![2.2. The Merton Model 9
We also assume that there exists a money-market account with a constant riskless
rate r whose price evolves deterministically as
βt = exp(rt).
We take it to be well known that in an economy consisting of these two assets, the
price C0 at time 0 of a contingent claim paying C(VT ) at time T is equal to
C0 = EQ
[exp(−rt)CT ],
where Q is the equivalent martingale measure1 under which the dynamics of V are
given as
Vt = V0 exp((r − 1
2 σ2
)t + σW
Q
t ).
Here, WQ is a Brownian motion and we see that the drift µ has been replaced by r.
To better understand this model of a firm, it is useful initially to think of assets
which are very liquid and tangible. For example, the firm could be a holding company
whose only asset is a ton of gold. The price of this asset is clearly the price of a
liquidly traded security. In general, the market value of a firm’s assets is the present
market value of the future cash flows which the firm will deliver—a quantity which
is far from observable in most cases. A critical assumption is that this asset-value
process is given and will not be changed by any financing decisions made by the
firm’s owners.
Now assume that the firm at time 0 has issued two types of claims: debt and
equity. In the simple model, debt is a zero-coupon bond with a face value of D and
maturity date T ⩽ T̄ . With this assumption, the payoffs to debt, BT , and equity, ST ,
at date T are given as
BT = min(D, VT ) = D − max(D − VT , 0), (2.1)
ST = max(VT − D, 0). (2.2)
We think of the firm as being run by the equity owners. At maturity of the bond,
equity holders pay the face value of the debt precisely when the asset value is higher
than the face value of the bond. To be consistent with our assumption that equity
owners cannot alter the given process for the firm’s assets, it is useful to think of
equity owners as paying D out of their own pockets to retain ownership of assets
worth more than D. If assets are worth less than D, equity owners do not want to
pay D, and since they have limited liability they do not have to either. Bond holders
then take over the remaining asset and receive a “recovery” of VT instead of the
promised payment D.
1We assume familiarity with the notion of an equivalent martingale measure, or risk-neutral measure,
and its relation to the notion of arbitrage-free markets. Appendix D contains further references.](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-32-320.jpg)
![14 2. Corporate Liabilities as Contingent Claims
Note also that while yields on corporate bonds increase when the riskless interest
rate increases, the yield spreads actually decrease. Representing the bond price as
B(r) = V − CBS(r), where we suppress all parameters other than r in the notation,
it is straightforward to check that
y
(r) =
−B(r)
T B(r)
∈ (0, 1)
and therefore s(r) = y(r) − 1 ∈ (−1, 0).
2.2.2 On Short Spreads in the Merton Model
The behavior of yield spreads at the short end of the spectrum in Merton-style models
plays an important role in motivating works which include jump risk. We therefore
now consider the behavior of the risk structure in the short end, i.e. as the time to
maturity goes to 0. The result we show is that when the value of assets is larger than
the face value of debt, the yield spreads go to zero as time to maturity goes to 0 in
the Merton model, i.e.
s(T ) → 0 for T → 0.
It is important to note that this is a consequence of the (fast) rate at which the
probability of ending below D goes to 0. Hence, merely noting that the default
probability itself goes to 0 is not enough.
More precisely, a diffusion process X has the property that for any ε 0,
P(|Xt+h − Xt | ⩾ ε)
h
−
−
−
→
h→0
0.
We will take this for granted here, but see Bhattacharya and Waymire (1990), for
example, for more on this. The result is easy to check for a Brownian motion
and hence also easy to believe for diffusions, which locally look like a Brownian
motion.
We now show why this fact implies 0 spreads in the short end. Note that a zero-
recovery bond paying 1 at maturity h if Vh D and 0 otherwise must have a lower
price and hence a higher yield than the bond with face value D in the Merton model.
Therefore, it is certainly enough to show that this bond’s spread goes to 0 as h → 0.
The price B0 of the zero-recovery bond is (suppressing the starting value V0)
B0
= EQ
[D exp(−rh)1{Vh⩾D}]
= D exp(−rh)Q(Vh ⩾ D),](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-37-320.jpg)
![2.4. The Merton Model with Jumps in Asset Value 21
and let this be the dynamics of the cumulative return for the underlying asset-value
process. As explained in Appendix D, we define the price as the semimartingale
exponential of the return and this gives us
Vt = V0 exp
r − 1
2 σ2
−
hi
λi
t + σWt
0⩽s⩽t
1 +
K
i=1
hi
Ni
s
.
Note that independent Poisson processes never jump simultaneously, so at a time s,
at most one of the Ni
s is different from 0.
Recall that we can get the Black–Scholes partial differential equation (PDE) by
performing the following steps (in the classical setup).
• Write the stochastic differential equation (SDE) of the price process V of the
underlying security under Q.
• Let f be a function of asset value and time representing the value of a con-
tingent claim.
• Use Itô to derive an SDE for f (Vt , t). Identify the drift term and the martingale
part.
• Set the drift equal to rf (Vt , t) dt.
We now perform the equivalent of these steps in our simple jump-diffusion case.
Define λ = λ1 + · · · + λK and let
h̄ =
1
λ
K
i=1
hi
λi
.
Then (under Q)
dVt = Vt {(r + h̄λ) dt + σ dWt } +
K
i=1
hi
Vt− dNi
t .
WenowapplyItôbyusingitseparatelyonthediffusioncomponentandtheindividual
jump components to get
f (Vt , t) − f (V0, 0)
=
t
0
[fV (Vs, s)rVs + ft (Vs, s) − fV (Vs, s)h̄λVs + 1
2 σ2
V 2
s fV V (Vs, s)] ds
+
t
0
fV (Vs, s)σVs dWs +
0⩽s⩽t
{f (Vs) − f (Vs−)}.](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-44-320.jpg)
![22 2. Corporate Liabilities as Contingent Claims
Now write
0⩽s⩽t
{f (Vs) − f (Vs−)} =
K
i=1
t
0
{f (Vs) − f (Vs−)} dNi
s
=
K
i=1
t
0
[f (Vs−(1 + hi
)) − f (Vs−)]λi
ds
+
K
i=1
t
0
[f (Vs−(1 + hi
)) − f (Vs−)] d[Ni
s − λi
s]
and note that we can write s instead of s− in the time index in the first integral
because we are integrating with respect to the Lebesgue measure. In total, we now
get the following drift term for f (Vt , t):4
(r − h̄λ)Vt fV + ft + 1
2 σ2
V 2
t fV V +
K
i=1
{f (Vt (1 + hi
)) − f (Vt )}λi
.
Letting
pi
:= λi
/λ
allows us to write
K
i=1
{f (Vt (1 + hi
)) − f (Vt )}λi
=
K
i=1
{pi
[f (Vt (1 + hi
)) − f (Vt )]}λ
≡ λE f (Vt ),
and our final expression for the term in front of dt is now
(r − h̄λ)Vt fV + ft + 1
2 σ2
V 2
t fV V + λE f (Vt ).
Thisisthetermwehavetosetequaltorf (Vt , t)andsolve(withboundaryconditions)
to get what is called an integro-differential equation. It is not a PDE since, unlike
a PDE, the expressions involve not only f ’s behavior at a point V (including the
behavior of its derivatives), it also takes into account values of f at points V “far
away” (at V (1+hi) for i = 1, . . . , K). Such equations can only be solved explicitly
in very special cases.
We have considered the evolution of V as having only finitely many jumps and
we have derived the integro-differential equation for the price of a contingent claim
in this case. It is straightforward to generalize to a case where jumps (still) arrive
as a Poisson process N with the rate λ but where the jump-size distribution has a
continuous distribution on the interval [−1, ∞) with mean k. If we let 1, 2, . . .
4We omit Vt and t in f .](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-45-320.jpg)
![24 2. Corporate Liabilities as Contingent Claims
0
0.002
0.004
0.006
0.008
Yield
spread
0.005
0.015
0.025
0.035
Yield
spread
0 10 15
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Time to maturity
Yield
spread
S(0) = 130
S(0) = 150
S(0) = 200
S(0) = 130
S(0) = 150
S(0) = 200
S(0) = 130
S(0) = 150
S(0) = 200
5
(a)
(b)
(c)
Figure 2.7. The effect of changing the mean jump size and the intensity in a Merton model
with jumps in asset value. From (a) to (b) we are changing the parameter determining the
mean jump size, γ , from log(0.9) to log(0.5). This makes recovery at an immediate default
lower and hence increases the spread in the short end. From (b) to (c) the intensity is doubled,
and we notice the exact doubling of spreads in the short end, since expected recovery is held
constant but the default probability over the next short time interval is doubled.
We focus first on the short end of the risk structure of interest rates. The price of
the risky bond with face value D maturing at time h (soon to be chosen small) is
B(0, h) = exp(−rh)E[D1{Vh⩾D} + Vh1{VhD}]
= exp(−rh)[DQ(Vh ⩾ D) + E[Vh | Vh D]Q(Vh D)]
= D exp(−rh) 1 − Q(Vh D) +
1
D
E[Vh | Vh D]Q(Vh D)
= D exp(−rh) 1 − Q(Vh D)
1 −
E[Vh | Vh D]
D
.](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-47-320.jpg)
![2.4. The Merton Model with Jumps in Asset Value 25
0 10 15
0
0.02
0.04
0.06
0.08
Yield
spread
S(0) = 130
S(0) = 150
S(0) = 200
0
0.02
0.04
0.06
Time to maturity
Yield
spread
5
(a)
(b)
S(0) = 130
S(0) = 150
S(0) = 200
Figure 2.8. The effect of the source of volatility in a Merton model with jumps in asset
value. (a) The diffusion part has volatility σ = 0.1, and the total quadratic variation is 0.4.
(b) The diffusion part has volatility σ = 0.3, but the total quadratic variation is kept at 0.4
by decreasing λ. In both cases, three different current asset values are considered. Changing
the source of volatility causes significant changes of the yield spreads in the short end for
the high-yield cases. The difference between the spread curves in the case of low leverage is
very small. Effects are also limited in the long end in all cases.
Now computing the yield spread limit as h → 0 and using log(1 − x) ≈ −x for x
close to 0, we find that for s(h) = y(0, h) − r,
lim
h↓0
s(h) = lim
h↓0
Q(Vh D)
h
1 −
E[Vh | Vh D]
D
.
Now as h ↓ 0 there is only at most one jump that can occur. The total jump intensity
is λ, but the probability of a jump being large enough to send Vh below D happens
with a smaller intensity λ∗ = λQ[(1 + ε)V0 D]. We recognize the second term
in the expression as the expected fractional loss given default. Altogether we obtain
lim
h↓0
s(h) = λ∗
E[(V0)],](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-48-320.jpg)
![26 2. Corporate Liabilities as Contingent Claims
where
(V0) = 1 −
E[V0(1 + ε) | V0(1 + ε) D]
D
.
An immediate consequence is that doubling the overall jump intensity should double
the instantaneous spread. Another consequence is, as is intuitively obvious, that
lowering the mean jump size should typically lead to higher spreads. Both facts are
illustrated in Figure 2.7.
When comparing the jump-diffusion model with the standard Merton model, it is
common to “level the playing field” by holding constant the “volatility” in a sense
that we now explain.
The optional quadratic variation of a semimartingale X can be obtained as a limit
[X]t = lim
n→∞
i∈N
[X(tn
i+1 ∧ t) − X(tn
i ∧ t)]2
,
where the grid size in the subdivision goes to 0 as n → ∞.5
From the definition of predictable quadratic variation, X found in Appendix D,
we know that when X has finite variance, E[Xt ] = E Xt , and since the jump-
diffusion process R studied here is a process with independent increments, Rt is
deterministic and we have that
E[Rt ] = Rt = σ2
t + λtEε2
i .
Holding Rt constant for a given t by offsetting changes in σ by changes in λ and/or
Eε2
i gives room for an experiment in which we change the source of volatility. This
is done in Figure 2.8. As is evident from that graph, the main effect is in the short
end of the risk structure of interest rates.
While it is tempting to think of quadratic variation as realized volatility, it is impor-
tant to understand the difference between the volatility arising from the diffusion
and the volatility arising from the jump part. For a fixed t we have
[R]t = Rc
t +
0⩽s⩽t
R2
s ,
where Rc is the continuous part of R. Therefore, when we compute (1/t)[R]t from
an approximating sum, we do not get a limit of σ2 + λEε2
i for fixed t and finer
subdivisions. When our time horizon is fixed, we will always have the random
component 0⩽s⩽t R2
s , and if jumps are rare this need not be close to λEε2
i .
5The limit is to be understood in the sense of uniform convergence in probability, i.e. on a finite
interval [0, t], if the enemy shows up with small 1 0 and 2 0 then we can choose N large enough
so that for n N the probability of the approximating sum deviating more than 1 anywhere on the
interval is smaller than 2.](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-49-320.jpg)
![2.5. Discrete Coupons in a Merton Model 27
However, as t → ∞ we have
1
t
0⩽s⩽t
R2
s → λEε2
i .
This highlights an important difference between diffusion-induced and jump-in-
duced volatility. We cannot obtain the jump-induced volatility, even theoretically, as
our observations get closer and closer in time. Observing the whole sample path in
[0, t]wouldallowustosingleoutthejumpsandthenobtain Rct exactly.Inpractice,
we do not have the exact jumps, and filtering out the jumps from the volatility based
on discrete observations is a difficult exercise. So, while the jump-diffusion model
is excellent for illustration and simulating the effects of jumps, the problems in
estimating the model make it less attractive in practical risk management.
2.5 Discrete Coupons in a Merton Model
As mentioned earlier we cannot use the Merton model for zero-coupon debt to price
coupon debt simply by pricing each component of the bond separately. The pricing
of the coupon bond needs to look at all coupon payments in a single model and in
this context our assumptions on asset sales become critical.
Tounderstandtheproblemandseehowtoimplementapricingalgorithm,consider
a coupon bond with two coupons D1 and D2 which have to be paid at dates t1 and t2.
For t t1, if the firm is still alive and the assets are worth Vt , we can value the
only remaining coupon D2 simply using the standard Merton model, so
B(Vt , t) = D2p(t, t2) − P BS
(Vt , D2, t2 − t)
for t t1. The situation at date t1 is more complicated and it critically depends on
the assumptions we make on what equity owners, who control the firm, are allowed
to do with the firm’s assets.
First, assume that equity owners are not allowed to use the firm’s assets to pay
debt. This means that they have to finance the debt payment either by paying “out
of their own pockets” or by issuing new equity to finance the coupon payment. In
this simple model with no information asymmetries, it does not matter which option
they choose. If they issue M new shares of stock in addition to the (say) N existing
shares, they will raise an amount equal to (M/(M + N))St1 , where St is the total
value of equity at time t. Hence, to finance D1 they need to choose M so that
M
M + N
St1 = D1,
thereby diluting the value of their own equity from St1 to (N/(M + N))St1 . This
dilution causes a fall in their equity value of St1 − D1, and so if they do not pay D1
out of their own pockets they lose D1 through dilution of equity. Hence it does not](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-50-320.jpg)
![30 2. Corporate Liabilities as Contingent Claims
due to liquidity constraints where we approximate frequent small coupon payments
by a continuous stream of payments.
First-passage times for diffusions have been heavily studied. If one is looking for
closed-form solutions, it is hard to go much beyond Brownian motion hitting a linear
boundary (although there are a few extensions, as mentioned in the bibliographical
notes). This mathematical fact almost dictates the type of boundary for asset value
that we are interested in, namely boundaries that bring us back into the familiar
case after we take logarithms. So, in their concrete model, Black and Cox consider
a process for asset value which under the risk-neutral measure is
dVt = (r − a)Vt dt + σVt dWt ,
where we have allowed for a continuous dividend payout ratio of a. The default
boundary is given as
C1(t) = C exp(−γ (T − t)).
Assume that the bond issued by the firm has principal D and that C D. Note that
since Vt = V0 exp((r − a)t − 1
2 σ2t + σWt ), the default time τ is given as
τ = inf{0 ⩽ t ⩽ T : log V0 + ((r − a) − 1
2 σ2
)t + σWt = log C − γ (T − t)}
= inf{0 ⩽ t ⩽ T : σWt + (r − a − 1
2 σ2
− γ )t = log C − log V0 − γ T },
i.e. the first time a Brownian motion with drift hits a certain level.
In the Black–Cox model the payoff to bond holders at the maturity date is
B(VT , T ) = min(VT , D)1{τT },
corresponding to the usual payoff when the boundary has not been crossed in [0, T ].
To simplify notation, let the current date be 0 so that the maturity date T is also time
to maturity. We let
Bm
(V, T, D, C, γ ) = E(exp(−rT ) min(VT , D)1{τT })
denote the value at time 0 of the payoff of the bond at maturity when the face value
is D and the function C1(·) is specified as a function of C and γ as above.
If the boundary is hit before the maturity of the bond, bond holders take over the
firm, i.e.
B(Vτ , τ) = C1(τ)1{τ⩽T }.
With the same conventions as above, we let
Bb
(V, T, D, C, γ ) = E(exp(−rτ)C1(τ)1{τ⩽T })
denote the value at time 0 of the payoff to the bond holders in the event that the
boundary is hit before maturity. We assume that the starting value V is above C1.
We will value the contribution from these two parts separately.](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-53-320.jpg)
![PRINCIPLES OF TAXATION.[6]
By the Late Hon. DAVID A. WELLS.
XX.—THE LAW OF THE DIFFUSION OF TAXES.
PART I.
No attempt ought to be made to construct or formulate an
economically correct, equitable, and efficient system of taxation
which does not give full consideration to the method or extent to
which taxes diffuse themselves after their first incidence. On this
subject there is a great difference of opinion, which has occasioned,
for more than a century, a vast and never-ending discussion on the
part of economic writers. All of this, however, has resulted in no
generally accepted practical conclusions; has been truthfully
characterized by a leading French economist (M. Parieu) as marked
in no small part by the simplicity of ignorance, and from a
somewhat complete review (recently published[7]) of the conflicting
theories advanced by participants one rises with a feeling of
weariness and disgust.
The majority of economists, legislators, and the public generally
incline to the opinion that taxes mainly rest where they are laid, and
are not shifted or diffused to an extent that requires any recognition
in the enactment of statutes for their assessment. Thus, a tax
commission of Massachusetts, as the result of their investigations,
arrived at the conclusion that the tendency of taxes is that they
must be paid by the actual persons on whom they are levied. But a
little thought must, however, make clear that unless the
advancement of taxes and their final and actual payment are one
and the same thing, the Massachusetts statement is simply an](https://image.slidesharecdn.com/17375582-250528182759-77227b20/85/Credit-Risk-Modeling-Theory-And-Applications-Theory-And-Applications-David-Lando-77-320.jpg)
Credit Risk Modeling Theory And Applications Theory And Applications David Lando
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- 7. Credit Risk Modeling: Theory and Applications is a part of the Princeton Series in Finance Series Editors Darrell Duffie Stephen Schaefer Stanford University London Business School Finance as a discipline has been growing rapidly. The numbers of researchers in academy and industry, of students, of methods and models have all proliferated in the past decade or so. This growth and diversity manifests itself in the emerging cross-disciplinary as well as cross-national mix of scholarship now driving the field of finance forward. The intellectual roots of modern finance, as well as the branches, will be represented in the Princeton Series in Finance. Titles in this series will be scholarly and professional books, intended to be read by a mixed audience of economists, mathematicians, operations research scien- tists, financial engineers, and other investment professionals. The goal is to pro- vide the finest cross-disciplinary work in all areas of finance by widely recognized researchers in the prime of their creative careers. Other Books in This Series Financial Econometrics: Problems, Models, and Methods by Christian Gourieroux and Joann Jasiak Credit Risk: Pricing, Measurement, and Management by Darrell Duffie and Kenneth J. Singleton Microfoundations of Financial Economics: An Introduction to General Equilibrium Asset Pricing by Yvan Lengwiler
- 8. Credit Risk Modeling Theory and Applications David Lando Princeton University Press Princeton and Oxford
- 9. Copyright c 2004 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All rights reserved Library of Congress Cataloguing-in-Publication Data Lando, David, 1964– Credit risk modeling: theory and applications / David Lando. p.cm.—(Princeton series in finance) Includes bibliographical references and index. ISBN 0-691-08929-9 (cl : alk. paper) 1. Credit—Management. 2. Risk management. 3. Financial management. I. Title. II. Series. HG3751.L36 2004 332.7011—dc22 2003068990 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library This book has been composed in Times and typeset by TT Productions Ltd, London Printed on acid-free paper ∞ www.pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
- 10. For Frederik
- 12. Contents Preface xi 1 An Overview 1 2 Corporate Liabilities as Contingent Claims 7 2.1 Introduction 7 2.2 The Merton Model 8 2.3 The Merton Model with Stochastic Interest Rates 17 2.4 The Merton Model with Jumps in Asset Value 20 2.5 Discrete Coupons in a Merton Model 27 2.6 Default Barriers: the Black–Cox Setup 29 2.7 Continuous Coupons and Perpetual Debt 34 2.8 Stochastic Interest Rates and Jumps with Barriers 36 2.9 A Numerical Scheme when Transition Densities are Known 40 2.10 Towards Dynamic Capital Structure: Stationary Leverage Ratios 41 2.11 Estimating Asset Value and Volatility 42 2.12 On the KMV Approach 48 2.13 The Trouble with the Credit Curve 51 2.14 Bibliographical Notes 54 3 Endogenous Default Boundaries and Optimal Capital Structure 59 3.1 Leland’s Model 60 3.2 A Model with a Maturity Structure 64 3.3 EBIT-Based Models 66 3.4 A Model with Strategic Debt Service 70 3.5 Bibliographical Notes 72 4 Statistical Techniques for Analyzing Defaults 75 4.1 Credit Scoring Using Logistic Regression 75 4.2 Credit Scoring Using Discriminant Analysis 77 4.3 Hazard Regressions: Discrete Case 81 4.4 Continuous-Time Survival Analysis Methods 83 4.5 Markov Chains and Transition-Probability Estimation 87 4.6 The Difference between Discrete and Continuous 93 4.7 A Word of Warning on the Markov Assumption 97
- 13. viii Contents 4.8 Ordered Probits and Ratings 102 4.9 Cumulative Accuracy Profiles 104 4.10 Bibliographical Notes 106 5 Intensity Modeling 109 5.1 What Is an Intensity Model? 111 5.2 The Cox Process Construction of a Single Jump Time 112 5.3 A Few Useful Technical Results 114 5.4 The Martingale Property 115 5.5 Extending the Scope of the Cox Specification 116 5.6 Recovery of Market Value 117 5.7 Notes on Recovery Assumptions 120 5.8 Correlation in Affine Specifications 122 5.9 Interacting Intensities 126 5.10 The Role of Incomplete Information 128 5.11 Risk Premiums in Intensity-Based Models 133 5.12 The Estimation of Intensity Models 139 5.13 The Trouble with the Term Structure of Credit Spreads 142 5.14 Bibliographical Notes 143 6 Rating-Based Term-Structure Models 145 6.1 Introduction 145 6.2 A Markovian Model for Rating-Based Term Structures 145 6.3 An Example of Calibration 152 6.4 Class-Dependent Recovery 155 6.5 Fractional Recovery of Market Value in the Markov Model 157 6.6 A Generalized Markovian Model 159 6.7 A System of PDEs for the General Specification 162 6.8 Using Thresholds Instead of a Markov Chain 164 6.9 The Trouble with Pricing Based on Ratings 166 6.10 Bibliographical Notes 166 7 Credit Risk and Interest-Rate Swaps 169 7.1 LIBOR 170 7.2 A Useful Starting Point 170 7.3 Fixed–Floating Spreads and the “Comparative-Advantage Story” 171 7.4 Why LIBOR and Counterparty Credit Risk Complicate Things 176 7.5 Valuation with Counterparty Risk 178 7.6 Netting and the Nonlinearity of Actual Cash Flows: a Simple Example 182 7.7 Back to Linearity: Using Different Discount Factors 183 7.8 The Swap Spread versus the Corporate-Bond Spread 189 7.9 On the Swap Rate, Repo Rates, and the Riskless Rate 192 7.10 Bibliographical Notes 194 8 Credit Default Swaps, CDOs, and Related Products 197 8.1 Some Basic Terminology 197 8.2 Decomposing the Credit Default Swap 201 8.3 Asset Swaps 204 8.4 Pricing the Default Swap 206
- 14. Contents ix 8.5 Some Differences between CDS Spreads and Bond Spreads 208 8.6 A First-to-Default Calculation 209 8.7 A Decomposition of m-of-n-to-Default Swaps 211 8.8 Bibliographical Notes 212 9 Modeling Dependent Defaults 213 9.1 Some Preliminary Remarks on Correlation and Dependence 214 9.2 Homogeneous Loan Portfolios 216 9.3 Asset-Value Correlation and Intensity Correlation 233 9.4 The Copula Approach 242 9.5 Network Dependence 245 9.6 Bibliographical Notes 249 Appendix A Discrete-Time Implementation 251 A.1 The Discrete-Time, Finite-State-Space Model 251 A.2 Equivalent Martingale Measures 252 A.3 The Binomial Implementation of Option-Based Models 255 A.4 Term-Structure Modeling Using Trees 256 A.5 Bibliographical Notes 257 Appendix B Some Results Related to Brownian Motion 259 B.1 Boundary Hitting Times 259 B.2 Valuing a Boundary Payment when the Contract Has Finite Maturity 260 B.3 Present Values Associated with Brownian Motion 261 B.4 Bibliographical Notes 265 Appendix C Markov Chains 267 C.1 Discrete-Time Markov Chains 267 C.2 Continuous-Time Markov Chains 268 C.3 Bibliographical Notes 273 Appendix D Stochastic Calculus for Jump-Diffusions 275 D.1 The Poisson Process 275 D.2 A Fundamental Martingale 276 D.3 The Stochastic Integral and Itô’s Formula for a Jump Process 276 D.4 The General Itô Formula for Semimartingales 278 D.5 The Semimartingale Exponential 278 D.6 Special Semimartingales 279 D.7 Local Characteristics and Equivalent Martingale Measures 282 D.8 Asset Pricing and Risk Premiums for Special Semimartingales 286 D.9 Two Examples 288 D.10 Bibliographical Notes 290 Appendix E A Term-Structure Workhorse 291 References 297 Index 307
- 16. Preface In September 2002 I was fortunate to be on the scientific committee of a confer- ence in Venice devoted to the analysis of corporate default and credit risk mod- eling in general. The conference put out a call for papers and received close to 100 submissions—an impressive amount for what is only a subfield of financial economics. The homepage www.defaultrisk.com, maintained by Greg Gupton, has close to 500 downloadable working papers related to credit risk. In addition to these papers, there are of course a very large number of published papers in this area. These observations serve two purposes. First, they are the basis of a disclaimer: this book is not an encyclopedic treatment of all contributions to credit risk. I am nervously aware that I must have overlooked important contributions. I hope that the overwhelming amount of material provides some excuse for this. But I have of course also chosen what to emphasize. The most important purpose of the book is to deliver what I think are the central themes of the literature, emphasizing “the basic idea,” or the mathematical structure, one must know to appreciate it. After this, I hope the reader will be better at tackling the literature on his or her own. The second purpose of my introductory statistics is of course to emphasize the increasing popularity of the research area. The most important reasons for this increase, I think, are found in the financial industry. First, the Basel Committee is in the process of formulating Basel II, the revision of the Basel Capital Accord, which among other things reforms the way in which the solvency requirements for financial institutions are defined and what good risk-management practices are. During this process there has been tremendous focus on what models are really able to do in the credit risk area at this time. Although it is unclear at this point precisely what Basel II will bring, there is little doubt that it will leave more room for financial institutions to develop “internal models” of the risk of their credit exposures. The hope that these models will better account for portfolio effects and direct hedges and therefore in turn lower the capital requirements has led banks to devote a significant proportion of their resources to credit risk modeling efforts. A second factor is the booming market for credit- related asset-backed securities and credit derivatives which present a new “land of opportunity” for structural finance. The development of these markets is also largely driven by the desire of financial institutions to hedge credit exposures. Finally, with (at least until recently) lower issuance rates for treasury securities and low yields, corporate bond issues have gained increased focus from fund managers.
- 17. xii Preface Thisdrivefromthepracticalsidetodevelopmodelshasattractedmanyacademics; a large number due to the fact that so many professions can (and do) contribute to the development of the field. The strong interaction between industry and academics is the real advantage of the area: it provides an important reality check and, contrary to what one might expect, not just for the academic side. While it is true that our models improve by being confronted with questions of implementability and estimability and observability, it is certainly also true that generally accepted, but wrong or inconsistent, ways of reasoning in the financial sector can be replaced by coherent ways of thinking. This interaction defines a guiding principle for my choice of which models to present. Some models are included because they can be implemented in practice, i.e. the parameters can be estimated from real data and the parameters have clear inter- pretations. Other models are included mainly because they are useful for thinking consistently about markets and prices. How can a book filled with mathematical symbols and equations be an attempt to strengthen the interaction between the academic and practitioner sides? The answer is simply that a good discussion of the issues must have a firm basis in the models. The importance of understanding models (including their limitations, of course) and having a model-based foundation cannot be overemphasized. It is impossible, for example, to discuss what we would expect the shape of the credit-spread curve to be as a function of varying credit quality without an arsenal of models. Of course, we need to worry about which are good models and which are bad models. This depends to some extent on the purpose of the model. In a perfect world, we obtain models which • have economic content, from which nontrivial consequences can be deducted; • are mathematically tractable, i.e. one can compute prices and other expres- sions analytically and derive sensitivities to changes in different parameters; • have inputs and parameters of the models which can be observed and esti- mated—the parameters are interpretable and reveal properties of the data which we can understand. Of course, it is rare that we achieve everything in one model. Some models are primarily useful for clarifying our conceptual thinking. These models are intended to define and understand phenomena more clearly without worrying too much about the exact quantitative predictions. By isolating a few critical phenomena in stylized models, we structure our thinking and pose sharper questions. The more practically oriented models serve mainly to assist us in quantitative analysis, which we need for pricing contracts and measuring risk. These models often make heroic assumptions on distributions of quantities, which are taken as
- 18. Preface xiii exogenous in the models. But even heroic assumptions provide insights as long as we vary them and analyze their consequences rigorously. The need for conceptual clarity and the need for practicality place different demands on models. An example from my own research, the model we will meet in Chapter 7, views an intensity model as a structural model with incomplete infor- mation, and clarifies the sense in which an intensity model can arise from a struc- tural model with incomplete information. Its practicality is limited at this stage. On the other hand, some of the rating-based models that we will encounter are of practical use but they do not change our way of thinking about corporate debt or derivatives. The fact is that in real markets there are rating triggers and other rating- related covenants in debt contracts and other financial contracts which necessitate an explicit modeling of default risk from a rating perspective. In these models, we make assumptions about ratings which are to a large extent motivated by the desire to be able to handle calculations. The ability to quickly set up a model which allows one to experiment with different assumptions calls for a good collection of workhorses. I have included a collection of tools here which I view as indispensable workhorses. This includes option-based techniques including the time-independent solutions to perpetual claims, techniques for Markov chains, Cox processes, and affine specifications. Mastering these tech- niques will provide a nice toolbox. Whenwewriteacademicpapers,wetrytofitourcontributionintoaperceivedvoid in the literature. The significance of the contribution is closely correlated with the amount of squeezing needed to achieve the fit. A book is of course a different game. Some monographs use the opportunity to show in detail all the stuff that editors would not allow (for reasons of space) to be published. These can be extremely valuable in teaching the reader all the details of proofs, thereby making sure that the subtleties of proof techniques are mastered. This monograph does almost the opposite: it takes the liberty of not proving very much and worrying mainly about model structure. Someone interested in mathematical rigor will either get upset with the format, which is about as far from theorem–proof as you can get, or, I am hoping, find here an application-driven motivation for studying the mathematical structure. In short, this book is my way of introducing the area to my intended audience. There are several other books in the area—such as Ammann (2002), Arvanitis and Gregory (2001), Bielecki and Rutkowski (2002), Bluhm et al. (2002), Cossin and Pirotte (2001), Duffie and Singleton (2003), and Schönbucher (2003)—and overlaps of material are inevitable, but I would not have written the book if I did not think it added another perspective on the topic. I hope of course that my readers will agree. The original papers on which the book are based are listed in the bibliography. I have attempted to relegate as many references as possible to the notes since the long quotes of who did what easily break the rhythm.
- 19. xiv Preface So who is my intended audience? In short, the book targets a level suitable for a follow-up course on fixed-income modeling dedicated to credit risk. Hence, the “core” reader is a person familiar with the Black–Scholes–Merton model of option- pricing, term-structure models such as those of Vasicek and Cox–Ingersoll–Ross, who has seen stochastic calculus for diffusion processes and for whom the notion of an equivalent martingale measure is familiar.Advanced Master’s level students in the many financial engineering and financial mathematics programs which have arisen over the last decade, PhD students with a quantitative focus, and “quants” working in the finance industry I hope fit this description. Stochastic calculus involving jump processes, including state price densities for processes with jumps, is not assumed to be familiar. It is my experience from teaching that there are many advanced students who are comfortable with Black–Scholes-type modeling but are much less comfortable with the mathematics of jump processes and their use in credit risk modeling. For this reader I have tried to include some stochastic calculus for jump processes as well as a small amount of general semimartingale theory, which I think is useful for studying the area further. For years I have been bothered by the fact that there are some extremely general results available for semimartingales which could be useful to people working with models, but whenever a concrete model is at work, it is extremely hard to see whether it is covered by the general theory. The powerful results are simply not that accessible. I have included a few rather wimpy results, compared with what can be done, but I hope they require much less time to grasp. I also hope that they help the reader identify some questions addressed by the general theory. Iamalsohopingthatthebookgivesausefulsurveytoriskmanagersandregulators who need to know which methods are in use but who are not as deeply involved in implementation of the models. There are many sections which require less technical background and which should be self-contained. Large parts of the section on rating estimation, and on dependent defaults, make no use of stochastic calculus. I have tried to boil down the technical sections to the key concepts and results. Often the reader will have to consult additional sources to appreciate the details. I find it useful in my own research to learn what a strand of work “essentially does” since this gives a good indication of whether one wants to dive in further. The book tries in many cases to give an overview of the essentials. This runs the risk of superficiality but at least readers who find the material technically demanding will see which core techniques they need to master. This can then guide the effort in learning the necessary techniques, or provide help in hiring assistants with the right qualifications. There are many people to whom I owe thanks for helping me learn about credit risk. The topic caught my interest when I was a PhD student at Cornell and heard talks by Bob Jarrow, Dilip Madan, and Robert Littermann at the Derivatives Sym-
- 20. Preface xv posium in Kingston, Ontario, in 1993. In the work which became my thesis I received a lot of encouragement from my thesis advisor, Bob Jarrow, who knew that credit risk would become an important area and kept saying so. The support from my committee members, Rick Durrett, Sid Resnick, and Marty Wells, was also highly appreciated. Since then, several co-authors and co-workers in addition to Bob have helped me understand the topic, including useful technical tools, better. They are Jens Christensen, Peter Ove Christensen, Darrell Duffie, Peter Fledelius, Peter Feldhütter, Jacob Gyntelberg, Christian Riis Flor, Ernst Hansen, Brian Huge, Søren Kyhl, Kristian Miltersen, Allan Mortensen, Jens Perch Nielsen, Torben Skødeberg, Stuart Turnbull, and Fan Yu. In the process of writing this book, I have received exceptional assistance from Jens Christensen. He produced the vast majority of graphs in this book; his explicit solution for the affine jump-diffusion model forms the core of the last appendix; and his assistance in reading, computing, checking, and criticizing earlier proofs has been phenomenal. I have also been allowed to use graphs produced by Peter Feldhütter, Peter Fledelius, and Rolf Poulsen. My friends associated with the CCEFM in Vienna—Stefan Pichler, Wolfgang Ausenegg, Stefan Kossmeier, and Joseph Zechner—have given me the opportunity to teach a week-long course in credit risk every year for the last four years. Both the teaching and the Heurigen visits have been a source of inspiration. The courses given for SimCorp Financial Training (now Financial Training Partner) have also helped me develop material. There are many other colleagues and friends who have contributed to my under- standing of the area over the years, by helping me understand what the important problems are and teaching me some of the useful techniques. This list of peo- ple includes Michael Ahm, Jesper Andreasen, Mark Carey, Mark Davis, Michael Gordy, Lane Hughston, Martin Jacobsen, Søren Johansen, David Jones, Søren Kyhl, Joe Langsam, Henrik O. Larsen, Jesper Lund, Lars Tyge Nielsen, Ludger Over- beck, Lasse Pedersen, Rolf Poulsen,Anders Rahbek, Philipp Schönbucher, Michael Sørensen, Gerhard Stahl, and all the members of the Danish Mathematical Finance Network. A special word of thanks to Richard Cantor, Roger Stein, and John Rutherford at Moody’s Investor’s Service for setting up and including me in Moody’s Academic AdvisoryandResearchCommittee.Thiscontinuestobeagreatsourceofinspiration. I would also like to thank my past and present co-members, Pierre Collin-Dufresne, Darrell Duffie, Steven Figlewski, Gary Gorton, David Heath, John Hull, William Perraudin, Jeremy Stein, and Alan White, for many stimulating discussions in this forum. Also, at Moody’s I have learned from Jeff Bohn, Greg Gupton, and David Hamilton, among many others.
- 21. xvi Preface I thank the many students who have supplied corrections over the years. I owe a special word of thanks to my current PhD students Jens Christensen, Peter Feldhütter andAllan Mortensen who have all supplied long lists of corrections and suggestions for improvement. Stephan Kossmeier, Jesper Lund, Philipp Schönbucher, Roger Stein, and an anonymous referee have also given very useful feedback on parts of the manuscript and for that I am very grateful. I have received help in typing parts of the manuscript from Dita Andersen, Jens Christensen, and Vibeke Hutchings. I gratefully acknowledge support from The Danish Social Science Research Foundation, which provided a much needed period of reduced teaching. Richard Baggaley at Princeton University Press has been extremely supportive and remarkably patient throughout the process. The Series Editors Darrell Duffie and Stephen Schaefer have also provided lots of encouragement. I owe a lot to Sam Clark, whose careful typesetting and meticulous proofreading have improved the finished product tremendously. I owe more to my wife Lise and my children Frederik and Christine than I can express. At some point, my son Frederik asked me if I was writing the book because I wanted to or because I had to. I fumbled my reply and I am still not sure what the precise answer should have been. This book is for him.
- 24. 1 An Overview The natural place to start the exposition is with the Black and Scholes (1973) and Merton (1974) milestones. The development of option-pricing techniques and the application to the study of corporate liabilities is where the modeling of credit risk has its foundations. While there was of course research out before this, the option- pricing literature, which views the bonds and stocks issued by a firm as contingent claims on the assets of the firm, is the first to give us a strong link between a statistical model describing default and an economic-pricing model. Obtaining such a link is a key problem of credit risk modeling. We make models describing the distribution of the default events and we try to deduce prices from these models. With pricing models in place we can then reverse the question and ask, given the market prices, what is the market’s perception of the default probabilities. To answer this we must understand the full description of the variables governing default and we must understand risk premiums. All of this is possible, at least theoretically, in the option-pricing framework. Chapter 2 starts by introducing the Merton model and discusses its implications for the risk structure of interest rates—an object which is not to be mistaken for a term structure of interest rates in the sense of the word known from modeling government bonds. We present an immediate application of the Merton model to bonds with different seniority. There are several natural ways of generalizing this, and to begin with we focus on extensions which allow for closed-form solutions. One direction is to work with different asset dynamics, and we present both a case with stochastic interest rates and one with jumps in asset value. A second direction is to introduce a default boundary which exists at all time points, representing some sort of safety covenant or perhaps liquidity shortfall. The Black–Cox model is the classic model in this respect.As we will see, its derivation has been greatly facilitated by the development of option-pricing techniques. Moreover, for a clever choice of default boundary, the model can be generalized to a case with stochastic interest rates. A third direction is to include coupons, and we discuss the extension both to discrete- time, lumpy dividends and to continuous flows of dividends and continuous coupon payments. Explicit solutions are only available if the time horizon is made infinite.
- 25. 2 1. An Overview Having the closed-form expressions in place, we look at a numerical scheme which works for any hitting time of a continuous boundary provided that we know the transition densities of the asset-value process. With a sense of what can be done with closed-form models, we take a look at some more practical issues. Coupon payments really distinguish corporate bond pricing from ordinary option pricing in the sense that the asset-sale assumptions play a critical role. The liquidity of assets would have no direct link to the value of options issued by third parties on the firm’s assets, but for corporate debt it is critical. We illustrate this by looking at the term-structure implications of different asset-sale assumptions. Another practical limitation of the models mentioned above is that they are all static, in the sense that no new debt issues are allowed. In practice, firms roll over debt and our models should try to capture that. A simple model is presented which takes a stationary leverage target as given and the consequences are felt at the long end of the term structure. This anticipates the models of Chapter 3, in which the choice of leverage is endogenized. One of the most practical uses of the option-based machinery is to derive implied asset values and implied asset volatilities from equity market data given knowledge of the debt structure. We discuss the maximum-likelihood approach to finding these implied values in the simple Merton model. We also discuss the philosophy behind the application of implied asset value and implied asset volatility as variables for quantifying the probability of default, as done, for example (in a more complicated and proprietary model), by Moody’s KMV. The models in Chapter 2 are all incapable of answering questions related to the optimal capital structure of firms. They all take the asset-value process and its division between different claims as given, and the challenge is to price the different claims given the setup. In essence we are pricing a given securitization of the firm’s assets. Chapter 3 looks at the standard approach to obtaining an optimal capital structure within an option-based model. This involves looking at a trade-off between having a tax shield advantage from issuing debt and having the disadvantage of bankruptcy costs, which are more likely to be incurred as debt is increased. We go through a model of Leland, which despite (perhaps because of) its simple setup gives a rich playingfieldforeconomicinterpretation.Itdoeshavesomeconceptualproblemsand these are also dealt with in this chapter. Turning to models in which the underlying state variable process is the EBIT (earnings before interest and taxes) of a firm instead of firm value can overcome these difficulties. These models can also capture the important phenomenon that equity holders can use the threat of bankruptcy to renegotiate, in times of low cash flow, the terms of the debt, forcing the debt holders to agree to a lower coupon payment. This so-called strategic debt service is more easily explained in a binomial setting and this is how we conclude this chapter.
- 26. 3 At this point we leave the option-pricing literature. Chapter 4 briefly reviews different statistical techniques for analyzing defaults. First, classical discriminant analysis is reviewed. While this model had great computational advantages before statistical computing became so powerful, it does not seem to be a natural statis- tical model for default prediction. Both logistic regression and hazard regressions have a more natural structure. They give parameters with natural interpretations and handle issues of censoring that we meet in practical data analysis all the time. Hazard regressions also provide natural nonparametric tools which are useful for exploring the data and for selecting parametric models. And very importantly, they give an extremely natural connection to pricing models. We start by reviewing the discrete-time hazard regression, since this gives a very easy way of understanding the occurrence/exposure ratios, which are the critical objects in estimation—both parametrically and nonparametrically. While on the topic of default probability estimation it is natural to discuss some techniques for analyzing rating transitions, using the so-called generator of a Markov chain, which are useful in practical risk management. Thinking about rating migra- tion in continuous time offers conceptual and in some respects computational im- provements over the discrete-time story. For example, we obtain better estimates of probabilities of rare events. We illustrate this using rating transition data from Moody’s. We also discuss the role of the Markov assumption when estimating tran- sition matrices from generator matrices. The natural link to pricing models brought by the continuous-time survival analy- sis techniques is explained in Chapter 5, which introduces the intensity setting in what is the most natural way to look at it, namely as a Cox process or doubly stochastic Poisson process. This captures the idea that certain background variables influence the rate of default for individual firms but with no feedback effects. The actual default of a firm does not influence the state variables. While there are impor- tant extensions of this framework, some of which we will review briefly, it is by far the most practical framework for credit risk modeling using intensities. The fact that it allows us to use many of the tools from default-free term-structure modeling, espe- cially with the affine and quadratic term-structure models, is an enormous bonus. Particularly elegant is the notion of recovery of market value, which we spend some time considering. We also outline how intensity models are estimated through the extended Kalman filter—a very useful technique for obtaining estimates of these heavily parametrized models. For the intensity model framework to be completely satisfactory, we should under- stand the link between estimated default intensities and credit spreads. Is there a way in which, at least in theory, estimated default intensities can be used for pricing? There is, and it is related to diversifiability but not to risk neutrality, as one might have expected. This requires a thorough understanding of the risk premiums, and
- 27. 4 1. An Overview an important part of this chapter is the description of what the sources of excess expected return are in an intensity model. An important moral of this chapter is that even if intensity models look like ordinary term-structure models, the structure of risk premiums is richer. How do default intensities arise? If one is a firm believer in the Merton setting, then the only way to get something resembling default intensities is to introduce jumps in asset value. However, this is not a very tractable approach from the point of view of either estimation or pricing credit derivatives. If we do not simply want to assume that intensities exist, can we still justify their existence? It turns out that we can by introducing incomplete information. It is shown that in a diffusion-based model, imperfect observation of a firm’s assets can lead to the existence of a default intensity for outsiders to the firm. Chapter 6 is about rating-based pricing models. This is a natural place to look at those, as we have the Markov formalism in place. The simplest illustration of intensity models with a nondeterministic intensity is a model in which the intensity is “modulated” by a finite-state-space Markov chain. We interpret this Markov chain as a rating, but the machinery we develop could be put to use for processes needing a more-fine-grained assessment of credit quality than that provided by the rating system. An important practical reason for looking at ratings is that there are a number of financial instruments that contain provisions linked to the issuer rating. Typi- cal examples are the step-up clauses of bond issues used, for example, to a large extent in the telecommunication sector in Europe. But step-up provisions also figure prominently in many types of loans offered by banks to companies. Furthermore, rating is a natural first candidate for grouping bond issues from different firms into a common category. When modeling the spreads for a given rating, it is desirable to model the joint evolution of the different term structures, recognizing that members of each category will have a probability of migrating to a different class. In this chapter we will see how such a joint modeling can be done. We consider a calibration technique which modifies empirical transition matrices in such a way that the transition matrix used for pricing obtains a fit of the observed term structures for different credit classes. We also present a model with stochastically varying spreads for different rating classes, which will become useful later in the chapter on interest-rate swaps. The problem with implementing these models in practice are not trivial. We look briefly at an alternative method using thresholds and affine process technology which has become possible (but is still problematic) due to recent progress using transform methods. The last three chapters contain applications of our machinery to some important areas in which credit risk analysis plays a role
- 28. 5 The analysis of interest-rate swap spreads has matured greatly due to the advances in credit risk modeling. The goal of this chapter is to get to the point at which the literature currently stands: counterparty credit risk on the swap contract is not a key factor in explaining interest-rate swap spreads. The key focus for understanding the joint evolution of swap curves, corporate curves, and treasury curves is the fact that the floating leg of the swap contract is tied to LIBOR rates. But before we can get there, we review the foundations for reaching that point. A starting point has been to analyze the apparent arbitrage which one can set up using swap markets to exchange fixed-rate payments for floating-rate payments. While there may very well be institutional features (such as differences in tax treatments) which permit such advantages to exist, we focus in Chapter 7 on the fact that the comparative-advantage story can be set up as a sort of puzzle even in an arbitrage- free model. This puzzle is completely resolved. But the interest in understanding the role of two-sided default risk in swaps remains.We look at this with a strong focus on the intensity-based models. The theory ends up pretty much confirming the intuitive result: that swap counterparties with symmetric credit risk have very little reason to worry about counterparty default risk. The asymmetries that exist between their payments—since one is floating and therefore not bounded in principle, whereas the other is fixed—only cause very small effects in the pricing. With netting agreements in place, the effect is negligible. This finally clears the way for analyzing swap spreads and their relationship to corporate bonds, focusing on the important problem mentioned above, namely that the floating payment in a swap is linked to a LIBOR rate, which is bigger than that of a short treasury rate. Viewing the LIBOR spread as coming from credit risk (something which is probably not completely true) we set up a model which determines the fixed leg of the swap assuming that LIBOR and AA are the same rate. The difference between the swap rate and the corporate AA curve is highlighted in this case. The difference is further illustrated by showing that theoretically there is no problem in having the AAA rate be above the swap rate—at least for long maturities. The result that counterparty risk is not an important factor in determining credit risk also means that swap curves do not contain much information on the credit qual- ity of its counterparties. Hence swaps between risky counterparties do not really help us with additional information for building term structures for corporate debt. To get such important information we need to look at default swaps and asset swaps. In idealized settings we explain in Chapter 8 the interpretation of both the asset- swap spread and the default swap spread. We also look at more complicated struc- tures involving baskets of issuers in the so-called first-to-default swaps and first m-of-n-to-default swaps. These derivatives are intimately connected with so-called collateralized debt obligations (CDOs), which we also define in this chapter.
- 29. 6 1. An Overview Pricing of CDOs and analysis of portfolios of loans and credit-risky securities lead to the question of modeling dependence of defaults, which is the topic of the whole of Chapter 9. This chapter contains many very simplified models which are developed for easy computation but which are less successful in preserving a realistic model structure. The curse is that techniques which offer elegant computation of default losses assume a lot of homogeneity among issuers. Factor structures can mitigate but not solve this problem. We discuss correlation of rating movements derived from asset-value correlations and look at correlation in intensity models. For intensity models we discuss the problem of obtaining correlation in affine specifications of the CIR type, the drastic covariation of intensities needed to generate strong default correlation and show with a stylized example how the updating of a latent variable can lead to default correlation. Recently, a lot of attention has been given to the notion of copulas, which are really just a way of generating multivariate distributions with a set of given marginals. We do not devote a lot of time to the topic here since it is, in the author’s view, a technique which still relies on parametrizations in which the parameters are hard to interpret. Instead, we choose to spend some time on default dependence in financial networks. Here we have a framework for understanding at a more fundamental level how the financial ties between firms cause dependence of default events. The interesting part is the clearing algorithm for defining settlement payments after a default of some members of a financial network in which firms have claims on each other. After this the rest is technical appendices.A small appendix reviews arbitrage-free pricing in a discrete-time setting and hints at how a discrete-time implementation of an intensity model can be carried out. Two appendices collect material on Brownian motion and Markov chains that is convenient to have readily accessible. They also contains a section on processes with jumps, including Itô’s formula and, just as important, finding the martingale part and the drift part of the contribution coming from the jumps. Finally, they look at some abstract results about (special) semi- martingales which I have found very useful. The main goal is to explain the struc- ture of risk premiums in a structure general enough to include all models included in this book. Part of this involves looking at excess returns of assets specified as special semimartingales. Another part involves getting a better grip on the quadratic variation processes. Finally, there is an appendix containing a workhorse for term-structure model- ing. I am sure that many readers have had use of the explicit forms of the Vasi- cek and the Cox–Ingersoll–Ross (CIR) bond-price models. The appendix provides closed-form solutions for different functionals and the characteristic function of a one-dimensional affine jump-diffusion with exponentially distributed jumps. These closed-form solutions cover all the pricing formulas that we need for the affine models considered in this book.
- 30. 2 Corporate Liabilities as Contingent Claims 2.1 Introduction This chapter reviews the valuation of corporate debt in a perfect market setting where the machinery of option pricing can be brought to use. The starting point of the models is to take as given the evolution of the market value of a firm’s assets and to view all corporate securities as contingent claims on these assets. This approach dates back to Black and Scholes (1973) and Merton (1974) and it remains the key reference point for the theory of defaultable bond pricing. Since these works appeared, the option-pricing machinery has expanded sig- nificantly. We now have a rich collection of models with more complicated asset price dynamics, with interest-rate-sensitive underlying assets, and with highly path- dependent option payoff profiles. Some of this progress will be used below to build a basic arsenal of models. However, the main focus is not to give a complete catalogue of the option-pricing models and explore their implications for pricing corporate bonds. Rather, the goal is to consider some special problems and questions which arise when using the machinery to price corporate debt. First of all, the extent to which owners of firms may use asset sales to finance couponpaymentsondebtisessentialtothepricingofcorporatebonds.Thisisclosely related to specifying what triggers default in models where default is assumed to be a possibility at all times. While ordinary barrier options have barriers which are stipulated in the contract, the barrier at which a company defaults is typically a modeling problem when looking at corporate bonds. Second, while we know the current liability structure of a firm, it is not clear that it will remain constant in the remaining life of the corporate debt that we are trying to model. In classical option pricing, the issuing of other options on the same underlying security is usually ignored, since these are not liabilities issued by the same firm that issued the stock. Of course, the future capital-structure choice of a firm also influences the future path of the firm’s equity price and therefore has an effect on equity options as well. Typically, however, the future capital-structure changes are subsumed as part of the dynamics of the stock. Here, when considering
- 31. 8 2. Corporate Liabilities as Contingent Claims corporate bonds, we will see models that take future capital-structure changes more explicitly into account. Finally, the fact that we do not observe the underlying asset value of the firm complicatesthedeterminationofimpliedvolatility.Instandardoptionpricing,where we observe the value of the underlying asset, implied volatility is determined by inverting an option-pricing formula. Here, we have to jointly estimate the underlying asset value and the asset volatility from the price of a derivative security with the asset value as underlying security. We will explain how this can be done in a Merton setting using maximum-likelihood estimation. A natural question in this context is to consider how this filtering can in principle be used for default prediction. This chapter sets up the basic Merton model and looks at price and yield impli- cations for corporate bonds in this framework. We then generalize asset dynamics (including those of default-free bonds) while retaining the zero-coupon bond struc- ture. Next, we look at the introduction of default barriers which can represent safety covenants or indicate decisions to change leverage in response to future movements in asset value.We also increase the realism by considering coupon payments. Finally, we look at estimation of asset value in a Merton model and discuss an application of the framework to default prediction. 2.2 The Merton Model Assume that we are in the setting of the standard Black–Scholes model, i.e. we analyze a market with continuous trading which is frictionless and competitive in the sense that • agents are price takers, i.e. trading in assets has no effect on prices, • there are no transactions costs, • there is unlimited access to short selling and no indivisibilities of assets, and • borrowing and lending through a money-market account can be done at the same riskless, continuously compounded rate r. AssumethatthetimehorizonisT̄ .Tobereasonablypreciseaboutassetdynamics,we fix a probability space (Ω, F , P) on which there is a standard Brownian motion W. The information set (or σ-algebra) generated by this Brownian motion up to time t is denoted Ft . We want to price bonds issued by a firm whose assets are assumed to follow a geometric Brownian motion: dVt = µVt dt + σVt dWt . Here, W is a standard Brownian motion under the probability measure P. Let the starting value of assets equal V0. Then this is the same as saying Vt = V0 exp((µ − 1 2 σ2 )t + σWt ).
- 32. 2.2. The Merton Model 9 We also assume that there exists a money-market account with a constant riskless rate r whose price evolves deterministically as βt = exp(rt). We take it to be well known that in an economy consisting of these two assets, the price C0 at time 0 of a contingent claim paying C(VT ) at time T is equal to C0 = EQ [exp(−rt)CT ], where Q is the equivalent martingale measure1 under which the dynamics of V are given as Vt = V0 exp((r − 1 2 σ2 )t + σW Q t ). Here, WQ is a Brownian motion and we see that the drift µ has been replaced by r. To better understand this model of a firm, it is useful initially to think of assets which are very liquid and tangible. For example, the firm could be a holding company whose only asset is a ton of gold. The price of this asset is clearly the price of a liquidly traded security. In general, the market value of a firm’s assets is the present market value of the future cash flows which the firm will deliver—a quantity which is far from observable in most cases. A critical assumption is that this asset-value process is given and will not be changed by any financing decisions made by the firm’s owners. Now assume that the firm at time 0 has issued two types of claims: debt and equity. In the simple model, debt is a zero-coupon bond with a face value of D and maturity date T ⩽ T̄ . With this assumption, the payoffs to debt, BT , and equity, ST , at date T are given as BT = min(D, VT ) = D − max(D − VT , 0), (2.1) ST = max(VT − D, 0). (2.2) We think of the firm as being run by the equity owners. At maturity of the bond, equity holders pay the face value of the debt precisely when the asset value is higher than the face value of the bond. To be consistent with our assumption that equity owners cannot alter the given process for the firm’s assets, it is useful to think of equity owners as paying D out of their own pockets to retain ownership of assets worth more than D. If assets are worth less than D, equity owners do not want to pay D, and since they have limited liability they do not have to either. Bond holders then take over the remaining asset and receive a “recovery” of VT instead of the promised payment D. 1We assume familiarity with the notion of an equivalent martingale measure, or risk-neutral measure, and its relation to the notion of arbitrage-free markets. Appendix D contains further references.
- 33. 10 2. Corporate Liabilities as Contingent Claims The question is then how the debt and equity are valued prior to the maturity date T . As we see from the structure of the payoffs, debt can be viewed as the difference between a riskless bond and a put option, and equity can be viewed as a call option on the firm’s assets. Note that no other parties receive any payments from V . In particular, there are no bankruptcy costs going to third parties in the case where equity owners do not pay their debt and there are no corporate taxes or tax advantages to issuing debt. A consequence of this is that VT = BT + ST , i.e. the firm’s assets are equal to the value of debt plus equity. Hence, the choice of D by assumption does not change VT , so in essence the Modigliani–Miller irrelevance of capital structure is hard-coded into the model. Given the current level V and volatility σ of assets, and the riskless rate r, we let CBS(V, D, σ, r, T ) denote the Black–Scholes price of a European call option with strike price D and time to maturity T , i.e. CBS (V, D, T, σ, r) = V N(d1) − D exp(−rT )N(d2), (2.3) where N is the standard normal distribution function and d1 = log(V/D) + rT + 1 2 σ2T σ √ T , d2 = d1 − σ √ T . We will sometimes suppress some of the parameters in C if it is obvious from the context what they are. Applying the Black–Scholes formula to price these options, we obtain the Merton model for risky debt. The values of debt and equity at time t are St = CBS (Vt , D, σ, r, T − t), Bt = D exp(−r(T − t)) − P BS (Vt , D, σ, r, T − t), where P BS is the Black–Scholes European put option formula, which is easily found from the put–call parity for European options on non-dividend paying stocks (which is a model-free relationship and therefore holds for call and put prices C and P in general): C(Vt ) − P(Vt ) = Vt − D exp(−r(T − t)). AnimportantconsequenceofthisparityrelationisthatwithD,r,T − t,andVt fixed, changing any other feature of the model will influence calls and puts in the same direction. Note, also, that since the sum of debt and equity values is the asset value, we have Bt = Vt − CBS(Vt ), and this relationship is sometimes easier to work with when doing comparative statics. Some consequences of the option representation are that the bond price Bt has the following characteristics.
- 34. 2.2. The Merton Model 11 • It is increasing in V . This is clear given the fact that the face value of debt remains unchanged. It is also seen from the fact that the put option decreases as V goes up. • It is increasing in D. Again not too surprising. Increasing the face value will produce a larger state-by-state payoff. It is also seen from the fact that the call option decreases in value, which implies that equity is less valuable. • It is decreasing in r. This is most easily seen by looking at equity. The call option increases, and hence debt must decrease since the sum of the two remains unchanged. • It is decreasing in time-to-maturity. The higher discounting of the riskless bond is the dominating effect here. • It is decreasing in volatility σ. The fact that volatility simultaneously increases the value of the call and the put options on the firm’s assets is the key to understanding the notion of “asset substitution.” Increasing the riskiness of a firm at time 0 (i.e. changing the volatility of V ) without changing V0 moves wealth from bond holders to shareholders. This could be achieved, for example, by selling the firm’s assets and investing the amount in higher-volatility assets. By definition, this will not change the total value of the firm. It will, however, shift wealth from bond holders to shareholders, since both the long call option held by the equity owners and the short put option held by the bond holders will increase in value. This possibility of wealth transfer is an important reason for covenants in bonds: bond holders need to exercise some control over the investment decisions. In the Merton model, this control is assumed, in the sense that nothing can be done to change the volatility of the firm’s assets. 2.2.1 The Risk Structure of Interest Rates Since corporate bonds typically have promised cash flows which mimic those of treasury bonds, it is natural to consider yields instead of prices when trying to compare the effects of different modeling assumptions. In this chapter we always look at the continuously compounded yield of bonds. The yield at date t of a bond with maturity T is defined as y(t, T ) = 1 T − t log D Bt , i.e. it is the quantity satisfying Bt exp(y(t, T )(T − t)) = D.
- 35. 12 2. Corporate Liabilities as Contingent Claims 0 10 Time to maturity Yield spread (bps) V = 150 V = 200 40 30 20 10 0 2 4 6 8 Figure 2.1. Yield spreads as a function of time to maturity in a Merton model for two different levels of the firm’s asset value. The face value of debt is 100. Asset volatility is fixed at 0.2 and the riskless interest rate is equal to 5%. Note that a more accurate term is really promised yield, since this yield is only realized when there is no default (and the bond is held to maturity). Hence the promised yield should not be confused with expected return of the bond. To see this, note that in a risk-neutral world where all assets must have an expected return of r, the promised yield on a defaultable bond is still larger than r. In this book, the difference between the yield of a defaultable bond and a corresponding treasury bond will always be referred to as the credit spread or yield spread, i.e. s(t, T ) = y(t, T ) − r. We reserve the term risk premium for the case where the taking of risk is rewarded so that the expected return of the bond is larger than r. Now let t = 0, and write s(T ) for s(0, T ). The risk structure of interest rates is obtained by viewing s(T ) as a function of T . In Figures 2.1 and 2.2 some examples of risk structures in the Merton model are shown. One should think of the risk structure as a transparent way of comparing prices of potential zero-coupon bond issues with different maturities assuming that the firm chooses only one maturity. It is also a natural way of comparing zero-coupon debt issues from different firms possibly with different maturities. The risk structure cannot be used as a term structure of interest rates for one issuer, however. We cannot price a coupon bond issued by a firm by
- 36. 2.2. The Merton Model 13 2 10 0 500 1000 1500 2000 Time to maturity Yield spread (bps) V = 90 V = 120 4 6 8 Figure 2.2. Yield spreads in a Merton model for two different (low) levels of the firm’s asset value. The face value of debt is 100. Asset volatility is fixed at 0.2 and the riskless interest rate is equal to 5%. When the asset value is lower than the face value of debt, the yield spread goes to infinity. valuing the individual coupons separately using the simple model and then adding the prices. It is easy to check that doing this quickly results in us having the values of the individual coupon bonds sum up to more than the firm’s asset value. Only in the limit with very high firm value does this method work as an approximation—and that is because we are then back to riskless bonds in which the repayment of one coupon does not change the dynamics needed to value the second coupon. We will return to this discussion in greater detail later. For now, consider the risk structure as a way of looking, as a function of time to maturity, at the yield that a particular issuer has to promise on a debt issue if the issue is the only debt issue and the debt is issued as zero-coupon bonds. Yields, and hence yield spreads, have comparative statics, which follow easily from those known from option prices, with one very important exception: the depen- dence on time to maturity is not monotone for the typical cases, as revealed in Fig- ures 2.1 and 2.2. The Merton model allows both a monotonically decreasing spread curve (in cases where the firm’s value is smaller than the face value of debt) and a humped shape. The maximum point of the spread curve can be at very short matu- rities and at very large maturities, so we can obtain both monotonically decreasing and monotonically increasing risk structures within the range of maturities typically observed.
- 37. 14 2. Corporate Liabilities as Contingent Claims Note also that while yields on corporate bonds increase when the riskless interest rate increases, the yield spreads actually decrease. Representing the bond price as B(r) = V − CBS(r), where we suppress all parameters other than r in the notation, it is straightforward to check that y (r) = −B(r) T B(r) ∈ (0, 1) and therefore s(r) = y(r) − 1 ∈ (−1, 0). 2.2.2 On Short Spreads in the Merton Model The behavior of yield spreads at the short end of the spectrum in Merton-style models plays an important role in motivating works which include jump risk. We therefore now consider the behavior of the risk structure in the short end, i.e. as the time to maturity goes to 0. The result we show is that when the value of assets is larger than the face value of debt, the yield spreads go to zero as time to maturity goes to 0 in the Merton model, i.e. s(T ) → 0 for T → 0. It is important to note that this is a consequence of the (fast) rate at which the probability of ending below D goes to 0. Hence, merely noting that the default probability itself goes to 0 is not enough. More precisely, a diffusion process X has the property that for any ε 0, P(|Xt+h − Xt | ⩾ ε) h − − − → h→0 0. We will take this for granted here, but see Bhattacharya and Waymire (1990), for example, for more on this. The result is easy to check for a Brownian motion and hence also easy to believe for diffusions, which locally look like a Brownian motion. We now show why this fact implies 0 spreads in the short end. Note that a zero- recovery bond paying 1 at maturity h if Vh D and 0 otherwise must have a lower price and hence a higher yield than the bond with face value D in the Merton model. Therefore, it is certainly enough to show that this bond’s spread goes to 0 as h → 0. The price B0 of the zero-recovery bond is (suppressing the starting value V0) B0 = EQ [D exp(−rh)1{Vh⩾D}] = D exp(−rh)Q(Vh ⩾ D),
- 38. 2.2. The Merton Model 15 and therefore the yield spread s(h) is s(h) = − 1 h log B0 D − r = − 1 h log Q(Vh ⩾ D) ≈ − 1 h (Q(Vh ⩾ D) − 1) = 1 h Q(Vh ⩽ D), and hence, for V0 D, s(h) → 0 for h → 0, and this is what we wanted to show. In the case where the firm is close to bankruptcy, i.e. V0 D, and the maturity is close to 0, yields are extremely large since the price at which the bond trades will be close to the current value of assets, and since the yield is a promised yield derived from current price and promised payment. A bond with a current price, say, of 80 whose face value is 100 will have an enormous annualized yield if it only has (say) a week to maturity. As a consequence, traders do not pay much attention to yields of bonds whose prices essentially reflect their expected recovery in an imminent default. 2.2.3 On Debt Return Distributions Debt instruments have a certain drama due to the presence of default risk, which raises the possibility that the issuer may not pay the promised principal (or coupons). Equity makes no promises, but it is worth remembering that the equity is, of course, far riskier than debt. We have illustrated this point in part to try and dispense with the notion that losses on bonds are “heavy tailed.” In Figure 2.3 we show the return distribution of a bond in a Merton model with one year to maturity and the listed parameters. This is to be compared with the much riskier return distribution of the stock shown in Figure 2.4. As can be seen, the bond has a large chance of seeing a return around 10% and almost no chance of seeing a return under −25%. The stock, in contrast, has a significant chance (almost 10%) of losing everything. 2.2.4 Subordinated Debt Before turning to generalizations of Merton’s model, note that the option framework easily handles subordination, i.e. the situation in which certain “senior” bonds have priority over “junior” bonds. To see this, note Table 2.1, which expresses payments to senior and junior debt and to equity in terms of call options. Senior debt can be priced as if it were the only debt issue and equity can be priced by viewing the entire debt as one class, so the most important change is really the valuation of junior debt.
- 39. 16 2. Corporate Liabilities as Contingent Claims −50 −40 −30 −20 −10 10 Rate of return (%) Probability 0 0.02 0.04 0.06 0.08 0.10 0.12 91.71% 0 Figure 2.3. A discretized distribution of corporate bond returns over 1 year in a model with very high leverage. The asset value is 120 and the face value is 100. The asset volatility is assumed to be 0.2, the riskless rate is 5%, and the return of the assets is 10%. −80 −40 0 40 80 120 160 200 240 280 320 360 420 Rate of return (%) Probability 0 0.02 0.04 0.06 0.08 0.10 Figure 2.4. A discretized distribution of corporate stock returns over 1 year with the same parameter values as in Figure 2.3.
- 40. 2.3. The Merton Model with Stochastic Interest Rates 17 Table 2.1. Payoffs to senior and junior debt and equity at maturity when the face values of senior and junior debt are DS and DJ, respectively. VT DS DS ⩽ VT DS + DJ DS + DJ VT Senior VT DS DS Junior 0 VT − DS DJ Equity 0 0 VT − (DS + DJ) Table 2.2. Option representations of senior and junior debt. C(V, D) is the payoff at expiration of a call-option with value of underlying equal to V and strike price D. Type of debt Option payoff Senior V − C(V, DS) Junior C(V, DS) − C(V, DS + DJ) Equity C(V, DS + DJ) 2.3 The Merton Model with Stochastic Interest Rates We now turn to a modification of the Merton setup which retains the assumption of a single zero-coupon debt issue but introduces stochastic default-free interest rates. First of all, interest rates on treasury bonds are stochastic, and secondly, there is evidence that they are correlated with credit spreads (see, for example, Duffee 1999). When we use a standard Vasicek model for the riskless rate, the pricing problem in a Merton model with zero-coupon debt is a (now) standard application of the numeraire-change technique. This technique will appear again later, so we describe the structure of the argument in some detail. Assume that under a martingale measure Q the dynamics of the asset value of the firm and the short rate are given by dVt = rt Vt dt + σV Vt (ρ dW1 t + 1 − ρ2 dW2 t ), drt = κ(θ − r) dt + σr dW1 t , where W1 t and W2 t are independent standard Brownian motions. From standard term-structure theory, we know that the price at time t of a default-free zero-coupon bond with maturity T is given as p(t, T ) = exp(a(T − t) − b(T − t)rt ), where b(T − t) = 1 κ (1 − exp(−κ(T − t))), a(T − t) = (b(T − t) − (T − t))(κ2θ − 1 2 σ2) κ2 − σ2b2(T − t) 4κ .
- 41. 18 2. Corporate Liabilities as Contingent Claims To derive the price of (say) equity in this model, whose only difference from the Merton model is due to the stochastic interest rate, we need to compute St = E Q t exp − T t rs ds (VT − D)+ , and this task is complicated by the fact that the stochastic variable we use for discounting and the option payoff are dependent random variables, both from the correlation in their driving Brownian motions and because of the drift in asset values being equal to the stochastic interest rate under Q. Fortunately, the (return) volatility σT (t) of maturity T bonds is deterministic.An application of Itô’s formula will show that σT (t) = −σrb(T − t). This means that if we define ZV,T (t) = V (t) p(t, T ) , then the volatility of Z is deterministic and through another application of Itô’s formula can be expressed as σV,T (t) = (ρσV + σrb(T − t))2 + σ2 V (1 − ρ2). Now define Σ2 V,T (T ) = T 0 σV,T (t)2 dt = T 0 (ρσV + σrb(T − t))2 + σ2 V (1 − ρ2 ) dt = T 0 (2ρσV σrb(T − t) + σ2 r b2 (T − t) + σ2 V ) dt. From Proposition 19.14 in Björk (1998), we therefore know that the price of the equity, which is a call option on the asset value, is given at time 0 by S(V, 0) = V N(d1) − Dp(0, T )N(d2), where d1 = log(V/Dp(0, T )) + 1 2 Σ2 V,T (T ) Σ2 V,T (T ) , d2 = d1 + Σ2 V,T (T ). This option price is all we need, since equity is then directly priced using this formula, and the value of debt then follows directly by subtracting the equity value
- 42. 2.3. The Merton Model with Stochastic Interest Rates 19 2 10 80 100 120 140 Time to maturity Yield spread (bps) Vol(r) = 0 Vol(r) = 0.015 Vol(r) = 0.030 4 6 8 Figure 2.5. The effect of interest-rate volatility in a Merton model with stochastic interest rates. The current level of assets is V0 = 120 and the starting level of interest rates is 5%. The face value is 100 and the parameters relevant for interest-rate dynamics are κ = 0.4 and θ = 0.05. The asset volatility is 0.2 and we assume ρ = 0 here. from current asset value. We are then ready to analyze credit spreads in this model as a function of the parameters. We focus on two aspects: the effect of stochastic interest rates when there is no correlation; and the effect of correlation for given levels of volatility. As seen in Figure 2.5, interest rates have to be very volatile to have a significant effect on credit spreads. Letting the volatility be 0 brings us back to the standard Merton model, whereas a volatility of 0.015 is comparable with that found in empir- ical studies. Increasing volatility to 0.03 is not compatible with the values that are typically found in empirical studies. A movement of one standard deviation in the driving Brownian motion would then lead (ignoring mean reversion) to a 3% fall in interest rates—a very large movement. The insensitivity of spreads to volatility is often viewed as a justification for ignoring effects of stochastic interest rates when modeling credit spreads. Correlation, as studied in Figure 2.6, seems to be a more significant factor, although the chosen level of 0.5 in absolute value is somewhat high. Note that higher correlation produces higher spreads. An intuitive explanation is that when asset value falls, interest rates have a tendency to fall as well, thereby decreasing the drift of assets, which strengthens the drift towards bankruptcy.
- 43. 20 2. Corporate Liabilities as Contingent Claims 2 10 60 80 100 120 140 Time to maturity Yield spread (bps) Correlation = 0.5 Correlation = 0 Correlation = −0.5 4 6 8 Figure 2.6. The effect of correlation between interest rates and asset value in a Merton model with stochastic interest rates. The current level of assets is V0 = 120 and the starting level of interest rates is 5%. The face value is 100 and the parameters relevant for interest-rate dynamics are κ = 0.4 and θ = 0.05. The asset volatility is 0.2 and the interest-rate volatility is σr = 0.015. 2.4 The Merton Model with Jumps in Asset Value We now take a look at a second extension of the simple Merton model in which the dynamics of the asset-value process contains jumps.2 The aim of this section is to derive an explicit pricing formula, again under the assumption that the only debt issue is a single zero-coupon bond. We will then use the pricing relationship to discuss the implications for the spreads in the short end and we will show how one compares the effect of volatility induced by jumps with that induced by diffusion volatility. We start by considering a setup in which there are only finitely many possible jump sizes. Let N1, . . . , NK be K independent Poisson processes with intensi- ties λ1, . . . , λK. Define the dynamics of the return process R under a martingale measure3 Q as a jump-diffusion dRt = r dt + σ dWt + K i=1 hi d(Ni t − λi t), 2The stochastic calculus you need for this section is recorded in Appendix D. This section can be skipped without loss of continuity. 3Unless otherwise stated, all expectations in this section are taken with respect to this measure Q.
- 44. 2.4. The Merton Model with Jumps in Asset Value 21 and let this be the dynamics of the cumulative return for the underlying asset-value process. As explained in Appendix D, we define the price as the semimartingale exponential of the return and this gives us Vt = V0 exp r − 1 2 σ2 − hi λi t + σWt 0⩽s⩽t 1 + K i=1 hi Ni s . Note that independent Poisson processes never jump simultaneously, so at a time s, at most one of the Ni s is different from 0. Recall that we can get the Black–Scholes partial differential equation (PDE) by performing the following steps (in the classical setup). • Write the stochastic differential equation (SDE) of the price process V of the underlying security under Q. • Let f be a function of asset value and time representing the value of a con- tingent claim. • Use Itô to derive an SDE for f (Vt , t). Identify the drift term and the martingale part. • Set the drift equal to rf (Vt , t) dt. We now perform the equivalent of these steps in our simple jump-diffusion case. Define λ = λ1 + · · · + λK and let h̄ = 1 λ K i=1 hi λi . Then (under Q) dVt = Vt {(r + h̄λ) dt + σ dWt } + K i=1 hi Vt− dNi t . WenowapplyItôbyusingitseparatelyonthediffusioncomponentandtheindividual jump components to get f (Vt , t) − f (V0, 0) = t 0 [fV (Vs, s)rVs + ft (Vs, s) − fV (Vs, s)h̄λVs + 1 2 σ2 V 2 s fV V (Vs, s)] ds + t 0 fV (Vs, s)σVs dWs + 0⩽s⩽t {f (Vs) − f (Vs−)}.
- 45. 22 2. Corporate Liabilities as Contingent Claims Now write 0⩽s⩽t {f (Vs) − f (Vs−)} = K i=1 t 0 {f (Vs) − f (Vs−)} dNi s = K i=1 t 0 [f (Vs−(1 + hi )) − f (Vs−)]λi ds + K i=1 t 0 [f (Vs−(1 + hi )) − f (Vs−)] d[Ni s − λi s] and note that we can write s instead of s− in the time index in the first integral because we are integrating with respect to the Lebesgue measure. In total, we now get the following drift term for f (Vt , t):4 (r − h̄λ)Vt fV + ft + 1 2 σ2 V 2 t fV V + K i=1 {f (Vt (1 + hi )) − f (Vt )}λi . Letting pi := λi /λ allows us to write K i=1 {f (Vt (1 + hi )) − f (Vt )}λi = K i=1 {pi [f (Vt (1 + hi )) − f (Vt )]}λ ≡ λE f (Vt ), and our final expression for the term in front of dt is now (r − h̄λ)Vt fV + ft + 1 2 σ2 V 2 t fV V + λE f (Vt ). Thisisthetermwehavetosetequaltorf (Vt , t)andsolve(withboundaryconditions) to get what is called an integro-differential equation. It is not a PDE since, unlike a PDE, the expressions involve not only f ’s behavior at a point V (including the behavior of its derivatives), it also takes into account values of f at points V “far away” (at V (1+hi) for i = 1, . . . , K). Such equations can only be solved explicitly in very special cases. We have considered the evolution of V as having only finitely many jumps and we have derived the integro-differential equation for the price of a contingent claim in this case. It is straightforward to generalize to a case where jumps (still) arrive as a Poisson process N with the rate λ but where the jump-size distribution has a continuous distribution on the interval [−1, ∞) with mean k. If we let 1, 2, . . . 4We omit Vt and t in f .
- 46. 2.4. The Merton Model with Jumps in Asset Value 23 denote a sequence of independent jump sizes with such a distribution, then we may consider the dynamics Vt = V0 exp((r − 1 2 σ2 + λk)t + σWt ) Nt i=1 (1 + εi). Between jumps in N, we thus have a geometric Brownian motion, but at jumps the price changes to 1 + εi times the pre-jump value. Hence 1 + εi 1 corresponds to a downward jump in price. In the example below with lognormally distributed jumps, use the following nota- tion for the distribution of the jumps: the basic lognormal distribution is specified as E log(1 + i) = γ − 1 2 δ2 , V log(1 + i) = δ2 , and hence E i = k = exp(γ ) − 1, E 2 i = exp(2γ + δ2 ) − 2 exp(γ ) + 1. One could try to solve the integro-differential equation for contingent-claims prices. It turns out that in the case where 1 + i is lognormal, there is an easier way: by conditioning on Nt and then using BS-type expressions. The result is an infinite sum of BS-type expressions. For the call option with price CJD we find (after some calculations) CJD (Vt , D, T, σ2 , r, δ2 , λ, k) = ∞ n=0 (λT )n n! exp(−λ T )CBS (Vt , D, T, σ2 n , rn), where CBS as usual is the standard Black–Scholes formula for a call and λ = λ(1 + k), rn = r + nγ T − λk, σ2 n = σ2 + nδ2 2T , γ = log(1 + k). To understand some of the important changes that are caused by introducing jumps in a Merton model, we focus on two aspects: the effect on credit spreads in the short end, and the role of the source of volatility, i.e. whether volatility comes from jumps or from the diffusion part.
- 47. 24 2. Corporate Liabilities as Contingent Claims 0 0.002 0.004 0.006 0.008 Yield spread 0.005 0.015 0.025 0.035 Yield spread 0 10 15 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Time to maturity Yield spread S(0) = 130 S(0) = 150 S(0) = 200 S(0) = 130 S(0) = 150 S(0) = 200 S(0) = 130 S(0) = 150 S(0) = 200 5 (a) (b) (c) Figure 2.7. The effect of changing the mean jump size and the intensity in a Merton model with jumps in asset value. From (a) to (b) we are changing the parameter determining the mean jump size, γ , from log(0.9) to log(0.5). This makes recovery at an immediate default lower and hence increases the spread in the short end. From (b) to (c) the intensity is doubled, and we notice the exact doubling of spreads in the short end, since expected recovery is held constant but the default probability over the next short time interval is doubled. We focus first on the short end of the risk structure of interest rates. The price of the risky bond with face value D maturing at time h (soon to be chosen small) is B(0, h) = exp(−rh)E[D1{Vh⩾D} + Vh1{VhD}] = exp(−rh)[DQ(Vh ⩾ D) + E[Vh | Vh D]Q(Vh D)] = D exp(−rh) 1 − Q(Vh D) + 1 D E[Vh | Vh D]Q(Vh D) = D exp(−rh) 1 − Q(Vh D) 1 − E[Vh | Vh D] D .
- 48. 2.4. The Merton Model with Jumps in Asset Value 25 0 10 15 0 0.02 0.04 0.06 0.08 Yield spread S(0) = 130 S(0) = 150 S(0) = 200 0 0.02 0.04 0.06 Time to maturity Yield spread 5 (a) (b) S(0) = 130 S(0) = 150 S(0) = 200 Figure 2.8. The effect of the source of volatility in a Merton model with jumps in asset value. (a) The diffusion part has volatility σ = 0.1, and the total quadratic variation is 0.4. (b) The diffusion part has volatility σ = 0.3, but the total quadratic variation is kept at 0.4 by decreasing λ. In both cases, three different current asset values are considered. Changing the source of volatility causes significant changes of the yield spreads in the short end for the high-yield cases. The difference between the spread curves in the case of low leverage is very small. Effects are also limited in the long end in all cases. Now computing the yield spread limit as h → 0 and using log(1 − x) ≈ −x for x close to 0, we find that for s(h) = y(0, h) − r, lim h↓0 s(h) = lim h↓0 Q(Vh D) h 1 − E[Vh | Vh D] D . Now as h ↓ 0 there is only at most one jump that can occur. The total jump intensity is λ, but the probability of a jump being large enough to send Vh below D happens with a smaller intensity λ∗ = λQ[(1 + ε)V0 D]. We recognize the second term in the expression as the expected fractional loss given default. Altogether we obtain lim h↓0 s(h) = λ∗ E[(V0)],
- 49. 26 2. Corporate Liabilities as Contingent Claims where (V0) = 1 − E[V0(1 + ε) | V0(1 + ε) D] D . An immediate consequence is that doubling the overall jump intensity should double the instantaneous spread. Another consequence is, as is intuitively obvious, that lowering the mean jump size should typically lead to higher spreads. Both facts are illustrated in Figure 2.7. When comparing the jump-diffusion model with the standard Merton model, it is common to “level the playing field” by holding constant the “volatility” in a sense that we now explain. The optional quadratic variation of a semimartingale X can be obtained as a limit [X]t = lim n→∞ i∈N [X(tn i+1 ∧ t) − X(tn i ∧ t)]2 , where the grid size in the subdivision goes to 0 as n → ∞.5 From the definition of predictable quadratic variation, X found in Appendix D, we know that when X has finite variance, E[Xt ] = E Xt , and since the jump- diffusion process R studied here is a process with independent increments, Rt is deterministic and we have that E[Rt ] = Rt = σ2 t + λtEε2 i . Holding Rt constant for a given t by offsetting changes in σ by changes in λ and/or Eε2 i gives room for an experiment in which we change the source of volatility. This is done in Figure 2.8. As is evident from that graph, the main effect is in the short end of the risk structure of interest rates. While it is tempting to think of quadratic variation as realized volatility, it is impor- tant to understand the difference between the volatility arising from the diffusion and the volatility arising from the jump part. For a fixed t we have [R]t = Rc t + 0⩽s⩽t R2 s , where Rc is the continuous part of R. Therefore, when we compute (1/t)[R]t from an approximating sum, we do not get a limit of σ2 + λEε2 i for fixed t and finer subdivisions. When our time horizon is fixed, we will always have the random component 0⩽s⩽t R2 s , and if jumps are rare this need not be close to λEε2 i . 5The limit is to be understood in the sense of uniform convergence in probability, i.e. on a finite interval [0, t], if the enemy shows up with small 1 0 and 2 0 then we can choose N large enough so that for n N the probability of the approximating sum deviating more than 1 anywhere on the interval is smaller than 2.
- 50. 2.5. Discrete Coupons in a Merton Model 27 However, as t → ∞ we have 1 t 0⩽s⩽t R2 s → λEε2 i . This highlights an important difference between diffusion-induced and jump-in- duced volatility. We cannot obtain the jump-induced volatility, even theoretically, as our observations get closer and closer in time. Observing the whole sample path in [0, t]wouldallowustosingleoutthejumpsandthenobtain Rct exactly.Inpractice, we do not have the exact jumps, and filtering out the jumps from the volatility based on discrete observations is a difficult exercise. So, while the jump-diffusion model is excellent for illustration and simulating the effects of jumps, the problems in estimating the model make it less attractive in practical risk management. 2.5 Discrete Coupons in a Merton Model As mentioned earlier we cannot use the Merton model for zero-coupon debt to price coupon debt simply by pricing each component of the bond separately. The pricing of the coupon bond needs to look at all coupon payments in a single model and in this context our assumptions on asset sales become critical. Tounderstandtheproblemandseehowtoimplementapricingalgorithm,consider a coupon bond with two coupons D1 and D2 which have to be paid at dates t1 and t2. For t t1, if the firm is still alive and the assets are worth Vt , we can value the only remaining coupon D2 simply using the standard Merton model, so B(Vt , t) = D2p(t, t2) − P BS (Vt , D2, t2 − t) for t t1. The situation at date t1 is more complicated and it critically depends on the assumptions we make on what equity owners, who control the firm, are allowed to do with the firm’s assets. First, assume that equity owners are not allowed to use the firm’s assets to pay debt. This means that they have to finance the debt payment either by paying “out of their own pockets” or by issuing new equity to finance the coupon payment. In this simple model with no information asymmetries, it does not matter which option they choose. If they issue M new shares of stock in addition to the (say) N existing shares, they will raise an amount equal to (M/(M + N))St1 , where St is the total value of equity at time t. Hence, to finance D1 they need to choose M so that M M + N St1 = D1, thereby diluting the value of their own equity from St1 to (N/(M + N))St1 . This dilution causes a fall in their equity value of St1 − D1, and so if they do not pay D1 out of their own pockets they lose D1 through dilution of equity. Hence it does not
- 51. 28 2. Corporate Liabilities as Contingent Claims matter which option we consider. The option to issue new debt is not considered here, where we assume that the debt structure is static. So, think of equity owners as deciding an instant before t1 whether to pay the coupon at date t1 out of their own pockets. Paying the coupon will leave them with equity worth C(Vt1 , D2, t2 − t1) and hence it is optimal to pay the coupon D1 if D1 C(Vt1 , D2, t2 − t1). If this is not true, they will default and debt holders will take over the firm. Applying this line of reasoning leads to the following recursion when pricing coupon debt assuming no asset sales. Given coupons D1, . . . , DN due at dates t1, . . . , tN , we now proceed as follows. (1) Price debt and equity at dates t tN−1 using the standard Merton setup. (2) At tN−1, find the value V̄N−1 for which DN−1 = C(V̄N−1, tN − tN−1, DN ). (3) At date tN−1, let S(V, tN−1) = C(V, DN , tN − tN−1) − DN−1 for V ⩾ V̄N−1, 0 for V V̄N−1, and B(V, tN−1) = DN−1 + V − C(V, DN , tN − tN−1) for V ⩾ V̄N−1, V for V V̄N−1; this gives us the boundary conditions for debt and equity at date tN−1. (4) From this we can, at least numerically, value equity right after a coupon payment at date tN−2. The value V̄N−2 is the value for which equity is worth DN−2 right after the coupon has been paid at date tN−2. (5) Use the same procedure then as in (3) to set the boundary conditions at date tN−2 and continue backwards to time 0. This will give us prices of debt and equity using an assumption of no asset sales. What if asset sales are allowed? In this case we still work recursively backwards but we need to adjust both the default boundary and the asset value.At date t tN−1 we are still in the classical Merton setup. To set the relevant boundary condition for debt and equity at date tN−1, we argue as follows. If assets are worth more than DN−1, it is never optimal for equity owners to default, since this leaves them with 0. Clearly, it is better to sell DN−1 worth of assets to cover the coupon and continue with assets worth V (tN−1) − DN−1, and hence equity is worth C(VtN−1 − DN−1, DN , tN − tN−1). They might also consider
- 52. 2.6. Default Barriers: the Black–Cox Setup 29 paying out of their own pockets, but in fact, it is optimal for equity owners to sell assets instead of covering the payment themselves. To see this, note that paying out of their own pockets leaves equity with C(VtN−1 , DN , tN − tN−1) − DN−1, but this is smaller than C(VtN−1 − DN−1, DN , tN − tN−1), since CV 1 for all V , and therefore C(VtN−1 ) − C(VtN−1 − DN−1) DN−1. This is also intuitively obvious, since the payment of the coupon by equity owners alone will benefit both equity and debt, but be an expense to equity only. To write down how to price the securities is a little more cumbersome even if the imple- mentation is not too hard. We leave the details to the reader. The asset value is a geometric Brownian motion between coupon dates, but if an asset sale takes place to finance a coupon, the value drops by an amount equal to the coupon. All the time, we set the equity equal to zero if asset value at a coupon date falls below the coupon payment at that date, and in that case we let debt holders take over the firm. If the assets are large enough, we subtract the coupon payment in the asset value. Pricing algorithms are easy to implement in a tree both in the case of asset sales and in the case of no asset sales, but note that only the first model permits a fully recombining tree, since asset value is unaffected by coupon payments. In the model with asset sales, we need to distinguish between the sequence of up-and-down moves, since we subtract an amount from the asset value at coupon dates that is not a constant fraction of asset value. The assumptions we make on asset sales are critical for our valuation and for term-structure implications. We return to this in a later section. Note that we have only considered one debt issue.When there are several debt issues we of course need to keep track of the recovery assigned to the different issues at liquidation dates. 2.6 Default Barriers: the Black–Cox Setup We now consider therefore the basic extension of the Merton model due to Black and Cox (1976). The idea is to let defaults occur prior to the maturity of the bond. In mathematical terms, default will happen when the level of the asset value hits a lower boundary, modelled as a deterministic function of time. In the original approach of Black and Cox, the boundary represents the point at which bond safety covenants cause a default.As we will see later, the technique is also useful for modeling default
- 53. 30 2. Corporate Liabilities as Contingent Claims due to liquidity constraints where we approximate frequent small coupon payments by a continuous stream of payments. First-passage times for diffusions have been heavily studied. If one is looking for closed-form solutions, it is hard to go much beyond Brownian motion hitting a linear boundary (although there are a few extensions, as mentioned in the bibliographical notes). This mathematical fact almost dictates the type of boundary for asset value that we are interested in, namely boundaries that bring us back into the familiar case after we take logarithms. So, in their concrete model, Black and Cox consider a process for asset value which under the risk-neutral measure is dVt = (r − a)Vt dt + σVt dWt , where we have allowed for a continuous dividend payout ratio of a. The default boundary is given as C1(t) = C exp(−γ (T − t)). Assume that the bond issued by the firm has principal D and that C D. Note that since Vt = V0 exp((r − a)t − 1 2 σ2t + σWt ), the default time τ is given as τ = inf{0 ⩽ t ⩽ T : log V0 + ((r − a) − 1 2 σ2 )t + σWt = log C − γ (T − t)} = inf{0 ⩽ t ⩽ T : σWt + (r − a − 1 2 σ2 − γ )t = log C − log V0 − γ T }, i.e. the first time a Brownian motion with drift hits a certain level. In the Black–Cox model the payoff to bond holders at the maturity date is B(VT , T ) = min(VT , D)1{τT }, corresponding to the usual payoff when the boundary has not been crossed in [0, T ]. To simplify notation, let the current date be 0 so that the maturity date T is also time to maturity. We let Bm (V, T, D, C, γ ) = E(exp(−rT ) min(VT , D)1{τT }) denote the value at time 0 of the payoff of the bond at maturity when the face value is D and the function C1(·) is specified as a function of C and γ as above. If the boundary is hit before the maturity of the bond, bond holders take over the firm, i.e. B(Vτ , τ) = C1(τ)1{τ⩽T }. With the same conventions as above, we let Bb (V, T, D, C, γ ) = E(exp(−rτ)C1(τ)1{τ⩽T }) denote the value at time 0 of the payoff to the bond holders in the event that the boundary is hit before maturity. We assume that the starting value V is above C1. We will value the contribution from these two parts separately.
- 54. 2.6. Default Barriers: the Black–Cox Setup 31 The contribution from the payment at maturity can be valued using techniques from barrier options and here we use the treatment of Björk (1998, Chapter 13, p. 182). Mimicking the expression of the payoff of the bond in the Merton model as a difference between a riskless bond and a put option, we note that the payoff at maturity here is D1{τT } − (D − VT )+ 1{τT }. (2.4) Hence we need to be able to value a “truncated bond” and a “truncated put option,” and the technique is available from the results on down-and-out barrier options. The only modification we have to take care of is that the barrier is exponential and that there is a dividend payout on the underlying asset. First, we consider the valuation in the case of a flat boundary and where the drift of the underlying asset is equal to the riskless rate under the risk-neutral measure (i.e. dividends are set to zero). Observe that a put–call parity for barrier options allows us to write D1{τT } − (D − VT )+ 1{τT } = (VT − D)+ 1{τT } − VT 1{τT }, (2.5) and so we can use price expressions for a barrier call and a contract paying the asset value at maturity if the boundary has not been hit. For these expressions we need to define the value of a contract paying 1 at maturity if the asset value is above L at maturity: H(V, T, L) = exp(−rT )N log(V/L) + (r − 1 2 σ2)T σ √ T , (2.6) where we assume V L. From Proposition 13.16 and Lemma 13.17 in Björk (1998) we obtain the value BL of the bond payout at maturity if the boundary is flat at the level L (corresponding to γ = 0 in our model) and there is no dividend payment on the underlying asset (corresponding to a = 0 in the model): BL(V, T, D, L) = LH(V, t, T, L) − L L V ((2r/σ2)−1) H L2 V , t, T, L + CBS (V, L, T ) − L V ((2r/σ2)−1) CBS L2 V , L, T − CBS (V, D, T ) + L V ((2r/σ2)−1) CBS L2 V , D, T , where we have suppressed the riskless rate and the volatility used in the Black– Scholes price of a European call CBS(V, D, T ). Now we will use this expression to get the price Bm of the bond payout at maturity of a bond in the Black–Cox model.
- 55. Discovering Diverse Content Through Random Scribd Documents
- 56. We cultivate such a use of our eyes, as indeed of all our faculties, as will on the whole lead to the most profitable results. As a rule, the particular impression is not so important as what it represents. Sense impressions are simply the symbols or signs of things or ideas, and the thing or the idea is more important than the sign. Accordingly, we are accustomed to interpret lines, whenever we can, as the representations of objects. We are well aware that the canvas or the etching or the photograph before us is a flat surface in two dimensions, but we see the picture as the representation of solid objects in three dimensions. This is the illusion of pictorial art. So strong is this tendency to view lines as the symbols of things that if there is the slightest chance of so viewing them, we invariably do so; for we have a great deal of experience with things that present their contours as lines, and very little with mere lines or surfaces. If we view outlines only, without shading or perspective or anything to definitely suggest what is foreground and what background, it becomes possible for the mind to supply these details and see foreground as background, and vice versa. A good example to begin with is Fig. 8. These outlines will probably suggest at first view a book, or better a book cover, seen with its back toward you and its sides sloping away from you; but it may also be viewed as a book opened out toward you and presenting to you an inside view of its contents. Should the change not come readily, it may be facilitated by thinking persistently of the appearance of an open book in this position. The upper portion of Fig. 9 is practically the same as Fig. 8, and if the rest of the figure be covered up, it will change as did the book cover; when, however, the whole figure is viewed as an arrow, a new conception enters, and the apparently solid book cover becomes the flat feathered part of the arrow. Look at the next figure (Fig. 10), which represents in outline a truncated pyramid with a square base. Is the smaller square nearer to you, and are the sides of the pyramid sloping away from you toward the larger square in the rear? Or are you looking into the hollow of a truncated pyramid with the smaller square in the background? Or is it now one and now the other, according as you
- 57. Fig. 8.—This drawing may be viewed as the representation of a book standing on its half-opened covers as seen from the back of the book; or as the inside view of an open book showing the pages. Fig. 9.—When this figure is viewed as an arrow, the upper or feathered end seems flat; when the rest of the arrow is covered, the feathered end may be made to project or recede like the decid e to see it? Here (Fig. 13) is a skelet on box which you may conce ive as made of wires outlini ng the sides. Now the front, or side neare st to me, seems directed downward and to the left; again, it has shifted its position and is no longer the front, and the side which appears to be the front seems directed upward and to the right. The presence of the diagonal line makes the change more striking: in one position it runs from the left-hand rear upper corner to the
- 58. book cover in Fig. 8. right-hand front lower corner; while in the other it connects the left-hand front upper corner with the right-hand rear lower corner. Fig. 10.—The smaller square may be regarded as either the nearer face of a projecting figure or as the more distant face of a hollow figure.
- 59. Fig. 11.—This represents an ordinary table-glass, the bottom of the glass and the entire rear side, except the upper portion, being seen through the transparent nearer side, and the rear apparently projecting above the front. But it fluctuates in appearance between this and a view of the glass in which the bottom is seen directly, partly from underneath, the whole of the rear side is seen through the transparent front, and the front projects above the back.
- 60. Fig. 12.—In this scroll the left half may at first seem concave and the right convex, it then seems to roll or advance like a wave, and the left seems convex and the right concave, as though the trough of the wave had become the crest, and vice versa.
- 61. Figs. 13, 13a, and 13b.—The two methods of viewing Fig. 13 are described in the text. Figs. 13a and 13b are added to make clearer the two methods of viewing Fig. 13. The heavier lines seem to represent the nearer surface. Fig. 13a more naturally suggests the nearer surface of the box in a position downward and to the left, and Fig. 13b makes the nearer side seem to be upward and to the right. But in spite of the heavier outlines of the one surface, it may be made to shift positions from foreground to background, although not so readily as in Fig. 13. Fig. 14.—Each member of this frieze represents a relief ornament, applied upon the background, which in cross- section would be an isosceles triangle with a large obtuse angle, or a space of similar shape hollowed out of the solid wood or stone. In running the eye along the pattern, it is interesting to observe how variously the patterns fluctuate from one of these aspects to the other.
- 62. Figs. 15, 15a, and 15b.—The two views of Fig. 15 described in the text are brought out more clearly in Figs. 15a and 15b. The shaded portion tends to be regarded as the nearer face. Fig. 15a is more apt to suggest the steps seen as we ascend them. Fig. 15b seems to represent the hollowed-out structure underneath the steps. But even with the shading the dual interpretation is possible, although less obvious. Fig. 15 will probably seen at first glimpse to be the view of a flight of steps which one is about to ascend from right to left. Imagine it, however, to be a view of the under side of a series of steps; the view representing the structure of overhanging solid masonwork seen from underneath. At first it may be difficult to see it thus, because the view of steps which we are about to mount is a more natural and frequent experience than the other; but by staring at it with the intention of seeing it differently the transition will come, and often quite unexpectedly.
- 63. Fig. 16.—This interesting figure (which is reproduced with modifications from Scripture— The New Psychology) is subject in a striking way to interchanges between foreground and background. Most persons find it difficult to maintain for any considerable time either aspect of the blocks (these aspects are described in the text); some can change them at will, others must accept the changes as they happen to come.
- 64. Figs. 17, 17a, and 17b.—How many blocks are there in this pile? Six or seven? Note the change in arrangement of the blocks as they change in number from six to seven. This change is illustrated in the text. Figs. 17a and 17b show the two phases of a group of any three of the blocks. The arrangement of a pyramid of six blocks seems the more stable and is usually first suggested; but hold the page inverted, and you will probably see the alternate arrangement (with, however, the black surfaces still forming the tops). And once knowing what to look for, you will very likely be
- 65. able to see either arrangement, whether the diagram be held inverted or not. This method of viewing the figures upside down and in other positions is also suggested to bring out the changes indicated in Figs. 13, 13a, 13b, and in Figs. 15, 15a, 15b. The blocks in Fig. 16 are subject to a marked fluctuation. Now the black surfaces represent the bottoms of the blocks, all pointing downward and to the left, and now the black surfaces have changed and have become the tops pointing upward and to the right. For some the changes come at will; for others they seem to come unexpectedly, but all are aided by anticipating mentally the nature of the transformation. The effect here is quite striking, the blocks seeming almost animated and moving through space. In Fig. 17 a similar arrangement serves to create an illusion as to the real number of blocks present. If viewed in one way—the black surface forming the tops of the blocks—there seem to be six arranged as in Fig. 18; but when the transformation has taken place and the black surfaces have become the overhanging bottoms of the boxes, there are seven, arranged as in Fig. 19. Somewhat different, but still belonging to the group of ambiguous figures, is the ingenious conceit of the duck-rabbit shown in Fig. 20. When it is a rabbit, the face looks to the right and a pair of ears are conspicuous behind; when it is a duck, the face looks to the left and the ears have been changed into the bill. Most observers find it difficult to hold either interpretation steadily, the fluctuations being frequent, and coming as a surprise.
- 66. Figs. 18 and 19. Fig. 20.—Do you see a duck or a rabbit, or either? (From Harper's Weekly, originally in Fliegende Blätter.) All these diagrams serve to illustrate the principle that when the objective features are ambiguous we see one thing or another according to the impression that is in the mind's eye; what the objective factors lack in definiteness the subjective ones supply, while familiarity, prepossession, as well as other circumstances influence the result. These illustrations show conclusively that seeing is not wholly an objective matter depending upon what there is to be seen, but is very considerably a subjective matter depending upon the eye that sees. To the same observer a given arrangement of lines now appears as the representation of one object and now of another; and from the same objective experience, especially in instances that demand a somewhat complicated exercise of the senses, different observers derive very different impressions. Not only when the sense-impressions are ambiguous or defective, but when they are vague—when the light is dim or the forms
- 67. obscure—does the mind's eye eke out the imperfections of physical vision. The vague conformations of drapery and make-up that are identified and recognized in spiritualistic séances illustrate extreme instances of this process. The whitewashed tree or post that momentarily startles us in a dark country lane takes on the guise that expectancy gives it. The mental predisposition here becomes the dominant factor, and the timid see as ghosts what their more sturdy companions recognize as whitewashed posts. Such experiences we ascribe to the action of suggestion and the imagination—the cloud that's almost in shape like a camel, or like a weasel, or like a whale. But throughout our visual experiences there runs this double strain, now mainly outward and now mainly inward, from the simplest excitements of the retina up to the realms where fancy soars freed from the confines of sense, and the objective finds its occupation gone.
- 68. NATURE STUDY IN THE PHILADELPHIA NORMAL SCHOOL. By L. L. W. WILSON, Ph. D. When it was first proposed to me to write for the Popular Science Monthly a brief account of the biological laboratories in the Philadelphia Normal School, and of the Nature work carried on under my direction in the School of Observation and Practice, I felt that I could not do justice either to the place or the work; for, in my judgment, the equipment of the laboratories and the work done in connection with them are finer than anything else of the kind either in this country or abroad—a statement which it seemed to me that I could not make with becoming modesty. But, after all, it is not great Babylon that I have built, but a Babylon builded for me, and to fail to express my sense of its worth is to fail to do justice to Dr. W. P. Wilson, formerly of the University of Pennsylvania, to whom their inception was due; to Mr. Simon Gratz, president of the Board of Education, who from the beginning appreciated their value, and without whose aid they never would have taken visible form; to the principals of the two schools, and, above all, to my five assistants, whose knowledge, zeal, and hard work have contributed more than anything else to the rapid building up of the work. The Laboratories and their Equipment.—The rooms occupied by the botanical and zoölogical departments of the normal school measure each seventy by twenty feet. A small workroom for the teachers cuts off about ten feet of this length from each room. In the middle of the remaining space stands a demonstration table furnished with hot and cold water. Each laboratory is lighted from the side by ten windows. From them extend the tables for the students. These give plenty of drawer space and closets for dissecting and compound
- 69. microscopes. Those in the zoölogical room are also provided with sinks. Each student is furnished with the two microscopes, stage and eyepiece micrometers, a drawing camera, a set of dissecting instruments, glassware, note-books, text-books, and general literature. The walls opposite the windows are in both rooms lined with cases, in which there is a fine synoptic series. In the botanical laboratory this systematic collection begins with models of bacteria and ends with trees. In other cases, placed in the adjoining corridor, are representatives, either in alcohol or by means of models, of most of the orders of flowering plants, as well as a series illustrating the history of the theory of cross-fertilization, and the various devices by which it is accomplished; another, showing the different methods of distribution of seeds and fruits; another, of parasitic plants; and still another showing the various devices by means of which plants catch animals. As an example of the graphic and thorough way in which these illustrations are worked out, the pines may be cited. There are fossils; fine specimens of pistillate and staminate flowers in alcohol; cones; a drawing of the pollen; large models of the flowers; models of the seeds, showing the embryo and the various stages of germination; cross and longitudinal sections of the wood; drawings showing its microscopic structure; pictures of adult trees; and samples illustrating their economic importance. For the last, the long-leaved pine of the South is used, and samples are exhibited of the turpentine, crude and refined; tar and the oil of tar; resin; the leaves; the same boiled in potash; the same hatcheled into wool; yarn, bagging and rope made from the wool; and its timber split, sawn, and dressed. The series illustrating the fertilization of flowers begins with a large drawing, adapted by one of the students from Gibson, showing the gradual evolution of the belief in cross-fertilization from 1682, when Nehemiah Grew first declared that seed would not set unless pollen
- 70. reached the stigma, down to Darwin, who first demonstrated the advantages of cross-fertilization and showed many of the devices of plants by which this is accomplished. The special devices are then illustrated with models and large drawings. First comes the dimorphic primrose; then follows trimorphic Lythrum, to the beautiful model of which is appended a copy of the letter in which Darwin wrote to Gray of his discovery: But I am almost stark, staring mad over Lythrum.... I should rather like seed of Mitchella. But, oh, Lythrum! Your utterly mad friend, C. Darwin. Models of the cucumber, showing the process of its formation, and the unisexual flowers complete this series. Supplementing this are models and drawings of a large number of flowers, illustrating special devices by which cross-fertilization is secured, such as the larkspur, butter and eggs, orchids, iris, salvia, several composites, the milkweed, and, most interesting of all, the Dutchman's pipe. This is a flower that entices flies into its curved trumpet and keeps them there until they become covered with the ripe pollen. Then the hairs wither, the tube changes its position, the fly is permitted to leave, carrying the pollen thus acquired to another flower with the same result. Pictures and small busts of many naturalists adorn both of the rooms. Of these the most notable is an artist proof of Mercier's beautiful etching of Darwin. Every available inch of wall space is thus occupied, or else, in the botanical laboratory, has on it mounted fungi, lichens, seaweeds, leaf cards, pictures of trees, grasses, and other botanical objects. The windows are beautiful with hanging plants from side brackets meeting the wealth of green on the sill. Here are found in one window ferns, in another the century plant; in others still, specimens of economic plants—cinnamon, olive, banana, camphor. On the
- 71. tables are magnificent specimens of palms, cycads, dracænas, and aspidistras, and numerous aquaria filled with various water plants. Most of these plants are four years old, and all of them are much handsomer than when they first became the property of the laboratory. How much intelligent and patient care this means only those who have attempted to raise plants in city houses can know. The zoölogical laboratory is quite as beautiful as the botanical, for it, too, has its plants and pictures. It is perhaps more interesting because of its living elements. Think of a schoolroom in which are represented alive types of animals as various as these: amœba, vorticella, hydra, worms, muscles, snails and slugs of various kinds, crayfish, various insects, including a hive of Italian bees, goldfish, minnows, dace, catfish, sunfish, eels, tadpoles, frogs, newts, salamanders, snakes, alligators, turtles, pigeons, canaries, mice, guinea-pigs, rabbits, squirrels, and a monkey! Imagine these living animals supplemented by models of their related antediluvian forms, or fossils, by carefully labeled dissections, by preparations and pictures illustrating their development and mode of life; imagine in addition to this books, pamphlets, magazines, and teachers further to put you in touch with this wonderful world about us, and you will then have some idea of the environment in which it is the great privilege of our students to live for five hours each week. In addition to these laboratories there is a lecture room furnished with an electric lantern. Here each week is given a lecture on general topics, such as evolution and its problems, connected with the work of the laboratories. The Course of Study pursued by the Normal Students.—Botany: In general, the plants and the phenomena of the changing seasons are studied as they occur in Nature. In the fall there are lessons on the composites and other autumn flowers, on fruits, on the ferns, mosses, fungi, and other cryptogams. In the winter months the students grow various seeds at home, carefully drawing and studying every stage in their development. Meanwhile, in the laboratory, they examine microscopically and macroscopically the
- 72. seeds themselves and the various food supplies stored within. By experimentation they get general ideas of plant physiology, beginning with the absorption of water by seeds, the change of the food supply to soluble sugar, the method of growth, the functions, the histology, and the modifications of stem, root, and leaves. In the spring they study the buds and trees, particularly the conifers, and the different orders of flowering plants. The particular merit of the work is that it is so planned that each laboratory lesson compels the students to reason. Having once thus obtained their information, they are required to drill themselves out of school hours until the facts become an integral part of their knowledge. For the study of fruits, for example, they are given large trays, each divided into sixteen compartments, plainly labeled with the name of the seed or fruit within. Then, by means of questions, the students are made to read for themselves the story which each fruit has to tell, to compare it with the others, and to deduce from this comparison certain general laws. After sufficient laboratory practice of this kind they are required to read parts of Lubbock's Flower, Fruit, and Leaves, Kerner's Natural History of Plants, Wallace's Tropical Nature, and Darwinism, etc. Finally, they are each given a type-written summary of the work, and after a week's notice are required to pass a written examination. Zoölogy: The course begins in the fall with a rather thorough study of the insects, partly because they are then so abundant, and partly because a knowledge of them is particularly useful to the grade teacher in the elementary schools. The locust is studied in detail. Tumblers and aquaria are utilized as vivaria, so that there is abundant opportunity for the individual study of living specimens. Freshly killed material is used for dissection, so that students have no difficulty in making out the internal anatomy, which is further elucidated with large, home-made charts, each of
- 73. which shows a single system, and serves for a text to teach them the functions of the various organs as worked out by modern physiologists. They then study, always with abundant material, the other insects belonging to the same group. They are given two such insects, a bug, and two beetles, and required to classify them, giving reasons for so doing. While this work is going on they have visited the beehive in small groups, sometimes seeing the queen and the drone, and always having the opportunity to see the workers pursuing their various occupations, and the eggs, larvæ, and pupæ in their different states of development. Beautiful models of the bees and of the comb, together with dry and alcoholic material, illustrate further this metamorphosis, by contrast making clearer the exactly opposite metamorphosis of the locust. At least one member of each of the other orders of insects is compared with these two type forms, and, although only important points are considered at all, yet from one to two hours of laboratory work are devoted to each specimen. This leisurely method of work is pursued to give the students the opportunity, at least, to think for themselves. When the subject is finished they are then given a searching test. This is never directly on their required reading, but planned to show to them and to their teachers whether they have really assimilated what they have seen and studied. After this the myriapods, the earthworm, and peripatus are studied, because of their resemblance to the probable ancestors of insects. In the meantime they have had a dozen or more fully illustrated lectures on evolution, so that at the close of this series of lessons they are expected to have gained a knowledge of the methods of studying insects, whether living or otherwise, a working hypothesis for the interpretation of facts so obtained, and a knowledge of one order, which will serve admirably as a basis for comparison in much of their future work.
- 74. They then take up, more briefly, the relatives of the insects, the spiders and crustaceans, following these with the higher invertebrates, reaching the fish in April. This, for obvious reasons, is their last dissection. But with living material, and the beautiful preparations and stuffed specimens with which the laboratory is filled, they get a very general idea of the reptiles, birds, and mammals. This work is of necessity largely done by the students out of school hours. For example, on a stand on one of the tables are placed the various birds in season, with accompanying nests containing the proper quota of eggs. Books and pamphlets relating to the subject are placed near. Each student is given a syllabus which will enable her to study these birds intelligently indoors and out, if she wishes to do so. In the spring are taken up the orders of animals below the insect, and for the last lesson a general survey of all the types studied gives them the relationships of each to the other. The Course of Study pursued in the School of Practice.—In addition to the plants and animals about them, the children study the weather, keeping a daily record of their observations, and summarizing their results at the end of the month. In connection with the weather and plants they study somewhat carefully the soil and, in this connection, the common rocks and minerals of Philadelphia—gneiss, mica schist, granite, sandstone, limestones, quartz, mica, and feldspar. As in the laboratories, so here the effort is made to teach the children to reason, to read the story told by the individual plant, or animal, or stone, or wind, or cloud. A special effort is made to teach them to interpret everyday Nature as it lies around them. For this reason frequent short excursions into the city streets are made. Those who smile and think that there is not much of Nature to be found in a city street are those who have never looked for it. Enough material for study has been gathered in these excursions to make them a feature of this work, even more than the longer ones which they take twice a year into the country.
- 75. Last year I made not less than eighty such short excursions, each time with classes of about thirty-five. They were children of from seven to fourteen years of age. Without their hats, taking with them note-books, pencils, and knives, they passed with me to the street. The passers-by stopped to gaze at us, some with expressions of amusement, others of astonishment; approval sometimes, quite frequently the reverse. But I never once saw on the part of the children a consciousness of the mild sensation that they were creating. They went for a definite purpose, which was always accomplished. The children of the first and second years study nearly the same objects. Those of the third and fourth years review this general work, studying more thoroughly some one type. When they enter the fifth year, they have considerable causal knowledge of the familiar plants and animals, of the stones, and of the weather. But, what is more precious to them, they are sufficiently trained to be able to look at new objects with a truly seeing eye. The course of study now requires general ideas of physiology, and, in consequences, the greater portion of their time for science is devoted to this subject. I am glad to be able to say, however, that it is not School Physiology which they study, but the guinea-pig and The Wandering Jew! In other words, I let them find out for themselves how and what the guinea-pig eats; how and what he expires and inspires; how and why he moves. Along with this they study also plant respiration, transpiration, assimilation, and reproduction, comparing these processes with those of animals, including themselves. The children's interest is aroused and their observation stimulated by the constant presence in the room with them of a mother guinea-pig and her child. Nevertheless, I have not hesitated to call in outside materials to help them to understand the work. A series of lessons on the lime carbonates, therefore, preceded the lessons on respiration; an elephant's tooth, which I happened to have, helped
- 76. to explain the guinea-pig's molars; and a microscope and a frog's leg made real to them the circulation of the blood. In spite of the time required for the physiology, the fifth-year children have about thirty lessons on minerals; the sixth-year, the same number on plants; and the seventh-year, on animals; and it would be difficult to decide which of these subjects rouses their greatest enthusiasm.
- 77. PRINCIPLES OF TAXATION.[6] By the Late Hon. DAVID A. WELLS. XX.—THE LAW OF THE DIFFUSION OF TAXES. PART I. No attempt ought to be made to construct or formulate an economically correct, equitable, and efficient system of taxation which does not give full consideration to the method or extent to which taxes diffuse themselves after their first incidence. On this subject there is a great difference of opinion, which has occasioned, for more than a century, a vast and never-ending discussion on the part of economic writers. All of this, however, has resulted in no generally accepted practical conclusions; has been truthfully characterized by a leading French economist (M. Parieu) as marked in no small part by the simplicity of ignorance, and from a somewhat complete review (recently published[7]) of the conflicting theories advanced by participants one rises with a feeling of weariness and disgust. The majority of economists, legislators, and the public generally incline to the opinion that taxes mainly rest where they are laid, and are not shifted or diffused to an extent that requires any recognition in the enactment of statutes for their assessment. Thus, a tax commission of Massachusetts, as the result of their investigations, arrived at the conclusion that the tendency of taxes is that they must be paid by the actual persons on whom they are levied. But a little thought must, however, make clear that unless the advancement of taxes and their final and actual payment are one and the same thing, the Massachusetts statement is simply an
- 78. evasion of the main question at issue, and that its authors had no intelligent conception of it. A better proposition, and one that may even be regarded as an economic axiom, is that, regarding taxation as a synonym for a force, as it really is, it follows the natural and invariable law of all forces, and distributes itself in the line of least resistance. It is also valuable as indicating the line of inquiry most likely to lead to exact and practical conclusions. But beyond this it lacks value, inasmuch as it fails to embody any suggestions as to the best method of making the involved principle a basis for any general system for correct taxation; inasmuch as the line of least resistance is not a positive factor, and may be and often is so arranged as to make levies on the part of the State under the name of taxation subservient to private rather than public interests. Under such circumstances the question naturally arises, What is the best method for determining, at least, the approximative truth in respect to this vexed subject? A manifestly correct answer would be: first, to avoid at the outset all theoretic assumptions as a basis for reasoning; second, to obtain and marshal all the facts and conditions incident to the inquiry or deducible from experience; third, recognize the interdependence of all such facts and conclusions; fourth, be practical in the highest degree in accepting things as they are, and dealing with them as they are found; and on such a basis attention is next asked to the following line of investigations. It is essential at the outset to correct reasoning that the distinction between taxation and spoliation be kept clearly in view. That only is entitled to be called a tax law which levies uniformly upon all the subjects of taxation; which does not of itself exempt any part of the property of the same class which is selected to bear the primary burden of taxation, or by its imperfections to any extent permits such exemptions. All levies or assessments made by the State on the persons, property, or business of its citizens that do not conform to such conditions are spoliations, concerning which nothing but irregularity can be predicated; nothing positive concerning their diffusion can be asserted; and the most complete collection of experiences in respect to them can not be properly dignified as a
- 79. science. And it may be properly claimed that from a nonrecognition or lack of appreciation of the broad distinction between taxation and spoliation, the disagreement among economists respecting the diffusion of taxes has mainly originated. With this premise, let us next consider what facts and experiences are pertinent to this subject, and available to assist in reaching sound conclusions; proceeding very carefully and cautiously in so doing, inasmuch as territory is to be entered upon that has not been generally or thoroughly explored. The facts and experiences of first importance in such inquiry are that the examination of the tax rolls in any State, city, or municipality of the United States will show that surprisingly small numbers of persons primarily pay or advance any kind of taxes. It is not probable that more than one tenth of the adult population or about one twentieth of the entire population of the United States ever come in contact officially with a tax assessor or tax collector. It is also estimated that less than two per cent of the total population of the United States advance the entire customs and internal revenue of the Federal Government. In the investigations made in 1871, by a commission created by the Legislature of the State of New York to revise its laws relative to the assessment and collection of taxes, it was found that in the city of New York, out of a population of over one million in the above year, only 8,920 names, or less than one per cent of this great multitude of people, had any household furniture, money, goods, chattels, debts due from solvent debtors, whether on account of contract, note, bond, or mortgage, or any public stocks, or stocks in moneyed corporations, or in general any personal property of which the assessors could take cognizance for taxation; and further, that not over four per cent, or, say, forty thousand persons out of the million, were subject to any primary tax in respect to the ownership of any property whatever, real or personal; while only a few years subsequent, or in 1875, the regular tax commissioners of New York estimated that of the property defined and described by the laws of
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