0 we can invest in an asset with price s0, or put money
in a savings account with a fixed interest rate. We model the asset price at
time 1 as a random variable S1 that can take only two values
(2.1)
P(S1 = us0) = p,
P(S1 = ds0) = 1 − p.
Here u (for “up”) and d (for “down”) are two known constants, with u > d,
and p is a number in [0, 1] that may be unknown. We assume that an
amount of one unit of money put in the savings account at time 0 grows to
a guaranteed amount of er
units at time 1.
We want to find the fair price at time 0 of a contract that pays the
amount C at time 1, where C = C(S1) may depend on the (unknown) value
of the asset at the payment date.

6 2: Binomial Tree Model
2.2 Example. A European call option corresponds to C = (S1 − K)+
,
for a given strike price K. The payment on a forward contract is equal to
C = S1 − K.
Suppose that at time 0 we buy φ0 assets and put an amount of ψ0
money units in the savings account. Then we have a “portfolio” (φ0, ψ0)
whose worth at time 0 is given by
(2.3) V0 = φ0s0 + ψ0 · 1.
If we do not trade between the times 0 and 1, then the value of the portfolio
changes to its value V1 at time 1, given by
V1 = φ0S1 + ψ0er
.
From the perspective of today (time 0) this is a random variable, that we
cannot know with certainty. However, the asset can only take the values
us0 and ds0 at time 1. In the ﬁrst case the contract is worth C(us0) at time
1, whereas in the second case it is worth C(ds0) at time 1. The value of the
portfolio is equal to φ0us0 +ψ0er
or φ0ds0 +ψ0er
in the two cases. Suppose
that we ﬁx the portfolio (φ0, ψ0) so that its value at time 1 agrees exactly
with the contract, for each of the two possibilities, i.e.
(2.4)
φ0us0 + ψ0er
= C(us0),
φ0ds0 + ψ0er
= C(ds0).
This portfolio will cost us V0 at time 0, and is guaranteed to have the same
value at time 1 as the contract with claim C(S1), whether the asset moves
up or down. We should therefore have no preference for the portfolio or the
contract, and hence a fair price for the contract at time 0 is the price of
the portfolio, i.e. V0 corresponding to the portfolio (φ0, ψ0) satisfying the
equations (2.4).
The equations (2.4) form a system of two linear equations in the un-
knowns φ0 and ψ0 and can be solved to give
φ0 =
C(us0) − C(ds0)
us0 − ds0
,
ψ0 = e−r uC(ds0) − dC(us0)
u − d
.
Inserting this in the equation (2.3), we see that this portfolio can be acquired
at time zero for the amount
(2.5) V0 = e−r
qC(us0) + (1 − q)C(ds0) , q =
er
− d
u − d
.
This is the fair price of the contract at time 0.
For d ≤ er
≤ u the number q is contained in the interval [0, 1] and
can be considered an alternative probability for the upward move of the

2.2: Two Period Model 7
asset process. In general, this probability is different from the probability
p, which turns out to be unimportant for the fair price. It can be seen that
q is the unique probability such that
(2.6) Eq(e−r
S1) = s0.
Here the subscript q in Eq tells us to evaluate the expectation of the random
variable S1 using the distribution given in (2.1) but with q replacing p.
Furthermore, the price of the contract can be written as
V0 = Eq e−r
C(S1) .
We can write equation (2.6) also in the form Eq(e−r
S1| S0) = S0 (with
S0 the random variable that is equal to the constant s0 with probability
one), which expresses that the expected value of the discounted asset price
e−r
S1 given S0 is equal to S0 = e−0
S0, or that the process S0, e−r
S1 is a
“martingale”.
2.7 Example (Forward). The value at time 0 of a forward is Eqe−r
(S1 −
K) = Eqe−r
S1 −e−r
K = S0 −e−r
K. The strike price that makes this value
equal to zero is K = er
S0, which is the value erT
S0 with T = 1 found in
Chapter 1.
2.2 Two Period Model
Suppose that at time 0 we have the same possibilities for investing as in
the preceding section, but we now consider a full trading horizon of three
times: 0, 1, 2. We wish to evaluate a claim on the asset process payable at
time 2.
Let the price of the asset at the three times be modelled by S0, S1, S2,
where we assume that S0 = s0 is fixed, S1 is equal to either dS0 or uS0,
and S2 is equal to either dS1 or uS1. Thus the asset prices follow a path
in a binary tree. We assume that at each node of the tree the decision to
move up or down is made with probabilities p and 1 − p, independently for
the different nodes.
Besides investing in the asset we may put money in a savings account
(also a negative amount, indicating that we borrow money) at a fixed in-
terest rate r. One unit in the savings account grows to er
units at time 1,
and to e2r
units at time 2.
The contract pays the amount C = C(S2) at time 2, and we wish to
find its fair price at time 0.
We can evaluate the claim recursively, backwards in time. At time 2
the claim is worth C = C(S2). At time 1 there are two possibilities: either

8 2: Binomial Tree Model
the asset price is dS0 or it is uS0. If we put ourselves at the perspective of
time 1, then we know which of the two possibilities is realized. If s1 is the
realized value of S1, then we can calculate the value of the claim (at time
1) using the one-period model as (cf. (2.5))
e−r
qC(us1) + (1 − q)C(ds1) .
For the two possibilities for s1 this gives
e−r
qC(uds0) + (1 − q)C(d2
s0) , if S1 = ds0,
e−r
qC(u2
s0) + (1 − q)C(dus0) , if S1 = us0.
This is the value of the contract at time 1, as a function of the asset price
S1 at time one. We can think of this value as the pay-oﬀ on our contract
at time 1, and next apply the one-period model a second time to see that
the value of the contract at time 0 is given by (cf. (2.5))
e−r
q e−r
qC(u2
s0) + (1 − q)C(dus0)
+ (1 − q) e−r
qC(uds0) + (1 − q)C(d2
s0) .
This equation can be rearranged as
e−2r
q2
C(u2
s0) + 2q(1 − q)C(uds0) + (1 − q)2
C(d2
s0) = Eq e−2r
C(S2) .
Hence once again the price is the expectation of the discounted claim,
presently e−2r
C(S2), under the probability measure on the tree given by
the branching probability q.
From the one-period model we know that Eq(e−r
S2| S1, S0) = S1 and
Eq(e−r
S1| S0) = S0. Together these equations show that, for n = 0, 1,
Eq(e−r(n+1)
Sn+1| Sn, . . . , S0) = e−nr
Sn.
We shall later summarize this by saying that the process S0, e−r
S1, e−2r
S2
is a martingale.
2.3 N Period Model
We can price a claim in a binomial tree model with N periods by extending
the backwards induction argument. The fair price of a claim C(SN ) is given
by
Eq e−Nr
C(SN ) .
Here SN is equal to uXN
dYN
S0 for XN and YN the number of up-moves and
down-moves, respectively. Of course YN = N − XN and the variable XN
possesses a binomial distribution with parameters N and success probability

2.3: N Period Model 9
q in the preceding display (and p in the real world). This allows to write
the fair price as a sum.
The induction argument to prove this formula is straightforward, but
tedious. We shall give a much prettier derivation after developing some
“martingale theory”, which will also give us the intuition needed to tackle
the continuous time models later on.
Note that the derivation of the formula involves portfolios (φ, ψ) that
are deﬁned in terms of the constants u and d. Thus intuitively the reason-
ing seems to require that we know these constants. In practice one might
observe the prices of some options (at least two for u and d and one more
for q) on the market, and next calibrate the constants u and d so that the
prices given by the formula agree with the market prices. We would do this
only if we believe the binomial tree model. Most people would consider the
continuous time models more believable. One unpleasant aspect of the bi-
nomial tree model is that it is essential that the splits in the tree are binary.
If there were three or more possible moves, everything would fall down.
2.8 EXERCISE. Verify this for the one-period model, replacing (2.1) by
the assumption that P(S1/S0 = x) > 0 for x ∈ {d, i, u} with probabilities
pd, pi and pu. Revisit (2.4), which becomes a system of three equations.
Why is there no solution if d < i < u?

3
Discrete Time
Stochastic Processes
3.1 Stochastic Processes
A stochastic process in discrete time is a (finite or infinite) sequence X =
(X0, X1, . . .) of random variables or vectors, defined on a given probability
space. Mathematically, random variables are maps Xn: Ω → R that map
outcomes ω ∈ Ω into numbers Xn(ω). The stochastic process X0, X1, . . . ,
maps every outcome ω into a sequence of numbers X0(ω), X1(ω), . . ., called
a sample path.
The best way to think of a stochastic process is to visualize the sample
paths as “random functions”. We generate an outcome ω according to some
probability measure on the set of all outcomes and next have a function
n → Xn(ω) on the domain N ∪ {0}. This domain is referred to as the set
of “discrete times”.
3.1 Example (Binomial tree model). The binomial tree model for the
stock price is a stochastic process S0, S1, . . . , SN , where each possible sam-
ple path is given by a path in the binomial tree, and the probability of a
sample path is the product of the probabilities on the branches along the
path.
As indicated before, the best way to think about this stochastic process
is as a random function, generated according to the probabilities attached
to the different paths on the tree. The preceding description gives an in-
tuitively clear description of the binomial tree process, but for later use it
is instructive to define the stochastic process also formally as a map on a
given outcome space. One possibility is to take Ω to be equal to the set
of N-tuples ω = (ω1, . . . , ωN ), where each ωi ∈ {0, 1}. The appropriate

3.2: Conditional Expectation 11
probability measure is
P({(ω1, . . . , ωN )}) = p#1≤i≤N:ωi=1
(1 − p)#1≤i≤N:ωi=0
,
and the stochastic process can be formally defined by setting S0 = s0 and,
for n = 1, 2, . . . , N,
Sn(ω1, . . . , ωN ) = S0u#1≤i≤n:ωi=1
d#1≤i≤n:ωi=0
.
Thus ωi = 1 indicates that the sample path goes up in the tree at time i,
whereas ωi = 0 indicates a down move. The value Sn at time n is determined
by the total number of moves up and down in the tree up till that time.
3.2 Conditional Expectation
For a discrete random variable X and a discrete random vector Y , the
conditional expectation of X given the event Y = y is given by
E(X| Y = y) =
x
xP(X = x| Y = y).
If we write this function of y as f(y) = E(X| Y = y), then we write E(X| Y )
for f(Y ). This is a random variable, called the conditional expectation of
X given Y . Some important rules are given in the following lemma.
3.2 Lemma.
(i) EE(X| Y ) = EX.
(ii) E(E(X| Y, Z)| Z) = E(X| Z) (tower property).
(iii) E(X| Y ) = X if X = f(Y ) for some function f.
(iv) E(X| Y ) = EX if X and Y are independent.
These rules can be proved from the definition, but are intuitively clear.
The first rule says that the expectation of a variable X can be computed
in two steps, first using the information on another variable Y , and next
taking the expectation of the result. Assertion (ii) gives exactly the same
property, with the difference that every of the expectations are computed
conditionally on a variable Z. Rule (iii) says that we can predict a variable
X exactly if X is a function of a known variable Y , which is obvious.

12 3: Discrete Time Stochastic Processes
3.3 EXERCISE. Suppose you generate N points in the interval [0, 1] as
follows. First you choose N from the Poisson distribution with mean 100.
Next given N = n you generate a random sample of n random variables
from a given distribution F on [0, 1]. What is the expected number of points
in an interval B ⊂ [0, 1]?
We shall use the notation E(X| Y ) also if X or Y are continuous ran-
dom variables or vectors. Then the preceding definition does not make sense,
because the probabilities P(X = x| Y = y) are not defined if P(Y = y) = 0,
which is the case for continuous random variable X. However, the condi-
tional expectation E(X| Y ) can still be defined as a function of Y , namely
as the function such that, for every function g,
E E(X| Y )g(Y ) = E Xg(Y ) .
The validity of this equality in the case of discrete random variables can
be checked in the same manner as the validity of the three rules in the
lemma. For general random variables X and Y we take this as a definition
of conditional expectation, where it is also understood that E(X| Y ) must
be a function of Y . The three rules of the lemma continue to hold for this
extended definition of conditional expectation.
In most cases this abstract definition agrees perfectly with your intu-
ition of the expected value of X given Y . However, in some cases where
there are many sets {Y = y}, all with probability zero, your intuition could
deceive you. The problem is then usually that there are several equally
“good”, but incompatible intuitions.
3.3 Filtration
A σ-field is a collection of events.†
A filtration in discrete time is an increas-
ing sequence F0 ⊂ F1 ⊂ · · · of σ-fields, one per time instant. The σ-field Fn
may be thought of as the events of which the occurrence is determined at
or before time n, the “known events” at time n. Filtrations are important
to us, because they allow to model the flow of information. Of course, the
information increases as time goes by.
The filtrations that are of interest to us are generated by stochastic
processes. The natural filtration of a stochastic process X0, X1, . . . is defined
by
Fn = {(X0, X1, . . . , Xn) ∈ B: B ⊂ Rn+1
}.
†
A rigorous mathematical definition includes the requirements (i) ∅ ∈ F; (ii) if F ∈ F,
then F c
∈ F; (iii) if F1, F2, . . . , ∈ F, then ∪iFi ∈ F. The Borel σ-field is the smallest σ-field
of subsets of Rn
that satisfies the properties and contains all intervals. In the following the
sets B should be required to be Borel sets.

3.4: Martingales 13
Thus Fn contains all events that depend on the first (n + 1) elements of
the stochastic process. It gives the “history” of the process up till time n.
A convenient notation to describe a σ-field corresponding to observing
a random vector X is σ(X). Thus σ(X), called the σ-field generated by
X, consists of all events that can be expressed in X: events of the type
{X ∈ B}. In this notation, the natural filtration of a stochastic process
X0, X1, . . . can be written as Fn = σ(X0, . . . , Xn).
We say that a process X0, X1, . . . is adapted to the filtration (Fn)n≥0 if
σ(Xn) ⊂ Fn for every n. Thus the events connected to an adapted process
up to time n are known at time n. The natural filtration corresponding
to a process is the smallest filtration to which it is adapted. If the pro-
cess Y0, Y1, . . . is adapted to the natural filtration of a stochastic process
X0, X1, . . ., then for each n the variable Yn is a function φn(X0, X1, . . . , Xn)
of the sample path of the process X up till time n.
We say that a process Y0, Y1, . . . is predictable relative to the filtra-
tion (Fn)n≥0 if σ(Yn) ⊂ Fn−1 for each n. Thus the events connected to
a predictable process are known one time instant before they happen.
If Fn is generated by (Xn), then this equivalent to Yn being a function
Yn = φn(X0, . . . , Xn−1) of the history of the process X0, X1, . . . up till time
n − 1, for some φn.
If F is the σ-field generated by Y , then we also write E(X| F) for the
random variable E(X| Y ). Thus E(X| F) is the expected value of X given
the information F. The trivial σ-field {∅, Ω} is the σ-field containing no
information.
3.4 Lemma.
(i) E(X| {∅, Ω}) = EX.
(ii) for two σ-fields F ⊆ G there holds E(E(X| G)| F) = E(X| F) (tower
property).
(iii) E(X| Y ) = X if σ(X) ⊂ σ(Y ).
3.4 Martingales
A stochastic process X0, X1, . . . is a martingale relative to a given filtration
(Fn)n≥0 if it is adapted to this filtration and E(Xn| Fm) = Xm for every
m < n.
The martingale property is equivalent to E(Xn − Xm| Fm) = 0 for
every m < n, expressing that the increment Xn − Xm given the “past”
Fm has expected value 0. A martingale is a stochastic process that, on the
average, given the past, does not grow or decrease.
3.5 Example (Random walk). Let X1, X2, . . . be independent random

variables with mean zero. Define S0 = 0 and Sn = X1 +· · ·+Xn for n ∈ N.
Then S is a martingale relative to its natural filtration. Indeed, S is adapted
by construction and E(Sn+1| Fn) = E(Sn +Xn+1| Fn) = Sn +EXn+1 = Sn,
since Sn ∈ Fn and Xn+1 is independent of Fn.
3.6 Example (Doob martingale). If Y is a random variable with E|Y | <
∞ and Fn an arbitrary filtration, then Xn = E(Y | Fn) defines a martingale.
This can be proved from the tower property of conditional expectations:
E E(Y | Fn)| Fm) = E(Y | Fm) for any m < n.
The martingale property can also be equivalently described through
one step ahead expectations. A process is a martingale if E(Xn+1| Fn) = Xn
for every n.
3.7 EXERCISE. Use the tower property to prove this.
3.8 EXERCISE. Let Z1, Z2, . . . be independent N(0, 1)-variables. Show
that the sequence Sn = exp(
n
i=1Zi − 1
2 n) forms a martingale.
3.9 EXERCISE. In a branching process we start with N0 = 1 individuals
at time 0, and at each time n each individual has a random number of
offspring independent of the other individuals, where the number is chosen
from a fixed distribution. The new generation consists of the offspring only.
Thus given that there are Nn = k individuals at time n the number of
individuals Nn+1 in the (n + 1)th generation is distributed as
k
i=1Xi for
i.i.d. random variables X1, . . . , Xk. Show that N0, N1, . . . , is a martingale
if and only if EXi = 1.
3.5 Change of Measure
If there are two possible probability measures P and Q on the set of out-
comes, and X is a martingale relative to Q, then typically it is not a mar-
tingale relative to P. This is because the martingale property involves the
expected values, and hence the probabilities of the various outcomes. An
important tool in finance is to change a given measure P into a measure Q
making a discounted asset process into a martingale.
3.10 Example. In the binomial tree model with Fn the natural filtration
of S0, S1, . . . and P(Sn+1 = uSn| Fn) = 1 − P(Sn+1 = dSn| Fn) = p, we
have that
E(Sn+1| Fn) = uSnp + dSn(1 − p) = Sn up + d(1 − p) .

3.6: Martingale Representation 15
This is equal to Sn if u, d and p satisfy the equation up + d(1 − p) = 1. For
instance, if u = 2 and d = 1/2, then this is true if and only if p = 1/3.
3.11 Example (Discounted stock). In the binomial tree model as in the
preceding example consider the discounted process S0, e−r
S1, e−2r
S2, . . ..
The one step ahead expectations are given by
E(e−(n+1)r
Sn+1| Fn) = ue−(n+1)r
Snp + de−(n+1)r
Sn(1 − p).
The discounted process is a martingale only if the right side is equal to
e−nr
Sn. This is the case only if
p =
er
− d
u − d
.
This value of p is contained in the unit interval and defines a probability
only if d ≤ er
≤ u. In that case the discounted process is a martingale.
3.6 Martingale Representation
In Example 3.11 we have seen that the process ˜S defined by ˜Sn = e−rn
Sn
in the binomial tree model is a martingale if the tree is equipped with the
probability
q =
er
− d
u − d
.
In this section we shall show that all other martingales in this setting can
be derived from ˜S in the sense that the increments ∆Mn = Mn − Mn−1
of any martingale M0, M1, . . . must be multiples φn∆ ˜Sn for a predictable
process φ = (φ0, φ1, . . .). In other words, the change ∆Mn of an arbitrary
martingale at time n−1 is proportional to the change in ˜S, with the propor-
tionality constant φn being a function of the preceding values ˜S0, . . . , ˜Sn−1
of the process ˜S. At time n−1 the only randomness to extend M0, . . . , Mn−1
into Mn is in the increment ∆ ˜Sn.
3.12 Theorem. If M is a martingle on the binomial tree model of Exam-
ple 3.1 with q = (er
− d)/(u − d) with filtration Fn = σ(S0, . . . , Sn), then
there exists a predictable process φ0, φ1, . . . such that, for every n ∈ N,
∆Mn = φn∆ ˜Sn.
Proof. Because M is adapted to the filtration generated by S0, S1, . . .,
for each n the variable Mn is a function of S0, . . . , Sn. Given Fn−1 the
values of S0, . . . , Sn−1 are fixed and hence Mn can assume only two possible

values, corresponding to a downward or upward move in the tree. By a
similar argument we see that the variable Mn−1 is fixed given Fn−1, and
hence ∆Mn has two possible values given Fn−1. If we fix S0, . . . , Sn−1,
then we can write ∆Mn = gn(Sn) for some function gn (which depends on
the fixed values of S0, . . . , Sn−1). Similarly we can write ∆ ˜Sn = fn(Sn).
The martingale properties of the processes M (by assumption) and ˜S (by
Example 3.11) give that Eq(∆Mn| Fn−1) = 0 = Eq(∆ ˜Sn| Fn−1), or
qgn(uSn−1) + (1 − q)gn(dSn−1) = 0,
qfn(uSn−1) + (1 − q)fn(dSn−1) = 0.
It follows from this that gn(uSn−1)/fn(uSn−1) = gn(dSn−1)/fn(dSn−1).
We can define φn as this common ratio.

4
Binomial Tree Model
Revisited
Suppose that the price St at time t is a stochastic process described by the
binomial tree model of Example 3.1, where it is assumed that the numbers
u and d are known. We choose the filtration equal to Fn = σ(S0, . . . , Sn),
so that the (only) information available at time n consists of observation of
the asset price process until that time.
In addition to the asset with price S we can save or borrow money at
a fixed rate of interest r. We assume that d ≤ er
≤ u. This is a reason-
able assumption, because if er
< d then the returns on the asset are with
certainty bigger than the return on the savings account, whereas if er
> u,
then the returns are with certainty smaller. Then the riskless savings ac-
count is never or always preferable over the risky asset, respectively, and a
reasonable portfolio will consist of only one type of investment.
We equip the branches of the binomial tree with the probability
q = (er
− d)/(u − d), rather than a possible real world probability based on
historical analysis. Example 3.11 shows that this gives the unique probabil-
ity measure on the tree that renders the discounted asset process ˜S0, ˜S1, . . .,
where ˜Sn = e−rn
Sn, into a martingale.
A claim is a nonnegative function of C = C(S0, . . . , SN ), where N is
the expiry time. Given a claim define the stochastic process
˜Vn = Eq(e−rN
C| Fn).
The index q on the expectation operator E indicates that we compute expec-
tations under the martingale measure q. In view of Example 3.6 the process
˜V is a martingale. Therefore, by Theorem 3.12 there exists a predictable
process φ such that
(4.1) ∆ ˜Vn = φn∆ ˜Sn, n = 1, . . . , N.
Given this process we define another process ψ by
(4.2) ψn = ˜Vn−1 − φn
˜Sn−1.

18 4: Binomial Tree Model Revisited
From the facts that φ is predictable and ˜V and ˜S are adapted, it follows
that the process ψ is predictable.
We now interpret (φn, ψn) as a portfolio at time n:
(i) φn is the number of assets held during the period (n − 1, n].
(ii) ψn is the number of units in the saving account during the period
(n − 1, n].
Because both processes are predictable, the portfolio (φn, ψn) can be cre-
ated at time n−1 based on information gathered up to time n−1, i.e. based
on observation of S0, S1, . . . , Sn−1. We shall think of the assets and savings
changing value (from Sn−1 to Sn and er(n−1)
to ern
) exactly at time n, and
of adapting our portfolio just after time n. Then the value of the portfolio
at time n is
Vn = φnSn + ψnern
.
Just after time n we change the content of the portfolio; the value of the
new portfolio is equal to
φn+1Sn + ψn+1ern
.
The following theorem shows that this amount is equal to Vn and hence
the new portfolio can be formed without additional money: the portfolio
process (φ, ψ) is self-financing.
Furthermore, the theorem shows that the value VN of the portfolio
is exactly equal to the value of the claim C at the expiry time N. As a
consequence, we should be indifferent to owning the contract with claim C
or the portfolio (φ0, ψ0) at time 0, and hence the “just price” of the contract
is the value V0 of the portfolio.
4.3 Theorem. The portfolio process (φ, ψ) defined by (4.1)-(4.2) is self-
financing. Furthermore, its value process V is nonnegative and satisfies
˜Vn = e−rn
Vn for every n. In particular VN = C with probability one.
Proof. The equation (4.2) that defines ψn+1 can be rewritten in the form
φn+1Sn + ψn+1ern
= ern ˜Vn, and ˜Vn = ˜Vn−1 + ∆ ˜Vn, where ∆ ˜Vn = φn∆ ˜Sn
by (4.1). Therefore,
φn+1Sn + ψn+1ern
− Vn = ern ˜Vn − Vn
= ern ˜Vn−1 + ern
φn∆ ˜Sn − (φnSn + ψnern
)
= ern ˜Vn−1 + ern
φn(∆ ˜Sn − ˜Sn) − ψnern
= ern ˜Vn−1 − ern
φn
˜Sn−1 − ψnern
.
The right side is zero by the definition of ψn in (4.2). Thus the portfolio is
self-financing, as claimed.
It also follows from these equations that ern ˜Vn − Vn = 0, whence
˜Vn = e−rn
Vn is the discounted value of the portfolio at time n, for every

4: Binomial Tree Model Revisited 19
n. Because ˜Vn = Eq(e−rN
C| Fn) is nonnegative, so is Vn. Furthermore, the
value of the portfolio at time N is
VN = erN ˜VN = erN
Eq(e−rN
C| FN ) = C,
since C is a function of S0, S1, . . . , SN , by assumption.
Because VN = C with certainty the portfolio (φ, ψ) is said to replicate
the claim C. The claim value C is a random variable that depends on
S0, . . . , SN . However, no matter which path the asset prices take in the
binomial tree, the portfolio always ends up having the same value as the
claim. This is achieved by reshuﬄing assets and savings at each time n,
based on the available information at that time, a strategy called hedging.
The fact that the value process of the claim is nonnegative ensures
that the portfolio management can be implemented in practice. If we have
suﬃcient funds to form the portfolio at time 0, then we never run into debt
when carrying out the hedging strategy.
We interpret the value of the portfolio at time 0, the amount of money
needed to create the portfolio (φ0, ψ0), as the “just price” of the claim at
time 0. This is
V0 = ˜V0 = Eq(e−rN
C| F0) = Eq(e−rN
C).
The second equality follows from the fact that ˜V is a martingale under q.
Note that the formula expresses the price in the claim C without interven-
tion of the portfolio processes. These were only a means to formulate the
economic argument.
4.4 Example (Forward). The claim of a forward with strike price K
is C = SN − K. The value at time 0 is equal to Eqe−rN
(SN − K) =
Eqe−rN
SN −e−rN
K. Because the process e−rn
Sn is a martingale under the
(martingale) measure q the expected value Ee−rn
Sn is constant in n. Hence
the value of the forward contract at time 0 is equal to Eqe−r0
S0 − e−rN
K.
The strike price K that makes the value equal to zero is K = erN
S0.
In Section 1.1 we obtained the same result by describing an explicit
hedging strategy.
4.5 Example (European call option). The claim of a European call op-
tion with strike price K is C = (SN −K)+
. The fair price at time 0 is equal
to Eqe−rN
(SN − K)+
. The variable SN is distributed as S0uXN
dN−XN
,
where XN is the number of upward moves in the tree. Because the variable
XN is binomially distributed with parameters N and q (under the mar-
tingale measure), it follows that the value of the option at time 0 is equal
to
e−rN
N
x=0
(S0ux
dN−x
− K)+ N
x
qx
(1 − q)N−x
.

This expression is somewhat complicated, but easy to evaluate on a com-
puter. An alternative method of computation is backwards induction, as
in Chapter 2. An approximation formula for large N is given in the next
section.
4.6 EXERCISE. Suppose that (φn, ψn) are predictable processes such that
the process V defined by Vn = φnSn + ψnern
satisfies Vn = φn+1Sn +
ψn+1ern
. (In other words (φ, ψ) defines a self-financing portfolio process
with value process V .) Show that the process ˜V defined by ˜Vn = e−rn
Vn
satisfies ∆ ˜Vn = φn∆ ˜Sn, and conclude that it is a martingale under q. [This
gives some motivation for the definitions used to prove the theorem.]
4.1 Towards Continuous Time
In the real world asset prices change almost continuously in time. The
binomial tree model can approximate this if the number of steps N is large.
Mathematically we can even compute limits as N → ∞, in the hope that
this gives a realistic model.
A limit exists only if we make special choices for the relative up and
down moves u and d. Unless u and d tend to 1 as the number of moves
N increases, the asset price will explode and our model does not tend to
a limit. We shall think of the N moves in the binomial tree taking place
in a fixed interval [0, T], at the times δ, 2δ, . . . , Nδ for δ = T/N. Then it
is reasonable to redefine the interest rate in one time instant as rδ, giving
a total interest of r over the interval [0, T]. We also assume that, for given
constants µ ∈ R and σ > 0,
d = eµδ−σ
√
δ
, u = eµδ+σ
√
δ
.
These definitions satisfy that they approach 1 as the length δ of a time
interval tends to zero. The exact definitions are somewhat special, but can
be motivated by the fact that the resulting model tends to continuous time
model considered later on.
The asset price at time N is equal to
SN = S0uXN
dN−XN
= S0 exp µT + σ
√
T
(2XN − N)
√
N
,
where XN is the number of times the stock price goes up in the time span
1, 2, . . . , N.
In a standard model for the stock market the jumps up and down have
equal probabilities. Then XN is binomially (N, 1
2 )-distributed and the “log

4.1: Towards Continuous Time 21
returns” satisfy
log
SN
S0
= µT + σ
√
T
XN − N/2
√
N/2
N(µT, σ2
T),
by the Central Limit theorem. Thus in the limit the log return at time T
is normally distributed with drift µT and variance σ2
T.
As we have seen the true distribution of the stock prices is irrelevant for
pricing the option. Rather we need to repeat the preceding calculation using
the martingale measure q = qN . Under this measure XN is binomially(N, q)
distributed, for
q =
erT/N
− eµT/N−σ
√
T/N
eµT/N+σ
√
T/N
− eµT/N−σ
√
T/N
= 1
2 − 1
2
T
N
µ + 1
2 σ2
− r
σ
+ O
1
N
,
by a Taylor expansion. Then qN (1 − qN ) → 1/4 and
log
SN
S0
= µT + σ
√
T
XN − NqN
√
N/2
−
√
T
µ + 1
2 σ2
− r
σ
+ O
1
√
N
N (r − 1
2 σ2
)T, σ2
T .
Thus, under the martingale measure, in the limit the stock at time T is log
normally distributed with drift (r − 1
2 σ2
)T and variance σ2
T.
Evaluating the (limiting) option price is now a matter of straightfor-
ward integration. For a claim C = C(SN ) with expiry time T the “fair
price” at time 0 is given by
e−rT
EqN
C(SN ) ≈ e−rT
EC( ¯ST ),
for log( ¯ST /S0) normally distributed with mean (r − 1
2 σ2
)T and variance
σ2
T.
4.7 Example (European call option). The (limiting) fair price of a Eu-
ropean call option with expriry time T and strike price K is the expectation
of e−rT
( ¯ST − K)+
, where log( ¯ST /S0) possesses the log normal distribution
with parameters (r − 1
2 σ2
)T and variance σ2
T. This can be computed to
be
S0Φ
log(S0/K) + (r + 1
2 σ2
)T
σ
√
T
− Ke−rT
Φ
log(S0/K) + (r − 1
2 σ2
)T
σ
√
T
.
This is the famous formula found by Black and Scholes in 1973 using a
continuous time model. We shall recover it later in a continuous time set-
up.

4.8 EXERCISE. Suppose that Z is standard normally distributed and S >
0, τ > 0, µ and K are constants. Show that, for z0 = τ−1
(log(K/S) − µ),
E(SeτZ+µ
− K)+
= Seµ+
1
2 τ2
Φ(τ − z0) − KΦ(−z0).

5
Continuous Time
Stochastic Processes
5.1 Stochastic Processes
A continuous-time stochastic process is an indexed collection of random
variables X = (Xt: t ≥ 0), defined on a given probability space. Thus every
Xt is a map Xt: Ω → R mapping outcomes ω ∈ Ω into numbers Xt(ω). The
functions t → Xt(ω) attached to the outcomes are called sample paths, and
the index t is referred to as “time”. The best way to think of a stochastic
process is to view it as a “random function” on the domain [0, ∞), with the
sample paths as its realizations.
For any finite set t1 < t2 < · · · < tk of time points the vector
(Xt1
, . . . , Xtk
) is an ordinary random vector in Rk
, and we can describe a
great deal of the process by describing the distributions of all such vectors.
On the other hand, qualitative properties such as continuity of differentia-
bility of a sample path depend on infinitely many time points.
5.2 Brownian Motion
Brownian motion is a special stochastic process, which is of much interest by
itself, but will also be used as a building block to construct other processes.
It can be thought of as the “standard normal” process. A Brownian motion
is often denoted by the letter W, after Wiener, who was among the first to
study Brownian motion in a mathematically rigorous way. The distribution
of Brownian motion is known as the “Wiener measure”.
A stochastic process W is a Brownian motion if

24 5: Continuous Time Stochastic Processes
(i) the increment Wt − Ws is normally distributed with mean 0 and vari-
ance t − s, for any 0 ≤ s < t.
(ii) the increment Wt − Ws is stochastically independent of (Wu: u ≤ s),
for any 0 ≤ s < t.
(iii) W0 = 0.
(iv) any sample path t → Wt(ω) is a continuous function.
It is certainly not clear from the deﬁnition that Brownian motion exists,
in the sense that there exists a probability space with random variables
Wt deﬁned on it that satisfy the requirements (i)-(iv). However, it is a
mathematical theorem that Brownian motion exists, and there are several
constructive ways of exhibiting one. We shall take the existence for granted.
Properties (i) and (ii) can be understood in the sense that, given the
sample path (Wu: u ≤ s) up to some point s, Brownian motion continues
from its “present value” Ws by adding independent (normal) variables. In
fact it can be shown that given (Wu: u ≤ s) the process t → Ws+t − Ws
is again a Brownian motion. Thus at every time instant Brownian motion
starts anew from its present location, independently of its past.
The properties of Brownian motion can be motivated by viewing Brow-
nian motion as the limit of the process in a binomial tree model, where
starting from S0 = 0 a process S1, S2, . . . is constructed by moving up or
down 1 in every step, each with probability 1/2, i.e.
Sn =
n
i=1
Xi,
for an i.i.d. sequence X1, X2, . . . with P(Xi = 1) = P(Xi = −1) = 1/2.
For a given N we could place the values of the process S0, S1, S2, . . . at
the time points 0, 1/N, 2/N, . . . and rescale the vertical axis so that the
resulting process remains stable. This leads to the process W(N)
given by
W
(N)
t =
1
√
N i:i≤tN
Xi.
By the Central Limit Theorem, as N → ∞, with denoting convergence
in distribution,
W
(N)
t − W(N)
s =
1
√
N i:sN<i≤tN
Xi N(0, t − s).
Furthermore, the variable on the left side is independent of the variables
Xi with index i not contained in the sum, and hence of W
(N)
s . Thus in the
limit as N → ∞ the processes W(N)
satisfy the properties (i)-(iii). It can
indeed be shown that the sequence W(N)
converges, in a suitable sense,
to a Brownian motion process. The main challenge in proving existence of
Brownian motion is the required continuity (iv) of the sample paths.

5.3: Filtrations 25
5.3 Filtrations
A filtration (Ft)t≥0 in continuous time is an increasing collection of σ-fields
indexed by [0, ∞). Thus Fs ⊂ Ft for every s < t. A stochastic process
X = (Xt: t ≥ 0) is adapted to a given filtration if σ(Xt) ⊂ Ft for every t.
In such case, all events concerning the sample paths of an adapted process
until time t are contained in Ft. The natural filtration of a stochastic process
X = (Xt: t ≥ 0) is
Ft = σ(Xs: s ≤ t).
This filtration corresponds exactly to observing the sample paths of X up
to time t, and is the smallest filtration to which X is adapted.
5.4 Martingales
A martingale in continuous time relative to a given filtration (Ft)t≥0 is an
adapted process X such that
E(Xt| Fs) = Xs, every s < t.
This property is equivalent to the increments Xt−Xs having expected value
0 given the past and present: E(Xt − Xs| Fs) = 0.
5.1 Example (Brownian motion). A Brownian motion W is a martingale
relative to its natural filtration. This follows since, the conditional expecta-
tion E(Wt − Ws| Fs) is equal to the unconditional expectation E(Wt − Ws)
by the independence property (ii), and the latter is zero by property (i).
5.2 EXERCISE. Let W be a Brownian motion and (Ft)t≥0 its natural
filtration. Show that the process W2
t − t is a martingale.
5.5 Generalized Brownian Motion
When working with a Brownian motion it is sometimes useful to include
more information into a filtration than given by observing the Brownian
sample paths. Given a filtration (Ft)t≥0 we replace property (ii) of a Brow-
nian motion by the alternative property
(ii’) W is adapted to (Ft)t≥0 and the increment Wt − Ws is stochastically
independent of Fs, for any 0 ≤ s < t.

By the requirement that W be adapted, the filtration (Ft)t≥0 is necessarily
larger than the natural filtration of W. Therefore, the property (ii’) requires
more than the corresponding (ii).
It can be checked that a “generalized Brownian motion” is still a mar-
tingale.
5.6 Variation
Brownian motion has strange sample paths. They are continuous by as-
sumption, but they are not differentiable. We can see this by studying the
variation of the sample paths.
Let 0 = tn
0 < tn
1 < · · · < tn
kn
be a sequence of partitions of partitions of
a given interval [0, t] such that the meshwidth maxi(tn
i −tn
i−1) tends to zero
as n → ∞. Then for a continuously differentiable function f: [0, t] → R we
have, as n → ∞,
kn
i=1
f(tn
i ) − f(tn
i−1) ≈
kn
i=1
f (tn
i−1) tn
i − tn
i−1 →
t
0
|f (s)| ds.
The left side of this equation is called the variation of f over the given
partition. The approximation can be shown to be correct in the sense that
the variation indeed converges to the integral on the right as the meshwidth
of the partition tends to zero. We conclude that the variation of a contin-
uously differentiable function is bounded if the meshwidth of the partition
decreases to zero. As a consequence the quadratic variation
kn
i=1
f(tn
i ) − f(tn
i−1)
2
≤ max
i
f(tn
i ) − f(tn
i−1)
kn
i=1
f(tn
i ) − f(tn
i−1)
tends to zero as n → ∞, since the maximum of the increments tends to
zero by the continuity of f and the variation is bounded.
The sample paths of Brownian motion do not possess this property. In
fact, the quadratic variation rather than the variation of the sample paths
of Brownian motion tends to a nontrivial limit.
This is true in a stochastic sense. It will be convenient to use the
notation L2
→ for “convergence in second mean”: a sequence of random
variables Xn is said to converge in second mean or converge in L2 to a
random variable X, notation Xn
L2
→ X, if
E(Xn − X)2
→ 0.
Because the second moment of a random variable is the sum of its variance
and the square of its expectation, convergence in L2 is equivalent to EXn →
EX and var(Xn − X) → 0.

5.6: Variation 27
5.3 Lemma. For any sequence of partitions of partitions 0 = tn
0 < tn
1 <
· · · < tn
kn
of the interval [0, t] with maxi(tn
i − tn
i−1) → 0 we have
kn
i=1
Wtn
i
− Wtn
i−1
2 L2
→ t.
Proof. The increments Wtn
i
− Wtn
i−1
of Brownian motion over the parti-
tion are independent random variables with N(0, tn
i − tn
i−1)-distributions.
Therefore,
E
kn
i=1
Wtn
i
− Wtn
i−1
2
=
kn
i=1
(tn
i − tn
i−1) = t,
var
kn
i=1
Wtn
i
− Wtn
i−1
2
=
kn
i=1
var |Wtn
i
− Wtn
i−1
|2
=
kn
i=1
2(tn
i − tn
i−1)2
.
Here we use that the variance of the square Z2
of a N(0, σ2
)-distributed
random variable (i.e. σ2
times a χ2
1-variable) is equal to 2σ4
. The expression
in the last line of the display goes to 0, because it is the quadratic variation
of the identity function.
Because the quadratic variation of a continuously differentiable func-
tion tends to zero as the meshwidth of the partition tends to zero, the sam-
ple paths of Brownian motion cannot be continuously differentiable. Other-
wise the limit in the lemma would have been 0, rather than t. This expresses
that the sample paths of Brownian motion possess a certain roughness. This
is nice if we want to use Brownian motion as a model for irregular processes,
such as the Brownian motion of particles in a fluid or gas, or a financial
process, but it complicates the use of ordinary calculus in connection to
Brownian motion.
For instance, if W were the price of an asset and we would have φti−1
assets in a our portfolio during the time interval (ti−1, ti], then our increase
in wealth due to changes in value of the asset during the full interval (0, t]
would be
i
φti−1
(Wti
− Wti−1
) =
i
φti−1
∆Wti
.
We would like to extend this to portfolios φt that change continuously in
time, and thus would like to be able to talk about something like
φt dWt.

For a continuously differentiable function f we would interpret φ(t) df(t)
as φ(t) f (t) dt. Because the sample paths of Brownian motion do not have
derivatives, we cannot define φt dWt in this way. “Stochastic integrals”
provide a way around this.
5.7 Stochastic Integrals
Let W be a Brownian motion relative to a given filtration (Ft)t≥0. We
define an integral
t
0
Xs dWs for given stochastic processes X defined on
the same probability space as W in steps:
(a) If Xt = 1(u,v](t)A for a random variable A ∈ Fu, then Xs dWs is the
random variable Wv − Wu A.
(b) If X = i X(i)
, then Xs dWs = i X
(i)
s dWs.
(c) If E (X
(n)
s − Xs)2
ds → 0 for some sequence X(n)
, then Xs dWs is
the L2-limit of the sequence X
(n)
s dWs.
(d)
t
0
Xs dWs = (1(0,t]Xs) dWs.
The following lemma shows that the integral
t
0
Xs dWs can be defined by
this procedure for any adapted process X with E
t
0
X2
s ds < ∞. If this
integral is finite for all t > 0, then we obtain a stochastic process denoted
by X · W and given by X · W = (
t
0
Xs dWs: t ≥ 0).
5.4 Theorem. Let X be adapted and satisfy E
t
0
X2
s ds < ∞ for every
t ≥ 0. Then
t
0
Xs dWs can be defined through steps (a)–(d) and
(i) E
t
0
Xs dWs = 0.
(ii) E
t
0
Xs dWs
2
= E
t
0
X2
s ds.
(iii) the process (
t
0
Xs dWs: t ≥ 0) is a martingale.
A sketch of the proof of the theorem is given in Section 5.15. The
interested reader is also referred to e.g. the book by Chung and Williams.
The intuition behind assertion (iii) is that the increments Xsi ∆Wsi of the
integral
t
0
Xs dWs satisfy
E Xsi
∆Wsi
| Fsi
= Xsi
E(∆Wsi
| Fsi
) = 0,
because the increments of Brownian motion have mean zero and are inde-
pendent of the past. This reasoning is insightful, perhaps more so than the
proof of the lemma. As a mathematical justification it is wrong, because the
integral is a much more complicated object than a sum of (infinitesimal)
increments.

5.8: Geometric Brownian Motion 29
5.8 Geometric Brownian Motion
The state Wt of a Brownian motion at time t is normally distributed with
mean zero and hence is negative with probability 1/2. This is an embarass-
ing property for a model of an asset price. One way out of this difficulty
would be to model the asset prices as the sum f(t) + Wt of a deterministic
function and Brownian motion. Brownian motion with a linear drift, the
process α+βt+Wt, is a special example. This type of model uses Brownian
motion as a noisy aberration of a deterministic asset price f(t). If the de-
terministic function satisfies f(t) 0, then the probability that f(t) + Wt
is negative is very small, but still positive.
Another way out is to model the asset price as a geometric Brownian
motion, which is given by
eσWt+α+βt
.
Putting the process in the exponential certainly keeps it positive.
5.9 Stochastic Differential Equations
A more general approach to modelling using Brownian motion is in terms of
differential equations. An asset price process could be postulated to satisfy,
for given stochastic processes µ and σ,
(5.5) St = S0 +
t
0
µs ds +
t
0
σs dWs.
This integral equation is usually written in differential form as
(5.6) dSt = µt dt + σt dWt.
In the case that σt = 0, this reduces to the ordinary differential equation
dSt = µt dt. Adding the term σt dWt introduces a random perturbation of
this differential equation. The infinitesimal change dSt in St is equal to µt dt
plus a noise term. Because the increments of Brownian motion are indepen-
dent, we interpret the elements dWt as “independent noise variables”.
The integral
t
0
σs dWs in (5.5) must of course be interpreted as a
stochastic integral in the sense of Section 5.7, whereas the integral
t
0
µs ds
is an ordinary integral, as in calculus. The stochastic differential equation
(5.6), or SDE, is merely another way of writing (5.5), the latter integral
equation being its only mathematical interpretation. The understanding of
dWt as a random noise variable is helpful for intuition, but does not make
mathematical sense.
For the integral
t
0
σs dWs to be well defined, the process σ must be
adapted.

In many examples the process µt and σt are defined in terms of the
process S. For instance, a diffusion equation takes the form
(5.7) dSt = µ(t, St) dt + σ(t, St) dWt,
for given functions µ and σ on [0, ∞) × R. Then the stochastic differential
equation is recursive and the process St is only implicitly defined, and in
fact there is no guarantee that it exists. Just as for ordinary differential
equations, existence of solutions for stochastic differential equations is an
important subject of study. There are several general theorems that guar-
antee the existence of solutions under certain conditions, but we omit a
discussion.
5.10 Markov Processes
A Markov Process X is a stochastic process with the property that for
every s < t the conditional distribution of Xt given (Xu: u ≤ s) is the same
as the conditional distribution of Xt given Xs. In other words, given the
“present” Xs the “past” (Xu: u ≤ s) gives no additional information about
the “future” Xt.
5.8 Example (Brownian Motion). Because Wt = Wt − Ws + Ws and
Wt − Ws is normal N(0, t − s) distributed and independent of (Wu: u ≤ s),
the conditional distribution of Wt given (Wu: u ≤ s) is normal N(Ws, t−s)
and hence depends on Ws only. Therefore, Brownian motion is a Markov
process.
5.9 Example (Diffusions). A diffusion process S, as given by the SDE
(5.7), does not possess independent increments as Brownian motion. How-
ever, in an infinitesimal sense the increments dSt depend only on St and
the infinitesimal increment dWt, which is independent of the past. This in-
tuitive understanding of the evolution suggests that a diffusion process may
be Markovian. This is indeed the case, under some technical conditions.
5.11 Quadratic variation - revisited
In Section 5.6 we have seen that the quadratic variation of Brownian motion
converges to a limit as the meshwidth of the partitions tends to zero. This
is true for general solutions to SDEs, except that in general the convergence

5.11: Quadratic variation - revisited 31
must be interpreted “in probability”. We say that a sequence of random
variables Xn converges in probability to a random variable X if
P |Xn − X| > ε → 0, for every ε > 0.
This is denoted by Xn
P
→ X.
5.10 Lemma. Consider a stochastic process S that satisﬁes the SDE (5.6)
for adapted processes µ and σ, then for any sequence of partitions 0 = tn
0 <
tn
1 < · · · < tn
kn
of the interval [0, t] with maxi(tn
i − tn
kn
i=1
Stn
i
− Stn
i−1
2 P
→
t
0
σ2
s ds.
Proof. (Sketch.) The increments can be written
Stn
i
− Stn
i−1
=
tn
i
tn
i−1
µs ds +
tn
i
tn
i−1
σs dWs.
The ﬁrst term on the right is an ordinary integral and gives no contribution
to the quadratic variation, since the sum of its squares is of the order
i(tn
i −tn
i−1)2
→ 0. The second term is approximately σtn
i−1
(Wtn
i
−Wtn
i−1
).
Now
E
kn
i=1
σ2
tn
i−1
(Wtn
i
− Wtn
i−1
)2
− (tn
i − tn
i−1) = 0,
var
kn
i=1
σ2
tn
i−1
(Wtn
i
− Wtn
i−1
)2
− (tn
i − tn
i−1)
=
kn
i=1
var σ2
tn
i−1
(Wtn
i
− Wtn
i−1
)2
− (tn
i − tn
i−1)
=
kn
i=1
Eσ4
tn
i−1
E (Wtn
i
− Wtn
i−1
)2
− (tn
i − tn
i−1)
2
=
kn
i=1
Eσ4
tn
i−1
(tn
i − tn
i−1)2
.
Here we used the independence of the increments of Brownian motion from
the past several times, for instance to see that the terms of the sum are
uncorrelated. Together these equations suggest that
E
kn
i=1
σ2
tn
i−1
(Wtn
i
− Wtn
i−1
)2
− (tn
i − tn
i−1)
2
→ 0.

Combined with the convergence
kn
i=1
σ2
tn
i−1
(tn
i − tn
i−1) →
t
0
σ2
s ds
this would give the result.
This proof can be made precise without much difficulty if the process
σ is bounded and left-continuous. For a complete proof we need to use a
truncation argument involving stopping times.
The limit of the sums of squares
kn
i=1 Stn
i
− Stn
i−1
2
is called the
quadratic variation of the process S. It is denoted by [S]t, and also known
as the “square bracket process”. For a solution S to the SDE (5.6) we have
[S]t =
t
0
σ2
s ds.
Besides the quadratic variation of a single process, there is also a cross
quadratic variation of a pair of processes R and S, defined as the limit (in
probability)
[R, S]t = lim
n→∞
kn
i=1
Rtn
i
− Rtn
i−1
Stn
i
− Stn
i−1
.
5.11 EXERCISE. Suppose that the processes R and S both satisfy an SDE
(5.6), but with different functions µ and σ. Guess [R, S] if
(i) R and S depend in (5.6) on the same Brownian motion.
(ii) the SDEs for R and S are driven by independent Brownian motions.
5.12 Itô Formula
The geometric Brownian motion is actually a special case of the SDE ap-
proach. By a celebrated formula of Itô it can be shown that geometric
Brownian motion satisfies a SDE.
Itô’s formula is a chain rule for stochastic processes, but due to the
special nature of stochastic integrals it takes a surprising form. The version
of Itô’s formula we present here says that a transformation f(St) of a process
that satisfies an SDE by a smooth function f again satisfies an SDE, and
gives an explicit expression for it.
Recall that for a stochastic process S as in (5.6), the quadratic variation
is the process [S] such that d[S]t = σ2
t dt.

5.12: Itô Formula 33
5.12 Theorem (Itô’s formula). If the stochastic process S satisfies the
SDE (5.6) and f: R → R is twice continuouly differentiable, then
df(St) = f (St) dSt + 1
2 f (St) d[S]t.
Proof. (Sketch.) For a sequence of sufficiently fine partitions 0 = tn
0 < tn
1 <
· · · < tn
kn
= t of the interval [0, t] with maxi(tn
i − tn
f(St) − f(S0) =
kn
i=1
f(Stn
i
) − f(Stn
i−1
)
≈
kn
i=1
f (Stn
i−1
)(Stn
i
) − Stn
i−1
) + 1
2
kn
i=1
f (Stn
i−1
)(Stn
i
− Stn
i−1
)2
.
The first term on the far right tends to the stochastic integral
t
0
f (Ss) dSs.
By the same arguments as used in Section 5.11 the second sum tends to
1
2
t
0
f (Ss)σ2
s ds.
We have written Itô’s formula in differential form, but as usually it
should be mathematically interpreted as a statement about integrals.
The striking aspect of Itô’s formula is the second term 1
2 f (St) d[S]t,
which would not appear if the sample path t → St were a differentiable
function. As the proof shows it does appear, because the variation of the
sample paths of S is not finite, whereas the quadratic variation tends to a
nontrivial limit.
5.13 Example. Brownian motion W itself certainly satisfies a stochastic
differential equation: the trivial one dWt = dWt.
Applied with the function f(x) = x2
Itô’s formula gives dW2
t =
2Wt dWt+ 1
2 2 dt, because [W]t = t. We conclude that W2
t = 2
t
0
Ws dWs+t.
Compare this to the formula f2
(t) = 2
t
0
f(s) df(s) for a continuously dif-
ferentiable function f with f(0) = 0.
5.14 Example (Geometric Brownian motion). As a consequence of Itô’s
formula, the geometric Brownian motion St = exp(σWt + α + βt) satisfies
the SDE
dSt = (β + 1
2 σ2
)St dt + σSt dWt.
To see this, apply Itô’s formula with the process Xt = α + βt + σWt and
the function f(x) = exp(x).
Itô’s theorem is also valid for functions of more than one process. For
instance, consider a process f(t, St) for a function f: [0, ∞) × R → R of
two arguments. Write ft, fs and fss for the partial derivatives ∂/∂tf(t, s),
∂/∂sf(t, s) and ∂2
/∂s2
f(t, s), respectively.

5.15 Theorem (Itô’s formula). If the stochastic process S satisfies the
SDE (5.6) and f: [0, ∞) × R → R is twice continuouly differentiable, then
df(t, St) = ft(t, St) dt + fs(t, St) dSt + 1
2 fss(t, St) d[S]t.
As a second example consider a process f(Rt, St) of two stochastic
processes R and S. If an index r or s denotes partial differentiation with
respect to r or s, then we obtain the following formula.
5.16 Theorem (Itô’s formula). If the stochastic processes R and S satisfy
the SDE (5.6) and f: R2
→ R is twice continuouly differentiable, then
df(Rt, St) = fr(Rt, St) dRt + fs(Rt, St) dSt + 1
2 frr(Rt, St) d[R]t
+ 1
2 fss(t, St) d[S]t + frs(Rt, St) d[R, S]t.
5.13 Girsanov’s Theorem
The stochastic integral
t
0
Xs dWs of an adapted process relative to Brow-
nian motion is a (local) martingale. Thus the solution S to the SDE (5.6)
is the sum of a local martingale
t
0
σs dWs and the process At =
t
0
µs ds.
The sample paths of the process A are the primitive functions, in the sense
of ordinary calculus, of the sample paths of the process µ, and are therefore
differentiable. They are referred to as “drift functions”. The presence of a
drift function destroys the martingale property: a solution of an SDE can
be a martingale only if the drift is zero.
5.17 Lemma. The process S defined by St =
t
0
µs ds +
t
0
σs dWs is a
local martingale if and only if µ = 0.
Proof. (Sketch.) If S is a local martingale, then so is the process At =
t
0
µs ds, because the process
t
0
σs dWs is a local martingale and the dif-
ference of two local martingales is a local martingale. Because the sample
paths of A are differentiable functions, the rules of ordinary calculus apply,
and yield that d(A2
t ) = 2At dAt, or
A2
t =
t
0
2As dAs.
The local martingale property of A carries over to every process of the
form
t
0
Xs dAs for an adapted process X. This can be proved by first
considering simple adapted processes, and next limits, along the same lines
as the martingale property of stochastic integrals was proved. In particular,

5.14: Brownian Representation 35
we may choose X = 2A, and we see that the process A2
is a local martingale.
If it is a martingale with finite second moments, then we can conclude that
EA2
t = EA2
0 = 0, whence A = 0. The general case can be handled by a
stopping time argument.
The martingale property refers to the underlying probability distribu-
tion on the outcome space. Therefore a process may well be a martingale
relative to a probability measure Q, whereas it is not a martingale if the
outcome space is equipped with another probability measure P. If the pro-
cess is given by an SDE under P, then this somehow means that the drift of
the process can be made to “disappear” by changing the probability distri-
bution on the space of outcomes. This observation turns out to be crucial
in the pricing theory.
It will be sufficient to consider this for the case that σ = 1, i.e. St =
Wt +
t
0
µs ds. If W is a Brownian motion under the probability measure P,
then W is a martingale under P and hence S cannot be a martingale under
P, (unless µ = 0). Girsanov’s theorem shows that for “most” processes µ,
there exists another probability measure Q such that S is a martingale, and
even a Brownian motion, under Q.
5.18 Theorem (Girsanov). If (Wt: 0 ≤ t ≤ T) is a Brownian mo-
tion under the probability measure P and µ is an adapted process with
E exp(1
2
T
0
µ2
s ds) < ∞, then there exists a probability measure Q such
that the process (Wt +
t
0
µs ds: 0 ≤ t ≤ T) is a Brownian motion under Q.
There is even a constructive formula for finding the “martingale mea-
sure” Q from P, given by
Q(A) = E 1Ae
−
T
0
µs dWs−
1
2
T
0
µ2
s ds
,
where the expectation on the right is computed under the probability mea-
sure P. The condition E exp(1
2
T
0
µ2
s ds) < ∞ ensures that the formula in
the preceding display indeed defines a probability measure Q. If the pro-
cess µ is bounded (e.g. constant), then the condition is clearly satisfied. In
general, the condition says that µ “should not grow too big”.
5.14 Brownian Representation
Let (Ft)t≥0 be the natural filtration of a given Brownian motion W.
Stochastic processes defined on the same outcome space that are martin-
gales relative to this “Brownian filtration” are referred to as Brownian

martingales. Brownian motion itself is an example, and so are all stochastic
integrals X · B for adapted processes X.
The following theorem shows that these are the only Brownian mar-
tingales.
5.19 Theorem. Let {Ft} be the (completion of the) natural filtration of a
Brownian motion process W. If M is a (cadlag) local martingale relative to
{Ft}, then there exists a predictable process X with
t
0
X2
s ds < ∞ almost
surely for every t ≥ 0 such that Mt = M0 +
t
0
Xs dWs.
This Brownian representation theorem remains true if the filtra-
tion is generated by multiple, independent Brownian motion processes
W(1)
, W(2)
, . . . , W(d)
. Then an arbitrary (cadlag) local martingale can be
written as Mt = M0 +
d
i=1
t
0
X
(i)
s dW
(i)
s .
* 5.15 Proof of Theorem 5.4
In this section we provide details for the construction of the stochastic
integral in Section 5.7. Because this material is mathematically quite in-
volved, we do not give a full proof of Theorem 5.4, but we indicate the most
essential steps.
A process of the type Xt = 1(u,v](t)A for a random variable A ∈ Fu as
in (a) is adapted and hence so is process of the type
Xt =
i
1(ui,vi](t)Aui
, ui < vi, Aui
∈ Fui
.
A process of this type is called simple adapted. By splitting up sets if neces-
sary, it is always possible to represent such a simple adapted process with
disjoint intervals (ui, vi]. For X as in the preceding display we define
(5.20) Xs dWs =
i
Aui (Wvi − Wui ).
Because the representation of X in terms of the intervals (ui, vi] and Aui
is not unique (we could for instance split up the intervals further), it must
be verified that this definition is consistent, but we omit this part of the
proof.
We next verify property (ii) for simple adapted processes X and
t = ∞. If, as we assume, the intervals (ui, vi] are disjoint, then X2
t =
i 1(ui,vi](t)A2
ui
. Therefore, the right side of (ii) with t = ∞ is equal to
E X2
s ds = E
i
1(ui,vi](t)A2
ui
ds =
i
EA2
ui
(vi − ui).

5.15: Proof of Theorem 5.4 37
The left side of (ii) with t = ∞ is given by, in view of (5.20),
E Xs dWs
2
= E
i j
Aui Auj (Wvi − Wui )(Wvj − Wuj ).
Because the intervals (ui, vi] are disjoint, we have that E(Wvi
−Wui
)(Wvj
−
Wuj
) = 0 for i = j, by the independence and the zero means of the incre-
ments of Brownian motion. It follows that the diagonal terms in the double
sum vanish, whence the preceding display is equal to
E
i
A2
ui
(Wvi
− Wui
)2
=
i
EA2
ui
E(Wvi
− Wui
)2
=
i
A2
ui
(vi − ui),
where in the second step we use the independence of the increment Wvi −
Wui
of Fui
and hence of Aui
. Thus we have verified that for simple adapted
processes X
E Xs dWs
2
= E X2
s ds.
In words we have shown that the integral is a “linear isometry”. A linear
isometry between normed spaces X and Y is a linear map I: X → Y such that
I(x) Y = x X for every x ∈ X. This isometry is the basis for the extension
of the integral to general adapted processes, by way of the following result
from analysis.
Any linear isometry I: X0 ⊂ X → Y from a linear subspace X0 of
a normed space X into a complete normed space Y possesses a unique
extension to an isometry defined on the closure ¯X0 = X ∈ X: ∃{Xn} ⊂
X0 with Xn − X → 0 of X0 in X.
In our situation we take the space X equal to all adapted processes
X with X 2
X = E
t
0
X2
s ds < ∞, and X0 equal to the collection of all
simple adapted processes. We have seen that the map I: X → Xs dWs is
an isometry into the set Y of random variables with finite second moments,
with Y 2
Y = EY 2
. Thus the integral can be extended to the closure of the
set of simple adapted processes. That this closure is the set of all adapted
processes with E
t
0
X2
s ds < ∞ can be shown by approximation by step
functions. We omit the details of this part of the proof.
Thus the integral is defined. The verification of its properties (i)-(iii)
proceeds by first verifying that these assertions hold on the set X0 of simple
processes and next showing that these properties are preserved under taking
limits.
For a simple adapted process of the form Xt = 1(u,v](t)A with σ(A) ⊂
Fu and s < t we have
t
0
Xr dWr −
s
0
Xr dWr =
A(Wt∧v − Ws∨u), if t ∧ v > s ∨ u,
0, otherwise.

Because A is known at time u ≤ s ∨ u and Brownian motion is a martin-
gale we have E(A(Wt∧v − Ws∨u)| Fs∨u) = AE(Wt∧v − Ws∨u| Fs∨u) = 0.
Therefore, with the help of the tower property of conditional expectation,
it follows that
E
t
0
Xr dWr −
s
0
Xr dWr| Fs = 0.
Thus the stochastic integral
t
0
Xs dWs is a martingale for X of this form.
Because the sum of two martingales is again a martingale, this conclusion
extends to all simple adapted processes. Because the martingale property is
preserved under taking L2-limits, it next extends to the stochastic integral
in general.
* 5.16 Stopping
Stopping times are intuitively meaningful objects that have interest on
their own, and are also essential for extension of the definition of stochastic
integrals, as given in the next section. However, we shall not need the
material in this and the following section in later chapters.
A stopping time relative to a filtration (Ft)t≥0 is a random variable T
with values in [0, ∞] such that {T ≤ t} ∈ Ft for every t ≥ 0. A stopping
time formalizes a strategy to play (or invest) on the market until a given
time, which need not be predetermined, but may be based on observing
the market. The requirement that the event {T ≤ t} is known at time t
says that the decision to stop trading must be made based on information
collected in past and present. If the filtration is generated by a process X,
then this requirement implies that the decision to stop at time t must be
based on the sample path of X until t.
5.21 Example (Hitting time). If X is an adapted process with continuous
sample paths, then T = inf{t ≥ 0: Xt ∈ B} is a stopping time for every
(Borel) set B. This is known as the hitting time of B.
Stopping times are important tools in the theory of stochastic pro-
cesses, but are also crucial to evaluate American options. These are con-
tracts that give the holder the right to collect a certain payment at a time
t in a given interval [0, T] of his own choosing. The amount of the payment
depends on the history of an asset price up to the time of payment. For
instance, an American call option on an asset with price process S gives
the right to buy the asset at a predetermined price K at any time t in an
interval [0, T]. This corresponds to a payment of (St − K)+
at the chosen

5.17: Extended Stochastic Integrals 39
time t. The financial problem is to determine an optimal stopping time for
the payment, and to evaluate the value of the resulting contract.
Given a stopping time T and a stochastic process the stopped process
XT
is defined as the stochastic process such that
(XT
)t = XT ∧t.
The sample paths of the stopped process are identical to the sample paths
X up to time T and take the constant value XT for t ≥ T.
5.22 Theorem. If X is a martingale, then so is XT
.
More explicitly, the theorem says that, if X is a martingale, then
E(XT ∧t| Fs) = XT ∧s, s < t.
In particular, we have EXT ∧t = EXT ∧s = EX0, because we can choose
s = 0. If T is a bounded stopping time, then we may choose t ≥ T and we
find
EXT = EX0.
This says that stopping does not help if the pay-off process X is a martin-
gale. No matter how clever the stopping strategy T, the expected pay-off
EXT is EX0.
5.23 Example. The process W2
t − t is a martingale. It can be shown that
T = inf{t ≥ 0: |Wt| = a} is finite almost surely, whence W2
T = a2
. The
identity E(W2
T − T) = E(W2
0 − 0) = 0 reduces to ET = a2
.
However, it is not permitted to apply this identity directly, as T is
not a bounded stopping time. A way around this is to apply the identity
with T ∧ n for a given n and next take limits as n → ∞. Because T ∧ n
is bounded we find EW2
T ∧n = E(T ∧ n). Because W2
T ∧n ≤ a2
, we have
EW2
T ∧n → EW2
T = a as n → ∞ by the dominated convergence theorem.
Also we have ET ∧ n ↑ ET by the monotone convergence theorem. Thus
the formula ET = a2
is correct.
5.24 EXERCISE. For given a > 0, let T = inf{t ≥ 0: Wt = a}.
(i) Show that Yt = eθWt−
1
2 θ2
t
is a martingale, for every θ ∈ R.
(ii) Show that E exp(−θT) = exp(−
√
2θa).
(iii) Show that ET = ∞.

* 5.17 Extended Stochastic Integrals
Using stopping times we can define a useful extension of the definition
of the stochastic integral. We have already defined the stochastic integral
t
0
Xs dWs for any adapted process X with E
t
0
X2
s ds < ∞. We shall now
extend this to all adapted processes X with
(5.25) X2
s ds < ∞, a.s..
This is a larger set of adapted processes, as finiteness of the expected value
of a positive random variable implies finiteness of the variable with proba-
bility one, but not the other way around.
We “truncate” a given adapted process by stopping it appropriately.
For a given n we define the stopping time
Tn = inf{t ≥ 0:
t
0
X2
s ds ≥ n}.
The finiteness (5.25) of the (nondecreasing) process
t
0
X2
s ds implies that
Tn ↑ ∞ as n → ∞. From the definition of Tn it follows immedi-
ately that
t
0
X2
s ds ≤ n if t ≤ Tn. Consequently E
t
0
(Xs1s≤Tn )2
ds =
E
t∧Tn
0
X2
s ds ≤ En = n < ∞. We can therefore define, for every n and t,
t
0
(Xs1s≤Tn ) dWs.
We define
t
0
Xs dWs as the limit of these variables, in the almost sure sense,
as n → ∞. It can be shown that this limit indeed exists.
Each of the processes in the preceding display is a martingale. The
stochastic integral Yt =
t
0
Xs Ws is the limit of these martingales, but need
not be a martingale itself. (The limit is only in an almost sure sense, and
this is not strong enough to preserve the martingale property.). However,
the stopped process Y Tn
is exactly the the integral
t
0
(Xs1s≤Tn
) dWs and
hence is a martingale. This has gained the stochastic integral
t
0
Xs dWs
the name of being a local martingale.

6
Black-Scholes Model
In this chapter we assume that we can trade continuously in a (riskless)
bond and some risky asset, for instance a stock. We assume that the bond-
price B evolves as
Bt = ert
,
where r is the riskless interest rate. The price process S of the risky asset
is assumed to be a geometric Brownian motion, i.e.
St = S0eµt+σWt
.
Here W is a Brownian motion, µ ∈ R is called the drift of the process, and
σ the volatility. We denote by (Ft) the filtration generated by the price
process S. Observe that (Ft) is also the natural filtration of the Brownian
motion W, since both processes generate the same flow of information.
For some fixed T > 0, let C ∈ FT be a non-negative random variable
whose value is determined by the information up till time T. We think of
C as the pay-off at time T of some contingent claim. For technical reasons,
we assume that EC2
< ∞. We want to answer the same question as in the
discrete-time setup: What is the fair price of the claim C at time zero?
To answer this question we follow the same route as in Chapter 4.
We first use Girsanov’s theorem to change the underlying probability mea-
sure in such a way that the discounted asset price ˜St = e−rt
St becomes a
martingale under the new measure Q. Then we consider the Q-martingale
˜Vt = EQ(e−rT
C| Ft) and use the representation theorem to write it as an
integral of a predictable process φ with respect to ˜S. This leads to the con-
struction of a self-financing trading strategy that replicates the pay-off C.
By an arbitrage argument, the value of this trading portfolio at time zero
must be the fair price of the claim. As in the binomial model, the fair price
will turn out to be EQe−rT
C, i.e. the expectation under the martingale
measure of the discounted pay-off.

42 6: Black-Scholes Model
6.1 Portfolios
Before we can carry out the programme outlined in the preceding section
we have to give a mathematically precise definition of a self-financing port-
folio in the present continuous-time setting. A portfolio is just a pair of
predictable processes (φt, ψt). We interpret φt as the number of risky assets
held at time t, and ψt as the number of bonds. Predictability roughly means
that to determine the positions φt and ψt, only the information available
before time t is used. For technical reasons we assume that almost surely,
T
0
|φt|2
dt +
T
0
|ψt| dt < ∞.
With the portfolio (φ, ψ) we associate the value process V defined by
Vt = φtSt + ψtBt.
For hedging strategies we need the notion of a self-financing portfolio.
Such a portfolio is created using some starting capital at time zero, and af-
ter time zero the portfolio is only changed by rebalancing, i.e. by replacing
bonds by the risky asset or vice versa. No additional injections or with-
drawals of money are allowed. Loosely speaking, such a portfolio has the
property that in an infinitesimally small time interval [t, t+dt], the changes
in the portfolio value are only caused by changes in the price processes S
and B, and not by changes in φt and ψt which are due to injections or
withdrawals of money. Therefore, we call a portfolio (φ, ψ) self-financing if
its price process V satisfies the SDE
dVt = φt dSt + ψt dBt.
A replicating, or hedging portfolio for the claim C is a self-financing portfolio
(φ, ψ) with a value process V which satisfies VT = C. If such a portfolio
exists, then an arbitrage argument shows that the “fair price” of the claim
at time t ∈ [0, T] equals the value Vt of the portfolio.
Of course, the arbitrage argument is an economic one, and not a math-
ematical argument. When we use the phrase “fair price” in mathematical
theorems below, the “fair price” or “value” will always be understood to
be defined as the value process of a replicating portfolio. (We shall be a
bit careless about the still open trap that there may be more than one
replicating portfolios, with different value processes.)
6.2 The Fair Price of a Derivative
Let us now derive the pricing formula for the derivative C announced in
the first section.

6.2: The Fair Price of a Derivative 43
For the discounted asset price we have ˜St = e−rt
St = S0 exp((µ −
r)t + σWt), whence it is a geometric Brownian motion with drift µ − r and
volatility σ. By Example 5.14 it satisfies the SDE
d ˜St = (µ − r + 1
2 σ2
) ˜St dt + σ ˜St dWt.
If we define ˜Wt = Wt + t(µ − r + 1
2 σ2
)/σ this simplifies to
(6.1) d ˜St = σ ˜St d ˜Wt.
By Girsanov’s theorem, there exists a new underlying probability measure
Q such that ˜W is a Brownian motion under Q. Hence, the preceding SDE
implies that the process ˜S is a Q-martingale.
Now consider the process ˜Vt = EQ(e−rT
C| Ft). By the tower property
of conditional expectations this is a Q-martingale relative to the filtration
(Ft). It is obvious that the natural filtration (Ft) of W is also the natural
filtration of the process ˜W (the processes generate the same flow of infor-
mation). Hence, by the Brownian representation theorem, Theorem 5.19,
there exists a predictable process ˜φ such that d ˜Vt = ˜φt d ˜Wt. So if we define
φt = ˜φt/σ ˜St, we obtain
(6.2) d ˜Vt = φt d ˜St.
Next we define the process ψt = ˜Vt − φt
˜St.
We claim that (φ, ψ) is a hedging portfolio for the derivative C. To
prove this, consider the value process V of the portfolio (φ, ψ). Then by
construction, we have
Vt = ert ˜Vt.
In particular VT = EQ(C| FT ) = C, so indeed the portfolio has the value
C at time T. To prove that it is self-financing, the result of the following
exercise is useful.
6.3 EXERCISE. Use Itô’s formula to show that if X satisfies an SDE and
F is a differentiable function, then d(F(t)Xt) = F(t) dXt + Xt dF(t).
Now we can compute dVt. By the result of the exercise we have
dVt = d(ert ˜Vt) = ˜Vt dert
+ ert
d ˜Vt.
If we use the definition of ψt to rewrite the first term on the right-hand
side, use (6.2) to rewrite the second term and recall that ert
= Bt, we find
that
dVt = (ψt + φt
˜St) dBt + φtBt d ˜St = φt( ˜St dBt + Bt d ˜St) + ψt dBt.
So by the result of the exercise again, we indeed have the relation
dVt = φt dSt + ψt dBt,

which shows that the portfolio (φ, ψ) is self-ﬁnancing.
In view of the standard arbitrage argument the fair price of the claim
C at time t is given by Vt = ert ˜Vt = ert
EQ(e−rT
C| Ft). Hence, we have
proved the following theorem.
6.4 Theorem. The value of the claim C ∈ FT at time t ∈ [0, T] is given
by ert
EQ(e−rT
C| Ft), where Q is the measure under which the discounted
price process e−rt
St is a martingale. In particular, the price at time t = 0
is given by EQe−rT
C.
6.3 European Options
If the claim C is European, meaning that it is of the form C = f(ST ) for
some function f, then we can derive a more explicit formula for its fair
price.
Recall that under the martingale measure Q we have that d ˜St =
σ ˜St d ˜Wt, where ˜W is a Q-Brownian motion. By the preceding exercise it
holds that d ˜St = e−rt
dSt + St de−rt
, and it follows that
dSt = rSt dt + σSt d ˜Wt.
This is the SDE of a geometric Brownian motion. By Example 5.14 it holds
that
St = S0e(r−
1
2 σ2
)t+σ ˜Wt
.
So under Q the asset price S is also a geometric Brownian motion, with
drift r − σ2
/2 and volatility σ. In particular we have, under Q,
ST = S0e(r−
1
2 σ2
)T +σ
√
T Z
,
where Z is a standard Gaussian random variable. It follows that for the
price EQe−rT
C of the claim we have the expression
EQe−rT
f(ST ) = e−rT
Ef S0e(r−
1
2 σ2
)T +σ
√
T Z
=
e−rT
√
2π R
f S0e(r−
1
2 σ2
)T +σ
√
T z
e−
1
2 z2
dz.
Thus, we have proved the following theorem.

6.4: The Black-Scholes PDE and Hedging 45
6.5 Theorem. The fair price of a European claim with pay-off C = f(ST )
is given by
e−rT
√
2π R
f S0e(r−
1
2 σ2
)T +σ
√
T z
e−
1
2 z2
dz.
For a given choice of the function f it is typically not possible to evalu-
ate this integral analytically and one has to resort to numerical integration.
For the prices of European calls and puts however, we can derive explicit
expressions. The European call option with strike K and maturity T cor-
responds to the function f(x) = (x − K)+
. For this special choice of f the
preceding formula can be simplified further and yields the expression
S0Φ(d1) − Ke−rT
Φ(d2)
for the price of the option, where Φ is the distribution function of the
standard Gaussian distribution and the constants are given by
d1 =
log(S0/K) + (r + σ2
/2)T
σ
√
T
, d2 = d1 − σ
√
T.
This is the celebrated Black-Scholes formula.
6.6 EXERCISE. Use the call-put parity to derive the Black-Scholes formula
for the price of a European put option.
6.4 The Black-Scholes PDE and Hedging
The derivation of the pricing formula for a European claim C = f(ST ) given
in Theorem 6.5 can easily be extended to a formula for the price at any
time t ≤ T. From the absence of arbitrage it follows that this price equals
the value Vt of the hedging portfolio at time t. It holds that Vt = F(t, St),
where the function F is given by
F(t, x) =
e−r(T −t)
√
2π R
f xe(r−
1
2 σ2
)(T −t)+σz
√
T −t
e−
1
2 z2
dz.
Observe that if we substitute t = 0, we indeed recover the result of Theo-
rem 6.5.
The pricing function F can also be obtained as the solution of the
so-called Black-Scholes partial differential equation (PDE). This provides
a second method for finding the price of the claim. For a given function f
this PDE can usually not be solved analytically, and one has to resort to
numerical methods.

6.7 Theorem. The value of a European claim C = f(ST ) at time t ≤ T
is given by Vt = F(t, St), where F is the solution of the partial differential
equation
Ft(t, x) + rxFx(t, x) + 1
2 σ2
x2
Fxx(t, x) − rF(t, x) = 0,
subject to the boundary condition F(T, x) = f(x).
Proof. The function F is smooth in both arguments. Therefore, we can
apply Itô’s formula to the the value process Vt = F(t, St) to see that this
satisfies the SDE
dVt = Ft(t, St) dt + Fx(t, St) dSt + 1
2 Fxx(t, St) d[S]t.
By Example 5.14 we have dSt = (µ+σ2
/2)St dt+σSt dWt. In particular its
quadratic variation satisfies d[S]t = σ2
S2
t dt. Substituting these identities
in the preceding display we see that the SDE for Vt reduces to
dVt = Ft(t, St) + (µ + 1
2 σ2
)Fx(t, St)St+1
2 σ2
Fxx(t, St)S2
t dt
+ σFx(t, St)St dWt.
Using the definition of ˜W we can also write this equation in the form
dVt = Ft(t, St) + rFx(t, St)St + 1
2 σ2
Fxx(t, St)S2
t dt + σFx(t, St)St d ˜Wt.
On the other hand, equations (6.1) and (6.2) imply that d ˜Vt = σφt
˜St d ˜Wt.
By the exercise above, it follows that Vt = ert ˜Vt satisfies
dVt = rF(t, St) dt + σφtSt d ˜Wt.
Comparison of the dt-terms of the last two equations for dVt yields the PDE
for the function F.
The boundary condition follows from the fact that f(ST ) = VT =
F(T, ST ).
It should be noted that the PDE for the value process is the same for
every type of European option. The type of option is only important for
the boundary condition.
In the proof of the preceding theorem we only compared the dt-terms
of the two SDE’s that we obtained for the value process V of the claim. By
comparing the d ˜Wt-terms we obtain the following explicit formulas for the
hedging portfolio of the claim.

6.5: The Greeks 47
6.8 Theorem. A European claim C = f(ST ) with value process Vt =
F(t, St) can be hedged by a self-ﬁnancing portfolio consisting at time t of
φt risky assets and ψt bonds, where
φt = Fx(t, St),
ψt = e−rt
F(t, St) − Fx(t, St)St .
Proof. The formula for φt follows from the comparison of the d ˜Wt-terms of
the two SDE’s for V that we obtained in the proof of the preceding theorem.
Recall that ψt = ˜Vt − φt
˜St. Substituting Vt = F(t, St) and φt = Fx(t, St)
yields the formula for ψt.
The hedging strategy exhibited in the preceding theorem is called the
delta hedge for the claim. Note that in general, the numbers of stocks and
bonds in the hedging portfolio change continuously. In practice it is of
course not possible to trade continuously. Moreover, very frequent trading
will not always be sensible in view of transaction costs. However, the delta
hedge can be used in practice to indicate what a hedging portfolio should
look like.
6.5 The Greeks
Parties which are trading a claim with associated value function Vt =
F(t, St) are often interested in the sensitivity of the price of the claim with
respect to the price of the underlying risky asset, and also with respect
to time, volatility, etc. Reasonable measures for these sensitivities are the
derivatives of the function F(t, x). These derivatives have special names.
The quantities
∆ = Fx, Γ = Fxx, Θ = Ft, V = Fσ
are called the delta, gamma, theta and vega of the claim, respectively.
Together they are called the Greeks.
For instance, the delta of a claim measures the ﬁrst order dependence
of the price of the claim relative to the price of the underlying asset. A very
high delta means that small changes in the asset price cause relatively large
changes in the value of the claim. Observe that the delta is precisely the
number of stocks in the hedging portfolio of Theorem 6.8.
6.9 EXERCISE. Calculate the delta for the European call option and give
the delta hedging strategy for the claim.

6.6 General Claims
If the claim C ∈ FT is not of European type we typically have no nice
closed-form expression for its price. In that case, one can use simulation
to find the approximate price. The price of the claim is given by EQe−rT
C
and in the preceding section we saw that under the martingale measure Q
St = S0e(r−
1
2 σ2
)t+σ ˜Wt
,
where ˜W is a Brownian motion. To approximate the price, the following
procedure can be followed:
1) Simulate a large number, say n, of realizations of the process S under
Q.
2) For each realization, compute the corresponding pay-off of the claim,
yielding n numbers C1, . . . , Cn. Compute the average
cn =
1
n
n
i=1
Ci.
3) Then by the law of large numbers, the discounted average e−rT
cn is a
good approximation for the price EQe−rT
C if n is large enough.
We can quantify the quality of the approximation by obtaining a con-
fidence interval. For this we also need the sample standard deviation sn,
which is defined by
s2
n =
1
n − 1
n
i=1
(Ci − cn)2
.
By the central limit theorem and the law of large numbers we have the
convergence in distribution
√
n
cn − EQC
sn
N(0, 1)
as n → ∞. Hence, for large n, we have the approximation
Q
√
n
cn − EQC
sn
> 1.96 ≤ 0.05.
It follows that [e−rT
(cn −1.96sn/
√
n), e−rT
(cn +1.96sn/
√
n)] is an approxi-
mate 95%-confidence interval for the price of the claim C. The length of the
interval tends to zero as n → ∞, which means that our simulation scheme
can achieve arbitrary accuracy if we simulate long enough. In practice we
shall be limited by computation time.

6.7: Exchange Rate Derivatives 49
6.7 Exchange Rate Derivatives
Companies who do business in a country with a different currency are often
interested in reducing the risk due to uncertainty in the exchange rate. One
possibility to reduce this risk is to buy a suitable ”exchange rate derivative”.
For instance, a Dutch company that will place a large order in the US one
month from now may want to have an option to buy a large number of
dollars for a specified price (in euros) at that time. In this section we use the
developed Black-Scholes theory to derive the fair price of such a derivative.
We assume that there exist dollar bonds in the US and we can trade
in euro bonds in The Netherlands. The prices of these bonds (in their
respective currencies) are supposed to be given by
Dt = eqt
, Bt = ert
respectively, where r is the European interest rate and q is the US interest
rate. The exchange rate Et, i.e. the euro value of one dollar, is modelled as
a geometric Brownian motion,
Et = E0eνt+σWt
,
for certain parameters ν, σ and a Brownian motion W.
From the Dutch perspective, we can now trade in two assets: the risk-
less euro bond and the ”risky” US bond, which have (euro) price processes
B and S = ED, respectively. The process S is given by
St = EtDt = S0e(q+ν)t+σWt
.
In other words, S is a geometric Brownian motion with drift q + ν and
volatility σ. From the point of view of a Dutch trader this is just a standard
Black-Scholes market and we know how to price derivatives.
Consider for instance a contract giving a Dutch trader the right to buy
one US dollar for K euros at time T > 0. The pay-off at time T of this
contract in euros is (ET − K)+
. By the standard theory the fair euro price
of the contract is e−rT
EQ(ET − K)+
, where Q is the martingale measure,
under which the discounted price process ˜St = e−rt
St is a martingale. Note
that
e−rT
EQ(ET − K)+
= e−qT
EQe−rT
(ST − KeqT
)+
.
This is exp(−qT) times the standard Black-Scholes price of a European call
option with maturity T and strike K exp(qT). For the latter we have an
explicit formula.

7
Extended
Black-Scholes Models
The classical Black-Scholes model that we considered in the preceding chap-
ter can be extended in several directions. So far we only considered mar-
kets in which a single bond and one risky asset are traded. We can also
study the more complex situation that there are several risky assets with
price processes that do not evolve independently. This allows the pricing of
derivatives which depend on the behaviour of several assets. The assump-
tion of a constant drift µ and volatility σ can also be relaxed. They can be
replaced by arbitrary, predictable stochastic processes.
In general we can consider a market in which a bond is traded with
price process B and n risky assets with price processes S1
, . . . , Sn
. We
assume that the bond price is of the form Bt = exp(
t
0
rs ds) for rt the
“interest rate” at time t, so that it satisfies the ordinary diffential equation
dBt = Btrt dt, B0 = 1.
The interest rate r may be an arbitrary predictable process and hence
depend on all information before time t. We assume that the asset price
processes satisfy the system of stochastic differential equations
(7.1) dSi
t = µi
tSi
t dt + Si
t
d
j=1
σij
t dWj
t , i = 1, . . . , n,
where W1
, . . . , Wd
are d independent Brownian motions, and the µi
and
σij
are predictable processes. Then the processes µi
model the drift, and
the σij
model both the volatility and the dependence structure of the price
processes.
Under certain conditions such general market models are also free of
arbitrage and have the property that each claim that is a function of the
asset prices can be hedged by a self-financing trading strategy. Explicit
pricing formulas are usually not available in such general models. However,

7.1: Market Price of Risk 51
if the model is free of arbitrage and complete, the general fact that ”price is
expectation of discounted pay-off under a martingale measure” is still true.
The SDE’s satisfied by the price processes under the martingale measure
are typically easily obtained, so the simulation method can be used to
approximate claim prices. This requires the simulation of solutions of multi-
dimensional SDE’s.
7.1 Market Price of Risk
The key structural condition needed to push through the theory is the
existence of a predictable, vector-valued process θ = (θ1
, . . . , θd
), called the
market price of risk, such that
(7.2)
d
j=1
σij
t θj
t = rt − µi
t, i = 1, 2, . . . , n.
We can write this system of equations in vector form as σtθt = rt1 − µt,
and hence the existence of the “market price of risk” process requires that
the vector rt1 − µt is contained in the range space of the (n × d)-matrix
σt. This is immediate if the rank of σt is equal to the number n of stocks
in the economy, as the range of σt is all of Rn
in that case. If the rank
of σt is smaller than the number of stocks, then existence of the market
price of risk process requires a relationship between the three parameters
σ, r and µ. This situation is certain to arise if the number of components
of the driving Brownian motion is smaller than the number of risky assets,
i.e. d < n. Hence we can interpret the condition of existence of a process
θ as in the preceding display as implying that the “random inputs W(i)
to the market should be at least as numerous as the (independent) risky
assets”. We shall see a somewhat different interpretation when discussing
models for the term structure of interest rates, where the market price of
risk assumption will come back in the natural, intuitive form that “a market
cannot have two different interest rates”.
7.2 Fair Prices
In the present extended situation a portfolio is still a pair (ψ, φ) of a pre-
dictable process ψt, giving the number of bonds, and a vector-valued pre-
dictable process φt = (φ1
t , . . . , φn
t ), giving the numbers of assets of the

52 7: Extended Black-Scholes Models
various types. To make the integrals well defined we assume that
n
i=1
T
0
|φi
t|2
dt +
T
0
|ψt| dt < ∞.
The portfolio is called self-financing if its value process Vt = ψtBt +
n
i=1φi
tSi
t satisfies
(7.3) dVt =
n
i=1
φi
t dSi
t + ψt dBt.
By definition the fair price at time t of a claim C is the value Vt of a
replicating strategy at time t, where a “replicating strategy” is exactly as
before a self-financing strategy whose value at T is equal to VT = C.
In the present situation we discount using the process B rather than
the exponential factors ert
. Thus the discounted stock processes are
˜Si
t = B−1
t Si
t = e
−
t
0
rs ds+
t
0
µi
s−
1
2
d
j=1
(σij
s )2
ds+
d
j=1
σij
s dW j
s
.
The second equality follows from the definition of B, the SDE (7.1) for the
asset prices and Itô’s formula, applied as in Example 5.14. In other words,
by another application of Itô’s formula,
d ˜Si
t = ˜Si
t(µi
t − rt) dt + ˜Si
t
d
j=1
σij
t dWj
t .
If there exists a market price of risk process θ, then this can be rewritten
in the form
d ˜Si
t = ˜Si
t
d
j=1
σij
t d ˜Wj
t ,
where ˜Wt = Wt −
t
0
θs ds. Unless θ = 0, the process ˜W will not be a P-
Brownian motion in view of Lemma 5.17. However, according to Girsanov’s
theorem there exists a probability measure Q under which ˜W is a Brownian
motion (if θ is appropriately integrable). Under this “martingale measure”
Q the discounted stock prices are local martingales.
We can now follow roughly the reasoning in Section 6.2 to construct a
replicating portfolio for a claim that pays an amount C at time T. A key
element in this construction is to find a process φt such that the martingale
˜Vt = EQ B−1
T C| Ft) is representable as d ˜Vt = φt d ˜St. In the present vector-
valued situation this is to be understood as
d ˜Vt =
n
i=1
φi
t d ˜Si
t.

7.3: Arbitrage 53
If the matrices σt are square and invertible, then this representation can be
easily obtained from the vector-valued version of the Brownian representa-
tion theorem, Theorem 5.19, by the same arguments as in Section 6.2. More
generally, the desired representation is typically possible if the ﬁltration Ft
is generated by the asset processes Si
t. In the following theorem we refer
to this assumption by assuming that “the stock price processes possess the
representation property”.
7.4 Theorem. Assume that there exists a predictable process θ satisfy-
ing (7.2), and that the stock price processes possess the representation
property. Furthermore, assume that E exp 1
2
T
0
σs
2
ds < ∞ and that
E exp 1
2
T
0
θs
2
ds < ∞. Then the value of the claim C ∈ FT at time
t ∈ [0, T] is given by BtEQ(B−1
T C| Ft), where Q is the measure under which
the discounted price processes B−1
t Si
t are local martingales.
7.3 Arbitrage
In the preceding it was seen that existence of the market-price-of-risk pro-
cess is essential for the construction of a martingale measure Q under which
the discounted stock price processes are local martingales. To underline the
necessity of the existence of the market-price-of-risk we shall now show that
without it, the market allows arbitrage.
The value process of a self-ﬁnancing strategy (ψ, φ) can be written as
(cf. (7.3))
Vt − V0 =
t
0
ψsBsrs ds +
n
i=1
t
0
φi
s dSi
s
=
t
0
Vsrs ds +
n
i=1
t
0
φi
s (dSi
s − Si
srs ds).
By the partial integration formula and the fact that dBt = rtBt dt,
(7.5) d
V
B t
= −Vt
1
B2
t
dBt +
1
Bt
dVt =
1
Bt
n
i=1
φi
t (dSi
t − Si
trt dt),
by the preceding display. Hence the discounted value process takes the form,
in view of (7.1),
(7.6) ˜Vt =
Vt
Bt
= V0 +
n
i=1
t
0
Si
sφi
s
Bs
d
j=1
σij
s dWj
s − (rs − µi
s) ds .

This formula does not make explicit reference to the amount ψ invested
in the bond, which has been eliminated. A “partial strategy” φ defines a
value process through the preceding displays (7.5)-(7.6), and given φ we can
define a process ψ from the equation Vt = ψtBt +
n
i=1φi
tSi
t. By retracing
the calculations the resulting strategy (ψ, φ) can be seen to be self-financing
and to possess value process Vt. Thus to see which value processes are
possible it suffices to construct the stock portfolio φ.
Nonexistence of a market price of risk process implies that the vector
rt1−µt is not contained in the range of σt, for a positive set of times t. Then
there exists a vector φt such that the vector (S1
t φ1
t , . . . , Sn
t φn
t ) is orthogonal
to this range such that the inner product with the vector rt1−µt is strictly
negative for a positive set of times t:
n
i=1
Si
tφi
tσij
t = 0, j = 1, . . . , d,
n
i=1
Si
t φi
t(rt − µi
t) < 0.
We can arrange it so that the latter inner product is never positive and
hence, by (7.6), the corresponding discounted gain process will be zero at
time 0 and strictly positive at time T. This is an example of arbitrage.
On the other hand, if the market price of risk process θ exists, then
the discounted gains process in (7.6) can be written as a stochastic integral
relative to the process σ · ˜W, for ˜Wt = Wt −
t
0
θs ds. Under the martingale
measure Q the process ˜W is a Brownian motion, and hence the discounted
gains process will be a local martingale. Under the integrability assumptions
of Theorem 7.4 it is a Q-martingale, and hence cannot become strictly
positive as its mean must remain zero. Thus existence of the market price
of risk is intimately connected to the nonexistence of arbitrage.
7.4 PDEs
Under the conditions of Theorem 7.4 the process ˜W = ( ˜W1
, . . . , ˜Wd
) de-
fined by ˜Wt = Wt −
t
0
θs ds is a Brownian motion under the martingale
measure Q. Because option prices can be written as expectations under
Q, it is useful to rewrite the stochastic differential equation (7.1) in terms
of the process ˜W. If we also assume that the processess r and σ take the
forms rt = r(t, Bt, St) and σt = σ(t, Bt, St), then the equations describing

7.4: PDEs 55
the asset prices become
(7.7)
dBt = Bt r(t, Bt, St) dt,
dSi
t = Si
t r(t, Bt, St) dt + Si
t
d
j=1
σij
(t, Bt, St) d ˜Wj
t , i = 1, . . . , n.
As usual we assume that (B, S) is adapted to the augmented natural filtra-
tion F
˜W
t of ˜W. Then, under regularity conditions on r and σ, the process
(B, S) will be Markovian relative to this filtration. If we assume in addi-
tion that σ is invertible, then ˜W can be expressed in (B, S) by inverting
the second equation, and hence the filtrations Ft and F
˜W
t generated by
(B, S) and ˜W are the same. The process (B, S) is then Markovian relative
to its own filtration Ft. In that case a conditional expectation of the type
EQ(X| Ft) of a random variable X that is a function of (Bs, Ss)s≥t can be
written as F(t, Bt, St) for a function F.
This observation can be used to characterize the value processes of
certain options through a partial differential equation. The value process
of a claim that is a function C = g(ST ) of the final value ST of the stocks
takes the form
Vt = BtEQ
g(ST )
BT
| Ft = EQ e
−
T
t
r(s,Bs,Ss) ds
g(ST )| Ft .
If the process (B, S) is Markovian as in the preceding paragraph, then we
can write Vt = F(t, Bt, St) for a function F. We assume that this function
possesses continuous partial derivatives up to the second order. For sim-
plicity of notation we also assume that S is one-dimensional. Then, by Itô’s
formula,
dVt = Ft dt + Fb dBt + Fs dSt + 1
2 Fss d[S]t.
Here Ft, Fb, Fs are the first order partial derivatives of F relative to its
three arguments, Fss is the second order partial derivative relative to its
third argument, and for brevity we have left off the argument (t, Bt, St) of
these functions. A second application of Itô’s formula and substitution of
the diffusion equation for (B, S) yields
d
Vt
Bt
=
1
Bt
−Fr + Ft + FbBtr + FsStr + 1
2 FssS2
t σ2
dt +
1
Bt
FsStσ d ˜Wt.
The process Vt/Bt was seen previously to be a Q-local martingale. Because
the process ˜W is a Brownian motion, this can only be true if the drift term
on the right side of the preceding display is zero, i.e.
− (Fr)(t, b, s) + Ft(t, b, s) + (rFb)(t, b, s)b
+ (rFs)(t, b, s)s + 1
2 (σ2
Fss)(t, b, s)s2
= 0.
This is an extension of the Black-Scholes partial differential equation.

This partial diﬀerential equation is useful for the numerical computa-
tion of option prices. Even though the equation is rarely explicitly solvable,
a variety of numerical methods permit to approximate the solution F. The
equation depends only on the functions r and σ deﬁning the stochastic dif-
ferential equation (7.7). Hence it is the same for every option with a claim
of the type C = g(ST ), the form of the claim only coming in to determine
the boundary condition. Because C = g(ST ) = F(T, BT , ST ), this takes the
form
F(T, r, s) = g(s).
For instance, for a European call option on the stock S, this becomes
F(T, r, s) = (s − K)+
.

8
Interest Rate Models
8.1 The Term Structure of Interest Rates
In the classical Black-Scholes model the interest rate is a deterministic
constant. In reality the situation is much more complicated of course. In
general, it is not even possible to talk about the interest rate, since short
term and long term rates are usually different. Moreover, the time evolution
of interest rates typically has a random component.
In this chapter we introduce interest rate models that capture these
properties of the time value of money. Such models are necessary for the
pricing of so-called interest rate derivatives. These are financial contracts
that are designed to trade and manage the risk that is caused by the un-
certainty about the time value of money.
8.1.1 Discount Bonds
Pure discount bonds are simple financial contracts that capture the time
value of money. A discount bond which matures at time T > 0, also called
a T-bond, is a contract which guarantees a pay-off of 1 euro at time T. The
price of a T-bond at time t ≤ T is denoted by P(t, T). It is the amount
we are willing to pay at time t to receive 1 euro at time T. The collection
{P(0, T): T > 0} of all bond prices at time t = 0 completely determines the
time-value of money at time 0. It is called the term structure of interest
rates.
For fixed t, the function T → P(t, T) is typically smooth, since, for
instance, the price of a bond that matures 9 years from now will be close to
the price of a bond that matures 10 years from now. For a fixed maturity
T > 0 however, the function t → P(t, T) will appear to fluctuate randomly.

58 8: Interest Rate Models
By construction it holds that P(T, T) = 1.
8.1.2 Yields
If we have 1 euro at time t, we can use it to buy 1/P(t, T) T-bonds. At
time T we then receive 1/P(t, T) euros. Hence, a euro at time t grows to
1/P(t, T) euros at time T. It the interest rate over the interval [t, T] had
been constant, say r, a euro at time t would have grown to exp(r(T − t))
at time T. If we compare this, we see that buying the T-bonds at time t
leads to a “constant interest rate” over the time interval [t, T] of
(8.1) Y (t, T) = −
log P(t, T)
T − t
.
We call this the yield over [t, T]. The collection of all yields of course contains
exactly the same information as the collection of all bond prices. However,
the yields have a somewhat easier interpretation in terms of interest rates.
8.1.3 Short Rate
Although the interest rate does not exist, we can construct an object that
can be interpreted in this way. We just saw that the yield Y (t, T) can be
interpreted as the constant interest rate valid in the time interval [t, T].
The number
rt = lim
T ↓t
Y (t, T) = −
∂
∂T
log P(t, T)
T =t
can therefore be viewed as the interest rate at time t (or in the infinitesimal
interval [t, t + dt]). We call rt the short rate at time t. From its definition
it is clear that in general, the short rate does not contain all information
about the time value of money.
8.1.4 Forward Rates
Let t < S < T and consider the following strategy. At time t, we sell one
S-bond, giving us P(t, S) euros. We immediately use this money to buy
P(t, S)/P(t, T) T-bonds. At time S the S-bond matures, which means we
have to pay one euro to its holder. At time T the T-bond matures, and we
receive P(t, S)/P(t, T) euros.
If we follow this strategy, the net effect is that one euro at time S grows
to P(t, S)/P(t, T) euros at time T. If the interest rate were a constant r
over the time interval [S, T], one euro at time S would grow to exp(r(T −S))
at time T. Hence, the “constant interest rate over [S, T] determined at time
t” is
−
log P(t, T) − log P(t, S)
T − S
.

8.2: Short Rate Models 59
This number is called the forward rate for [S, T], contracted at time t. If we
let S ↑ T we get
(8.2) f(t, T) = −
∂
∂T
log P(t, T),
which is the forward rate at time T, contracted at time t. Note that the
short rate is a particular forward rate, we have f(t, t) = rt. Moreover, it is
easy to see that
P(t, T) = e
−
T
t
f(t,s) ds
,
so the collection of all forward rates contains all information about the term
structure of interest rates.
8.2 Short Rate Models
The classical approach to interest rate models is to specify a stochastic
model for the short rate rt and to assume that the bond price P(t, T) is
some smooth function of rt. A model of this type is called a short rate
model.
So let us suppose that under the “real world” probability measure P,
the short rate satisfies the SDE
(8.3) drt = µ(t, rt) dt + σ(t, rt) dWt,
where W is a P-Brownian motion, and µ and σ are certain functions on
[0, ∞)×R. Let (Ft) be the filtration generated by the process r. We assume
that we can put money in a bank which pays the interest rate rt, in the
sense that one euro at time zero grows to Bt euros at time t, where Bt =
exp(
t
0
rs ds). In differential notation, the process B satisfies
dBt = rtBt dt.
For the bond prices we assume that P(t, T) = FT
(t, rt), where FT
is some smooth function on [0, ∞) × R which may depend on the time to
maturity T. Clearly, the functions should satisfy FT
(T, r) = 1 for all T and
r. In the preceding section we noted that the short rate does not determine
the whole term structure of interest rates, so we can expect that we have
some freedom in choosing the functions FT
. On the other hand we do not
want to allow arbitrage opportunities in the bond market. It is intuitively
clear that this implies certain restrictions on the relation between the prices
of the T-bonds for various T, leading to restrictions on the functions FT
.
In the remainder of this section we explain how we can construct arbitrage
free short rate models.
The first step is the observation that by the absence of arbitrage there
cannot be banks with different rates of interest.

8.4 Lemma. Suppose there exists a self-financing portfolio with value pro-
cess V which satisfies dVt = qtVt dt for some adapted process q. Then qt = rt
for all t ≥ 0.
Proof. We sketch the proof. Suppose for simplicity that q and r are con-
stant and that q > r. Then we can borrow 1 euro at rate r and invest it in
the portfolio which pays “interest” q. At time T, say, we sell the portfolio,
giving us exp(qT) euros. We pay back our loan, which is now exp(rT), and
are left with a risk-free profit of exp(qT) − exp(rT) euros. This is clearly
an arbitrage, which is not allowed.
The general case of random, nonconstant processes q and r can be
handled similarly.
By assumption the price P(t, T) of a T-bond is given by P(t, T) =
FT
(t, rt). This is a smooth function of t and a process which satisfies an
SDE. Hence, by Itô’s formula, we have that
dP(t, T) = FT
t (t, rt) dt + FT
r (t, rt) drt + 1
2 FT
rr(t, rt) d[r]t.
If we combine this with the SDE (8.3) for the short rate rt we obtain
(8.5) dP(t, T) = αT
(t, rt)P(t, T) dt + σT
(t, rt)P(t, T) dWt,
where the functions αT
and σT
are given by
(8.6) αT
=
FT
t + µFT
r + 1
2 σ2
FT
rr
FT
,
(8.7) σT
=
σFT
r
FT
.
Below we write αT
t and σT
t instead of αT
(t, rt) and σT
(t, rt).
To study the relation between the prices of bonds with different ma-
turities we are now going to consider a self-financing portfolio consisting of
S-bonds and T-bonds, for S < T. Suppose we are given such a portfolio,
consisting at time t < S of φT
t T-bonds and φS
t S-bonds, and let V denote
its value process. Since the portfolio is self-financing we have
dVt = φT
t dP(t, T) + φS
t dP(t, S)
= uT
t Vt
dP(t, T)
P(t, T)
+ uS
t Vt
dP(t, S)
P(t, S)
,
where uT
and uS
are the fractions of the portfolio consisting respectively
of T-bonds and S-bonds, defined by
uT
t =
φT
t P(t, T)
Vt
, uS
t =
φS
t P(t, S)
Vt
.

8.2: Short Rate Models 61
If we combine this with the SDE (8.5) for P(t, T) we get
dVt = uT
t αT
t + uS
t αS
t Vt dt + uT
t σT
t + uS
t σS
t Vt dWt.
This SDE holds for every self-financing portfolio consisting of S-bonds
and T-bonds. Conversely, we can construct a particular portfolio by speci-
fying fractions uT
and uS
satisfying uT
t + uS
t = 1. The choice
uT
t = −
σS
t
σT
t − σS
t
,
uS
t =
σT
t
σT
t − σS
t
leads to a self-financing portfolio with value process V satisfying
dVt =
αS
t σT
t − αT
t σS
t
σT
t − σS
t
Vt dt.
The dWt-term has disappeared, so by Lemma 8.4 it must hold that
αS
t σT
t − αT
t σS
t
σT
t − σS
t
= rt
for all t ≥ 0. We can rewrite this relation as
αS
t − rt
σS
t
=
αT
t − rt
σT
t
.
In other words, the ratio (αT
t − rt)/σT
t must be independent of T. Thus,
we have proved the following lemma.
8.8 Lemma. There exists a function λ on [0, ∞) × R, independent of T,
such that, for all t, T,
λ(t, rt) =
αT
(t, rt) − rt
σT (t, rt)
Recall that αT
t and σT
t are the local rate of return and volatility of the
T-bond, respectively (cf. (8.5)). Hence, the difference αT
t −rt can viewed as
a risk premium. It is the excess return that we get if we invest in the risky T-
bond instead of putting our money in the bank. The quantity (αT
t −rt)/σT
t ,
i.e. the risk premium per unit of volatility, is called the market price of risk
of the T-bond. In this terminology the preceding lemma states that in an
arbitrage free bond market, all bonds have the same market price of risk.
If we combine the result of Lemma 8.8 with the definitions (8.6) and
(8.7) of the processes αT
and σT
, we arrive at a PDE for the pricing func-
tions of T-bonds, called the term structure equation.

8.9 Theorem. Let λ(t, rt) denote the market price of risk. Then for every
T > 0 the function FT
satisfies the PDE
FT
t + (µ − λσ)FT
r + 1
2 σ2
FT
rr − rFT
= 0,
subject to the boundary condition FT
(T, r) = 1.
Under certain regularity conditions the term structure equation has a
unique solution for every T > 0, so the bond prices P(t, T) = FT
(t, rt) are
completely determined by the functions µ, σ and λ.
It is now clear how we can construct a short rate model leading to an
arbitrage free bond market:
1) Specify the drift µ and volatility σ for the short rate rt (under P)
and assume that rt satisfies the SDE (8.3), where W is a P-Brownian
motion.
2) Choose a function λ on [0, ∞)×R and for T > 0, let FT
be the solution
of the term structure equation corresponding to µ, σ and λ.
3) Finally, define the price of a T-bond as P(t, T) = FT
(t, rt).
Observe that the term structure equation for the price of a T-bond is
very similar to the Black-Scholes PDE for the pricing function of a European
claim, cf. Theorem 6.7. In the preceding chapter we saw that the price of a
European claim also equals the expectation of the discounted pay-off under
a new measure Q. We have the following analogous theorem for the price
of a T-bond in a short rate model.
8.10 Theorem. If µ and σ are the drift and volatility of the short rate
under P and λ is the market price of risk, the price of a T-bond at time t
is given by
P(t, T) = BtEQ B−1
T | Ft = EQ e
−
T
t
rs ds
| Ft ,
where Q is the measure under which the short rate satisfies the SDE
drt = µ(t, rt) − σ(t, rt)λ(t, rt) dt + σ(t, rt) d ˜Wt,
and ˜W is a Q-Brownian motion.
Note that for every T > 0 it holds that the discounted price ˜P(t, T) =
B−1
t P(t, T) of a T-bound satisfies
B−1
t P(t, T) = EQ B−1
T | Ft ,
so for every T > 0 the process ( ˜P(t, T))t≤T is a martingale under Q. There-
fore the measure Q appearing in the statement of the theorem is called the
martingale measure of the model. Observe that the formula
P(0, T) = EQB−1
T

8.3: The Hull-White Model 63
for the current price of a T-bond is a statement of the usual form “price of
a derivative is expectation of the discounted pay-off under the martingale
measure”, since a T-bond can be viewed as a claim which pays off 1 euro
at time T.
Theorem 8.10 gives us a second method for the construction of a model
for an arbitrage free bond market:
1) Specify an SDE for the short rate rt under the martingale measure Q
and let (Ft) be the natural filtration of the process r.
2) Define the price P(t, T) of a T-bond by
P(t, T) = EQ e
−
T
t
rs ds
| Ft .
This second procedure for the construction of short rate models is
known as martingale modeling and has the obvious advantage that we do
not have to specify the market price of risk explicitly. In the next section
we introduce a popular short rate model that is constructed in this way.
8.3 The Hull-White Model
The Hull-White model for the term structure of interest rates assumes that
under the martingale measure Q the short rate rt satisfies the SDE
(8.11) drt = (θ(t) − art) dt + σ dWt,
where a and σ are certain numbers, θ is a deterministic function and W is
a Q-Brownian motion. The natural filtration of r (and W) is denoted by
(Ft) and the price P(t, T) of a T-bond at time t is defined by
P(t, T) = EQ e
−
T
t
rs ds
| Ft .
By the preceding section this defines an arbitrage free model for the bond
market.
It is possible to obtain concrete formulas for the bond prices in this
model. The main reason is that we have an explicit expression for the solu-
tion of the SDE (8.11). This allows us to calculate the conditional distribu-
tion of the integral
T
t
rs ds given Ft, which we need to calculate P(t, T).
8.12 Lemma. Given Ft the integral
T
t
rs ds possesses a Gaussian distri-
bution with mean
B(t, T)rt +
T
t
B(u, T)θ(u) du

and variance
σ2
T
t
B2
(u, T) du,
where
(8.13) B(t, T) =
1 − e−a(T −t)
a
.
Proof. First we apply the Itô formula to calculate d(exp(at)rt) and use
(8.11) to find that
(8.14) rs = e−as
r0 + e−as
s
0
θ(u)eau
du + σe−as
s
0
eau
dWu.
Integrating this from t to T and interchanging integrals gives
T
t
rs ds =r0e−at
B(t, T) +e−at
B(t, T)
t
0
eau
θ(u) du +
T
t
B(u, T)θ(u) du
+ B(t, T)σe−αt
t
0
eαu
dWu + σ
T
t
B(u, T) dWu
= B(t, T)rt +
T
t
B(u, T)θ(u) du + σ
T
t
B(u, T) dWu.
Given Ft the first two terms on the right-hand side are known. The third
one is independent of Ft and is Gaussian with mean zero and variance
σ2
T
t
B2
(u, T) du.
This completes the proof.
We can now derive the bond price formula for the Hull-White model.
8.15 Theorem. In the Hull-White model the price of a T-bond is given
by
(8.16) P(t, T) = eA(t,T )−B(t,T )rt
,
where B is defined by (8.13) and
A(t, T) =
T
t
1
2 σ2
B2
(u, T) − θ(u)B(u, T) du.
Proof. We have to calculate the expectation of exp(−
T
t
rs ds) given Ft.
By the preceding lemma this boils down to computing the expectation of
the exponential of a Gaussian random variable. If Z is a Gaussian random
variable with mean m and variance s2
it holds that E exp(Z) = exp(m +
s2
/2). Together with the lemma this yields the statement of the theorem.

8.3: The Hull-White Model 65
Short rate models in which the bond price is of the form (8.16) are
called affine models. The reason for this name is that the yields and forward
rates are affine in rt in that case. The yield Y (t, T) (see (8.1) is given by
Y (t, T) =
B(t, T)
T − t
rt −
A(t, T)
T − t
and for the forward rate (see (8.2) we have
f(t, T) = BT (t, T)rt − AT (t, T).
Now consider a specific bond market in which bonds of all maturities
are traded. Then at time zero, we can observe the bond prices and forward
rates with all maturities. We denote the observed prices and rates in the
market by P∗
(0, T) and f∗
(0, T), respectively. On the other hand, the Hull-
White model gives the formula
(8.17) f(0, T) = BT (0, T)r0 − AT (0, T)
for the forward rates. Obviously, we would like to match the theoretical
rates f(0, T) with the observed rates f∗
(0, T). We will now show that this
is possible by choosing an appropriate function θ in (8.11). This procedure
is called fitting the model to the term structure of interest rates.
8.18 Theorem. Let the parameters a, σ in (8.11) be given. Then with the
choice
(8.19) θ(T) = af∗
(0, T) + f∗
T (0, T) + σ2
B(0, T)(e−aT
+ 1
2 aB(0, T))
the theoretical Hull-White bond prices and forward rates coincide with the
observed prices and rates. The price of a T-bond is then given by
P(t, T) =
P(0, T)
P(0, t)
exp B(t, T)f∗
(0, t)−
σ2
4a
B2
(t, T)(1−e−2at
)−B(t, T)rt .
Proof. If we insert the expressions for A and B into (8.17) we see that we
have to solve the equation
f∗
(0, T) = e−aT
r0 +
T
0
e−a(T −u)
θ(u) du −
σ2
2a2
(1 − e−aT
)2
= g(T) − h(T),
where g is the solution of the differential equation g + ag = θ, g(0) = r0
and h(t) = σ2
B2
(0, t)/2. Then g(T) = f∗
(0, T) + h(T), which shows that
the solution of the equation for θ is given by
θ(T) = g (T) + ag(T) = f∗
T (0, T) + hT (T) + a(f∗
(0, T) + h(T)).
This proves the first part of the theorem.
The second statement can be obtained by inserting the expression for
θ in formula (8.16).

So after fitting the term structure of interest rates there are only two
free parameters left in the Hull-White model, a and σ. In practice these
are determined by matching the theoretical prices for certain interest rate
derivatives with observed prices. This procedure is called the calibration of
the model. The pricing of interest rate derivatives is the topic of the next
section.
8.4 Pricing Interest Rate Derivatives
The result of Theorem 8.10 can be viewed as a pricing formula for the
simplest possible claim which has a pay-off of one euro at time T. Using
the same arguments as above it can be extended to a general claim which
pays some random amount C ∈ FT at time T.
8.20 Theorem. Let C ∈ FT be a claim. Its value at time t ≤ T is given
by
Vt = EQ e
−
T
t
rs ds
C| Ft ,
where Q is the martingale measure.
Note that for C ≡ 1 we indeed recover the formula for the bond price
P(t, T).
Many interest rate derivatives do not only have a payment at the time
T of maturity, but also at certain fixed intermediate times. Holding such
a product is equivalent to holding a portfolio of derivatives with different
maturities. Hence, Theorem 8.20 implies the following.
8.21 Theorem. Let 0 < T1 < · · · < Tn = T and Ci ∈ FTi
for i = 1, . . . , n.
Consider a derivative with a payment of Ci at time Ti for i = 1, . . . , n. Its
value at time 0 is given by
V0 = EQ
n
i=1
e
−
Ti
0
rs ds
Ci,
where Q is the martingale measure.
The result of Theorem 8.21 can now be used to determine the price of a
given derivative by simulation methods, just as we discussed in Section 6.6
for the Black-Scholes model. The procedure is as follows:
1) Simulate a large number of realizations, say n, of the short rate process
r under the martingale measure Q.

8.4: Pricing Interest Rate Derivatives 67
2) For realization number j, compute the pay-off of the claim and de-
termine an approximation cj for the discounted pay-off given by
i exp(−
Ti
0
rs ds)Ci.
3) Then by the law of large numbers,
cn =
1
n
n
j=1
cj
is a good approximation of the claim price, provided that n is large
enough.
4) By the central limit theorem the interval [cn − 1.96sn/
√
n, cn +
1.96sn/
√
n] is an approximate 95% confidence interval for the price
of the derivative, where
s2
n =
1
n − 1
n
j=1
(cj − cn)2
.
The usual approach to simulating realizations of the short rate is to
discretize the SDE for rt under Q. Suppose that under Q the short rate
satisfies (8.3), where µ and σ are given functions. Then for small ∆ > 0 we
have the approximation
r(k+1)∆ − rk∆ ≈ µ(k∆, rk∆)∆ + σ(k∆, rk∆)(W(k+1)∆ − Wk∆)
for k = 0, 1, 2, . . .. The increments W(k+1)∆ − Wk∆ are independent Gaus-
sian random variables with mean zero and variance ∆. So the approximation
can be written as
r(k+1)∆ − rk∆ ≈ ∆µ(k∆, rk∆) +
√
∆σ(k∆, rk∆)Zk,
where Z0, Z1, Z2, . . . are independent, standard Gaussian random variables.
This so-called Euler approximation of the SDE (8.3) can be used to
simulate sample paths and to determine the corresponding approximation
of the discounted pay-off. A single realization is constructed as follows:
1) Partition the interval [0, T] into m intervals of length ∆ = T/m and
simulate m i.i.d. standard Gaussian random variables Z0, Z1, . . . , Zm∆.
2) The short rate r0 at time 0 is known. The future rates r∆, r2∆, . . . , rm∆
are computed recursively by the formula
r(k+1)∆ = rk∆ + ∆µ(k∆, rk∆) +
√
∆σ(k∆, rk∆)Zk.
3) For this realization the pay-off is calculated and the discount factors
exp(−
T
0
rs ds) are approximated by exp(−∆ k rk∆).
The outlined procedure works for any short rate model. In the special
case of the Hull-White model we have in fact an exact recurrence formula
for the discretized short rate process. This means that realizations can be
simulated without introducing approximation errors due to discretization
of the SDE.

8.22 Theorem. Suppose the short rate satisﬁes (8.11) under the martin-
gale measure Q, where θ is given by (8.19). Then rt = α(t) + yt, where
α(t) = e−αt
r0 + f∗
(0, t) + 1
2 σ2
B2
(0, t)
and yt satisﬁes y0 = 0 and for ∆ > 0 and k = 0, 1, . . .,
y(k+1)∆ − e−a∆
yk∆ ∼ 1
2 σ2B(0, 2∆)Zk,
with Z0, Z1, . . . i.i.d. standard Gaussian and B given by (8.13).
Proof. It follows from (8.14) that rt = α(t) + yt, where
α(t) = e−at
r0 +
t
0
θ(u)eau
du
and
yt = σe−at
t
0
eau
dWu.
The expression for α in the statement of the theorem now follows after
inserting (8.19) and some straightforward calculations. To prove the recur-
rence formula for the process y, observe that the random variables
ea(k+1)∆
y(k+1)∆ − eak∆
yk∆ = σ
(k+1)∆
k∆
eau
dWu
are independent, Gaussian, have zero mean and variance
σ2
(k+1)∆
k∆
e2au
du = σ2
e2ak∆ e2a∆
− 1
2a
.
Hence
ea(k+1)∆
y(k+1)∆ − eak∆
yk∆ ∼ eak∆
σ2
e2a∆ − 1
2a
Zk,
with Z0, Z1, . . . i.i.d., standard Gaussian. The proof is completed by dividing
this by exp(a(k + 1)∆).
8.5 Examples of Interest Rate Derivatives
In this section we discuss some common interest rate products, and their
valuation.

8.5: Examples of Interest Rate Derivatives 69
8.5.1 Bonds with Coupons
In practice pure discount bonds are not often traded. Instead, bonds typi-
cally do not only have a pay-off at maturity, the so-called principal value,
but also make smaller regular payments before maturity. Such a bond is
called a coupon bond. A 10-year, 5% coupon bond with a principle value
of 100 euros for instance, pays 5 euros every year until maturity and 100
euros at maturity (after ten year).
More generally, suppose that a bond makes a payment of k euros at
times T1 < · · · < Tn = T, and pays off its principle value of 1 euro at time
T. Then holding this coupon bond is equivalent to holding k pure discount
bonds with maturity Ti for i = 1, . . . , n, and one T-bond. Hence, the value
of the coupon bond is
P(0, T) + k
n
i=1
P(0, Ti).
We remark that conversely, the prices of pure discount bonds may be
expressed in terms of the prices of coupon bonds. In practice this is the
usual way in which the prices of discount bonds are inferred from market
data.
8.5.2 Floating Rate Bonds
There also exist bonds with intermediate payments that depend on the
interest rates at the time of the payment. The LIBOR rate for the time
interval [S, T], set at time S is defined as
L(S, T) = −
P(S, T) − 1
(T − S)P(S, T)
.
This is simply the return per time unit on an investment at time S in a
T-bond. A floating rate bond is a bond with additional payments at times
T1 < · · · < Tn = T. The payment Ci at time Ti is
(Ti − Ti−1)L(Ti−1, Ti) =
1
P(Ti−1, Ti)
− 1.
This is precisely the gain we would have had at time Ti if we had bought
one euro worth of Ti-bonds at time Ti−1. The principle value of one euro is
payed at time T.
By Theorem 8.21, the price of this asset at time 0 is given by
P(0, T) +
i
EQB−1
Ti
(P(Ti−1, Ti)−1
− 1).

By the tower property of conditional expectation and Theorem 8.10 the ith
term in the sum equals
EQEQ B−1
Ti
(P(Ti−1, Ti)−1
− 1)| FTi−1
= EQ(P(Ti−1, Ti)−1
− 1)B−1
Ti−1
P(Ti−1, Ti)
= EQB−1
Ti−1
− EQB−1
Ti−1
P(Ti−1, Ti)
= P(0, Ti−1) − P(0, Ti).
Hence, the value of the bond is
P(0, T) +
n
i=1
P(0, Ti−1) − P(0, Ti) = P(0, 0) = 1
So the price of a floating rate bond equals its principal value.
8.23 EXERCISE. Derive the pricing formula for the floating rate bond by
showing that there exists a self-financing portfolio with initial value of one
euro which has the same pay-off as the bond.
8.5.3 Swaps
A swap is contract that exchanges a stream of varying, interest rate de-
pendent payments for a stream of fixed amount payments, or vice versa.
Consider for example time points 0 < T1 < · · · < Tn. At time Ti we make
a payment of (Ti − Ti−1)L(Ti−1, Ti) euros and we receive k euros. In other
words, we swap the gain obtained from a one euro investment in Ti-bonds
at time Ti−1 for the constant “gain” k. Buying this contract is equivalent to
selling a floating rate bond and buying a coupon bond which pays k euros
at each time Ti. Hence, the price of the swap at time 0 is
P(0, Tn) + k
n
i=1
P(0, Ti) − 1.
8.5.4 Swaptions
A swaption is a contract giving the holder the right to enter into a swap
at a future date. Suppose for instance that the contract gives the right to
enter at time T0 > 0 into a swap described in the preceding section. Then
the pay-off at time T0 of the option is
P(T0, Tn) + k
n
i=1
P(T0, Ti) − 1
+
.

8.5: Examples of Interest Rate Derivatives 71
By Theorem 8.20, the price at time 0 of the swaption is therefore given by
EQe
T0
0
rs ds
P(T0, Tn) + k
n
i=1
P(T0, Ti) − 1
+
.
In general this expectation can not be evaluated analytically and one has to
resort to numerical methods. Note by the way that the latter formula shows
that a swaption can also be viewed as a call option on a coupon bond.

9
Risk Measurement
Financial institutions deal in risk of various types. Market risk is the expo-
sure to the changing prices of assets on the market, and can be limited by
using appropriate portfolios that include instruments such as options and
swaps. Managing risk is important for:
- Internal management, e.g. optimizing profit subject to restrictions on
risk.
- To fulfill the requirements of regulatory authorities, such as national
banks.
- Credit ratings.
The management of risk requires that risk be measured. In this chapter we
discuss the most popular measure of risk: Value-at-Risk, abbreviated VaR.
9.1 Value-At-Risk
Let Vt be the value of a portfolio at time t, and Ft the information available
at time t. We fix a given time t+δt in the future and confidence level 1−α.
The variable Vt −Vt+δt is the loss that we shall incur in the period [t, t+δt].
VaR is defined as a number such that the loss is with high probability
smaller than this number.
9.1 Definition. VaR is the upper α-quantile of the conditional distribution
of Vt − Vt+δt given Ft, i.e.
P(Vt − Vt+δt < VaR| Ft) ≤ 1 − α ≤ P(Vt − Vt+δt ≤ VaR| Ft).
The Value-at-Risk depends of course on both δt and α. For regulators
a period of 10 days and α = 1% is usual, whereas for other purposes periods
from one day to a year or α = 0.05% may be considered appropriate. VaR

9.1: Value-At-Risk 73
as defined here refers to the absolute value of the portfolio, but for some
purposes it can be useful to consider the “Value-at-Risk per capital” VaR/Vt
instead.
The conditioning on the past information Ft in our definition appears
natural, but is not always made explicit. Although we shall consider VaR
in the following only at one fixed time t, Value-at-Risk is in our definition
a stochastic process in time. For larger lags δt the dependence on t will
typically be small.
One criticism to the use of VaR for risk measurement is that it may say
little about expected losses. In particular, it says nothing about the sizes of
the losses bigger than VaR, which occur with frequency α, and which could
be arbitrarily big. Related to this is that VaR is in general not subconvex
under combination of portfolios. If two subunits of a financial institution
both control their VaR, then there is no guarantee that the VaR of the
financial institution as a whole is also under control. This is illustrated in
Example 9.2. Other measures of risk, which are not open to these criticisms
are the conditional variance (or “volatility”) and the expected excess loss,
given by
var(Vt − Vt+δt| Ft), and E (Vt − Vt+δt − c)+
| Ft).
Here c is some given threshold. The volatility is a classical measure, dating
back to Markowitz, who first proposed to optimize profit under the side
condition of bounded volatility. It is somewhat unstable for heavy-tailed
distributions and perhaps can be critized for being symmetric.
9.2 Example (Nonconvexity). Suppose that the portfolio consists of two
parts, with values Xt and Yt at time t, so that the total value is Vt = Xt+Yt.
Then Vt − Vt+δt = ∆Xt + ∆Yt for ∆Xt and ∆Yt the losses on the two
subportfolios in the interval [t, t + δt], and the relative contributions of
the two subportfolios in the total are wX = Xt/Vt and wY = Yt/Vt. If
VaR(V ), VaR(X), VaR(Y ) are the Value-at-Risks of the three portfolios,
then it may happen that the VaR(V ) is bigger than the convex combination
wXVaR(X) + wY VaR(Y ).
For an example choose some fixed α and let the vector (∆Xt, ∆Yt) be
distributed (conditionally on Ft) on the set of points {(0, 0), (c, 0), (0, c)}
according to the probabilities 1−2α, α, α, for a given c > 0. Then VaR(X) =
VaR(Y ) = 0, but VaR(V ) = c. We can make the discrepancy c between the
total Value-at-Risk and the convex combination arbitrarily large.
To determine VaR we need a model for the conditional distribution
of Vt − Vt+δt given Ft. There are many possibilities, such as the ARMA
and GARCH models from time series, or the Black-Scholes or Hull-White
models from derivative pricing. It is important to note that we need the
distribution of the value process under the real-world measure, not the mar-
tingale measure. Thus given a model the parameters are estimated from

74 9: Risk Measurement
time series’ giving the actual prices of the assets over time, so-called “his-
torical analysis”. Because some of the parameters may be common to the
real-world measure and martingale measure, some parameters could also be
calibrated using observed derivative prices.
In rare cases, depending on the model, it is possible to determine an
analytic expression for the conditional distribution of Vt+δt − Vt given Ft.
More often the VaR must be calculated numerically, for instance by stochas-
tic simulation. If we can generate a large number N (e.g. at least N = 10000
if α = 1%) of realizations from the conditional distribution of Vt − Vt+δt
given Ft, then VaR is approximately the (1 − α)N largest value among the
realizations. Remember that, unlike when using simulation to determine a
derivative price, this time we must simulate from the real-world measure.
9.2 Normal Returns
In practice it is often assumed that the return Vt+δt/Vt − 1 is conditionally
normally distributed given Ft. (It may even be assumed that the returns
are independent of the past and form an i.i.d. sequence if restricted to a
discrete time grid, but that is even more unrealistic and unimportant for
the following.) If the conditional mean and variance are µt and σ2
t , then
the conditional distribution of Vt − Vt+δt is normal with mean −Vtµt and
standard deviation Vtσt, and the Value-at-Risk is given by
(9.3) VaR = Vt σtΦ−1
(1 − α) − µt .
Note that it is proportional to the current capital Vt and linearly increasing
in the volatility σt. A positive drift µt decreases VaR.
If a portfolio consists of n assets or subportfolios, with value processes
V 1
, . . . , V n
, then it is often assumed that the vector of returns (V 1
t+δt/V 1
t −
1, . . . , V n
t+δt/V n
t −1) is conditionally multivariate-normally distributed given
Ft. The value of the whole portfolio Vt =
n
i=1V i
t can be written as
Vt+δt − Vt = Vt
n
i=1
wi
t
V i
t+δt
V i
t
− 1 ,
where wi
t = V i
t /Vt is the relative contribution of asset i to the whole port-
folio. The sum is a linear combination of a Gaussian vector and hence is
normally distributed. If the return vector possesses conditional mean vector
(µ1
t , . . . , µn
t ) and covariance matrix (σi,j
t ), then the conditional distribution
of Vt − Vt+δt given Ft is normal with mean −Vtµt and standard deviation

9.3: Equity Portfolios 75
Vtσt, for
(9.4)
µt =
n
i=1
wi
tµi
t,
σ2
t =
n
i=1
n
j=1
wi
twj
t σi,j
t .
The Value-at-Risk again takes the same form (9.3), but with the new values
of µt and σt substituted.
The Cauchy-Schwarz inequality says that the covariances satisfy
|σi,j
t | ≤ σi
tσj
t , where σi
t = σi,i
t are the variances of the components. This
shows that
σt ≤
n
i=1
wi
tσi
t.
Because the VaR is linear in the standard deviation, this shows that the
combined portfolio has a smaller VaR than a similar single portfolio of the
same volatility. This expresses the well-known guideline that diversiﬁcation
of a portfolio helps to control the risk.
As Example 9.2 shows, diversiﬁcation is not always useful to control
VaR, but the preceding shows that for portfolios with normal returns it is.
Empirical studies have now well established that economic time series
are not Gaussian random walks, an assumption of the past that still lives
on in many VaR-methods. Returns are not i.i.d. and their marginal distri-
butions deviate from normal distributions in that they are typically heavier
tailed and sometimes skewed. Conditional normality of the returns given
the past, as is assumed in this section, is also debatable, but not always re-
jected by statistical tests on empirical data. In particular, GARCH models
are often combined with normal conditional distribution, which automati-
cally leads to heavier-tailed unconditional distributions.
9.3 Equity Portfolios
The value process of a portfolio of one stock with price St is equal to
Vt = St. If we adopt the Black-Scholes model, then St = S0 exp(µt + σWt)
for a Brownian motion process W, and hence the log returns satisfy
Rt: = log
St+δt
St
= µδt + σ(Wt+δt − Wt).
Because the increments of Brownian motion are independent of the past,
it follows that the log returns are conditionally normally distributed with
mean µδt and variance σ2
δt. The loss can be expressed in the log returns

as Vt −Vt+δt = St(1−eRt
). Solving the equation P(Vt(1−eR
) ≤ v) = 1−α
for a N(µδt, σ2
δt) distributed variable R and a fixed value Vt, yields the
Value-at-Risk
(9.5) VaR = Vt 1 − eσ
√
δtΦ−1
(α)+µδt
.
This has similar features as the equation (9.3): the risk is proportional to
the current value Vt, increasing in the volatility σ and decreasing in the
drift µ.
Because log x ≈ x−1 for x ≈ 1, the log return Rt is close to the “ordi-
nary” return St+δt/St − 1, if δ is small. If we make this approximation and
still assume the normal model for the return, then we are in the situation
of Section 9.2, with µt = µδt and σt = σ
√
δt. The resulting formula (9.3) is
identical to the formula obtained by replacing the exponential in (9.5) by
its linear approximation.
The value process of a combined portfolio consisting of φi
t assets of
price Si
t is (i = 1, 2) is given by Vt = φ1
t S1
t + φ2
t S2
t . If we assume that the
numbers φt
i do not change in the interval [t, t + δt], then the gain in this
interval is given by
Vt+δt − Vt = φ1
t (S1
t+δt − S1
t ) + φ2
t (S2
t+δt − S2
t ).
To determine VaR we need a model for the conditional distribution of the
vector (S1
t+δt − S1
t , S2
t+δt − S2
t ). There are many possibilities.
A natural generalization of the Black-Scholes model would be to as-
sume that both asset price processes follow Black-Scholes models Si
t =
Si
0 exp(µi
t + σi
Wi
t ). Here W1
and W2
are Brownian motions, of which it
would be natural to assume that they are also jointly Gaussian with some
correlation. Then the joint returns (R1
t , R2
t ), for Ri
t = log Si
t+δt/Si
t, will be
bivariate Gaussian, and we can compute the VaR in terms of the parameters
µi
, σi
and the correlation of R1
t and R2
t , at least by computer simulation.
If, as before, we simplify by assuming that the returns, and not the
log returns, are bivariate Gaussian, then we shall be in the situation of
Section 9.2. The VaR is then given by (9.3) with µt and σt given by (9.4),
where wi
t = φi
tSi
t/Vt.
Alternatively, we may apply more realistic, but more complicated mod-
els. The conditional distribution of the loss will then typically be non-
Gaussian, and not analytically tractable, but the Value-at-Risk can often
be obtained easily by simulation.
Deriving VaR of portfolios of more than two stocks does not cause
conceptual difficulties. However, making realistic models for the joint dis-
tribution of many equities is difficult. Gaussian models are a possibility,
but unrealistic. Other standard models may include many parameters, that
may be difficult to estimate.

9.4: Portfolios with Stock Options 77
9.4 Portfolios with Stock Options
In the Black-Scholes model a European call option with strike K and expiry
time T has value
Ct = StΦ
log(St/K) + (r + σ2
/2)(T − t)
σ
√
T − t
− Ke−r(T −t)
Φ
log(St/K) + (r − σ2
/2)(T − t)
σ
√
T − t
.
The distribution of this random variable, or, more appropriately, the con-
ditional distribution of Ct+δt given Ft, is not easily obtainable by analytic
methods, but it is easy to simulate, being an explicit function of the stock
price St (or St+δt) and a number of nonrandom quantities. Thus the VaR
of a portfolio consisting of a European call option can be easily obtained
numerically.
The value of some other European options on a given stock with price
process St can also be written as an explicit function Ct = F(t, St) of the
stock price. The Value-at-Risk can then also be obtained by one round of
computer simulation. Given Ft the stock price St is known and hence the
gain Ct+δt −Ct = F(t+δt, St+δt)−F(t, St) is stochastic only through St+δt.
We can simulate the conditional distribution of the gain by simulating a
sample from the conditional distribution of St+δt given Ft. In the Black-
Scholes model we have that
St+δt = Steµδt+σ(Wt+δt−Wt)
,
and hence an appropriate simulation scheme is to simulate a standard nor-
mal variable Z and next compute Steµδt+σ
√
δtZ
.
Even though this seems easy enough, in practice one often uses ap-
proximations of the form
Ct+δt−Ct = F(t + δt, St+δt) − F(t, St)
≈ Ft(t, St)δt + Fs(t, St)(St+δt − St) + 1
2 Fss(t, St)(St+δt − St)2
.
The three partial derivatives on the right side are exactly the “Greeks” Θ,
∆, and Γ, already encountered in Section 6.5. For small δt, the increment
St+δt − St is typically of the order O(
√
δt) in probability, and hence the
middle term on the right dominates. If we neglect the other two terms,
then we arrive in the pleasant situation that the gain Ct+δt − Ct is a linear
transformation ∆ StRt of the return Rt = St+δt/St − 1. If we also assume
that this return is conditionally normally distributed, then we are back in
the situation of Section 9.2, and VaR takes a familiar form.
Options for which there is no explicit pricing formula are more diﬃcult
to incorporate. According to the general pricing theory, the value of an
option with payment C at time T in the Black-Scholes model is equal to
Ct = EQ e−r(T −t)
C| Ft .

For complicated claims C this could be determined numerically by sim-
ulation, this time under the martingale measure Q. Combined with the
preceding this leads to a double (nested) simulation scheme, sometimes re-
ferred to as the “full monte”. We shall consider the most complicated case,
that of a claim that depends on the full path (St: 0 ≤ t ≤ T) of the stock
price.
Given the information Ft, the beginning (Ss: 0 ≤ s ≤ t) of the sample
path is known and hence given Ft the claim can be written as a function
C = ht(Ss: t < s ≤ T) of the future of the path. Therefore we can simulate
the value of the option given Ft by simulating many times the future path
(Ss: t < s ≤ T) from its conditional distribution given Ft, evaluating each
time ht(Ss: t < s ≤ T), and finally taking the average of these values. In
this round of simulations we simulate from the martingale measure. If the
stock price process S is Markovian, then the conditional distribution of
(Ss: t < s ≤ T) given Ft is the same as the conditional distribution of this
process given St. For instance, for the Black-Scholes model we have
Ss = Ste(r−σ2
/2)(s−t)+σ( ˜Ws− ˜Wt)
,
for a Brownian motion ˜W. The conditional distribution of the right side is
determined from the conditional distribution of ˜Ws − ˜Wt given Ft, which is
simply that of a Brownian motion. Note that we have taken the drift equal
to r − σ2
/2, because we must simulate under the martingale measure.
We can now perform a nested sequence of simulations to determine the
Value-at-Risk of a portfolio consisting of options and stocks as follows. We
denote the true-world and martingale measures by P and Q, respectively,
and abbreviate the value of the claim C given the initial path (Ss: 0 ≤ s ≤ t)
by ht(Ss: t < s ≤ T).
FOR (i in 1:MANY)
{
SIMULATE (Si
s: 0 ≤ s ≤ t + δt) ACCORDING TO P
FOR (j in 1:MANY)
{
GIVEN Si
t SIMULATE (Sj
s: t < s ≤ T) ACCORDING TO Q
GIVEN Si
t+δt SIMULATE (Sj
s: t + δt < s ≤ T) ACCORDING TO Q
}
COMPUTE Ci
t AS AVERAGE ht(Sj
s: t < s ≤ T) OVER j
COMPUTE Ci
t+δt AS AVERAGE ht+δt(Sj
s: t + δt < s ≤ T) OVER j
}
VaR IS 1 − α LARGEST OF THE VALUES Ci
t − Ci
t+δt.
This scheme is sufficiently involved that it will pay to use special com-
putational techniques to make the simulations more efficient and more ac-
curate.

9.5: Bond Portfolios 79
9.5 Bond Portfolios
The value of a bond portfolio, consisting of bonds of different maturity, with
or without coupons, is a linear combination Vt =
n
i=1φi
tPt,Ti
of discount
bond prices Pt,Ti
. A term structure model gives exactly the joint distri-
bution of the zero-coupon bonds. Thus in principle every term structure
model allows to calculate the VaR of the portfolio, if necessary by simula-
tion. In the present situation we need the term structure model under the
true world measure P!
As a particular example, consider the Hull-White model. In this model
the bond prices are given by an explicit formula of the form
Pt,T = eA(t,T )−B(t,T )rt
.
Thus we can simulate the bond price at time t by simulating the short
rate rt. Under the martingale measure Q the short rate is the sum of a
deterministic function and an Ornstein-Uhlenbeck process. Unfortunately,
to compute VaR we need to simulate under the true-world measure P,
which may add a random drift to the Ornstein-Uhlenbeck process. This
may destroy the Gaussianity and other nice properties, which the short
rate process possesses under the martingale measure.
More specifically, in the Hull-White model the short rate satisfies for-
mula (8.14), which can be written in the form
rt = α(t) + σe−at
t
0
eas
dWs,
for α the deterministic function α(t) = e−at
r0 + e−at t
0
θ(s)eas
ds. The
function θ in this expression can be found by calibration on option prices
observed in the market, and so can the parameters a and σ. However,
the process W in the preceding display is a Brownian motion under the
martingale measure Q, and not under P. In agreement with Girsanov’s
theorem, under P the process W is a Brownian motion with drift, and
Wt −
t
0
λ(s, rs) ds for λ the “market price of risk” is a P-Brownian motion.
There is no way the market price of risk can be calibrated from derivative
prices alone, but it can be determined by historical analysis. If λ(t, rt) does
not depend on rt, then the change of drift only changes the deterministic
function α in the preceding display, and we can use the simulation scheme
for Ornstein-Uhlenbeck processes discussed in Section 8.4 and Theorem 8.22
to generate rt, the bond prices, and hence the VaR. If λ(t, rt) is random,
then the drift of the short rate process under P is random, and we must fall
back on the more complicated simulation schemes for diffusion processes,
such as the Euler scheme discussed in Section 8.4. Rather than calibrate
a and θ from observed derivative prices on the market, we may then also
choose to fit the diffusion model
drt = µ(t, rt) dt + σ dWt,

with W a P-Brownian motion, directly to historical data. Note that the
volatility parameter σ is common to both the P and Q models and hence
can both be calibrated and estimated.
This discussion of the Hull-White model extends to any model where
the bond prices are simple functions Pt,T = FT
(t, rt) of the short rate,
and more generally to multi-factor models in which the bond prices Pt,T =
FT
(t, Xt) can be written as a simple function of multivariate diﬀusion pro-
cess X.
In practice one often uses simpler approaches based on approxima-
tions and the assumption that the yields are multivariate normal. Given a
bond portfolio with value process Vt =
n
i=1φi
tPt,Ti
, where φi
t is assumed
constant in [t, t + δt], we ﬁrst approximate
Vt+δt − Vt =
n
i=1
φi
t(Pt+δt,Ti
− Pt,Ti
)
=
n
i=1
φi
t e−(Ti−t−δt)Yt+δt,Ti − e−(Ti−t)Yt,Ti
≈
n
i=1
φi
tPt,Ti −(Ti − t)(Yt+δt,Ti − Yt,Ti ) + δtYt+δt,Ti
+ 1
2 (Ti − t)2
(Yt+δt,Ti
− Yt,Ti
)2
.
In the last step we use the approximation ey
−ex
≈ ex
(y−x+ 1
2 (y−x)2
) for
y ≈ x. The derivatives in the linear and quadratic parts are known as the
duration and the convexity, respectively. The conditional distribution of the
right side given Ft can be evaluated once we have a model for the conditional
joint distribution of the vector of yield increments (Yt+δt,Ti − Yt,Ti ) given
Ft. In practice this is often modelled by a mean zero multivariate normal
distribution. If we neglect the quadratic term (“use only duration”), then
we fall back in the situation of Section 9.2. VaR then takes the form as
given in (9.3):
VaR = Φ−1
(1 − α)
n
i=1
n
j=1
φi
tφj
t Pt,Ti
Pt,Tj
(Ti − t)(Tj − t)σi,j
t .
VaR based on using both duration and convexity in the approximation can
be determined by simulation.
9.6 Portfolios of Bonds and Swaptions
A swaption relative to the swap times T1 < T2 < · · · < Tn pays the amount
PT0,Tn + K
n
i=1PT0,Ti − 1
+
at time T0 < T1. Depending on the term

9.6: Portfolios of Bonds and Swaptions 81
structure model used, there may or may not be an explicit formula for the
value of a swaption at time t < t0. If there is, then determining the VaR
does not present great diﬃculties.
In general, the value at time t < T0 can be evaluated as
Ct = EQ e
−
T0
t
rs ds
PT0,Tn
+ K
n
i=1
PT0,Ti
− 1
+
| Ft .
In short rate models, such as the Hull-White model, the variable inside the
expectation can be written as a function of the short rate process (rs: 0 ≤
t ≤ Tn). Given Ft the initial path (rs: 0 ≤ s ≤ t) is known and hence we can
evaluate the price Ct by computing an expectation under Q of a function
ht(rs: t ≤ s ≤ T) of the future sample path, given its present state rt. For
instance, in the Hull-White model Ct is given by
EQ e
−
T0
t
rs ds
eA(T0,Tn)−B(T0,Tn)rT0 +K
n
i=1
eA(T0,Ti)−B(T0,Ti)rT0 −1
+
| r0 .
In this particular case it is possible to evaluate the expectation analytically
(at least for n = 1), using the approach of Lemma 8.12. In general, we need
to use simulation or approximations.
To compute the VaR of a swaption portfolio, we need a double, nested
round of simulations, one under the true-world measure and one under the
martingale measure. If the bond price Pt,Ti
can be expressed in rTi
and the
value of the swaption at time t can be written as
EQ ht(rs: t < s ≤ T0)| rt ,
then a simple (but computationally ineﬃcient) scheme is as follows.
FOR (i in 1:MANY)
{
SIMULATE (rs: 0 ≤ s ≤ t + δt) ACCORDING TO P
FOR (j in 1:MANY)
{
GIVEN ri
t SIMULATE (rj
s: t < s ≤ T0) ACCORDING TO Q
GIVEN ri
t+δt SIMULATE (rj
s: t + δt < s ≤ Tn) ACCORDING TO Q
}
COMPUTE Ci
t AS AVERAGE ht(rj
s: t < s ≤ T0) OVER j
COMPUTE Ci
t+δt AS AVERAGE ht+δt(rj
s: t + δt < s ≤ T0) OVER j
}
VaR IS 1 − α LARGEST OF THE VALUES Ci
t − Ci
t+δt.
Rather than using this double simulation scheme we may here also
apply approximations. However, in general it is not easy to compute partial
derivatives of the value process relative to e.g. the yield (“duration” and

“convexity”), and we may need to use numerical (i.e. discretized) derivatives
instead.
9.7 Diversiﬁed Portfolios
In the preceding section we have considered a variety of portfolios. The
balance sheet of a large ﬁnancial institution will typically include a com-
bination of the assets considered so far. To compute the Value-at-Risk we
can use the same arguments, with the important complication that we need
models for the joint distribution of the various assets. For instance, the joint
distribution of stocks on Philips and IBM, and bonds of various maturities.
There is no standard approach to this.

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