Credit risk modeling with affine processes - Darrell Duffie

52 Pages·2005·0.4 MB·English

JournalofBanking&Finance29(2005)2751–2802
www.elsevier.com/locate/jbf
q
Credit risk modeling with affine processes
Darrell Duffie
GraduateSchoolofBusiness,StanfordUniversity,Stanford,CA94305,UnitedStates
ScuolaNormaleSuperiore,Pisa,Italy
Availableonline25March2005
Abstract
Thisarticlecombinesanorientationtocreditriskmodelingwithanintroductiontoaffine
Markovprocesses,whichareparticularlyusefulforfinancialmodeling.Weemphasizecorpo-
ratecredit risk andthe pricingofcredit derivatives.Applications of affineprocesses thatare
mentionedincludesurvivalanalysis,dynamicterm-structuremodels,andoptionpricingwith
stochasticvolatilityandjumps.Thedefault-riskapplicationsincludedefaultcorrelation,par-
ticularlyinfirst-to-defaultsettings.Thereaderisassumedtohavesomebackgroundinfinan-
cial modelingand stochasticcalculus.
(cid:1)2005Elsevier B.V.All rightsreserved.
JELclassification: G12;G33;C41
Keywords: Creditrisk;Affineprocesses;Defaultcorrelation;Firsttodefault
1. Introduction
This is a written version of the Cattedra Galileana lectures, presented in 2002 at
theScuolaNormaleinPisa.Theobjectiveistocombineanorientationtocredit-risk
modeling (emphasizing the valuation of corporate debt and credit derivatives) with
q ThisisthewrittenversionoftheCattedraGalileanalectures,ScuolaNormaleSuperiore,inPisa,2002,
made possible through the wonderful organizational work of Maurizio Pratelli, to whom I am most
grateful. I am also grateful for support for the course offered by the Associazione Amici della Scuola
NormaleSuperiore,whoweregenerouslyrepresentedbyMr.CarloGulminelli.
E-mailaddresses:duffi[email protected],duffi[email protected]
0378-4266/$-seefrontmatter (cid:1)2005ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jbankfin.2005.02.006
2752 D.Duffie/JournalofBanking&Finance29(2005)2751–2802
an introduction to the analytical tractability and richness of affine state processes.
This is not a general survey of either topic, but rather is designed to introduce
researchers with some background in mathematics to a useful set of modeling tech-
niques and an interesting set of applications.
Appendix A contains a brief overview of structural credit risk models, based on
defaultcausedbyaninsufficiencyofassetsrelativetoliabilities,includingtheclassic
Black–Scholes–Merton model of corporate debt pricing as well as a standard struc-
turalmodel,proposedbyFisheretal.(1989)andsolvedbyLeland(1994),forwhich
default occurs when the issuer(cid:1)s assets reach a level so small that the issuer finds it
optimal to declare bankruptcy. The alternative, and our main objective, is to treat
defaultbya‘‘reduced-form’’approach,thatis,atanexogenouslyspecified intensity
process. As a special tractable case, we often suppose that the default intensity and
interest rate processes are linear with respect to an ‘‘affine’’ Markov state process.
Section2beginswiththenotionofdefaultintensity,andtherelatedcalculationof
survival probabilities in doubly-stochastic settings. The underlying mathematical
foundationsarefoundinAppendixE.Section3introducesthenotionofaffinepro-
cesses,themainsourceofexamplecalculationsfortheremainder.Technicalfounda-
tions for affine processes are found in Appendix C. Section 4 explains the notion of
risk-neutralprobabilities,andprovidesthechangeofprobabilitymeasureassociated
withagivenchangeofdefaultintensity(aversionofGirsanov(cid:1)sTheorem).Technical
details for this are found in Appendix E.
BySection5,weseethebasicmodelforpricingdefaultabledebtinasettingwith
stochastic interest rates and stochastic risk-neutral default intensities, but assuming
no recovery at default. The following section extends the pricing models to handle
default recovery under alternative parameterizations. Section 7 introduces multi-
entity default modeling with correlation. Section 8 turns to applications such as de-
fault swaps, credit guarantees, irrevocable lines of credit, and ratings-based step-up
bonds. Appendix F provides some directions for further reading.
2. Intensity-based modeling of default
Thissection introduces a model for a defaulttime as a stopping time s with a gi-
venintensityprocess,asdefinedbelow. Fromthejoint behavior ofthedefaulttime,
interest-rates, the promised payment of the security, and the model of recovery at
default, as well as risk premia, one can characterize the stochastic behavior of the
term structure of yields on defaultable bonds.
Inapplications,defaultintensitiesmaybeallowedtodependonobservablevariables
that are linked with the likelihood of default, such as debt-to-equity ratios, volatility
measures,otheraccountingmeasuresofindebtedness,marketequityprices,bondyield
spreads,industryperformancemeasures,andmacroeconomicvariablesrelatedtothe
business cycle,as inDuffie andWang (2003). Thisdependence could, but in practice
doesnotusually,ariseendogenouslyfromamodeloftheabilityorincentivesofthefirm
tomakepaymentsonitsdebt.Becausetheapproachpresentedheredoesnotdependon
thespecificsettingofafirm,ithasalsobeenappliedtothevaluationofdefaultablesov-
D.Duffie/JournalofBanking&Finance29(2005)2751–2802 2753
ereigndebt,asinDuffieetal.(2003b)andPage`s(2000).(Formoreonsovereigndebt
valuation,seeGibsonandSundaresan(1999)andMerrick(1999).)
WefixacompleteprobabilityspaceðX;F;PÞandafiltrationfG :tP0gofsub-
t
r-algebras of F satisfying the usual conditions, which are listed in Appendix B.
AppendixEdefinesanon-explosivecountingprocess.SuchacountingprocessKre-
cordsbytimetthenumberK ofoccurrencesofeventsofconcern.AppendixEalso
t
defines the notion of a predictable process, which is, intuitively speaking, a process
whose value at any time t depends only on the information in the underlying filtra-
tion fG :tP0g that is available up to, but not including, time t.
t
AcountingprocessKhasanintensitykifkisapredictablenon-negativeprocess
satisfying Rtk ds<1 almost surely for all t, with the property that a local martin-
0 s
gale M, the compensated counting process, is given by
Z t
M ¼K (cid:1) k ds: ð2:1Þ
t t s
0
DetailsarefoundinAppendixE.Theaccompanyingintuitionisthat,atanytimet,
theG-conditionalprobabilityofaneventbetweentandt+DisapproximatelykD,
t t
for small D. This intuition is justified in the sense of derivatives if k is bounded and
continuous, and under weaker conditions.
A counting process with a deterministic intensity process is a Poisson process. If
the intensity of a Poisson process is some constant a, then the times between events
areindependentexponentiallydistributedtimeswithmean1/a.Astandardreference
on counting processes is Bre´maud (1981). Additional sources include Daley and
Vere-Jones (1988) and Karr (1991).
Wewillsaythatastoppingtimeshasanintensitykifsisthefirsteventtimeofa
non-explosive counting process whose intensity process is k.
A stopping time s is non-trivial if Pðs2ð0;1ÞÞ>0. If a stopping time s is non-
trivial and if the filtration fG :tP0g is the standard filtration of some Brownian
t
motionBinRd,thenscouldnothaveanintensity.Weknowthisfromthefactthat
iffG :tP0gisthestandardfiltrationofB,thentheassociatedcompensatedcount-
t
ing process M of (2.1) (indeed, any local martingale) could be represented as a sto-
chasticintegralwithrespecttoB,andthereforecannotjump,butMmustjumpats.
Inordertohaveanintensity,astoppingtimemustbetotallyinaccessible,aproperty
whose definition (for example, in Meyer (1966)) suggests arrival as a ‘‘sudden sur-
prise’’, but there are no such surprises on a Brownian filtration!
Asanillustration,wecouldimaginethattheequityholdersormanagersofafirm
are equipped with some Brownian filtration for purposes of determining their opti-
maldefaulttimes,asinAppendixA,butthatbondholdershaveimperfectmonitor-
ing, and may view s as having an intensity with respect to the bondholders(cid:1) own
filtrationfG :tP0g,whichcontainslessinformationthantheBrownianfiltration.
t
DuffieandLando(2001)provide,underconditions,theassociateddefaultintensity.1
1 Elliottetal.(2000)giveanewproofofthisintensityresult,whichisgeneralizedbySong(1998)tothe
multi-dimensional case. Kusuoka (1999) provides an example of this intensity result that is based on
unobservabledriftofassets.
2754 D.Duffie/JournalofBanking&Finance29(2005)2751–2802
Wesaythatastoppingtimesisdoublystochasticwithintensitykiftheunderly-
ing counting processwhose first jump time iss isdoublystochasticwith intensityk,
asdefinedinAppendixE.Thedoubly-stochasticpropertyimpliesthat,foranytime
t,ontheeventthatthedefaulttimesisaftert,theprobabilityofsurvivaltoagiven
future time s is
(cid:1) Rs (cid:2) (cid:3)
Pðs>sjGtÞ¼E e(cid:1) t kðuÞdu(cid:2)(cid:2)Gt : ð2:2Þ
Property (2.2) is convenient for calculations, because evaluating the expectation in
(2.2)iscomputationallyequivalenttothestandardfinancialcalculationofdefault-free
zero-couponbondprice,treatingkasashort-terminterest-rateprocess.Indeed,this
analogyisalsoquitehelpfulforintuitionwhenextending(2.2)topricingapplications.
It is sufficient for the convenient survival-time formula (2.2) that k =K(X) for
t t
some measurable K:Rd !½0;1Þ, where X in Rd solves a stochastic differential
equation of the form
dX ¼lðX ÞdtþrðX ÞdB; ð2:3Þ
t t t t
for some ðGÞ-standard Brownian motion B in Rd. Here, l(Æ) and r(Æ) are functions
t
on the state space of X that satisfy enough regularity for (2.3) to have a unique
(strong) solution. With this, the survival probability calculation (2.2) is of the form
(cid:1) Rs (cid:2) (cid:3)
Pðs>sjGtÞ¼E e(cid:1) t KðXðuÞÞdu(cid:2)(cid:2)XðtÞ ð2:4Þ
¼fðXðtÞ;tÞ; ð2:5Þ
where, under the usual regularity for the Feynman–Kac approach, f(Æ) solves the
partial differential equation (PDE)
Afðx;tÞ(cid:1)fðx;tÞ(cid:1)KðxÞfðx;tÞ¼0; ð2:6Þ
t
for the generator A of X, given by
X o 1 X o2
Afðx;tÞ¼ fðx;tÞlðxÞþ fðx;tÞc ðxÞ;
ox i 2 ox ox ij
i i i;j i j
and where c(x)=r(x)r(x)0, with the boundary condition
fðx;sÞ¼1: ð2:7Þ
ParametricassumptionsareoftenusedtogetanexplicitsolutiontothisPDE,aswe
shall see.
More generally, (2.2) follows from assuming that the doubly-stochastic counting
processKwhosefirstjumptimeissisdrivenbysomefiltrationfF :tP0g,acon-
t
ceptdefinedinAppendixE.(IncludedinthedefinitionistheconditionthatF (cid:2)G,
t t
andthatfF :tP0gsatisfiestheusualconditions.)Theintuitionofthedoubly-sto-
t
chasticassumptionisthatF containsenoughinformationtorevealtheintensityk,
t t
but not enough information to reveal the event times of the counting process K. In
particular,atanycurrent time tandforanyfuturetimes,afterconditioningon the
D.Duffie/JournalofBanking&Finance29(2005)2751–2802 2755
r-algebra G _F generated by the events in G [F , K is a Poisson process up to
t s t s
timeswith(conditionallydeterministic)time-varyingintensity{k :06t6s},sothe
t
numberK (cid:1)K ofarrivalsbetweentandsisthereforeconditionallydistributedasa
s t
PoissonrandomvariablewithparameterRsk du.(ArandomvariableqhasthePois-
t u
son distribution with parameter b if Pðq¼kÞ¼e(cid:1)bbk=k! for any non-negative inte-
ger k.) Thus, letting A be the event {K (cid:1)K =0} of no arrivals, the law of iterated
s t
expectations implies that, for t<s,
Pðs>sjGÞ¼Eð1 jGÞ¼E½Eð1 jG _F ÞjG(cid:3)
t A t A t s t
(cid:1) Rs (cid:2) (cid:3)
¼E½PðKs(cid:1)Kt¼0jGt_FsÞjGt(cid:3)¼E e t (cid:1)kðuÞdu(cid:2)(cid:2)Gt ;
ð2:8Þ
consistentwith(2.2).AppendixEconnectstheintensityofswithitsprobabilityden-
sity function and its hazard rate.
3. Affine processes
Inmanyfinancialapplicationsthatarebasedonastateprocess,suchasthesolu-
tion X of (2.3), a useful assumption is that the state process X is ‘‘affine’’. An affine
processXwithsomestatespaceD(cid:2)Rd isaMarkovprocesswhoseconditionalchar-
acteristic function is of the form, for any u2Rd,
Eðeiu(cid:4)XðtÞjXðsÞÞ¼euðt(cid:1)s;iuÞþwðt(cid:1)s;iuÞ(cid:4)XðsÞ ð3:1Þ
for some coefficients u(Æ,iu) and w(Æ,iu). We will take the state space D to be of the
standardformRn (cid:5)Rd(cid:1)n,for06n6d.WesaythatXis‘‘regular’’ifthecoefficients
þ
u(Æ,iu) and w(Æ,iu) of the characteristic function are differentiable and if their deriv-
ativesarecontinuousat0.Thisregularityimpliesthatthesecoefficientssatisfyagen-
eralizedRiccatiordinarydifferentialequation(ODE)giveninAppendixC.Theform
ofthisODEinturnimplies,roughlyspeaking,thatXmustbeajump-diffusionpro-
cess, in that
dX ¼lðX ÞdtþrðX ÞdB þdJ ð3:2Þ
t t t t t
for a standard Brownian motion B in Rd and a pure-jump process J, such that the
drift l(X), the ‘‘instantaneous’’ covariance matrix r(X)r(X)0, and the jump mea-
t t t
sure associated with J all have affine dependence on the state X. Conversely,
t
jump-diffusions of this form (3.2) are affine processes in the sense of (3.1). A more
careful statement of this result is found in Appendix C.
Simple examples of affine processes used in financial modeling are the Gaussian
Ornstein–Uhlenbeck model, applied to interest rates by Vasicek (1977), and the
Feller(1951)diffusion,appliedtointerest-ratemodelingbyCoxetal.(1985).Agen-
eral multivariate class of affine jump-diffusion models was introduced by Duffie
and Kan (1996) for term-structure modeling. Using three-dimensional affine diffu-
sion models, for example, Dai and Singleton (2000) found that both time-varying
2756 D.Duffie/JournalofBanking&Finance29(2005)2751–2802
conditional variances and negatively correlated state variables are essential ingredi-
ents to explaining the historical behavior of term structures of US interest rates.
For option pricing, there is a substantial literature building on the particular af-
fine stochastic-volatility model for currency and equity prices proposed by Heston
(1993). Bates (1997), Bakshi et al. (1997), Bakshi and Madan (2000), and Duffie
etal.(2000)broughtmoregeneralaffinemodelstobearinordertoallowforstochas-
tic volatility and jumps, while maintaining and exploiting the simple property (3.1).
Akeypropertyrelatedto(3.1)isthat,foranyaffinefunctionK:D!Randany
w2Rd, subject only to technical conditions reviewed in Duffie et al. (2003a),
(cid:1) Rs (cid:3)
Et e t (cid:1)KðXðuÞÞduþw(cid:4)XðsÞ ¼eaðs(cid:1)tÞþbðs(cid:1)tÞ(cid:4)XðtÞ ð3:3Þ
for coefficients a(Æ) and b(Æ) that satisfy generalized Riccati ODEs (with real bound-
ary conditions) of the same type solved by u and w of (3.1), respectively.
Inordertogetaquicksenseofhow(3.3)andtheassociatedRiccatiequationsfor
the solution coefficients a(Æ) and b(Æ) arise, we consider the special case of an affine
diffusionprocessXsolvingthestochasticdifferentialequation(2.3),withstatespace
D¼R ,andwithl(x)=a+bxandr2(x)=cx,forconstantcoefficientsa,b,andc.
þ
(This is the continuous branching process of Feller (1951).) We let K(x)=q +q x,
0 1
for constants q and q , and apply the (Feynman–Kac) PDE (2.6) to the candidate
0 1
solution(3.3).AftercalculatingalltermsofthePDEandthendividingeachtermof
the PDE by the common factor f(x,t), we arrive at
1
(cid:1)a0ðzÞ(cid:1)b0ðzÞxþbðzÞðaþbxÞþ bðzÞ2c2x(cid:1)q (cid:1)q x¼0; ð3:4Þ
2 0 1
for all zP0. Collecting terms in x, we have
uðzÞxþvðzÞ¼0; ð3:5Þ
where
1
uðzÞ¼(cid:1)b0ðzÞþbðzÞbþ bðzÞ2c2(cid:1)q ð3:6Þ
2 1
vðzÞ¼(cid:1)a0ðzÞþbðzÞa(cid:1)q : ð3:7Þ
0
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 Riccati equations:
1
b0ðzÞ¼bðzÞbþ bðzÞ2c2(cid:1)q ; ð3:8Þ
2 1
a0ðzÞ¼bðzÞa(cid:1)q ; ð3:9Þ
0
with the boundary conditions a(0)=0 and b(0)=w, from the boundary condition
f(x,s)=w for all x. The explicit solutions for a(z) and b(z), developed by Cox
et al. (1985) for bond pricing (that is, for w=0), is repeated in Appendix D, in
the context a slightly more general model with jumps.
D.Duffie/JournalofBanking&Finance29(2005)2751–2802 2757
The calculation (3.3) arises in many financial applications, some of which will be
reviewed momentarily. An obvious example is discounted expected cash flow (with
discountrateK(X)),aswellasthesurvival-probabilitycalculation(2.2)foranaffine
t
state process X and a default intensity K(X), taking w=0 in (3.3).
t
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 l(x) and r(x)r(x)0 are affine in x. This class includes
theGaussian(Ornstein–Uhlenbeck)case,forwhichr(x)isconstant(usedbyVasicek
(1977) to model interest rates), as well as the Feller (1951) diffusion model, used by
Cox et al. (1985) to model interest rates. These two examples are one-dimensional;
that is, d=1. For the case in which X is a Feller diffusion, we can write
pffiffiffiffiffi
dX ¼jð(cid:1)x(cid:1)X Þdtþc X dB; ð3:10Þ
t t t t
for constant positive parameters2 c, j, and(cid:1)x. The parameter(cid:1)x is called a ‘‘long-run
mean’’,andtheparameter jiscalledthemean-reversionrate.Indeedfor(3.10),the
meanofX convergesfromanyinitialconditionto(cid:1)xattheratejastgoesto1.The
t
Feller diffusion, originally conceived as a continuous branching process in order to
model randomly fluctating population sizes, has become popularized in finance as
the ‘‘Cox–Ingersoll–Ross’’ (CIR) process.
BeyondtheGaussiancase,anyOrnstein–Uhlenbeckprocess,whetherdrivenbya
Brownianmotion(asfortheVasicekmodel)orbyamoregeneralLe´vyprocess,asin
Sato (1999), is affine. Moreover, any continuous-branching process with immigra-
tion (CBI process), including multi-type extensions of the Feller process, is affine.
(See Kawazu andWatanabe (1971).) Conversely, as stated in AppendixC,an affine
process in Rd is a CBI process.
þ
A special example of (3.2) is the ‘‘basic affine process’’, with state space D¼R ,
þ
satisfying
pffiffiffiffiffi
dX ¼jð(cid:1)x(cid:1)X Þdtþc X dB þdJ ; ð3:11Þ
t t t t t
where J is a compound Poisson process,3 independent of B, with exponential jump
sizes.ThePoissonarrivalintensitykofjumpsandthemeancofthejumpsizescom-
pletesthelistðj;(cid:1)x;c;k;cÞofparametersofabasicaffineprocess.Specialcasesofthe
basic affine model include the model with no diffusion (c=0) and the diffusion of
Feller (1951) (for k¼0). The basic affine process is especially tractable, in that the
coefficients a(t) and b(t) of (3.3) are known explicitly, and recorded in Appendix
D.4. The coefficients u(t,iu) and w(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 =(X ,...,X ), for independent affine coordinate processes X ,...,X . The
t 1t dt 1 d
2 ThesolutionX of(3.10)will never reachzerofromastrictlypositive initialconditionif j(cid:1)x>c2=2,
whichissometimescalledthe‘‘Fellercondition’’.
3 AcompoundPoissonprocesshasjumpsatiidexponentialeventtimes,withiidjumpsizes.
2758 D.Duffie/JournalofBanking&Finance29(2005)2751–2802
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-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).
Animportanttwo-dimensionalaffinemodelwasusedbyHeston(1993)tomodel
optionpricesinsettingswithstochasticvolatility.Here,onesupposesthattheunder-
lying price process U of an asset satisfies
pffiffiffiffiffi
dU ¼U ðc þc V ÞdtþU V dB ; ð3:12Þ
t t 0 1 t t t 1t
where c and c are constants and V is a stochastic-volatility process, which is a
0 1
Feller diffusion satisfying
pffiffiffiffiffi
dV ¼jð(cid:1)v(cid:1)V Þdtþc V dZ ; ð3:13Þ
t t t t
pffiffiffiffiffiffiffiffiffiffiffiffiffi
for constant coefficients j, (cid:1)v, and c, where Z ¼qB þ 1(cid:1)q2B is a standard
1 2
Brownian motion that is constructed as a linear combination of independent stan-
dard Brownian motions B and B . The correlation coefficient q generates what is
1 2
known as ‘‘volatility asymmetry’’, and is usually measured to be negative for major
marketstockindices.Optionimplied-volatility‘‘smilecurves’’are,roughlyspeaking,
rotated clockwise into ‘‘smirks’’ as q becomes negative. Letting Y=logU, a calcu-
lation based on Itoˆ(cid:1)s Formula (see Appendix B) yields
(cid:5) (cid:5) (cid:6)(cid:6)
1 pffiffiffiffiffi
dY ¼ c þ c (cid:1) V dtþ V dB ; ð3:14Þ
t 0 1 2 t t 1t
whichimpliesthatthetwo-dimensionalprocessX=(V,Y)isaffine,withstatespace
D¼R (cid:5)R.ByvirtueoftheexplicitcharacteristicfunctionoflogU,thisleadstoa
þ t
simplemethodforpricingoptions,asexplainedinSection8.Extensionsallowingfor
jumps have also been useful for the statistical analysis of stock returns from time-
series data on underlying asset returns and of option prices, as in Bates (1996)
and Pan (2002).4
4. Risk-neutral probability and intensity
Basictothetheoryofthemarketvaluationoffinancialsecuritiesare‘‘risk-neutral
probabilities’’, artificially chosen probabilities under which the price of any security
is the expectation of the discounted cash flow of the security, as will be made more
precise shortly.
We will assume the existence of a short-rate process, a progressively measurable
processrwiththepropertythatRtjrðuÞjdu<1forallt,andsuchthat,foranytimes
0
sandt>s,aninvestmentofoneunitofaccount(say,oneEuro)atanytimes,rein-
vested continually in short-term lending until any time t after s, will yield a market
4 AmonganalysesofoptionpricingforthecaseofaffinestatevariablesareBates(1997),Bakshietal.
(1997),BakshiandMadan(2000),Duffieetal.(2000),andScott(1997).
D.Duffie/JournalofBanking&Finance29(2005)2751–2802 2759
Rt
valueofe srðuÞdu.Whenwesay‘‘underQ’’,foranequivalent5probabilitymeasureQ,
wemeanwithrespecttotheprobabilityspaceðX;F;QÞandthesamegivenfiltration
fG :tP0g.
t
For thepurpose ofmarket valuation, we fix some equivalentmartingalemeasure
Q,basedondiscountingattheshortrater.Thismeansthat,asofanytimet,forany
stoppingtimeTandboundedG -measurablerandomvariableF,asecuritypayingF
T RT
atThasamodeledprice,ontheevent{T>t},ofEQt ½e t (cid:1)rðuÞduF(cid:3),where,forconve-
nience, we writeEQ for expectation under Q, given G. Uniquenessof an equivalent
t t
martingale measure would be unexpected in a setting of default risk. an equivalent
martingale measure.
HarrisonandKreps(1979)showedthattheexistenceofanequivalentmartingale
measure is equivalent (up to technical conditions) to the absence of arbitrage. Del-
baen and Schachermayer (1999) gave definitive technical definitions and conditions
forthisresult.Theremaybemorethanoneequivalentmartingalemeasure,however,
andformodelingpurposes,one would workunderonesuchmeasure.Commonde-
vices for estimating an equivalent martingale measure include statistical analysis of
historicalpricedata,orthemodelingofmarketequilibrium.Ifmarketsarecomplete,
meaning roughly that any contingent cash flow can be replicated by trading the
availablesecurities,theequivalentmartingalemeasureisunique(inacertaintechni-
cal sense), and can be deduced from the price processes of the available securities.
For further treatment, see, for example, Duffie (2001).
A risk-neutral intensity process for a default time s is an intensity process kQ for
the default time s, under Q. We also call kQ the Q-intensity of s. Artzner and
Delbaen (1995) gave us the following convenient result.
Proposition. Suppose that a non-explosive 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 kQ=k (for k strictly positive) represents a risk premium for uncertainty
associated with the timing of default, in the sense of the following version of Girsa-
nov(cid:1)s Theorem, which provides conditions suitable for calculating the change of
probability measure associated with a change of intensity, by analogy with the
‘‘changeindrift’’ofaBrownianmotion.SupposeKisanon-explosivecountingpro-
cesswithintensityk,andthatuisastrictlypositivepredictableprocesssuchthat,for
some fixed time horizon T, RT u k ds is finite almost surely. A local martingale n is
0 s s
then well defined by
(cid:5)Z t (cid:6) Y
n ¼exp ð1(cid:1)u Þk ds u ; t6T: ð4:1Þ
t s s TðiÞ
0 fi:TðiÞ6tg
5 AprobabilitymeasureQisequivalenttoPifPandQassignzeroprobabilitiestothesameeventsin
Gt,foreacht.
2760 D.Duffie/JournalofBanking&Finance29(2005)2751–2802
Girsanov(cid:1)sTheorem. Suppose thelocal martingalen isactuallyamartingale.Then an
equivalent probability measure Q is defined by dQ¼nðTÞ. Restricted to the time
dP
interval [0,T], the counting process K has Q-intensity ku.
AproofmaybefoundinBre´maud(1981).Caremustbetakenwithassumptions,
fortheconvenientdoubly-stochasticpropertyneednotbepreservedwithachangeto
an equivalent probability measure. Kusuoka (1999) gives examples of this failure.
Appendix E gives sufficient conditions for the martingale property of n, and for K
to be doubly stochastic under both P and Q.
Under certain conditions on the filtration fG :tP0g outlined in Appendix E,
t
the martingale representation property applies, and for any equivalent probability
measureQ,onecanobtaintheassociatedQ-intensityofKfromthemartingalerep-
resentation of the associated density process.
5. Zero-recovery bond pricing
We consider the valuation of a security that pays F1 at a given time s>0,
{s>s}
where F is a G -measurable bounded random variable. As 1 is the random var-
s {s>s}
iablethatis1intheeventofnodefaultbysandzerootherwise,wemayviewFasthe
contractuallypromisedpaymentofasecuritywiththepropertythat,intheeventof
defaultbefore the contractual maturitydates,there isno payment (that is, zero de-
fault recovery). The case of a defaultable zero-coupon bond is treated by letting
F=1. In the next section, we will consider non-zero recovery at default.
FromthedefinitionofQasanequivalentmartingalemeasure,thepriceS ofthis
t
security at any time t<s is given by
(cid:1) Rs (cid:3)
St ¼EQt e(cid:1) t rðuÞdu1fs>sgF : ð5:1Þ
From (5.1) and the fact that s is a stopping time, S must be zero for all tPs. The
t
followingresultisbasedonLando(1994).(See,also,Duffieetal.(1996)andLando
(1998).)
Theorem 1. Suppose that F, r, and kQ are bounded and that, under Q, s is doubly
stochastic driven by a filtration fF :tP0g, with intensity process kQ. Suppose,
t
moreover, that r is ðFÞ-adapted and F is F -measurable. Fix any t<s. Then, for
t s
tPs, we have S =0, and for t<s,
t
St ¼EQt (cid:1)e(cid:1)RtsðrðuÞþkQðuÞÞduF(cid:3): ð5:2Þ
The idea of (5.2) is that discounting for default that occurs at an intensity is analo-
gous to discounting at the short rate r.
Proof. From (5.1), the law of iterated expectations, and the assumption that r is
ðFÞ-adapted and F is F -measurable,
t s