A Short Course on Credit Risk Modeling

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A Short Course on Credit Risk Modeling

with Affine ProcessesDarrell Duffie1

Stanford University and Scuola Normale Superiore, Pisa

Rough and Incomplete Draft: April, 2002

Contents1 Introduction 3

2 Intensity-Based Modeling of Default 3

3 Affine Processes 73.1 Examples of Affine Processes . . . . . . . . . . . . . . . . . . . 9

4 Risk-Neutral Probability and Intensity 11

5 Zero-Recovery Bond Pricing 13

6 Pricing with Recovery at Default 15

7 Default-Adjusted Short Rate 18

8 Correlated Default 191I am extremely grateful for the wonderful work done by Maurizio Pratelli in arranging

my visit to Pisa and for organizing this course, for the hospitality of the Scuola NormaleSuperiore, and for the support for the course offered by the Associazione Amici dellaScuola Normale Superiore, who were generously represented by Mr. Carlo Gulminelli.

These notes are currently in the form of a rough draft. Future improvements can beobtained at www.stanford.edu/ duffie/.

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9 Credit Derivatives 21

9.1 Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 Credit Guarantees . . . . . . . . . . . . . . . . . . . . . . . . 229.3 Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . 249.4 Irrevocable Lines of Credit . . . . . . . . . . . . . . . . . . . . 269.5 Ratings-Based Step-Up Bonds . . . . . . . . . . . . . . . . . . 28

A Structural Models of Default 31A.1 The Black-Scholes-Merton Model . . . . . . . . . . . . . . . . 31A.2 First-Passage Models of Default . . . . . . . . . . . . . . . . . 33A.3 Example: Brownian Dividend Growth . . . . . . . . . . . . . . 36

B Itos Formula 40

C Foundations of Affine Processes 42C.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 43C.2 Denition and Characterization . . . . . . . . . . . . . . . . . 44

D Toolbox for Affine Processes 50D.1 Statistical Estimation of Affine Models . . . . . . . . . . . . . 50D.2 Laplace Transforms and Moments . . . . . . . . . . . . . . . . 51D.3 Inversion of the Transform . . . . . . . . . . . . . . . . . . . . 52

D.4 Solution for The Basic Affine Model . . . . . . . . . . . . . . . 53E Doubly Stochastic Counting Processes 54

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1 Introduction

These are lecture notes for a two-week course at Scuola Normale Superiore,Pisa, during April-May, 2002, on credit-risk modeling, emphasizing the valua-tion of corporate debt and credit derivatives, using the analytical tractabilityand richness of affine state processes.

Appendix A containts a brief overview of structural models that are basedon default caused by an insufficiency of assets relative to liabilities, includingthe classic Black-Scholes-Merton model of corporate debt pricing as well as astandard structural model, proposed by Fisher, Heinkel, and Zechner [1989]and solved by Leland [1994], for which default occurs when the issuers assets

reach a level so small that the issuer nds it optimal to declare bankruptcy.The main part of the course, however, treats default at an exogenouslyspecied intensity process, exploiting as a special case affine Markov stateprocesses.

Section 2 begins with the notion of default intensity, and the related calcu-lation of survival probabilities in doubly stochastic settings. The underlyingmathematical foundations are found in Appendix E. Section 3 introduces thenotion of affine processes, the main source of example calculations for the re-mainder of the course. Technical foundations for affine processes are found inAppendix C. Section 4 explains the notion of risk-neutral probabilities, andprovides the change of probability measure associated with a given changeof default intensity (a version of Girsanovs Theorem). Technical details forthis are found in Appendix E.

By Section 5, we see the basic model for pricing defaultable debt in a set-ting with stochastic interest rates and stochastic risk-neutral default intensi-ties, but assuming no recovery at default. The following two sections extendthe pricing models to handle default recovery under alternative parameteri-zations. Section 8 introduces multi-entity default modeling with correlation.Section 9 turns to applications such as default swaps, credit guarantees, ir-revocable lines of credit, and ratings-based step-up bonds. Exercises addressadditional applications. Some notes provide direction to further reading.

2 Intensity-Based Modeling of Default

This section introduces a model for a default time as a stopping time witha given intensity process, as dened below. From the joint behavior of the

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default time, interest-rates, the promised payment of the security, and the

model of recovery at default, as well as risk premia, one can characterize thestochastic behavior of the term structure of yields on defaultable bonds.

In applications, default intensities may be allowed to depend on observ-able variables that are linked with the likelihood of default, such as debt-to-equity ratios, volatility measures, other accounting measures of indebtedness,market equity prices, bond yield spreads, industry performance measures,and macroeconomic variables related to the business cycle. For details, seeDuffie and Singleton [2002]. This dependence could, but in practice does notusually, arise endogenously from a model of the ability or incentives of therm to make payments on its debt. Because the approach presented here

does not depend on the specic setting of a rm, it has also been appliedto the valuation of defaultable sovereign debt, as in Duffie, Pedersen, andSingleton [2000] and Pages [2000]. (For more on sovereign debt valuation,see Gibson and Sundaresan [1999] and Merrick [1999].)

We x a complete probability space ( , F , P ) and a ltration {Gt : t 0}of sub--algebras of F satisfying the usual conditions, which are listed inAppendix B. Also dened in Appendix B is the notion of a predictableprocess, which is, intuitively speaking, a process whose value at any time tdepends only on the information in the underlying ltration {Gt : t 0} thatis available up to, but not including, time t.

Appendix E denes a nonexplosive counting process. Such a countingprocess K records by time t the number K t of occurences of events of con-cern. A basic example of a counting process is a Poisson process. A countingprocess K has an intensity if is a predictable non-negative process sat-isfying

t0 s ds < almost surely for all t, with the property that a localmartingale M , the compensated counting process , is given by

M t = K t t

0s ds. (2.1)

Details are found in Appendix E. The accompanying intuition is that, atany time t, the

Gt -conditional probability of an event between t and t +

is approximately t , for small . This intuition is justied in the sense of derivatives if is bounded and continuous, and under weaker conditions.

For example, a Poisson process has a deterministic intensity process. If the intensity of a Poisson process is some constant , then the times be-tween events are independent exponentially distributed times with mean 1 / .

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A standard reference on counting processes is Bremaud [1981]. Additional

sources include Daley and Vere-Jones [1988] and Karr [1991].We will say that a stopping time has an intensity if is the rst event

time of a nonexplosive counting process whose intensity process is .A stopping time is nontrivial if P( (0, )) > 0. If a stopping time is nontrivial and if the ltration {Gt : t 0} is the standard ltration of some Brownian motion B in Rd, then could not have an intensity. We

know this from the fact that, if {Gt : t 0} is the standard ltration of B,then the associated compensated counting process M of (2.1) (indeed, anylocal martingale) could be represented as a stochastic integral with respectto B, and therefore cannot jump, but M must jump at . In order to

have an intensity, a stopping time must be totally inaccessible , a propertywhose denition (for example, in Meyer [1966]) suggests arrival as a suddensurprise, but there are no such surprises on a Brownian ltration!

As an illustration, we could imagine that the equityholders or managers of a rm are equipped with some Brownian ltration for purposes of determiningtheir optimal default time , as in Appendix A, but that bondholders haveimperfect monitoring, and may view as having an intensity with respect tothe bondholders own ltration {Gt : t 0}, which contains less informationthan the Brownian ltration. Such a situation gives rise to a default intensity,as in Duffie and Lando [2001], who show how to calculate the associateddefault intensity. 2

We say that a stopping time is doubly stochastic with intensity if theunderlying counting process whose rst jump time is is doubly stochasticwith intensity , as dened in Appendix E. The doubly-stochastic propertyimplies that, for any time t, on the event that the default time is after t,the probability of survival to a given future time s is

P ( > s | Gt ) = E e s

t (u) du Gt . (2.2)This property (2.2) is convenient for calculations, as evaluating the expec-tation in (2.2) is computationally equivalent to the standard calculation in

nance applications of pricing a default-free zero-coupon bond, treating asa short-term interest-rate process. Indeed, this analogy is also quite helpfulfor intuition when extending (2.2) to pricing applications.

2Elliott, Jeanblanc, and Yor [1999] give a new proof of this intensity result, which isgeneralized by Song [1998] to the multi-dimensional case. Kusuoka [1999] provides anexample of this intensity result that is based on unobservable drift of assets.

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It is sufficient for the convenient survival-time formula (2.2) that t =

(X t ) for some measurable : R d [0, ), where X in R d solves a stochas-tic differential equation of the formdX t = (X t ) dt + (X t ) dB t , (2.3)

for some (Gt )-standard Brownian motion B in Rd, where ( ) and ( ) arefunctions on the state space of X that satisfy enough regularity for (2.3) tohave a unique (strong) solution. With this, the survival probability calcula-tion (2.2) is of the form

P ( > s

| Gt ) = E e

st (X (u)) du X (t) (2.4)

= f (X (t), t ), (2.5)

where, under the usual regularity for the Feynman-Kac approach, f ( ) solvesthe partial differential equation (PDE)Af (x, t ) f t (x, t ) (x)f (x, t ) = 0 , (2.6)

for the generator A of X ,

Af (x, t ) =i

x i

f (x, t )i(x) + 12

i,j

2

x i x jf (x, t ) ij (x),

and where (x) = (x)(x) , with the boundary condition

f (x, s ) = 1 . (2.7)

Parametric assumptions are often used to get an explicit solution to thisPDE, as we shall see.

More generally, (2.2) follows from assuming that the doubly stochasticcounting process K whose rst jump time is is driven by some ltration

{F t : t 0}, a concept dened in Appendix E. (Included in the den-ition is the condition that F

t

Gt , and that

{F t : t

0

} satises the

usual conditions.) The intuition of the doubly-stochastic assumption is thatF t contains enough information to reveal the intensity t , but not enoughinformation to reveal the event times of the counting process K . In par-ticular, at any current time t and for any future time s, after conditioningon the -algebra Gt F s generated by the events in Gt F s , K is a Poissonprocess up to time s with (conditionally deterministic) time-varying intensity

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sense of (3.1). A more careful statement of this result is found in Appendix

C.Simple examples of affine processes used in nancial modeling are the

Gaussian Ornstein-Uhlenbeck model, applied to interest rates by Vasicek[1977], and the Feller [1951] diffusion, applied to interest-rate modeling byCox, Ingersoll, and Ross [1985]. A general multivariate class of affine jump-diffusion models was introduced by Duffie and Kan [1996] for term-structuremodeling. Using 3-dimensional affine diffusion models, for example, Daiand Singleton [2000] found that both time-varying conditional variances andnegatively correlated state variables are essential ingredients to explainingthe historical behavior of term structures of U.S. interest rates.

For option pricing, there is a substantial literature building on the partic-ular affine stochastic-volatility model for currency and equity prices proposedby Heston [1993]. Bates [1997], Bakshi, Cao, and Chen [1997], Bakshi andMadan [2000], and Duffie, Pan, and Singleton [2000] brought more generalaffine models to bear in order to allow for stochastic volatility and jumps,while maintaining and exploiting the simple property (3.1).

A key property related to (3.1) is that, for any affine function : D Rand any w Rd, subject only to technical conditions reviewed in Duffie,Filipovic, and Schachermayer [2001],E t e

st (X (u)) du + wX (s) = e (s t )+ (s t ) X (t) , (3.3)

for coefficients ( ) and ( ) that satisfy generalized Riccati ODEs (withreal boundary conditions) of the same type solved by and of (3.1),respectively.

In order to get a quick sense of how (3.3) and the associated Riccatiequations for the solution coefficients ( ) and ( ) arise, we consider thespecial case of an affine diffusion process X solving the stochastic differentialequation (2.3), with state space D = R+ , and with (x) = a + bx and2(x) = cx, for constant coefficients a, b, and c. (This is the continuousbranching process of Feller [1951].) We let (x) = 0 + 1x, for constants 0and 1, and apply the (Feynman-Kac) PDE (2.6) to the candidate solution(3.3). After calculating all terms of the PDE and then dividing each term of the PDE by the common factor f (x, t ), we arrive at

(z ) (z )x + (z )(a + bx) + 12

(z )2c2x 0 1x = 0, (3.4)

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for all z

0. Collecting terms in x, we have

u(z )x + v(z ) = 0 , (3.5)

where

u(z ) = (z ) + (z )b + 12

(z )2c2 1 (3.6)v(z ) = (z ) + (z )a 0. (3.7)

Because (3.5) must hold for all x, it must be the case that u(z ) = v(z ) =0. (This is known as separation of variables.) This leaves the Riccatiequations:

(z ) = (z )b + 12

(z )2c2 1 (3.8) (z ) = (z )a 0, (3.9)

with the boundary conditions (0) = 0 and (0) = w, from the fact thatf (x, s ) = w for all x. The explicit solutions for (z ) and (z ), developedby Cox, Ingersoll, and Ross [1985] for purposes of bond pricing (that is, forw = 0), is repeated in Appendix D, in the context a slightly more generalmodel with jumps.

The calculation (3.3) arises in many nancial applications, some of whichwill be reviewed momentarily. An obvious example is discounted expectedcash ow (with discount rate ( X t )), as well as the survival-probability cal-cuation (2.2) for an affine state process X and a default intensity ( X t ),taking w = 0 in (3.3).

3.1 Examples of Affine Processes

An affine diffusion is a solution X of the stochastic differential equation of the form (2.3) for which both (x) and (x)(x) are affine in x. This classincludes the Gaussian (Ornstein-Uhlenbeck) case, for which (x) is constant

(used by Vasicek [1977] to model interest rates), as well as the Feller [1951]diffusion model, used by Cox, Ingersoll, and Ross [1985] to model interestrates. These two examples are one-dimensional; that is, d = 1. The Fellerdiffusion satises

dX t = (x X t ) dt + c X t dB t , (3.10)9


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for constant positive parameters 3 c, , and x. The parameter x is called

a long-run mean, and the parameter is called the mean-reversion rate.Indeed for (3.10), the mean of X t converges from any initial condition tox at the rate as t goes to . The Feller diffusion, originally conceivedas a continuous branching process in order to model randomly uctatingpopulation sizes, has become popularized in nance as the Cox-Ingersoll-Ross (CIR) process.

Beyond the Gaussian case, any Ornstein-Uhlenbeck process, whetherdriven by a Brownian motion (as for the Vasicek model) or by a more generalLevy process, as in Sato [1999], is affine. Moreover, any continuous-branchingprocess with immigration (CBI process), including multi-type extensions of

the Feller process, is affine. (See Kawazu and Watanabe [1971].) Conversely,as stated in Appendix C, an affine process in Rd+ is a CBI process.For state spaces of the form D = R n+ R d n , Appendix C gives a smooth-ness condition on the coefficients ( ) and ( ) of the characteristic function(3.1) under which any affine process is of the jump-diffusion formA special example of (3.2) is the basic affine process, with state space

D = R + , satisfying

dX t = (x X t ) dt + c X t dB t + dJ t , (3.11)where J is a compound Poisson process,4 independent of B, with exponential jump sizes. The Poisson arrival intensity of jumps and the mean of the jump sizes completes the list ( , x,c, , ) of parameters of a basic affineprocess. Special cases of the basic affine model include the model with nodiffusion (c = 0) and the diffusion of Feller [1951] (for = 0). The basicaffine process is especially tractable, in that the coefficients (t) and (t) of (3.3) are known explicitly, and recorded in Appendix D.4. The coefficients

(t,iu ) and (t,iu ) of the characteristic function (3.1) are of the same form,albeit complex.

A simple class of multivariate affine processes is obtained by letting X t =(X 1t , . . . , X dt ), for independent affine coordinate processes X 1, . . . , X d. The

independence assumption implies that we can break the calculation (3.3)down as a product of terms of the same form as (3.3), but for the one-3The solution X of (3.10) will never reach zero from a strictly positive initial condition

if x > c 2 / 2, which is sometimes called the Feller condition.4A compound Poisson process has jumps at iid exponential event times, with iid jump

sizes.

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dimensional coordinate processes. This is the basis of the multi-factor CIR

model, often used to model interest rates, as in Chen and Scott [1995].An important 2-dimensional affine model was used by Heston [1993] to

model option prices in settings with stochastic volatility. Here, one supposesthat the underlying price process U of an asset satises

dU t = U t ( 0 + 1V t ) dt + U t V t dB1t , (3.12)where 0 and 1 are constants and V is a stochastic-volatility process, whichis a Feller diffusion satisfying

dV t = (v V t ) dt + c

V t dZ t , (3.13)

for constant coefficients , v, and c, where Z = B1 + 1 2 B2 is a stan-

dard Brownian motion that is constructed as a linear combination of inde-pendent standard Brownian motions B1 and B2. The correlation coefficient generates what is known as volatility asymmetry, and is usually measuredto be negative for major market stock indices. Option implied-volatilitysmile curves are, roughly speaking, rotated clockwise into smirks as becomes negative. Letting Y = log U , a calculation based on It os Formula(see Appendix B) yields

dY t = 0 + 1 12

V t dt +

V t dB1t , (3.14)

which implies that the 2-dimensional process X = ( V, Y ) is affine, with statespace D = R+ R . This leads to a simple method for pricing options, asexplained in Section 9. Extensions allowing for jumps have also useful forthe statistical analysis of stock returns from time-series data on underlyingasset returns and of option prices, as in Bates [1996] and Pan [1999]. 5

4 Risk-Neutral Probability and Intensity

Basic to the theory of the market valuation of nancial securities are risk-neutral probabilities, articially chosen probabilities under which the priceof any security is the expectation of the discounted cash ow of the security,as will be made more precise shortly.

5Among many other papers examining option prices for the case of affine state variablesare Bates [1997], Bakshi, Cao, and Chen [1997], Bakshi and Madan [2000], Duffie, Pan,and Singleton [2000], and Scott [1997].

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Risk-neutral probabilities are assigned by an equivalent martingale mea-

sure, a probability measure Q equivalent to P, in the sense that P and Qassign zero probabilities to the same events in Gt , for each t. Harrison andKreps [1979] showed that the existence of an equivalent martingale measureis equivalent (up to technical conditions) to the absence of arbitrage. Delbaenand Schachermayer [1999] gave denitive technical conditions for this result.There may be more than one equivalent martingale measure, however, andfor modeling purposes, one would work under one such measure. Commondevices for estimating an equivalent martingale measure include statisticalanalysis of historical price data, or the modeling of market equilibrium. Whenmarkets are complete, meaning that any contingent cash ow can be repli-

cated by trading the available securities, the equivalent martingale measureis unique (in a certain technical sense), and can be deduced from the priceprocesses of the available securities. For further treatment,see, for example,Duffie [2001].

We will assume the existence of a short-rate process, a progressively mea-surable process r with the property that

t0 |r (u)|du < for all t, and suchthat, for any times s and t > s , an investment of one unit of account (say,

one Euro) at any time s, reinvested continually in short-term lending untilany time t after s, will yield a market value of e

ts r (u) du . When we say under

Q , for some equivalent probability measure Q , we mean with respect to the

probability space ( , F ,Q

) and the same given ltration {Gt : t 0}. Forthe purpose of market valuation, we x some equivalent martingale measureQ , based on discounting at the short rate r . This means that, as of any timet, a security paying at any bounded stopping time T an amount given bya bounded GT -measurable random variable F has a modeled price, on theevent {T > t}, of E

t [e T

t r (u) du F ], where, for convenience, we write E t forexpectation under Q , given Gt .A risk-neutral intensity process for a default time is an intensity process for the default time , under Q . We also call the Q-intensity of .Artzner and Delbaen [1995] gave us the following convenient result.

Proposition. Suppose that a nonexplosive counting process K has a P-intensity process, and that Q is any probability measure equivalent to P . Then K has a Q-intensity process.

The ratio / (for strictly positive) represents a risk premium foruncertainty associated with the timing of default, in the sense of the follow-ing version of Girsanovs Theorem, which provides conditions suitable for

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From the denition of Q as an equivalent martingale measure, the price

S t of this security at any time t < s is given by

S t = E

t e s

t r (u) du 1{>s }F . (5.1)

From (5.1) and the fact that is a stopping time, S t must be zero for allt .Theorem 1. Suppose that F , r , and are bounded and that, under Q , is doubly stochastic driven by a ltration {F t : t 0}, with intensity process . Suppose, moreover, that r is (F t )-adapted and F is F s -measurable. Fix any t < s . Then, for t , we have S t = 0, and for t < ,

S t = E

t e s

t ( r (u)+ (u)) du F . (5.2)

The idea of (5.2) of is that discounting for default that occurs at anintensity is analogous to discounting at the short rate r . This result is basedon Lando [1994]. (See, also, Duffie, Schroder, and Skiadas [1996] and Lando[1998].)

Proof: From (5.1), the law of iterated expectations, and the assumptionthat r is (F t )-adapted and F is F s -measurable,

S t = E

E

e s

t r (u) du 1{>s }F F s Gt Gt

= E e s

t r (u ) du F E 1{>s } F s Gt Gt .The result then follows from the implication of double stochasticity that, onthe event { > t }, we have Q( > s |F s Gt ) = e

st

(u) du .

As a special case, suppose the driving ltration {F t : t 0} is thatgenerated by a process X that is affine under Q , with state space D. It isthen natural to allow dependence of , r , and F on the state process X inthe sense that

t = (X t ), r t = (X t ), F = eu X (s) , (5.3)

where and are real-valued affine functions on D. We have already adoptedthe convention that an intensity process is predictable, and have thereforedened t to depend in (5.3) on the left limit X t , rather than X t itself,because X need not itself be a predictable process. For the calculation (5.2),

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however, this makes no difference, because

(X t ) dt =

(X t ) dt, given

that X (, t) = X (, t) for almost every t.With (5.2) and (5.3), we can apply the basic property (3.3) of affineprocesses, so that for t < , under mild regularity,

S t = e (s t )+ (s t ) X (t) ,

for coefficients ( ) and ( ) satisfying the integro-differential associatedwith (5.2), namelyAe (s t )+ (s t ) xe (s t )+ (s t ) x (s t) (s t) x (x) (x) = 0 , (5.4)

where A is the generator (under Q) of X , and the boundary conditione (0)+ (0) x = 1. This in turn implies, by the same separation-of-variablesargument used in the simple example of Section 3, an associated general-ized Riccati ODE for ( ) and ( ), with boundary condition (0) = 0and (0) = 0 Rd. Solution of the ODE is explicit in certain cases, andotherwise a routine numerical computation, say by a Runge-Kutta method.

A sufficient regularity condition for this solution is that X is a regularaffine process and that the short-rate process r is non-negative. (See Duffie,Filipovic, and Schachermayer [2001] for details.)

6 Pricing with Recovery at DefaultThe next step is to consider the recovery of some random payoff W at thedefault time , if default occurs before the maturity date s of the security. Weadopt the assumptions of Theorem 1, and add the assumption that W = w ,where w is a bounded predictable process that is also adapted to the drivingltration {F t : t 0} of the doubly stochastic assumption of the theoremstatement.

The market value at any time t before default of any default recovery is,by the denition of the equivalent martingale measure Q ,

J t = E e

t r (u) du 1{ s}w Gt . (6.1)

The assumption that is doubly-stochastic implies that it has a proba-bility density under Q , at any time u in [t, s ], conditional on Gt F s , on theevent that > t , of

q (t, u ) = e u

t (z) dz u .

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(For details, see Appendix E.) Thus, using the same iterated-expectations

argument of the proof of Theorem 5, we have, on the event that > t ,

J t = E E e

t r (z) dz 1{ s}w F s Gt Gt= E

s

te

ut r (z) dz q (t, u )wu du Gt

= s

t(t, u ) du,

using Fubinis Theorem, where

(t, u ) = E

t e

u

t ( (z)+ r (z)) dz

u wu . (6.2)We summarize the main defaultable valuation result as follows.

Theorem 2. Consider a security that pays F at s if > s , and otherwise pays w at . Suppose that w, F , , and r are bounded. Suppose that is doubly stochastic under Q driven by a ltration {F t : t 0} with the property that r and w are (F t)-adapted and F is F s -measurable. Then, for t , we have S t = (0) , and for t < ,

S t = E

t e s

t ( r (u)+ (u)) du F +

s

t

(t, u ) du. (6.3)

Schonbucher [1998] extends to the case of a default recovery W that is notof the form w for some predictable process w, but rather allows the recoveryto be revealed just at the default time . We now allow this, taking howevera different construction. We let T be the stopping time min( , s ), and letW = E (W 1{


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= E t e T

t r (u) du W (6.5)

= E t e T

t r (u) du w(T ) (6.6)

= E t s

te

ut [r (z)+

(z)] dz (u)w(u) du , (6.7)

and we are back in the setting of the previous Theorem. Intuitively speaking,w(u) is the risk-neutral expected recovery given the information available inthe ltration {Gt : t 0} up until, but not including, time u, and given thatdefault will occur in the next instant, at time u.

In the affine state-space setting described at the end of the previous sec-

tion, (t, u ) can be computed by our usual affine methods, provided thatw(t) is of the form ea(t)+ b(t) X (t ) for deterministic a(t) and b(t). (In applica-tions, a common assumption is that w(t) is deterministic, but the availabledata appear to suggest otherwise for loan and bond recovery; see Duffie andSingleton [2002].) In this case, it is an exercise to show that, under technicalregularity,

(t, u ) = e (t,u )+ (t,u ) X (t)[c(t, u ) + C (t, u ) X (t)], (6.8)for readily computed deterministic coefficients , , c, and C . (For this ex-tended affine calculation, see Duffie, Pan, and Singleton [2000].) This stillleaves the simple but typically numerical task of computing of the integral

s

t (t, u ) du, say by quadrature.For the price of a typical defaultable bond promising periodic coupons

followed by its principal at maturity, one may sum the prices of the couponsand of the principal, treating each of these payments as though it were aseparate zero-coupon bond. An often-used assumption, although one thatneed not apply in practice, is that there is no default recovery for coupons,and that all bonds of the same seniority (priority in default) have the samerecovery of principal, regardless of maturity. In any case, convenient para-metric assumptions, based for example on an affine driving process X , leadto straightforward computation of a term structure of defaultable bond yieldsthat may be applied in practical situations.

For the case of defaultable bonds with embedded American options, themost typical cases being callable bonds or convertible bonds, the usual resortis valuation by some numerical implementation of the associated dynamicprogramming problems for optimal exercise timing. Acharya and Carpenter[1999] and Berndt [2002] treat callable defaultable bonds. On the related

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problem of convertible bond valuation, see Davis and Lischka [1999], Loshak

[1996], Nyborg [1996], and Tsiveriotis and Fernandes [1998]. On the timingof call and conversion options on convertible bonds, see Ederington, Caton,and Campbell [1997].

7 Default-Adjusted Short Rate

In the setting of Theorem 6, a particularly simple pricing representation canbe based on the denition of a predictable process for the fractional loss inmarket value at default, dened by

(1 )(S ) = w . (7.1)Manipulation that is left as an exercise shows that, under the conditions of Theorem 2, for t before the default time,

S t = E

t e s

t [r (u )+ (u) (u)] du F . (7.2)

This valuation model (7.2) is from Duffie and Singleton [1999], based on pre-cursors due to Pye [1974] and Litterman and Iben [1991]. This is particularlyconvenient if we take as an exogenously given fractional loss process, as itallows for the application of standard valuation methods, treating the pay-

off F as default-free, but accounting for the intensity and severity of defaultlosses through the default-adjusted short-rate process r + . The adjust-ment is in fact the risk-neutral mean rate of proportional loss in marketvalue due to default.

Notably, the dependence of the bond price on the intensity and frac-tional loss at default is only through the product . Thus, for anybounded strictly positive predictable process , the price process S of (7.2)is invariant (before default), to a substitution for and of and / ,respectively. For example, doubling and halving has no effect on theprice process before default.

Suppose, for example, that is doubly stochastic driven by X , and wetake r t + t

t = R(X t ) and F = f (X T ), for a Markov state process X . Forexample, X could be given as the solution to (2.3), or be an affine process.Then, under typical Feynman-Kac regularity conditions, we obtain at eachtime t before default the bond price S t = g(X t , t ), for a solution g of the

Ag(x, t ) gt (x, t ) R(x)g(x, t ) = 0 , (x, t ) D [0, s), (7.3)18


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doubly stochastic assumption, for any current time t < t (1), on the event

that i > t for all i (no defaults yet), we have

P ( 1 t1, . . . , k tk | Gt ) = E t e t ( k )

t (s) ds , (8.1)

where

(t) ={i: t (i) > t }

i(t). (8.2)

Now, in order to place the joint survivorship calculation into a computa-tionally tractable setting, we suppose that the driving ltration {F t : t 0}is that generated by an affine process X , and that i(t) = i(X t ), for anaffine function i( ) on the state space D of X . The state X t could includeindustry or economy-wide business-cycle variables, or market yield-spreadinformation, as well as rm-specic data. In this case, (t) = M (X t , t ),where,

M (x, t ) =i: t (i) > t

i(x). (8.3)

Because i( ) is affine for each i, so is M ( , t ) for each t. Thus, beginningour calculation at time t(k 1), and working recursively backward using thelaw of iterated expectations, we have, at each t(i), a solution of the formE t (i) e

t ( i +1)

t ( i ) M (X (s),s )) ds e (i+1)+ (i+1) X (t(i+1)) = e (i)+ (i) X (t(i)) , (8.4)

where the solution coefficients (i) and (i) are obtained from the generalizedRiccati Equation associated with the discount rate M ( , s), a xed7 affinefunction for t(i) s t(i + 1). By taking t(0) = t, we thus have

P ( 1 > t 1, . . . , k > t k | Gt ) = e (0)+ (0) X (t) . (8.5)Related calculations are explored in Duffie [1998b]. Applications in-

clude portfolio credit risk calculations and collateralized debt obligations (seeDuffie and Garleanu [2001]), credit-linked notes based on the rst to default,loans guaranteed by a defaultable guarantor, and default swaps signed by adefaultable counterparty.

7One could also solve in one step with a generalized Riccato equation having time-dependent coefficients, but in practice one might prefer to use a time-homogeneous affinemodel for which the solution coefficients (i) and (i) are known explicitly, arguing forthe recursive calculation of the joint survival probability.

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9 Credit Derivatives

We end the course with for some examples of credit-derivative pricing, be-ginning with the most basic and popular, the default swap. We then turnto credit guarantees, spread options on defaultable bonds, irrevocable linesof credit, and ratings-based step-up bonds. For more examples and analysisof credit derivatives, see Chen and Sopranzetti [1999], Cooper and Mar-tin [1996], Davis and Mavroidis [1997], Duffie [1998b], Duffie and Singleton[2002], Longstaff and Schwartz [1995b], and Pierides [1997].

9.1 Default Swaps

The simplest form of credit derivative is a default swap, which pays the buyerof protection, at the default time of a stipulated loan or bond, the differencebetween its face value and its recovery value, provided the default occursbefore a stated expiration date T . The buyer of protection makes periodiccoupon payments of some amount U each, until default or T , whichever isrst. This is, in effect, an insurance contract for the event of default.

The initial pricing problem is to determine the credit default swap rate(CDS rate) U (which is normally expressed at an annualized rate, so thatsemi-annual payments at a rate U = 0.03 per unit of face value means a CDSrate of 6%). Some discussion and contractual details are provided in Duffieand Singleton [2002].

Given a short-term interest rate process r and xing an equivalent mar-tingale measure Q , the total market value of the protection offered, per unitof face value, is

B = E e min( T, )

0 r (u) du (1 W )1{ t } (9.3)

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is the price of a zero-coupon no-recovery bond whose maturity date is t.

For the default-swap contract to be of zero market value to each counter-party, it must be the case that A = B, and therefore that

U = B

ni=1 V (t i)

. (9.4)

For example, suppose that we are in the setting of Theorem 2, and thatthe underlying bond has a risk-neutral default intensity process . Then

V (t) = E e t

0 [r (s)+ (s )] ds (9.5)

and, based on the same calculations used in Section 6,

B = T

0q (t) dt, (9.6)

where

q (t) = E e t

0 [ (s )+ r (s)] ds (t)(1 w(t)) , (9.7)

and where w(t), is the expected recovery conditional on information availableat time t, assuming that default is about to occur, in the sense dened morecarefully in Section 6. In practice, w(t) is normally taken to be a constantrisk-neutral mean recovery level.

In an affine setting, V (t), q (t), and thus the credit default swap rate U of (9.4) can be calculated routinely.

9.2 Credit Guarantees

We will suppose that a loan to a defaultable borrower with a default time Bhas a guarantor whose default time is G . We make the assumption that, inthe event of default by the borrower before the maturity date T of the loan,the guarantor simply takes over the obligation to pay the notional amount

of the loan at the original maturity date. This is somewhat unrealistic, butsimplies the exposition. The guaranteed loan thus defaults, in effect, onceboth the borrower and guarantor default, if both do before the loans ma-turity. In practice, the contractual obligation of the guarantor is normallyto pay the full principal on the loan withing a short time period after the

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borrower defaults. Our modeled price of the of the guaranteed loan is there-

fore conservative, that is, lowered by the assumption that the guarantor maydelay paying the obligation until the original maturity date of the loan.

For any given date t, the event that at least one of the borrower andthe guarantor survive until t is At = { B > t} { G > t} that at least onesurvives to t. We have

Q (At ) = Q ({ B > t}) + Q ({ G > t}) Q ({ B > t} { G > t}). (9.8)We will assume, in order to obtain concrete calculations, a doubly sto-

chastic model for B and G , with respective risk-neutral intensities B andG . From (9.8),

Q (At ) = E e t

0 B (s) ds + E e

t0

G (s) ds

E e t

0 [B (s)+ G (s)] ds , (9.9)

each term of which can be easily calculated if both intensities, B and G ,are affine with respect to a state process X that is affine under Q .

From this, depending on the recovery model and the probabilistic rela-tionship between interest rates and default times, one has relatively straight-forward pricing of the guaranteed loan. For example, suppose that interestrates are independent under Q of the default times B and G , and assumeconstant recovery of a fraction w of the face value of the loan. We let (t) = E (e

t0 r (u) du ) denote the price of a default-free zero-coupon bond

of maturity t, of unit face value. Then the market value of the guaranteedloan, per unit of face value, is

V = (T )q (T ) w T

0 (t)q (t) dt, (9.10)

where q (t) = Q(At ) is the risk-neutral probability that at least one of theborrower and the guarantor survive to t, so that q (t) is the risk-neutraldensity of the time of default of the guaranteed loan. In an affine setting,q (t) is easily computed, so the computation of (9.10) is straightforward.

Many variations are possible, including a recovery from guarantor defaultthat differs from that of the original borrower.

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9.3 Spread Options

The yield of a zero-coupon bond of unit face value, of price V , and of maturityt is (log V )/t . The yield spread of one bond relative to another of a loweryield is simply the difference in their yields. The prices of defaultable bondsare often quoted in terms of their yield spreads relative to some benchmark,such as government bonds. (Swap rates are often used as a benchmark, butwe defer that issue to Duffie and Singleton [2002].)

We will consider the price of an option to put (that is, to sell) at agiven time t a defaultable zero-coupon bond at a given spread s relative to abenchmark yield Y (t). The remaining maturity of the bond at time t is m.This option therefore has a payoff at time t of

Z = e (Y (t )+ s )m e (Y (t )+ S (t )) m+

, (9.11)

where S (t) is the spread of the defaultable bond at time t. For simplicity,we will suppose that a contractual provision prevents exercise of the optionin the event that default occurs before the exercise date t. A slightly morecomplicated result arises if the option can be exercised even after the defaultof the underlying bond. We will also suppose for convenience that the bench-mark yield Y (t) is the default-free yield of the same time m to maturity. Themarket value of this spread option is therefore

V = E

e t

0 r (u ) du 1{ > t }Z . (9.12)

We will consider a setting with a constant fractional loss of market valueat default, as in Section 7, so that the defaultable bond has a price at timet on the event that > t , of

e (Y (t )+ S (t )) m = E t e t + m

t R(u) du , (9.13)

where R(u) = r (u) + Q (u) is the default-adjusted short rate.In order to simplify the calculations, we will suppose that is doubly

stochastic driven by a state process X that is affine under Q , with a risk-neutral intensity

t = a0 + a1 X t . We also suppose that the default freeshort-rate process r is affine with respect to X , so that rt = 0 + 1 X t ,and the default-free reference yield to maturity m is of thus of the formY (t) = 0 + 1 X (t).The default adjusted short rate is R t = 0 + 1 X t , for 0 = 0 + a 0 and1 = 1 + a 1, so we have a defaultable yield spread, in the event of survival

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to t, of the form S (t) = 0 + 1

X (t). The option payoff, in the event of no

default by t, is thus of the formZ = ec0 + c1 X (t) ef 0 + f 1 X (t) 1{dX y}, (9.14)

where

f 0 = ( 0 + 0)mf 1 = ( 1 + 1)mc0 = ( 0 + s)mc1 = 1md = f 1

c1

y = c0 f 0.From (9.12) and the doubly-stochastic assumption, we therefore have

V (t) = V 0(t) + V 1(t), where

V 0(t) = E e t

0 [ 0 + 1 X (t)] du ec0 + c1 X (t)1{dX y} (9.15)

= Gt,,c,d (y) (9.16)

V 1(t) = E e t

0 [ 0 + 1 X (t)] du ef 0 + f 1 X (t)1{dX y} , (9.17)

= Gt,,f,d (y) (9.18)

where 0 = 0 + a0, 1 = 1 + a1, and

Gt,,c,d (y) = E e t

0 0 + 1 X (t) ec0 + c1 X (t) 1{dX (t) y} . (9.19)

Using the usual approach for option pricing in an affine setting, Gt,,c,d ( ) canbe calculated by inverting its Fourier Transform Gt,,c,d ( ), which is denedby

Gt,,c,d (z ) = eizy dGt,,c,d (y)= E e

t0 0 + 1 X (t ) ec0 +( c1 + izd ) X (t ) ,

using Fubinis Theorem.That is, under regularity, we can apply the Levy Inversion Formula,

Gt,,c,d (y) =Gt,,c,d (0)

2 1

0

Im Gt,,c,d (z )e izy

z dz,

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where Im(w) denotes the imaginary part of any complex number w.

The affine model is convenient, for it provides an easily computed solutionof the Fourier transform of the form

Gt,,c,d (z ) = e (t ;z)+ (t ;z) X (0) , (9.20)

where we have indicated explicitly the dependence on z of the boundarycondition of the generalized Riccati equations for (t; z ) and (t, z ).

In practice, one might chooses a parameterization of X for which and are explicit (for example multivariate versions of the basic affine model),or one may solve the generalized Riccati equations by a numerical method,such as Runge-Kutta.

9.4 Irrevocable Lines of Credit

Banks often provide a credit facility under which a borrower is offered theoption to enter into short-term loans for up to a given notional amount N ,until a given time T , all at a contractually xed spread. The short-term loansare of some term m. That is, at any time t among the potential borrowingdates m, 2m, 3m , . . . , T , the borrower may enter into a loan maturing at timet + m, of maximum size N , at a xed spread s over the current reference yieldY (t) for m-period loans.

Such a credit facility may be viewed as a portfolio of defaultable put

options sold to the borrower on the borrowers own debt. The borrower isusually charged a fee, however, for the unused portion of the credit facility,say F per unit of unused notional.

Our objective is to calculate the market value to the borrower of thecredit facility, including the effect of any fees paid. We will ignore the factthat use of the credit facility may either signal or affect the borrowers creditquality. This endogeneity is not easily treated directly by the option pricingmethod that we will use. Instead, we will assume the same doubly-stochsticaffine model of default given in the previous application to defaultable bondoptions. We will also assume that the bank is default free. (Otherwise, access

to the facility would be somewhat less valuable.)For each unit of notional, the option to use the facility at time t actuallymeans the option to sell, at a price equal to 1, a bond promising to pay thebank e(Y (t )+ s )m at time t + m. The market value of this obligation at time t,assuming that the borrower has survived to time t, is therefore

e (Y (t )+ S (t )) m e(Y (t )+ s )m = e(s S (t )) m ,

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recalling that S (t) is the borrowers spread on loans at time t maturing at

time t + m.If surviving to time t, the value of the option to the borrower, per unit of

notional, reecting also the benet of the reduction in the fee F per unit of unused credit, is, in the affine setting of the previous treatment of defaultabledebt options,

H (t) = 1 + F e(s S (t )) m+

= 1 + F eh0 + h1 X (t) 1gX (t) v, (9.21)where h0 = ( s 0)m, h1 = m 1, g = m 1, and v = log(1 + F ) + 0 s .Here, we used our previous spread calculation, S (t) = 0 + 1

X (t).

The optionality value of the credit facility for the portion of the periodbeginning at time t is

U (t) = E e t

0 r (s ) ds H (t)1{>t }

= E e t

0 [r (s)+ (s)] ds H (t)

= E e t

0 [ 0 + 1 X (s)] ds 1 + F eh0 + h1 X (t) 1{gX (t) v}= (1 + F )Gt,, 0,g(v) Gt,,h,g (v).

We can therefore calculate U (t) by the Fourier inversion method of the pre-vious application to defaultable debt options.

The fee F is paid at time t only when the borrower has not defaulted byt, and only when the borrower is not drawing on the facility at t. We havealready accounted in our calculation of U (t) for the benet of not paying thefee when the facility is used, so the total market value of the facilty to theborrower is

V (F ) =T/m

i=1

U (mi ) F e (mi )+ (mi ) X (0) , (9.22)

where ( ) and ( ) solve the the generalized Riccati equation associatedwith the survival-contingent discountE e

t0 [ 0 + 1 X (s)] ds = e (t )+ (t ) X (0) . (9.23)

If the credit facility is priced on a stand-alone zero-prot basis, the asso-ciated fee F would be set to solve the equation V (F ) = 0. In practice, the fee

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has not necessarily set in this fashion. On May 3, 2002, The Financial Times

indicated concern by banks over their pricing policy on credit facilities, andindicated fees on undrawn lines have recently been ranging from roughly 9basis points for A-rated rms to roughly 20 basis points for BBB-rated rms.Banks are also, apparently, begining to build some protection against the op-tionality in the exercise of these lines, by using fees (F ) or spreads (s) thatdepend on the fraction of the line that is drawn.

9.5 Ratings-Based Step-Up Bonds

Most publicly traded debt, and much privately issued debt, is assigned a

credit rating, essentially a credit quality score, by one or more of the majorcredit rating agencies. For example, the standard letter credit ratings forMoodys are Aaa, Aa, A, Baa, Ba, B, and D (for default). There are 3 renedratings for each letter rating, as in Ba1, Ba2, and Ba3. The term investmentgrade means rated Aaa, Aa, A, or Baa. For Moodys, speculative-graderatings are those below Baa. For Standard and Poors, whose letter ratingsare AAA, AA, A, BBB, BB, B, C, and D, speculative-grade means belowBBB.

It has become increasingly common for bond issuers to link the size of thecoupon rate on their debt with their credit rating, offering a higher couponrate at lower ratings, perhaps in an attempt to appeal to investors based onsome degree of hedging against a decline in credit quality. This embeddedderivative is called a ratings-based step-up. For example, The Financial Times reported on April 9, 2002, that a notional amount of approximately120 billion Euros of such ratings-based step-up bonds had been issued bytelecommunications rms, one of which, a 25-billion-Euro Deutsche Telekombond, would begin paying at extra 50 basis points (0.5%) in interest per yearwith the downgrade by Standard and Poors of Deutsche Telekom debt fromA to BBB+ on April 8, 2002, following8 a similar downgrade by Moodys.There is a potentially adverse effect of such a step-up feature, however, fora downgrade brings with it an additional interest expense, which, dependingon the capital structure and cash ow of the issuer, may actually reduce thetotal market value of the debt, and even bring on further ratings downgrades,

8Most ratings-based step-ups occur with a stipulated reduction in rating by any of themajor ratings agencies, but this particular bond required a downgrade by both Moodysand Standard and Poors before the step-up provision could occur.

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higher coupon rates, and so on. We will ignore this feedback effect in our

calculation of the pricing of ratings-based step-up bonds.We will simplify the pricing problem by considering the common case in

which the only ratings changes that cause a change in coupon rate are intoand out of the investment-grade ratings categories. That is, we assume thatthe coupon rate is cA whenever the issuers rating is investment grade, andthat the speculative-grade coupon rate is cB > cA .

Our pricing model is the same doubly-stochastic model used in earlierapplications. In order to treat ratings transitions risk, we will assume thatthat the risk-neutral default intensity t is higher for speculative grade rat-ings than for investment grade ratings. This is natural. In practice, however,

the maximum yield spread associated with investment grade uctuates withuncertainty over time. In order to maintain tractability, we will suppose theissuer has an investment-grade rating whenever t t , and is otherwise of speculative grade, where ( t) = 0 + 1 X (t). It would be equivalent forpurposes of tractability, since yield spreads are affine in X (t) in this setting,to suppose that the maximal level of investment-grade straight-debt yieldspreads at a given maturity is affine with respect to X (t).

We will use, for simplicity, a model with zero recovery of coupons atdefault, and recovery of a given risk-neutral expected fraction w of principalat default.

The coupon paid at coupon date t, in the event of survival to that date,is

c(t) = cA + ( cB cA)1{ (t ) ( t)}= cA + ( cB cA)1{h X (t ) u}, (9.24)

where h = 1 a1 and u = a0 0. The initial market value of this couponisF (t) = E e

t0 r (s ) ds c(t)1{>t }

= E e t

0 [r (s)+ (s)] ds c(t)

= E e

t0 [ 0 + 1 X (s)] ds [cA + ( cB cA)1{h X (t ) u}] ,

= cAe (t)+ (t ) X (0) + ( cB cA)Gt,, 0,h (u),where e (t)+ (t ) X (0) = E e

t0 [ 0 + 1 X (s)] ds . Calculation of Gt,, 0,h (u) is by

the Fourier inversion method used previously.

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For principal payment date T and coupon dates t1, t2, . . . , t n = T , the

initial price of the ratings-based step-up bond, under our assumptions, isthen

V 0 = e (T )+ (T ) X (0) + T

0(0, t) dt +

n

i=1

F (t i), (9.25)

where (0, t) is the market-value density for the recovery of principal, calcu-lated in an affine setting as in (6.8).

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Appendices

A Structural Models of Default

This appendix reviews the most basic classes of structural models of defaultrisk, which are built on a direct model of survival based on the sufficiency of assets to meet liabilities.

For this appendix, we let B be a standard Brownian motion in Rd ona complete probability space ( , F , P ), and we x the standard ltration

{F t : t

0

} of B.

A.1 The Black-Scholes-Merton Model

For the Black-Scholes-Merton model, based on Black and Scholes [1973] andMerton [1974], we may think of equity and debt as derivatives with respect tothe total market value of the rm, and priced accordingly. In the literature,considerable attention has been paid to market imperfections and to controlthat may be exercised by holders of equity and debt, as well as managers.With these market imperfections, the theory becomes more complex and lesslike a derivative valuation model.

With the classic Black-Scholes-Merton model of corporate debt and eq-uity valuation, one supposes that the rms future cash ows have a totalmarket value at time t given by At , where A is a geometric Brownian motion,satisfying

dAt = A t dt + A t dB t ,

for constants and > 0, and where we have taken d = 1 as the dimension of the underlying Brownian motion B. One sometimes refers to At as the assets of the rm. We will suppose for simplicity that the rm produces no cashows before a given time T . In order to justify this valuation of the rm,one could assume there are other securities available for trade that create

the effect of complete markets, namely that, within the technical limitationsof the theory, any future cash ows can be generated as the dividends of a trading strategy with respect to the available securities. There is thena unique price at which those cash ows would trade without allowing anarbitrage.

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We take it that the original owners of the rm have chosen a capital

structure consisting of pure equity and of debt in the form of a single zero-coupon bond maturing at time T , of face value L. In the event that thetotal value AT of the rm at maturity is less than the contractual paymentL due on the debt, the rm defaults, giving its future cash ows, worth AT ,to debtholders. That is, debtholders receive min( L, AT ) at T . Equityholdersreceive the residual max( AT L, 0). We suppose for simplicity that there areno other distributions (such as dividends) to debt or equity. We will shortlyconrm the natural conjecture that the market value of equity is given bythe Black-Scholes option-pricing formula, treating the rms asset value asthe price of the underlying security.

Bond and equity investors have already paid the original owners of therm for their respective securities. The absence of well-behaved arbitrageimplies that at, any time t < T , the total of the market values S t of equityand Y t of debt must be the market value At of the assets. This is one of themain points made by Modigliani and Miller [1958], in their demonstration of the irrelevance of capital structure in perfect markets.

Markets are complete given riskless borrowing or lending at a constantrate r and given access to a self-nancing trading strategy whose value processis A. (See, for example, Duffie [2001], Chapter 6.) This implies that there isat most one equivalent martingale measure.

Letting B t = B t + t, where = (

r )/ , we have

dAt = rA t dt + A t dB

t .

Girsanovs Theorem states that B is a standard Brownian motion underthe equivalent probability measure Q dened by

dQdP

= e B (T ) 2 T/ 2.

By Itos Formula, {e rt At : t [0, T ]} is a Q-martingale. It follows that,after discounting by e rt , Q is the equivalent martingale measure. As Q isunique in this regard, we have the unique price process S of equity in the

absence of well-behaved arbitrage (see, for example, Duffie [2001], Chapter6), given byS t = E

t e r (T t ) max( AT L, 0) .Thus, the equity price S t is computed by the Black-Scholes option-pricingformula, treating At as the underlying asset price, as the volatility coef-cient, the face value L of debt as the strike price, and T t as the time

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remaining to exercise. The market value of debt at time t is the residual,

At S t .When the original owners of the rm sold the debt with face value L andthe equity, they realized a total initial market value of S 0 + Y 0 = A0, whichdoes not depend on the chosen face value L of debt. This is again an aspect of the Modigliani-Miller Theorem. The same irrelevance of capital structure forthe total valuation of the rm applies much more generally, and has nothingto do with geometric Brownian motion, nor with the specic nature of debtand equity. With market imperfections, however, the design of the capitalstructure can be important in this regard.

Fixing the current value A t of the assets, the market value S t of equity is

increasing in the asset volatility parameter , due to the usual Jensen effectin the Black-Scholes formula. Thus, equity owners, were they to be giventhe opportunity to make a switch to a riskier technology, one with a largerasset volatility parameter, would increase their market valuation by doingso, at the expense of bondholders, provided the total initial market value of the rm is not reduced too much by the switch. This is a simple example of what is sometimes called asset substitution.

Given the time value of the option embedded in equity, bondholders wouldprefer to advance the maturity date of the debt; equityholders would preferto extend it.

Equityholders (or managers acting as their agents) typically hold thepower to make decisions on behalf of the rm, subject to legal and contrac-tual restrictions such as debt covenants. This is natural in light of equitysposition as the residual claim on the rms cash ows.

Geske [1977] used compound option modeling so as to extend to debt atvarious maturities.

A.2 First-Passage Models of Default

A rst-passage model of default is one for which the default time is therst time that the market value of the assets of the issuer have reacheda sufficiently low level. Black and Cox [1976] developed the idea of rst-passage-based default timing, but used an exogenous default boundary. Weshift now to a slightly more elaborate setting for the valuation of debt andequity, and consider the endogenous timing of default, using an approachformulated by Fisher, Heinkel, and Zechner [1989] and solved and extendedby Leland [1994], and subsequently, by others.

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We take as given an equivalent martingale measure Q .

The resources of a given rm are assumed to consist of cash ows atthe rate t for each time t. We suppose that is an adapted process with

t

0 | s|ds < almost surely for all t. The market value of the assets of therm at time t is dened as the market value At of the future cash ows. Thatis,

At = E

t

te r (s t ) s ds . (A.1)

We assume that At is well dened and nite for all t. The martingale repre-sentation theorem, which also applies under Q for the Brownian motion B ,

then implies thatdAt = ( rA t t ) dt + t dB

t , (A.2)

where is an adapted Rd-valued process such that T

0 t t dt < for allT [0, ), and where B is the standard Brownian motion in Rd under Qobtained from B and Girsanovs Theorem.We suppose that the original owners of the rm chose its capital structure

to consist of a single bond as its debt, and pure equity, dened in detail below.The bond and equity investors have already paid the original owners for thesesecurities. Before we consider the effects of market imperfections, the totalof the market values of equity and debt must be the market value A of theassets, which is a given process, so the design of the capital structure is againirrelevant from the viewpoint of maximizing the total value received by theoriginal owners of the rm.

For simplicity, we suppose that the bond promises to pay coupons at aconstant total rate c, continually in time, until default. This sort of bondis sometimes called a consol . Equityholders receive the residual cash ow inthe form of dividends at the rate t c at time t, until default. At default,the rms future cash ows are assigned to debtholders.

The equityholders dividend rate, t c, may have negative outcomes. Itis commonly stipulated, however, that equity claimants have limited liabil-ity , meaning that they should not experience negative cash ows. One canarrange for limited liability by dilution of equity. That is, so long as themarket value of equity remains strictly positive, newly issued equity can besold into the market so as to continually nance the negative portion ( c t )+of the residual cash ow. (Alternatively, the rm could issue debt, or other

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forms of securities, to nance itself.) When the price of equity reaches zero,

and the nancing of the rm through equity dilution is no longer possible,the rm is in any case in default, as we shall see. While dilution increasesthe quantity of shares outstanding, it does not alter the total market valueof all shares, and so is a relatively simple modeling device. Moreover, dilu-tion is irrelevant to individual shareholders, who would in any case be in aposition to avoid negative cash ows by selling their own shares as necessaryto nance the negative portion of their dividends, with the same effect asif the rm had diluted their shares for this purpose. We are ignoring hereany frictional costs of equity issuance or trading. This is another aspect of the Modigliani-Miller theory, that part of it dealing with the irrelevance of

dividend policy.Equityholders are assumed to have the contractual right to declare defaultat any stopping time T , at which time equityholders give up to debtholdersthe rights to all future cash ows, a contractual arrangement termed strict priority, or sometimes absolute priority . We assume that equityholders arenot permitted to delay liquidation after the value A of the rm reaches 0,so we ignore the possibility that AT < 0. We could also consider the optionof equityholders to change the rms production technology, or to call in thedebt for some price.

The bond contract conveys to debtholders, under a protective covenant ,the right to force liquidation at any stopping time at which the asset valueA is as low or lower than some stipulated level, which we take for now to bethe face value L of the debt. Debtholders would receive A at such a time ;equityholders would receive nothing.

Assuming that A0 > L , we rst consider the total coupon payment ratec that would be chosen at time 0 in order that the initial market value of the bond is its face value L. Such a bond is said to be at par, and thecorresponding coupon rate per unit of face value, c/L , is the par yield . If bondholders rationally enforce their protective covenant, we claim that thepar yield must be the riskless rate r. We also claim that, until default,the bond paying coupons at the total rate c = rL is always priced at itsface value L, and that equity is always priced at the residual value, A L.Finally, equityholders have no strict preference to declare default on a par-coupon bond before (L) = inf {t : At L}, which is the rst time allowedfor in the protective covenant, and bondholders rationally force liquidationat (L).

If the total coupon rate c is strictly less than the par rate rL , then eq-

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uityholders never gain by exercising the right to declare default (or, if they

have it, the right to call the debt at its face value) at any stopping time T with AT L, because the market value at time T of the future cash owsto the bond is strictly less than L if liquidation occurs at a stopping timeU > T with AU L. Avoiding liquidation at T would therefore leave amarket value for equity that is strictly greater than AT L. With c < rL ,bondholders would liquidate at the rst time (L) allowed for in their pro-tective covenant, for by doing so they receive L at (L) for a bond that, if left alive, would be worth less than L. In summary, with c < rL , the bondis liquidated at (L), and trades at a discount price at any time t beforeliquidation, given by

Y t = E

t (L)

te r (s t )c ds + e r ( (L) t )L (A.3)

= c

r + E t e r ( (L) t ) L

cr

< L. (A.4)

A.3 Example: Brownian Dividend Growth

As an example, suppose the cash-ow rate process is a geometric Brownianmotion under Q , in that

d t = t dt + t dB

t ,for constants and , where B is a standard Brownian motion under Q .We assume throughout that < r , so that, from (A.1), A is nite and

dAt = At dt + A t dB

t .

We calculate that t = ( r )At .For any given constant K (0, A0), the market value of a security thatclaims one unit of account at the hitting time (K ) = inf {t : At K } is, atany time t < (K ),E t e r ( (K ) t ) =

AtK

, (A.5)

where

= m + m2 + 2 r 2

2 , (A.6)

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and where m =

2/ 2. One can verify (A.5) as an exercise, applying It os

Formula.Let us consider for simplicity the case in which bondholders have no

protective covenant. Then equityholders declare default at a stopping timethat solves the maximum equity valuation problem

w(A0) supT T E

T

0e rt ( t c) dt , (A.7)

where T is the set of stopping times.We naturally conjecture that the maximization problem (A.7) is solvedby a hitting time of the form (AB ) = inf

{t : At

AB

}, for some default-

triggering level AB of assets, to be determined. Given this conjecture, wefurther conjecture from Itos Formula that the function w : (0, ) [0, )dened by (A.7) solves the ODE

Aw(x) rw (x) + ( r )x c = 0, x > A B , (A.8)where

Aw(x) = w (x)x + 12

w (x)2x2, (A.9)

with the absolute-priority boundary condition

w(x) = 0 , x AB . (A.10)Finally, we conjecture the smooth-pasting condition

w (AB ) = 0 , (A.11)

based on (A.10) and continuity of the rst derivative w ( ) at AB . Althoughnot an obvious requirement for optimality, the smooth-pasting condition,sometimes called the high-order-contact condition, has proven to be a fruitfulmethod by which to conjecture solutions, as follows.

If we are correct in conjecturing that the optimal default time is of theform (AB ) = inf {t : A t AB }, then, given an initial asset level A0 = x >AB , the value of equity must be

w(x) = x AB xAB

cr

1 xAB

. (A.12)

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at c, where W x (c,c) = 0. Under these conditions, we obtain the result, as

though from a standard application of Itos Formula, 9

W (As , c) = W (A0, c)+ s

0 AW (At , c) dt + s

0W x (At , c)A t dB

t ,(A.17)

where

AW (x, c) = W x(x, c)x + 12

W xx (x, c)2x2, (A.18)

except at x = c, where we may replace W xx (c,c) with zero.For each time t, let

q t = e rt W (At , c) + t

0e rs ((r )As c) ds.

From Itos Formula,

dq t = e rt f (At ) dt + e rt W x (At , c)A t dB

t , (A.19)

wheref (x) = AW (x, c) rW (x, c) + ( r )x c.

Because W x is bounded, the last term of (A.19) denes a Q-martingale. For

x c, we have both W (x, c) = 0 and ( r )x c 0, so f (x) 0.For x > c , we have (A.8), and therefore f (x) = 0. The drift of q istherefore never positive, and for any stopping time T we have q 0 E (q T ),or equivalently,

W (A0, c) E T

0e rs ( s c) ds + e rT W (AT , c) . (A.20)

For the particular stopping time (c), we have

W (A0, c) = E

(c )

0

e rs ( s

c) ds , (A.21)

9We use a version of Itos Formula that can be applied to a real-valued function that isC 1 and is C 2 except at a point, as, for example, in Karatzas and Shreve [1988], page 219.

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using the boundary condition (A.15) and the fact that f (x) = 0 for x > c .

So, for any stopping time T ,

W (A0, c) = E (c )

0e rs ( s c) ds (A.22)

E T

0e rs ( s c) ds + e rT W (AT , c)

E T

0e rs ( s c) ds ,

using the nonnegativity of W for the last inequality. This implies the opti-mality of the stopping time (c) and verication of the proposed solutionW (A0, c) of (A.7).

This model was further elaborated to treat taxes in ?], coupon debt of nite maturity in Leland and Toft [1996], endogenous calling of debt and re-capitalization in Leland [1998] and Uhrig-Homburg [1998], incomplete obser-vation by bond investors, with default intensity, in Duffie and Lando [2001],and alternative approaches to default recovery in Anderson and Sundaresan[1996], Anderson, Pan, and Sundaresan [1995], Fan and Sundaresan [1997],Mella-Barral [1999], and Mella-Barral and Perraudin [1997].

Longstaff and Schwartz [1995a] developed a similar rst-passage default-able bond pricing model with stochastic default-free interest rates. (See alsoNielsen, Saa-Requejo, and Santa-Clara [1993] and Collin-Dufresne and Gold-stein [1999].) Zhou [2000] bases pricing on rst passage of a jump-diffusion.

B Itos Formula

This appendix states Itos Formula, allowing for jumps, and including somebackground properties of semimartingales. A standard source is Protter[1990].

We rst establish some preliminary denitions. We x a complete prob-ability space ( , F , P ) and a ltration {Gt : t 0} satisfying the usual conditions :

For all t, Gt contains all of the null sets of F . For all t, Gt = s>t Gs , a property called right-continuity .

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A function Z : [0,

)

R is left-continuous if, for all t, we have Z t =

limst Z s ; the process has left limits if Z t = lim st Z s exists; and nally theprocess is right-continuous if Z t = lim st Z s . The jump Z of Z at time tis Z t = Z t Z t . The class of processes that are right-continuous withleft limits is called RCLL, or sometimes cadlag, for continue ` a, limite agauche.

Under the usual conditions, we can without loss of generality for ourapplications assume that a martingale has sample paths that are almostsurely right-continuous with left limits. See, for example, Protter [1990],page 8. This is sometimes taken as a dening property of martingales, forexample by Jacod and Shiryaev [1987b].

Lemma 1. Suppose Q is equivalent to P , with density process . Then an adapted process Y that is right-continuous with left limits is a Q-martingale if and only if Y is a P-martingale.

A process X is a nite-variation process if X = U V , where U and V areright-continuous increasing adapted processes with left limits. For example,X is nite-variation if X t =

t0 s ds, where is an adapted process such that

the integral exists. The next lemma is a variant of Itos Formula.

Lemma 2. Suppose X is a nite-variation process and f : R R is con-tinuously differentiable. Then

f (X t ) = f (X 0) + t

0+f (X s ) dX s +

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The class of regular affine processes include the entire class of continuous-

state branching processes with immigration (CBI) (for example, Kawazu andWatanabe [1971]), as well as the class of processes of the Ornstein-Uhlenbeck(OU) type (for example, Sato [1999]). Roughly speaking, the regular affineprocesses with state space Rm+ are CBI, and those with state space Rn areof OU type. For any regular affine process X = ( Y, Z ) in Rm+ R n , Therst component Y is necessarily a CBI process. Any CBI or OU typeprocess is innitely decomposable, as was well known and apparent fromthe exponential-affine form of the characteristic function of its transition dis-tribution. A regular (to be dened below) Markov process with state space Dis innitely decomposable if and only if it is a regular affine process. Regular

affine processes are also semimartingales, a crucial property in most nancialapplications because the standard model of the nancial gain generated bytrading a security is a stochastic integral with respect to the underlying priceprocess.

We will restrict our attention to the case of time-homogeneous conser-vative processes (no killing) throughout. For the case that allows for forkilling, see Duffie, Filipovic, and Schachermayer [2001]. For the case of time-inhomogeneous affine processes, see Filipovic [2001].

The remainder of the appendix is organized as follows. In Section C.2we provide the denition of a regular affine process X (Denitions C.1 andC.3) and the main characterization result of affine processes. Three otherequivalent characterizations of regular affine processes are reviewed: (i) interms of the generator (Theorem C.5), (ii) in terms of the the semimartingalecharacteristics (Theorem C.8), and (iii) in terms of the property of innitedecomposability (Theorem C.10). Proofs of these results are found in Duffie,Filipovic, and Schachermayer [2001].

C.1 Basic Notation

For background and notation we refer to Jacod and Shiryaev [1987b] andRevuz and Yor [1994]. Let k

N. For and in Ck , we write

, :=

1 1 + + k k (notice that this is not the scalar product on Ck). We letSemk be the convex cone of symmetric positive semi-denite k k matrices.If U is an open set or the closure of an open set in Ck , we write U for theclosure, U 0 for the interior, and U = U \U 0 for the boundary.We use the following notation for function spaces:

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Observe that, with a slight abuse of notation,

U u P t f u (x)is the characteristic function of the measure pt (x, ), that is, the characteristicfunction of X t with respect to Px .Denition C.1 The Markov process (X, (P x )x D ), and (P t ), is called affineif, for every t R+ , the characteristic function of pt (x, ) has exponential-affine dependence on x. That is, if for every (t, u ) R+ U there exist (t, u ) C and (t, u ) = ( Y (t, u ), Z (t, u )) C m C n such that

P t f u (x) = e (t,u )+ (t,u ),x (C.2)= e (t,u )

Y (t,u ),y + i Z (t,u ),z , x = ( y, z ) D. (C.3)Because P t f u is in bD for all (t, u ) R+ U , we infer from (C.2) that,a fortiori (t, u ) C+ and (t, u ) = ( Y (t, u ), Z (t, u )) U for all (t, u ) R + U .We note that (t, u ) is uniquely specied by (C.2). But Im (t, u ) is

determined only up to multiples of 2 . Nevertheless, by denition we haveP t f u (0) = 0 for all (t, u ) R + U . Since U is simply connected, P t f u (0)admits a unique representation of the form (C.2) and we shall use the symbol

(t, u ) in this sense from now on such that (t,

) is continuous on

U and

(t, 0) = 0.

Denition C.2 The Markov process (X, (P x )x D ), and (P t ), is stochasti-cally continuous if ps (x, ) pt (x, ) weakly on D, for s t, for every (t, x ) R + D.If (X, (P x )x D ) is affine then, by the continuity theorem of Levy, ( X, (P x )x D )is stochastically continuous if and only if (t, u ) and (t, u ) from (C.2) arecontinuous in t R + , for every u U .Denition C.3 The Markov process (X, (P x )x D ), and (P t ), is called regu-

lar if it is stochastically continuous and the right-hand derivative Af u (x) := +t P t f u (x)|t=0

exists, for all (x, u ) D U , and is continuous at u = 0, for all x D.We call (X, (P x )x D ), and (P t ), simply regular affine if it is regular and affine.

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If there is no ambiguity, we shall write indifferently X or (Y, Z ) for the

Markov process (X, (P x )x D ), and say shortly that X is affine , stochasti-cally continuous , regular , regular affine if (X, (P x )x D ) shares the respectiveproperty.

In order to state the main characterization results, we require certainnotation and terminology. We denote by {e1, . . . , e d} the standard basis inR d, and write I := {1, . . . , m} and J := {m + 1, . . . , d}. We dene thecontinuous truncation function = ( 1, . . . , d) : R d [1, 1]d by

k( ) = 0 , k = 0,

= 1 | k|

| k| k , otherwise. (C.4)

Let = ( ij ) be a d d-matrix, = ( 1, . . . , d) a d-tuple and I, J {1, . . . , d}. Then we write T for the transpose of , and IJ := ( ij )i I, j J and I := ( i)i I . Examples are I ( ) = ( k( ))k I or I := ( xk )k I . Ac-cordingly, we have Y (t, u ) = I (t, u ) and Z (t, u ) = J (t, u ) (since thesemappings play a distinguished role we introduced the former, coordinate-free notation). We also write 1 := (1 , . . . , 1) without specifying the dimen-sion whenever there is no ambiguity. For i I we dene I (i) := I \ {i}and J (i) := {i} J , and let Id( i) denote the m m-matrix given byId( i)kl = ik kl , where kl is the Kronecker Delta ( kl equals 1 if k = l and 0otherwise).Denition C.4 The parameters (a,,b,,m, ) are called admissible if

a Semd with a II = 0 (hence a IJ = 0 and aJ I = 0). = ( 1, . . . , m ) with i Semd and i, II = i,ii Id( i), for all i I . b D. Rd d such that IJ = 0 and i I (i) Rm 1+ , for all i I . (Hence, II has nonnegative off-diagonal elements). m is a Borel measure on D \ {0} satisfying

D \{ 0} I ( ), 1 + J ( ) 2 m(d ) < .

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= ( 1, . . . , m ), where every i is a Borel measure on D

\ {0

} satis-

fying D \{ 0} I (i)( ), 1 + J (i)( ) 2 i(d ) < .

Now we have the main characterization results from Duffie, Filipovic, andSchachermayer [2001]. First, we state an analytic characterization result forregular affine processes.

Theorem C.5 Suppose X is regular affine. Then X is a Feller process.Let A be its innitesimal generator. Then C c (D) is a core of A, C 2c (D) D(A), and there exist admissible parameters (a,,b,,m, ) such that, for f C 2c (D),

Af (x) =d

k,l =1 (akl + I ,kl , y ) 2f (x)x kx l + b + x, f (x)

+ D \{ 0} Gf 0(x, ) m(d )+

m

i=1 D \{ 0} Gf i(x, ) yii(d ), (C.5)where

Gf 0(x, ) = f (x + ) f (x) J f (x), J ( )Gf i(x, ) = f (x + )

f (x)

J (i)f (x), J (i)( ) .

Moreover, (C.2) holds for all (t, u ) R + U where (t, u ) and (t, u ) solve the generalized Riccati equations ,(t, u ) =

t

0F ((s, u )) ds (C.6)

t Y (t, u ) = RY Y (t, u ), e Z t w , Y (0, u) = v (C.7)

Z (t, u ) = e Z t w (C.8)

with

F (u) =

au, u

b, u

D \{ 0} e u, 1 uJ , J ( ) m(d ), (C.9)RY i (u) = i u, u Y i , u

D \{ 0} e u, 1 uJ (i) , J (i)( ) i(d ), (C.10)47


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Then the ltrations (

F

(x)t ) and (

F t ) are right-continuous, and X is still a

Markov process with respect to ( F t ). That is, (C.1) holds for F 0t replaced byF t , for all x D.By convention, we call X a semimartingale if X is a semimartingale on(, F , (F t ), P x ), for every x D. For the denition of the characteristics of a semimartingale with refer to (Jacod and Shiryaev 1987b, Section II.2).We emphasize that the characteristics below are associated to the truncationfunction , dened in (C.4).

Let X be a D-valued stochastic process dened on some probability space( , F , P ). Then P X 1 denotes the law of X , that is, the image of P bythe mapping X ( ) : ( , F ) (, F 0).

The following is a characterization result for regular affine processes inthe class of semimartingales.

Theorem C.8 Let X be regular affine and (a,,b,,m, ) the related ad-missible parameters. Then X is a semimartingale and admits the character-istics (B,C, ), where

B t = t

0b + X s ds (C.13)

C t = 2

t

0

a +m

i=1

iY is ds (C.14)

(dt,d ) = m(d ) +m

i=1

Y it i(d ) dt (C.15)

for every Px , where b D and R d d are given by b = b + D \{ 0} ( I ( ), 0) m(d ), (C.16)

kl = kl + (1 kl ) D \{ 0} k( ) l(d ), if l I , (C.17)= kl , if l J , for 1 k d. (C.18)

Moreover, let X = ( Y , Z ) be a D-valued semimartingale dened on some ltered probability space ( , F , (F t ), P ) with P [X 0 = x] = 1. Suppose X admits the characteristics (B , C , ), given by (C.13)(C.15) where X is replaced by X . Then P X 1 = P x .

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There is a third way of characterizing regular affine processes, which

generalizes Shiga and Watanabe [1973]. Let P and Q be two probabilitymeasures on (, F 0). We write P Q for the image of P Q by the measurablemapping ( , ) + : (, F 0 F 0) (, F 0). Let P RM be the set of all families (P x )x D of probability measures on (, F 0) such that ( X, (P x )x D )is a regular Markov process with Px [X 0 = x] = 1, for all x D.Denition C.9 We call (P x )x D innitely decomposable if, for every k N ,there exists (P (k)x )x D P RM such that

P x (1) + ... + x ( k ) = P(k)x (1) P

(k)x ( k ) , for all x

(1) , . . . , x (k) D. (C.19)

Theorem C.10 The Markov process (X, (P x )x D ) is regular affine if and only if (P x )x D is innitely decomposable.

Without going much into detail, we remark that in (C.5) we can distin-guish the three building blocks of any jump-diffusion process, the diffusion matrix A(x) = a + y11 + + ym m , the drift B(x) = b+ x , and the Levy measure (the compensator of the jumps) M (x,d ) = m(d ) + y11(d ) + + ym m (d ). An informal denition of an affine process could consist of the requirement that A(x), B (x), and M (x,d ) have affine dependence on x.(See, for example, Duffie, Pan, and Singleton [2000].) The particular kind of

this affine dependence in the present setup is implied by the geometry of thestate space D.

D Toolbox for Affine Processes

This appendix contains some tools for affine processes.

D.1 Statistical Estimation of Affine Models

For a given (regular) affine process X with state space D R d, we reconsiderthe conditional characteristic function of X T given X t , dened from (3.1)by

(u, X t , t ,T ) = E eiu X T |X t , (D.1)for real u in Rd. Because knowledge of is equivalent to knowledge of the joint conditional transition distribution function of X , this result is useful

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in estimation and all other applications involving the transition densities of

affine processes.For instance, Singleton [2001] exploits knowledge of to derive maximum

likelihood estimators for the coefficients of an affine process, based on theconditional density f ( | X t ) of X t+1 given X t , obtained by Fourier inversionof as

f (X t+1 |X t ) = 1(2)N N e iu X t +1 (u, X t , t , t + 1) du. (D.2)

Das [1998] exploits (D.2) for special case of an affine process to computemethod-of-momen

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