ftd valuation stanford

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First-to-Default ValuationDarrell Duffie

Graduate School of Business, Stanford Universityand Universite de Paris, Dauphine

Current draft: May 10, 1998First draft: April 23, 1998

PRELIMINARY AND INCOMPLETE

Abstract: This paper! provides simple models and applications for thevaluation and simulation of contingent claims that depend on the time andidentity of the first to occur of a given list of credit events, such as de-faults. Examples include credit derivatives with a first-to-default feature,credit derivatives signed with a default able counterparty, credit-enhancementor guarantees, and other related financial positions.

1I am grateful for motivating discussions with Ken Singleton of Stanford University;David Friedman and Tony Morris of SBC Warburg Dillon Read; Paul Vartsosis of ChaseSecurities; and Mark Preston, Josh Danziger, Stanley Myint, and Gerson Riddy of CIBC.Address for correspondence: GSB Stanford CA 94305-5015 (Phone: 650-723-1976; Fax:650-725-7979; Email: duffiebaht.stanford.edu).This work was supported in partby the Coopers-Lybrand Risk Institute, by the Financial Research Initiative, and by theGifford Fong Associates Fund at The Graduate School of Business, Stanford University. Iam also grateful for hospitality from the Institut de Finance, Universite de Paris, Dauphine,where this research was undertaken, and especially from my host there, Ivar Ekeland. Forcomments, corrections, and suggestions, I am particularly grateful to Freddy Delbaen.

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1 IntroductionThis paper provides simple models and applications for the valuation andsimulation of contingent claims that depend on the time and identity of thefirst to occur of a given list of credit events, such as defaults. Examplesinclude credit derivatives with a first-to-default feature, credit derivativessigned with a default able counterparty, credit-enhancement or guarantees,and other related financial positions.

2 Model SettingWe suppose there are n credit events to be considered. A credit event istypically default by a particular entity, such as a counterparty, borrower, orguarantor. There are interesting applications, however, in which credit eventsmay be defined instead in terms of downgrades, events that may instigate(with some uncertainty perhaps) the default of one or more counterparties,or other credit-related occurrences.P The key to modeling the joint timingof these credit events is a collection h = (hi, ... ,hn) of stochastic intensityprocesses, where hi is the intensity process of credit event i. In general, anintensity process A for a stopping time T is characterized by the propertythat, for N(t) =1T2:t , a martingale is defined by

N t - l t (1 - Ns )As ds, t ~ o .For details, see Bremaud (1980).3 With constant intensity A , the event hasa Poisson arrival at intensity A . More generally, for t before a stopping time

2At a presentation at the March, 1998 ISDA conference in Rome, Daniel Cunnighamof Cravath, Swaine, and Moore reviwed the documentation of credit swaps, includingthe specification of credit event types such as "bankruptcy, credit event upon merger,cross acceleration, cross default, downgrade, failure to pay, repudiation, or restructuring."The credit event is to be documented with a notice, suported with evidence of publicannouncement of the event, for example in the international press. The amount to bepaid at the time of the credit event is determined by one or more third parties, and basedon physical or cash settlement, as indicated in the confirmation form of the OTC creditswap transaction, a standard contract form with alternatives to be indicated.

3All random variables are defined on a fixed probability space (0, F, P). A filtration{Ft : t 2 O } of a-algebras, satisfying the usual conditions, is fixed and defines the informa-tion available at each time. An intensity process A . is assumed to be non-negative and pre-dictable (a natural measurability restriction) and to satisfy, for each t > 0, J~A .s ds < 00

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T with intensity process A, we may view At as the conditional rate of arrivalof the event at time t, given all information available up to that time. Inother words, for a small time interval of length ~, the conditional probabilityat time t that the event occurs between t and t + ~, given survival to t, isapproximately'' At~.

Intensity processes h1, ... , hn for credit event times Tl, ... , Tn may insome cases be derived from information concerning the incentives and abil-ities of counterparties to meet their obligations, as in Duffie and Lando(1997), or might be fitted to market yield spreads, firm-specific financialratios, sovereign risk indicators, or macroeconomic variables, as in Altman(199?), Bijnen and Wijn (1994), McDonald and Van de Gucht (1996), andShumway (1996). We simply take a joint intensity process h = (hi, ... ,hn)for (Tl' ... ,Tn) as given, as in the "reduced-form" default able term-structureliterature."

3 First-Arrival IntensityA simplifying and natural assumption is that there is zero probability thatmore than one credit event occurs at the same time. Under this assumption,the intensity associated with the first" credit event is easily obtained.Lemma 1. Suppose, for each iE {I,... ,n}, that event time Ti has intensityprocess hi. Suppose that, P( Ti=Tj) =0 for i 1 = j. Then hi + ... + hn is anintensity process for T =min(Tl, ... , Tn).Proof: Let N, denote the point process associated with Ti, and N denotealmost surely. Some authors prefer to to describe "the" intensity as (1 - Ns's ratherthan A .s, and this indeed, under technical continuity conditions, allows for a uniqueness-of-intensity property. Our definition is weaker and does not suggest uniqueness, but hasother definitional advantages, as in Lemma 1 and Proposition 1.4This is true in the usual sense of derivatives if, for example, A . is a bounded continuous

process, and otherwise can be interpreted in an almost-everywhere sense.5Reduced-form models include those of Artzner and Delbaen (1995), Duffie and Sin-

gleton (1995), Duffie, Schroder, and Skiadas (1996), Jarrowand Turnbull (1996), Jarrow,Lando, and Turnbull (1997), Lando (1993, 1997, 1998), Madan and Unal (1995), Martin(1997), Nielsen and Ronn (1995), Pye (1974), Sch6nbucher (1997), and others.6We may have Ti = 00 with positive probability, and we take the definition T = 00 on

the event that Ti = 00 for all i.

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the point process associated with T. Then, because P( Ti =Tj) =0,

By the definition of the intensities hi, .. . ,hm martingales M1 , ... ,Mn aredefined byMi(t ) =Ni( t ) - i t (1 - N i(s) )hi(s) ds.

Let a process M be defined byM(t) =N( t ) - I t ( 1 - N(s ) ) [h1( s ) + ... + hn(s)] ds.

We have M(t ) =Ml(t ) + ... + Mn(t ) for t : :; T , and M(t ) =M(T) for t ~ T ,so M is also a martingale. Thus hi + ... + hn is, by definition, the intensityprocess for T, as asserted.We would next like to characterize the "survival" probability P(T ~ t) ,

for a given t. The jump ~Y of a semi-martingale Y is defined by ~Y( t ) =Y(t ) -limstt Y(s) . The following proposition is likely to be well known amongspecialists; a proof is in any case provided for completeness.Proposition 1. Suppose T is a stopping time with a bounded intensity pro-cess A . Fixing some time T > 0, le t

If the jump ~Y(T) is zero almost surely, thent < T,

a lm ost surely.Proof: Let Z be the martingale defined by

z; =E [exp ( - i T Au d U ) I Ft], t : : ; T.Because4


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Ito's Formula implies that

By definition of A, there is a martingale M such that

Let U be defined by U; =Y(t)(1 - Nt). By Ito's Formula,dUt =-Y(t-) dNt + (1 - N(t-)) dyt - ~Y(t)~N(t).

By the assumption that ~Y(T) =0, we know that ~Y(t)~N(t) =O. Usingour expressions above for dyt and dNt,

au , =-Y(t-) dMt + (1 - N(t-)) exp ( I t A u dU) dZt.We thus find that U is a martingale with U(T) =1 - N(T). Thus, for anyt < T,

Y(t) =U(t) =E(1 - NT 1Ft) =P(T ~ TIFt),as claimed.

The assumption that A is bounded can be replaced with integrabilityconditions, as usual. 7Remark 1: For any given predictable non-negative process A satisfying,for each t, J ; A s ds < 00, we can always define a stopping time T with theproperty that A is its intensity and with the property assumed in Proposition1, that ~Y(T) =0 almost surely. This can be done, for example, by letting7For U to be a martingale, we want J Y(t-) dM t and (1 - N(t-)) exp (I~u dU ) az,

to be martingales. The first is a martingale as Y is bounded. For the second, letting,{}(t) = exp (I~As d s ) , it is enough that E [ J o T {}(t) d [ Z l t ] < 00, where [ Z l is the "square-brackets" process for Z. For details, see, for example, Protter (1990).

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A be an exponentially distributed random variable with m ean 1, independentoj Y', and by let ting T =inf{t: J ; A s ds =A}.4 Formulation of Credit EventsIf the hypotheses of Lemma 1 or Proposition 1 are not satisfied, it may bepossible to re-formulate the definitions of the credit events so as to retainthese properties. For instance, their failures can occur because, at somestopping time U, one or more designated credit events may perhaps occuronly with probabilities (conditional on the information Fu _ available justbefore U) that are neither zero nor one. For example, the onset of defaultby one counterparty can generate simultaneous default by another, perhapscontractually connected, with a conditional probability that is between 0 and1.

A generally recommended principle is to reduce to primitive credit events,and to "build up" credit events derived from the primitives, by defining 0-or-l random variables that indicate whether or not a derived credit event isinstigated or not by the primitive event, as follows. The primitive event timesare denoted Tl, ... ,Tn. These should, if possible, have the "good" propertiesdefined above. A derived event time f is then modeled by taking its pointprocess N (that is, N( t ) = I t : : : o : f ) to be of the form

ndN( t ) =(1 - N(t - ) ) L A (i) dN it,

i=l(1 )

where A(I) , ... ,A (n) are random variables valued in {0,1}, with A(i) mea-surable with respect to FT(i). We let ai denote a predictable process withthe property that, for t < T, ai(t) =E(A(i )IFt ) . The idea is that, at eachstopping time Ti, if the credit-event time f has not already occured, then8In order to do this, it may be necessary to define a new probability space {A, A,o},

for example by letting A = n x [0,00) with the usual product a-algebra and productmeasure 0: = P X u, where t/ is the distribution of an exponential density on [0,00). Thena new filtration {At} is defined by letting At be the a-algebra generated by the unionof Ft and Us< t{w : T ( W ) 2 s}. In terms of arriving at a reasonable economic modelwith a given intensity process, expanding the probability space and information in thismanner, if necessary, seems innocuous. Our construction of T allows for the possibilitythat P(T = 00) > O .

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it occurs at Ti with proba~ility ai(Ti-), conditional on the information FTi-"just before" Ti. If we let h be defined by

nh(t) =L ai(t)hi(t),

i=l

!hen, under the property that P( Ti =Tj) =0 for i 1 = i, it can be shown thath is an intensity process for f. If , however, we let

Y(t) =E [exp ( [ 8 -h(U) dU) 1Ft],it is no t generally true that Y(t) =P(T > siFt) for t < i,ven if the "good"hypotheses of Lemma 1 and Proposition 1 are satisfied for each (Ti' hi) . In-deed, Y can jump at f with positive probability, because ai(t) jumps to 0 or1 at Ti , with may turn out to be f. Depending on the context, this failureof good properties for f may not bo so inconvenient, as one may have theability to model prices or probabilities in terms of the primitive credit events.

5 Valuation ModelingThis section presents our basic valuation models. Subsequent sections presentapplications and calculation methods.We take as given some bounded short-rate process" r , and some associ-ated equivalent martingale measure Q , defined, following Harrison and Kreps(1979), as follows.

First, there is a given collection of securities available for trade, with eachsecurity defined by its cumulative dividend process D. This means that, foreach time t, the total cumulative payment of the security up to and includingtime t is D(t) . For our purposes, a dividend process will always be taken tobe of the form D =A - B, where A and B are bounded increasing adaptedright-continuous left-limits (RCLL) processes, and we suppose that there areno dividends after a fixed time T > 0, in that D(t) =D(T) for all t larger9The probability measure Q is equivalent to P, in that the two measures have

the same events of probability zero. The process r is assumed to be predictable.Most results go through without a bound on r, for example under conditions such asEQ [exp (- J O T rtdt)] < 00 for all T.

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than T. The fact that Q is an equivalent martingale measure means that,for any such security D, the ex-dividend price process S for the security isgiven by

(2)where EQ denotes expectation under Q. The ex-dividend terminal priceS(T) is of course zero. An example is a security whose price is always 1,and paying the short-rate as a dividend, in that D( t ) =J ; Ts ds . As pointedout by Harrison and Kreps (1979) and Harrison and Pliska (1981), the exis-tence of an equivalent martingale measure implies the absence of arbitrageand, under technical conditions, is equivalent to the absence of arbitrage.For weak technical conditions supporting this equivalence, see Delbaen andSchachermeyer (1994, 1997).

In some cases, markets are incomplete, for example one may not be ableto perfectly hedge losses in market value that may occur at default, and thiswould mean that there need not be a unique equivalent martingale measure.

Given an intensity process h for a default time T, Artzner and Delbaen(1995, Appendix AI) show that there is also an intensity process A for thesame time T under the equivalent martingale measure Q, and show how toobtain A in terms of hand Q. One should beware of the fact that even ifProposition 1 applies under the original measure P, it may not apply underan equivalent martingale measure Q, as the conjectured solution yQ, de-fined by ~Q =EQ [exp ( J / - Au d u ) ] , for the process describing the survivalprobability under Q may in fact jump at the event time T, even though thecorresponding process Y associated with the original measure P does notnot. Kusuoka (1998) provides such an example.

Fixing an equivalent martingale measure Q, let us consider the valuationof a security that pays an amount Z at time T in the event that a givencredit event time T is after T, and otherwise pays an amount!" W at theevent time T.

The cumulative dividend process D of this security can therefore be de-fined by

dD( t ) =W dN t, t < T,lOWe take W to be a bounded oFT-measurable random variable, and Z to be a bounded

oFT-measurable random variable.

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andD(T) =WNT + (1 - NT)Z,where N = 1T2:t is the point process associated with T. We suppose that,

under Q , the event time T has a bounded intensity process A , so that(3 )

where MN is a Q-martingale.One can refer to Bremaud (1980), for example, to see that

(4 )for a Q-martingale MD, where f is the compensator for W, in the followingsense. Ifwe let 9 =EQ (W I FT-), we may think of 9 as the expected paymentconditional on all information up to, but not including, the time T of thecredit event. According!' to Dellacherie and Meyer (1978), Result IV.67(b),there exists a predictable process f such that f (T ) =g. For each fixed timet, we may view f(t) as the risk-neutral expected payment, conditional on allinformation up to but not including time t, and given no default before timet, that would apply if default were to occur at time t.

As an example, one could take the amount paid at default to be W =F(T), for an adapted process F. If F has continuous sample paths.P thenf(t) = F(t). For another example, suppose that the amount paid is theloss, relative to par, on a bond at default. Suppose that, until default, thisloss has a conditional distribution T J (under Q) that is fixed, and given by astatistically estimated distribution based on the history of losses of relatedbonds, and perhaps some risk premia parameters. (For example, T J couldbe the empirical distribution for the seniority class of the bond, with anassumption of no risk premia.) Then f is a constant on [0,T], and simplythe mean (under Q) of this distribution T J . Likewise, if the contractuallystipulated payment W is some bounded measurable function w : 1R -+ 1Rof the loss 100 - Y(T) relative to par, given recovery Y. Under the same11I am grateful to Freddy Delbaen for showing me this construction. Please note that

there is a typographical error in Dellacherie and Meyer (1978), Theorem IV.67(b), inthat the second sentence should read: "Conversely, if Y is an :F~_ -measurable ... " ratherthan "Conversely, ifY is an :F~-measurable ... ," as can be verified from the proof, or, forexample, from their Remark 68(b).

12If F is a semi-martingale with jumps only at non-predictable stopping times, thenf(t) = F(t-).

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independence assumption, we would have f a constant equal to the expectedpayment J w ( x ) d T J ( x ) . For example, w ( x ) =max(x, 50) would stipulate apayment of the loss or 50, whichever is greater.

If an issuer has multiple obligations with priorities defined by maturity orsubordination, then, for each obligation, f ( t ) changes over time based on theprioritized allocation of assets to liabilities, in expectation (under Q), condi-tional on survival to date. For example, if an issuer has one short-maturityand one long-maturity bond, as t passes through the earlier maturity date,the Q-expected default-contingent payment f ( t ) on the long-maturity bondwould jump, possibly downward if the two bonds are of equal priority andassets are close in distribution to the level necessary to pay down the short-maturity bond. Jumps at predictable times such as a maturity date causeno difficulty.In general, the price process S of this security is given from (2 ) and (4 )bys, =EQ [iT O :'s(1 - N s)Asf(s) ds + O:'T(l - N T)Z 1 F t ] , t < T, (5)where, for s ~ t and each predictable process a with JO Tl a t l dt < 00 a.s.,

This expression (5) for the price process S can in practice be difficultto evaluate, as N, appears directly. For example, brute-force simulation,by discretization, of (N t, rt , A t, ft) can be extrmely tedious, and relativelyprecise estimates of the initial market value So may call for a large numberof independent scenarios if A is small, which is typical in practice. Thefollowing result, along the lines of results found in Duffie, Schroder, Skiadas(1996), Lando (1997), and Schonbucher (1997), is more convenient.Proposition 2. Let

V(t) =EQ [iT O ;,;-A Asf(s) ds + o;JAZ 1 F t ] , t < T,and V (T ) = o . If the jump ~ V ( T ) of V at T is zero almost surely, thenS(t) =V(t) for t < T.

(6 )

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The condition that ~V(T) =0 (analogous to ~Y(T) =0 in Proposition 1)is not restrictive in settings for which there are no sudden (jump) surprises,other than default, in the joint conditional distribution under Q of interestrates, arrival intensities, and the payoff variables j and Z. For example, ifT, A, I,and Z are functions of some diffusion process'" then V is continuousup to T, and thus ~ V(t ) =0 for any t < T. More generally, one can allowjumps in V provided they cannot happen at default times. As provided inRemark 1, for any such candidate value process V, one can always constructa model with ~ V (T ) = 0 and price process S = (1 - N) V . (In this case,the artificially introduced random variable A of the Remark is defined to beindependent and exponential under Q.) If~V ( T ) need not be zero, a some-what more complicated formula, provided in Duffie, Schroder, and Skiadas(1996), can be applied. Duffie, Schroder, and Skiadas (1996) also extend themodel to treat cases in which the expected loss j ( t ) or the intensity A maydepend on V itself, under additional technical regularity.Proof: The proof is an extension of that of Proposition 1. Let M be theQ-martingale defined by

Mt=EQ[lTc5~~AAsj(S)dS+c5~YZ I F t J .We have

Mt =c5~!Avt + l t c5~~AAs!s ds, t < T,and we may therefore write

(7 )

vt =l t MV(S) ds + Mv(t), t < T, (8 )where Mv is a Q-martingale and, applying the fact that M is a martingaleto (7),

13That is, if . x , r, j, and Z are measurable with respect to some process that is a d-dimensional Brownian motion with respect to (0, F, {Ft : t 2 O},Q), then, on the timeinterval [0 , T), V is an Ito process, and therefore has no jumps.

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We will complete the proof by showing that S, =(1 - Nt)vt, which is true ifand only if H =J, where

andt;=Stc5~,t + I t c5~,uu;

Because ST =VT =0, we have JT =HT. Because !:::. .S(T) = (1-NT)!: : : . .V(T) =! : : :"D(T) = (1 - N T )Z , we know that ! : : :"H(T) = !:::. .J(T) =o . We know from(2) that J is a Q-martingale. It therefore suffices to show that H is also aQ-martingale. This follows from an application of Ito's Formula.l'' (3), (4),and (8):

-rtc5~ t(1- N t)vtdt - vt_c5~ tdN t + (1- Nt_)c5~tdvt + c5~tdDt, , "- (rt + At)c5~ t(1 - N t) vt dt + (1 - N t- )c5~tMv( t) dt,+c5~t(1 - N t)Adt dt + dMH(t)

dMH (t), t < T,

where MH is the Q-martingale, defined by

(We used the assumption that ! : : : . .V ( T ) = 0 when disregarding the term!:::. .V(t)!:::. .N(t).) The result therefore follows as stated.

We can reinterpret Proposition 2 so as to apply to the valuation of a lossat a given default (or other credit event) that may be brought on, or not,by a primitive credit events. That is, suppose a given credit event is derivedfrom T in the sense that its stopping time f is modeled, as with, (1), byN(t) =1f2:t with

dN(t) =A dN(t),where N, =1T2: t is the indicator for a stopping time T with Q-intensity A, andA is an FT-measurable random variable valued in {O,1} indicating whether14We use the fact that, for any bounded semi-martingale U, we have J O T U(t) dt =

J O T U(t-) dt almost surely, as U can have at most a countable number of jumps.12


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or not the arrival of T causes the event in question to occur. We let a bea process valued in l O , 1 ] such that, for tA < T, we have a(t) = EQ(A 1Ft).Then a Q-intensity A of f is defined by A(t) = a(t)A(t) . As noted earlier,(f,5.) may not have the "good" property exploited in Proposition 1. Wecan nevertheless proceed by modeling in terms of ( T , A ) . Suppose that W isthe amount paid at f. (We assume that W is bounded and FT-measurable.)Let W =AW and f be a predictable process such that, for t < T, we havef ( t ) =E(W I Ft). It follows that, under the hypotheses of Proposition 2, themarket value of receiving W at f (and Z at T if f > T) is given at any timet < min(T,T) by V(t) of (9), and at any time t ~ min(T,T) by zero.

6 First-to-Default ValuationNow we consider the valuation of a contingent claim that pays off at T =min(T1, ... , Tn), the first of n credit events, a contingent amount Wi if T =Ti.That is, the amount paid is explicitly dependent on the identity of the firstcredit event to occur.!" We suppose that the n credit events have stoppingtimes T1, ... ,Tn with respective bounded intensity processes A1 , ... ,An underQ , and that Ti i = Ti almost surely for ii = j. By Lemma 1, the intensityprocess of T under Q is A = A1 + ... + An. Our candidate security pays arandom variable!" Z at T if T > T. The dividend process D of this securityis therefore defined by

dD(t)dD(T)

(1 - N t-) L W i dN i(t),(1 - NT)Z,

t < T,

where N, =1T2:t . We let fi denote the predictable projection of a measure-valued adapted process TJ iwith the property that, for t < T, TJi(t) is theconditional distribution of Wi given Ft. Using the fact that

where MD is a Q-martingale, we have the following.15The random variables W1, ... , Wn are bounded, and Wi is measurable with respect

to Fr(i).16Again, we take Z to be bounded and FT-measurable.

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o : : ; t < T, (9 )and V (T) = o . If the jump ~ V ( T ) of V at T is zero almost surely, thenS, =vt for t < T.The proof is identical to that of Proposition 2. One merely exploits the factthat dv t =j.jv( t) dt + dMv(t) , where Mv is a Q-martingale and

j.jv(t) =vt(rt + At) - L fi(t)Ai(t).i

This implies that0;vt(1 - N t) + I t 0;(1 - N s) ~ Ai(s)fi(s) ds, t < T,

zis, under the given hypotheses, a Q-martingale, which leaves V(1 - N) =S,as with the proof of Proposition 2.

Kusuoka (1998) gives examples in which the timing risk-premium process(loosely, the difference between the intensities of the default arrivals underthe original measure P and the equivalent martingale measure Q) may jumpunexpectedly at a default arrival. This would be the case, for example, ifdefault by one obligor causes a sudden re-assessment of the equilibrium riskpremium for another default by another obligor. In such cases, a somewhatmore complicated pricing formula arises, as in Duffie, Schroder, and Skiadas(1996).

7 Analytical Solutions in Affine SettingsThis section proposes parametric examples in which the first-to-default pric-ing formula (9) can be computed either explicitly, or by numerically solvingrelatively simple ordinary differential equations, in an "affine" setting. Thatis, these examples are based on a a "state" process X valued in IRk that(under Q) is a k-dimensional affine jump-diffusion, in the sense of Duffie andKan (1996). That is, X is valued in some appropriate domain Dc IRk, with

d.X; =j.j(X t) dt + O"(X t) d.B , + dJ t,14


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where B is a standard brownian motion in IR d, J is a pure jump processwith jump-arrival intensity { J ) ; ( X t) : t ~ O} and jump distribution v on IRk,and where J);: D - - - + [0,00), u : D - - - + IRk, and C (mjT): D - - - + IRkxdare affine functionsY We delete time dependencies to the coefficients fornotational simplicity only; the approach outlined below extends to that casein a stragihtforward manner. A classical special case is the "multi-factorCIR state process" X , for which X(l), X(2), .. . , X (k) are independent (orQ-independent, in a valuation context) processes of the "square-root" type18introduced into term-structure modeling by Cox, Ingersoll, and Ross (1985).For many other examples of affine models, see Duffie, Pan, and Singleton(1997).

We can take advantage of this setting if we suppose short rates, intensities,and payoffs are of the affine'" form

r(t) a r(t) + br(t) . X tA i(t) a> .(i, t ) + b> .(i, t) . X,fi(t) exp (a f(i, t) + b f(i, t) . X (t-))

Z exp (az + bz . XT) ,where o z E IR and bz E IRk are given constants and where, for each iin{1, ... , n}:

aT) a> .(i, .) , and af(i, . ) are bounded measurable real-valued determin-istic functions on [0,T] .

17The generator D for X is defined byD f(x, t) = !t(x, t)+ fx(x, t)JL (x)+ ~ ~ C ij(x)fxi Xj (x, t)+K ,(x) j[f(X+Z , t)- f(x, t)] dv(z).

2J

One can add time dependencies to these coefficients. Conditions must be imposed forexistence and uniqueness of solutions, as indicated by Duffie and Kan (1996).18That is,

dXt(i) = K ,i(X i - X t(i) ) dt + aiM dB~i),for some given constants K, i > 0, Xi > 0, and a,190ne can proceed in more or less the same fashion if one generalizes by allowing

quadratic terms for r(t) and the jump intensity K" subject of course to technical con-ditions. In such cases, higher-order terms will appear in the solution polynomial. One canalso extend in a similar fashion to yet higher-order polynomials.

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b., b>. (i, . ), and bf(i, .) are bounded measurable deterministic iRk-valuedfunctions on [0,T] .

Except in degenerate cases, the boundedness assumptions used in Proposi-tions 1 through 3 do not apply, and integrability conditions must be assumedor established. For the special case in which fi(t) represents the risk-neutralexpected loss (relative to par, say) on a floating-rate note (a typical applica-tion in practice, say first-to-default swaps), one might reasonably take fi ( t )to be deterministic. (For this case, b f =0.)

For analytical approaches based on the affine structure just described, onecan repeatedly use the following calculation, regularity conditions for whichare provided in Duffie, Pan, and Singleton (1997).Let X be an affine jump-diffusion. For a given time s, and for each t :: ; slet R (t) =aR(t ) + bR(t) . X ( t) , for bounded measurable aR : [0 , s] - - - + iR an dbR : [O ,s]---+ iRk. For given coefficients a in iR, and b in iRk, le t

(10)Under technical conditions, there are specified OD Es for a : [0 , s] - - - + iR an d(J : [0 , s] - - - + iRk such that

g(x, t) =exp (a( t ) + (J (t) . x ) ,with boundary conditions a( s) =a and (J( s) =b. In addition, for given A iniR, and B in iRk, le t

Then, under technical conditions, there are specified OD Es for & : [0 , s] - - - + iRand ~ : [0 , s] - - - + iRk such that

G (x, t) =ea( t )+ !1 ( t ) 'X (&( t ) + ~ ( t) . x) ,with boundary conditions & (s) =a and ~ (s) =b .Details, with illustrative numerical examples and empirical applications, canbe obtained in Duffie, Pan, and Singleton (1997). For our application to

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(9), we would be assuming that X is an affine jump-diffusion under Q , andthe expectations in (10) and (11) would be under Q. With solutions for C Y ,/ 3 , X , and ~ in hand, the pricing formula (9) reduces to a one-dimensionalnumerical integral, which is a relatively fast exercise. Monte-Carlo basedapproaches are illustrated in the following section.

For the special multi-factor CIR case, explicit closed-form solutions forC Y and / 3 can be deduced from Cox, Ingersoll, and Ross (1985) (for the casea = a = 0 and b = b = 0), and Duffie, Pan, and Singleton (1997), whoalso provide analytical solutions for the Fourier transforms of X in the gen-eral affine setting, and related calculations that lead to analytical solutionsfor option pricing via Levy inversion of the transforms. The Fourier-basedoption-pricing results can be applied in this setting for cases in which

for some exercise price K. Some explicit results for option pricing are avail-able in certain cases, as shown by by Bakshi, Cao, and Chen (1996), Bakshiand Madan (1997), Bates (1996), and Chen and Scott (1995). These resultscan be applied in the present setting for valuation of default able optionswith affine structure, or with recovery determined by collateralization withan instrument whose price can be described in an exponential-affine form.Collateralization with equities, foreign currency, or notes (in domestic orforeign currency) would be natural examples for this.

8 Simulating The First to DefaultIn some cases, explicit solutions may be complicated, but Monte Carlo sim-ulation may be straightforward. This section presents an algorithm for sim-ulation based on explicit, or easily computed, distributions for the time tothe first credit event, and for the identity of that event. The setup is that ofSection 5.

That is, T =min(Tl, ... , T n ) , and A =2 . : i Ai . For any t ~ 0 and s ~ t, welet

We suppose that the jump Y(T, s ) - Y(T-, s ) is zero almost surely. FromProposition 1,

t < T,17


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almost surely.Let us suppose that we have a effective method for computing Y(t, s ) ,for each s > t . A special case is deterministic intensities, for which Y(t, s)

is of course given explicitly or by numerical integration. More generally, theaffine structure proposed in Section 6 provides such a method. We may thensimulate a random variable R that is equivalent in distribution to 7 (to beprecise, equivalent in Frconditional distribution under Q) by independentlysimulating a random variable U that is uniformly distributed on [0,1], and(provided Y ( t, s) ranges from 1 to as s ranges from t to 00) by then lettingR be chosen" so that Y(t, R) =U. In this caseQ(R ~ siFt) =Q(Y(t, R) ::; Y(t , s) 1Ft) =Q (U ::; Y(t , s) 1Ft) =Y(t, s) .

If Y(t) = lims-too Y(t, s ) > 0, then we simply let R =00 (the credit eventnever happens) in the event that U ::;Y(t) .

Having simulated the first arrival time 7, we wish to simulate which of then events occured at that time. Given an outcome t for 7, we will simulate arandom variable 1ith outcomes in {I, ... ,n}, and with probability q(i, 7 ) =Q(1 =i 7 ) for the outcome iequal to the conditional probability, under Q,that 7(i) = 7 given 7.21 We apply the following calculation, relying on aformal applications of Bayes' Rule:

" ,(i,t) dtq (i, t) =Q (7 =7i I 7 E ( t, t + dt)) =Q ( ( d )) ,7E t , t + t(12)

where22,(i, t) dt =Q(7 =7i and 7 E ( t, t + dt)) . (13)

Because q(l, t) + ... + q (n, t) =1, it is enough to calculate ,(i,t) for each iand t, leaving

(" ) ,(i,t)q 'l t =---,--.,-------------,------:-, ,(I, t) + ... + ,(n, t) (14)20As Y(t, ") is continuous and monotone, some such measurable R is well defined. If

Y(t, ") is not strictly monotone, any measurable selection of its inverse will suffice forchoosing R(w) given U(w).21To be precise, we are looking for a regular version q (i, ") : [0,00) -+ [0,1] of this

conditional probability.22That is, ,),(i,") is the density of the measure I', on [0,00) defined by fi(A) = Q(T =

Ti and TEA).

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Now, for bounded and right-continuous ( A l , . . . , A n ) and using dominatedconvergence and the assumption that T i # - T j a.s.,1,(i,t ) =lim - r ( t , i,E ) ,

E . j , . Q E (15)where

r ( t , i, E ) =Q ( T ~ t - E and T i : : ; t ) .Next, we calculate thatr ( t , i, E ) E Q [ l T 2 : t - E 1T ( i ) : S t ]

E Q [ l T 2 : t - E E Q [ I T ( i ) : S t I F t - E ] ]E Q [ l T 2 : t - E ( l - l T ( i ) 2 :t - E E Q [exp ( l ~ E- A i ( S ) d S ) I F t - E ] ) ]E Q [ l T 2 : t - E ( 1 - E Q [exp ( l ~ E- A i ( s ) d S ) I F t - E ] ) ] .

(16)

By dominated convergence, the assumption that A i is bounded and right-continuous (which can be weakened), and the fact that, letting H ( x ) =1- e-8x, for a given constant 6 , we have H ' ( O ) =6 , it then follows that

E Q [ 1 ! f o 1 1 T 2 : t - E l ! f o 1 ~ ( 1 - E Q [exp ( l ~ E - A i ( s ) d S ) I F t - E ] ) ]E Q [ lT 2 :t A i ( t) ] .

,(t, i)

Finally, we can apply Proposition 2 to see that, under our usual "no-jump-at-r" regularity.f'

For the affine structure described earlier, we have an efficient method forcomputing ,(i,t) , and thereby q(i, t) . Simulation of the first to default, boththe time T and the identity 1 ( T ) of the first to default, is therefore straight-forward in an affine setting, including of course deterministic intensities.23Letting

'Y(t,i;s) = EQ [exp .: A(S)dS) Ai(t) 1 . 1 " 8 ] 'we would impose the hypothesis that, for each fixed t and i, 'Y(t , i; T) - 'Y(t, i; T- ) = 0almost surely.

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9 Simulating with Mortality for DiscountsThis section shows how to compute the market value of the first to default,including the effect of discounting for interest rates. In order to do this, wewill introduce a fictitious event with "risk-neutral" arrival intensity AO =T,the short rate. (For this section, we assume that T is non-negative.) Thecontingent payment at the associated." stopping time TO is Wo = 0. Thefirst-to-arrive event time, now including TO, is T* =min(TO,Tl, ... ,Tn), withQ-intensity process A* =AO + Al + ... + An. Then, under the hypotheses ofProposition 3, we have the price of the first to default given by

Yo = E Q [ [ exp ([ -(r, + A ,) d S ) t, A id " d t ]E Q [ [ exp ([ - A :d S ) ~ ? id " d t ]EQ[IT*:STWJ* ],

where 1* is the random variable valued in {O , 1, ... ,n} that is the identity ofthe credit event that happens at T*. That is, T* =TJ*.

By our previous calculations, we can estimate Y o by simulation of T* andthen 1*. We can draw T* by simulating a uniformly distributed random vari-able and then, as above, using the inverse cumultative distribution functionfor T*, defined, under the hypotheses of Proposition 1 (under Q) by

As for 1*, once we have the outcome t of T*, we simulate a number from{O , 1, ... ,n}, drawing iwith probability

*(. t) ,*(i, t)q i, =,*(0, t) + ,*(1, t) + ... + ,*(n, t) ' (17)

where(18)

24Inorder to complete the story, we could introduce a standard (unit intensity) Poissonprocess A, independent under Q , of all previously defined variables, and let TO = inf {t :A ( J ~r8 d S ) = I}. There is of course no need to actually construct such a process.

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Again, affine dynamics (including deterministic intensities) make computa-tions straightforward.Finally, given the outcomes of T* and 1*, we would let F =0 if T* > Tand otherwise let F =[t- (T*). Let F1,F2, F3, ... denote an ii d sequence ofrandom variables, all (simulated, in practice) with the distribution of F. Wehave, by the law of large numbers,

1 Nlim - ""' Fi =Y o a.s. ,N-too N c:n=l (19)giving us the desired price, or an approximation for a large number N ofdraws. This procedure is certainly effective for deterministic f i ( . ) , or, for thecase in which f i ( t ) =g i ( X t), if we can compute the conditional distributionof X(T* ) given T* and 1*, to which we next turn.

10 Conditional State Distribution at EventsIn a state-space setting such as the affine setting described earlier, we willnow focus on the conditional distribution of the state X ( T ) given the firstevent time T. This can be applied, for example, to the computation of theconditional expected projected payoff EQ[Ji(T) I T J , or to the simulation ofX ( T ) conditional on T , which is in turn useful for simulating event timesafter T . In some cases, we are interested in the joint distribution of X ( T )and I(the identity of the first credit event), after conditioning on T. We willbegin by working under the measure P, but after re-formulation, the resultscan be applied under Q.

Suppose the state process X is valued in some domain D C IRk. We arelooking for some 7f : D x [0 , T] -+ [0,00) with the property that, for anybounded measurable F : D -+ IR, we have

E[F(X(T) ) I T ] =I n F(x)7f(X , T ) dx a .s.We begin by supposing that T has intensity process {H(X t) : t ~ O},where H is bounded, continuous, and strictly positive. We adopt the hy-

pothesis of "no jumping at T" for conditional expectations, wherever calledfor. Again, by a formal application of Bayes's Rule, one finds that, for each

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t,E [ F ( X ( T ) ) I T =t ] = ~ * ( X ( t ) , t ) ,E [exp ( f o - H ( X s ) d S ) H ( X t)]

where

We haveF * ( x , t ) =lF (x ) H ( x ) 7 r * ( X , t ) d x ,

where n" is the fundamental solution ofD g ( x , t ) - H ( x ) g ( x , t ) =0, (20)

for D the infinitesimal generator associated with X. Under regularity, itfollows that

( ) _ 7 r * ( x , t ) H ( x )7 r X , t - p ( t ) , (21)where p is the density of T, given by

There are several special cases of the affine models, for affine H ( . ) , forwhich the fundamental solution n" is known explicitly, including the multi-factor CIR model and the model of Chen (1994). (Such n" are sometimescalled "Green's functions.") As for p ( t ) , it can be readily computed in anaffine setting, as discussed previously.

From (25), we are in a position to "re-start" the state process at a simu-lated first-credit-event time T , by simulating the re-started state X ( T ) withdensity 7 r ( . , t).

Simulation of X ( T ) , conditional on T , might be simpler if the coordi-nate processes X(l), ... ,X(k) were independent, as with the multi-factor CIRmodel. Even if this is true, however, it is not generally true after conditioningon T, as can be seen from the form of the joint density 7 r ( . , t) , which is notof a product form because of the appearance of H ( x ) in (25).

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Simplification is possible, however, in the case that X(l), ... , X (k) areindependent and H (x) ((Xj) for some ((.) and some particular j E{l, ... ,n}. In this case,

where, for i 1 = i, 1 f;( ., t) is the fundamental solution ofVig(Xi, t) =0, ( 22 )

for Vi the infinitesimal generator associated with X(i), and where 1fj( . ,t ) isthe fundamental solution of

(23)for Vj the infinitesimal generator associated with XU) . Here, one can simu-late X(T) given T by simulating, given T, {X(i)(T) : 1 ::; i::;} conditionallyindependently, with T -conditional density 1 f ; ( . , T ) for i 1 = j and, for i=i,with T-conditional density 1fj( ., T)(( . )/p(T), based on simulating n indepen-dent uniform-[O,l] variables.

As for the distribution of X(T) given both T =min(Tl, ... , Tn), and theidentity Iof the first of n credit event times Tl, ... , Tn, we take again asetting in which Ti has intensity process {H i (X t) : t ~ O}, for H i boundedand continuous, and with H = 2 . : 7 = 1 H i strictly positive. We suppose thatP( Ti =Tj) = for i 1 = i, and continue to assume the principal of "no jumpingat T" of the appropriate conditional expectations, wherever called for.

We have the calculation of 1 f : D x [0 , T] x {l, ... , n} -+ [0,00) with theproperty that, for any bounded measurable F : D -+ lR, we have

E[F(XT) I T,I] =lF(x)1 f(x, T,I) dx a.s.By a formal application of Bayes's Rule, one finds that, for each t and i,

where

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4 . Otherwise, simulate [ from { O , 1, ... ,n} by drawing iwith probabilityq*(i, t) given by (17) and (18). These probabilities are, in an affinesetting, either explicitly computed or obtained by numerical solutionof ODEs, as indicated previously.

5. If [ (w) =0, let F(w) =0, and stop.6. Otherwise, if [(w ) =i, let

. 7rQ (x, t ) A i ( X )7rdx,t,'l) = C ) 'q* i, t (25)where 7 r Q is the fundamental solution of

DQg(x, t) - [a R + bR . x]g(x, t) =0,and DQ denotes the infinitesimal generator associated with X underQ.

7 . LetF(w) =l,:R(w), [(w ))G (x, R (w), [(w )) dx, (26)

estimated by numerical methods if necessary, and stop. One shouldbear in mind that the Fourier transform of 7 r Q A i can be computed byrelatively straightforward methods, or in several cases explicitly, as ex-plained in Duffie, Pan, and Singleton (1997), and approximate Fouriermethods may be more efficient than direct numerical integration of(26) .

References[1] P. Artzner and F. Delbaen (1995), "Default Risk Insurance and Incom-

plete Markets," Mathematical Finance 5: 187-195.[2] M. Beneish and E. Press (1995) "Interrelation Among Events of De-

fault," Working Paper, Duke University, forthcoming: ContemporaryAccounting Research.

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[3] E. J. Bijnen and M.F.C.M. Wijn (1994), "Corporate Prediction Mod-els, Ratios or Regression Analysis?" Working Paper, Faculty of SocialSciences, Netherlands.

[4] P. Bremaud (1980) Point Processes and Queues - Martingale Dynam-ics, New York: Springer-Verlag.

[5] M. Brennan and E. Schwartz (1980), "Analyzing Convertible Bonds,"Journal of Financial and Quantitative Analysis 15: 907-929.

[6] J. Cox, J. Ingersoll, and S. Ross (1985) "A Theory of the Term Struc-ture of Interest Rates," Econometrica 53: 385-408.

[7] F. Delbaen and W. Schachermayer (1997) "A General Version of theFundamental Theorem of Asset Pricing," Math. Ann. 300: 463-520.[8] F. Delbaen and W. Schachermayer (1997) "A General Version of the

Fundamental Theorem of Asset Pricing for Unbounded Stochastic Pro-cesses," forthcoming: Math. Ann.

[9] C. Dellacherie and P.-A. Meyer (1978) Probabilities and Potential Am-sterdam: North-Holland.

[10] D. Duffie, M. Schroder, and C. Skiadas (1996), "Recursive Valuation ofDefaultable Securities and the Timing of Resolution of Uncertainty,"Annals of Applied Probability, 6: 1075-1090.

[11] D. Duffie and D. Lando (1997), "Term Structures of Credit Spreadswith Incomplete Accounting Information," Working Paper, GraduateSchool of Business, Stanford University.

[12] D. Duffie and K. Singleton (1997), "Modeling Term Structures of De-fault able Bonds," Working Paper, Graduate School of Business, Stan-ford University.

[13] J. Fons (1994), "Using Default Rates to Model the Term Structureof Credit Risk," Financial Analysts Journal, September-October, pp.25-32.

[14] A. Friedman (1975) Stochastic Differential Equations and Applications,Volume I, New York: Academic Press.

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[15] M. Harrison and D. Kreps (1979), "Martingales and Arbitrage in Mul-tiperiod Security Markets," Journal of Economic Theory 20: 381-408.

[16] R. Jarrow, D. Lando, and S. Turnbull (1997), "A Markov Model forthe Term Structure of Credit Spreads," Review of Financial Studies10: 481-523

[17] R. Jarrow and S. Turnbull (1995), "Pricing Options on Financial Se-curities Subject to Default Risk," Journal of Finance 50: 53-86.

[18] I. Karatzas and S. Shreve (1988) Brownian Motion and Stochastic Cal-culus, New York: Springer-Verlag.

[19] S. Kusuoka (1998), "A Remark on Default Risk Models," WorkingPaper, Graduate School of Mathematical Sciences, University of Tokyo.[20] D. Lando (1993), "A Continuous-Time Markov Model of the Term

Structure of Credit Risk Spreads," Working Paper, Statistics Center,Cornell University.

[21] D. Lando (1997), "Modeling Bonds and Derivatives with Default Risk,"in M. Dempster and S. Pliska (editors) Mathematics of Derivative Se-curities, pp. 369-393, Cambridge University Press.

[22] D. Lando (1998), "On Cox Processes and Credit Risky Securities,"Working Paper, Department of Operations Research, University ofCopenhagen, Forthcoming in Review of Derivatives Research.

[23] D. Madan and H. Unal (1993), "Pricing the Risks of Default," WorkingPaper, College of Business, University of Maryland.

[24] M. Martin (1997) "Credit Risk in Derivative Products," Ph.D. Disser-tation, London Busines School, University of London.

[25] C.G. McDonald and L.M. Van de Gucht (1996), "The Default Risk ofHigh-Yield Bonds," Working Paper, Louisiana State University.

[26] S. Nielsen and E. Ronn (1995), "The Valuation of Default Risk inCorporate Bonds and Interest Rate Swaps," Working Paper, Universityof Texas at Austin.

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[27] P. Protter (1990) Stochastic Integration and Differential Equations,New York: Springer-Verlag.

[28] G. Pye (1974), "Gauging The Default Premium," Financial Analyst'sJournal (January-February), pp. 49-50.

[29] P. Schonbucher (1997), "The Term Structure of Defaultable BondPrices," Working Paper, University of Bonn.

[30] T. Shumway (1996) "Forecasting Bankruptcy More Efficiently: A Sim-ple Hazard Model," Working Paper, University of Michigan BusinessSchool

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