Credit spread derivatives are credit. No-Arbitrage Approach to Pricing Credit Spread Derivatives CHI CHIU CHU AND YUE KUEN KWOK

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1 No-Arirage Approach o Pricing Credi Spread Derivaives CHI CHIU CHU AND YUE KUEN KWOK CHI CHIU CHU is a PhD suden in aheaics a Hong Kong Universiy of Science and Technology in Hong Kong, China accc@ushk YUE KUEN KWOK is an associae professor of aheaics a Hong Kong Universiy of Science and Technology aykwok@ushk No arirage in he case of pricing credi spread derivaives refers o deerinaion of he ie-dependen drif ers in he ean reversion sochasic processes of he insananeous spo rae and spo spread y fiing he curren er srucures of defaul-free and defaulale ond prices The riskless rae and he credi spread of a reference eniy are aken o e correlaed sochasic sae variales in his pricing odel When he spo rae and spo spread oh follow he Hull and Whie odel, one can derive an analyic represenaion for he ie-dependen drif ers and analyic price forulas for credi spread opions Algorihs for he nuerical valuaion of credi spread derivaives are developed, and he pricing ehaviors of credi spread opions are exained Credi spread derivaives are credi rae-sensiive financial insruens designed o hedge agains or capialize on changes in he credi spread of a reference eniy such as a risky ond The credi spread of a risky ond is he difference eween he curren yield of he risky ond and a enchark rae of coparale auriy The spread represens he risk preiu he arke deands for holding he risky ond Unlike defaul swaps, credi spread derivaives do no depend upon any specific credi even occurring The uyer of a credi spread opion pays an up-fron preiu, and in reurn he wrier agrees o pay an aoun should he acual spo spread reach soe srike level Alernaively, he payoff ay depend on he difference eween he spo spreads of wo reference eniies Credi spread opions ay e used eiher o earn incoe fro he opion preiu, o e one s view on credi spread change, or o arge he purchase of a reference eniy on a forward asis a a favored price By following a reduced-for approach ha odels he occasion of defaul as a poin process, Jarrow, Lando, and Turnull [1997 consruc a Markov chain odel for valuaion of risky de and credi derivaives ha incorporaes he credi raings of a fir as an indicaor of he likelihood of defaul The Markov odel provides he evoluion of an ariragefree er srucure of he credi spread Proles arise in nuerical ipleenaion of he odel ecause onds wih high credi raings ay experience no defaul wihin he saple period, so he risk preiu adjusens ecoe ill-defined as he esiaed defaul proailiies are zero Kijia and Kooriayashi [1998 propose a echnique of risk preiu adjusen o overcoe his difficuly wih high-raed onds In relaed work, Duffie and Singleon [1999 and Schönucher [1998 use inensiy odels o analyze he er srucure of defaulale onds and copue he price of credi derivaives Schönucher [1999 consrucs a wo-facor ree odel for fiing oh defaulale and defaul-free er srucures of SPRING 2003 THE JOURNAL OF DERIVATIVES 51

2 ond prices, following he Hull and Whie fraework wih ie-dependen drif ers He uses he Hull and Whie [1990 odel for he dynaics of he risk-free ineres rae and defaul inensiy Garcia, van Gindersen, and Garcia [2001 use a wo-facor ree algorih ha is coposed of a Hull and Whie ree for he risk-free ineres rae dynaics and a Black and Karasinski [1991 ree for he inensiy dynaics Das and Sundara [2000 propose a discree-ie Heah-Jarrow-Moron [1992 odel for valuaion of credi derivaives They derive risk-neural drifs for he processes of he risk-free forward rae and forward spread Coining a recovery of arke value condiion, hey oain a recursive represenaion of drifs ha are pahdependen This forulaion has he advanage ha i can easily handle he Aerican exercise feaure in he pricing of he credi derivaives, and he corresponding algorih also includes inforaion aou defaul proailiies inferred fro arke daa Is ajor disadvanage is ha is copuaional coplexiy grows exponenially wih he nuer of ie seps, since he corresponding inoial ree is non-recoining A srucural approach derives he er srucure of he credi spread of a credi eniy y exaining he credi qualiy of he issuer In srucural odels, he credi spread is no one of he sae variales in he odel, u raher is a derived quaniy Defaul occurs when he issuer fir value falls elow soe hreshold value The srucural approach has een used y Das [1995 o price credi risk derivaives Longsaff and Schwarz [1995 develop a valuaion odel for pricing credi spread derivaives y assuing ha he riskless ineres rae and he spread follow correlaed ean reversion diffusion processes Fro epirical analysis of oserved credi spreads, hey argue ha he logarih of he credi spread follows a ean reversion process They assue he paraeers in he characerizaion of he processes o e consan, and derive closed-for price forulas for he credi spread opions A disadvanage of he process is ha heir odel does no incorporae inforaion aou he curren er srucures of ond prices Chacko and Das [2002 have since hen derived price forulas for credi spread opions ased on he uncorrelaed Cox-Ingersoll-Ross [1985 processes for he spo rae and spo spread, and again he paraeers in he sochasic processes are assued o e consan Tahani [2000 odels he process of he logarih of credi spread using he GARCH odel and oains closed-for price forulas for credi spread opions Mougeo [2000 exends he wo-facor Longsaff-Schwarz odel o a hree-facor odel for pricing opions on he spread of he values of wo reference asses We exend he Longsaff and Schwarz odel y assuing ie-dependen drif funcions in he ean reversion diffusion processes of he riskless ineres rae and credi spread The spo riskless ineres rae is aken o follow he Hull-Whie odel, while he spo credi spread can e aken o follow eiher he Hull-Whie odel [1990 or he Black-Karasinski odel [1991 When oh sae variales follow he Hull-Whie odel, we can derive analyic expressions for he ie-dependen drif funcions ha fi he curren er srucures of defaulfree and defaulale ond prices Also, we can oain analyic pricing forulas for credi spread opions Unforunaely, he ie-dependen drif funcion in he spo spread process is no analyically racale when he logarih of he spo spread is assued o follow a ean-revering diffusion (his is he assupion in he Black-Karasinski odel) By exending he Hull-Whie rinoial ree fiing ehodologies [1994, we exend he fiing algorihs o allow nuerical valuaion of credi spread opions I CONTINUOUS MODELS We egin wih he pricing of credi spread derivaives under he Hull-Whie [1990 odel This is an exension of he one-facor Hull-Whie odel o he wo-facor version, where oh he insananeous spo riskless ineres rae and he credi spread follow a ean-revering odel wih ie-dependen drif ers Like he one-facor version, he wo-facor odel exhiis nice analyical racailiy and ease of nuerical ipleenaion Pricing Fraework Under Two-Facor Hull-Whie Model We specify he processes of he insananeous spo riskless ineres rae r and he spo spread s o follow he exended Vasicek [1977 odel, where he drif ers are assued o e ie-dependen funcions Under a riskneural valuaion fraework, we assue ha he sochasic processes followed y r and s are given y dr =[θ() ard σ r dw r, (1-A) 52 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003

3 ds =[φ() sd dw s, (1-B) hen where θ() and φ() are ie-dependen funcions, and a and are posiive consan paraeers ha give he raes of ean reversion of r and s, respecively Furher, σ r and are he consan volailiy paraeers for he spo rae r and spo spread s, respecively, and dw r and dw s are sandard Brownian oions under he risk-neural easure P Le ρ denoe he correlaion coefficien eween dw r and dw s ; ha is, dw r dw s = ρd Deerinaion of he Tie-Dependen Drif Ters Hull and Whie [1990 deonsrae how o deerine he ie-dependen funcion θ() in he drif er of he defaul-free spo rae y fiing he curren er srucure of defaul-free ond prices Their procedure can e suarized as follows Le P(, T) denoe he price a ie of a discoun ond auring a ie T, which is given y: P(, T )=E [exp( P T r(u) du), where E P denoes he expecaion under he risk-neural easure P By aking he risk-neural expecaion in Equaion (2), where r is defined in Equaion (1-A), one oains: (2) σ P (, T )= σ r a [1 e a(t ) By following a siilar procedure, i is possile o deerine he oher ie-dependen funcion φ() in he ean reversion process of s y fiing he curren er srucure of defaulale ond prices This is achieved y changing he proailiy easure o he forward-adjused easure Q T wih delivery a fuure ie T Le D(, T) denoe he price a ie of a defaulale ond auring a ie T By he Girsanov heore, we have: D(, T )=E P [ exp ( T = P(, T )E QT [exp ( T [r(u) s(u) du Accordingly, he process followed y he spo spread s under he forward easure Q T is given y ds =[φ() s ρ σ P (, T )d dw T, s(u) du ) ) (6) (7) (8) T P (, T )=exp Differeniaing Equaion (3) wih respec o T wice, we oain: θ(t )= σ2 r ( r() a [1 e a(t ) θ(u) 1 e a(t u) a 2a [1 e 2a(T ) 2 du T 2 ln P(, T) a This equaion gives he dependence of he drif er θ() using he inforaion of he curren er srucure of he defaul-free ond price P(, T) over he period [, T Suppose we wrie he process followed y P(, T) as: ) σ 2 r 2a [1 2 e a(t u) 2 du (3) ln P (, T ) (4) where dw T is a sandard Brownian oion under Q T Under he process defined in Equaion (8), he quaniy s(u) du follows he noral disriuion wih ean: [ (T ) E QT T s(u) du = s() [1 e and variance var[ T [φ(u)ρ σ P (u, T ) 1 e (T u) du (9-A) T σ s(u) du = 2 s [1 2 e (T u) 2 du (9-B) dp (, T ) P(, T ) = rd σ P (, T )dw r, (5) Since SPRING 2003 THE JOURNAL OF DERIVATIVES 53

4 ( ) E [exp QT T s(u) du = we oain [ exp ( E QT T s(u) du ( D(, T ) P (, T ) =exp s() [1 e (T ) (10-A) (10-B) Again, y differeniaing Equaion (10-B) wih respec o T wice and oserving Equaion (6), we oain: 2 φ(t )= σ2 s 2 [1 e 2(T ) ρ σ r[ 1 e a(t ) a σ 2 s 2 2 [1 e (T u) 2 du [ln D(, T ) P(, T ) (11) Credi Spread Pu Opion Le p sp (r, s, ) denoe he price of he credi spread pu opion, whose erinal payoff is given y p sp (r, s, T )=ax(s K, 0) a(t ) 1 e (T ) e (12) Here K is he srike spread a which he opion holder has he righ o pu he risky ond o he opion wrier I is called a credi spread pu opion alhough he erinal payoff reseles a call payoff When he spo credi spread is higher han he srike spread, his signifies ha he price of he underlying risky ond drops elow he srike price of he ond corresponding o he srike spread Since he erinal payoff depends only on s, we ay use he defaul free ond price as he nueraire and oain: p sp (r, s, ) =P(, T )E QT [ax(s K, 0) [φ(u)ρ σ P (u, T )( 1 e (T u) [ln D(, T ) 2 P(, T ) (13) Suppose we define ξ() = s()exp( [T ) Then ξ follows he sochasic process: ) 12 var [ ) s(u) du ) du dξ =[φ()ρ σ P (, T )s()( (T ))d s( )( (T ))dw T (14) Noe ha ξ T is condiionally norally disriued wih ean µ ξ and variance σ 2 ξ, where: [ 1 e (a)(t ) µ ξ = s()e (T ) ρσ r a ρσ r a φ(u)e (T u) du = σ2 s 2 { [1 2 e (T ) [1 e 2 e (T ) 1 e (a)(t ) a e (a) e (T ) [1 e a(t ) σ 2 ξ = σ2 s 2 [1 e 2(T ) (15-A) (15-B) Noe ha µ ξ depends no only on he ie o expiraion T u also on he curren ie Once he ean and variance of ξ T are availale, he price forula for he credi spread pu is given y: [ σξ p sp (r, s, ) =P (, T ) a 1 e (T ) 2π e (K µ ξ) 2 /2σ 2 ξ (µξ K)N(µ ξ K σ ξ Coined Hull-Whie and Black-Karasinski Model (16) Longsaff and Schwarz [1995) propose ha he ineres rae and he logarih of he credi spread follow ean-revering Vasicek processes We exend heir odel y assuing ie-dependen funcions in he drif ers; ha is, we choose he Hull-Whie odel for he ineres rae process and he Black-Karasinski odel for he credi spread process Assue ha y = ln s, and he sochasic processes followed y r and y are governed y: }, 1 e (T ) [ln D(, T ) P(, T ) e a [e a(t ) e (T ) ) dr =[θ() ard σ r dw r, (17-A) 54 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003

5 dy =[ψ() cyd σ y dw y, (17-B) σ 2 ζ = σ2 y 2c [1 e 2c(T ) (20-B) where c is he rae of ean reversion of y, σ y is he consan volailiy paraeer for y, and dw r dw y = ρ^d Unforunaely, he Black-Karasinski odel lacks he level of analyic racailiy exhiied y he Hull-Whie odel Under he Black-Karasinski assupion of he credi spread process, we find ha Equaion (7) has o e odified as: (18) Unlike s(u) du, however, he quaniy T exp[y(u)du is no longer norally disriued Hence, here is no a siple relaion eween [D(, T )/[P(, T ) and he ie-dependen drif funcion ψ(), like ha shown in Equaion (11) If θ() and ψ() are known funcions, we ay derive he price forula for he credi spread pu opion In ers of y, he erinal payoff of he credi spread pu is given y ax(e y K, 0) The pu price forula akes he usual Black-Scholes for: where ( T D(, T ) P (, T ) = EQT[exp exp(y(u)) du d = ln K µ ζ σ 2 ζ σ ζ ) [ p sp (s, r, ) =P(, T ) e µ ζσζ 2 /2 N(d) KN(d σ ζ ) (19-A) (19-B) Siilar o µ ξ and σ 2 ξ as given in Equaions (15-A) and (15-B), he ean µ ξ and variance σ 2 ξ are given y: Heah-Jarrow-Moron Model The wo-facor exended Vasicek odel is a special case of he Heah-Jarrow-Moron (HJM) odel corresponding o a specific choice of he volailiy srucures Firs, we presen he general HJM forulaion of he pricing odel wih he wo sae variales r and s Le f(, T ) denoe he riskless insananeous forward rae Following he HJM fraework, he dynaic of f(, T ) is assued o e df (, T )=α f (, T )d σ f (, T )dw 1, (21) where α f is he drif, and σ f is he volailiy Also, dw 1 is a sandard Brownian oion under he risk-neural easure P Under he no-arirage condiion, he drif α f and volailiy σ f us oserve: α f (, T)=σ f (, T) σ f (, u) du (22) Fro he dynaics of he risky ond prices, we can define he risky insananeous forward rae f d (, T ) in he sae anner as he risk-free insananeous forward rae Nex, he insananeous forward spread h(, T) is defined o e f d (, T ) f(, T ); accordingly, he spo spread s() = f d (, ) f(, ) Assue ha h(, T ) follows he sochasic process: dh(, T )=α h (, T )d σ h 1(, T)dW 1 σ h 2(, T )dw 2, (23) µ ζ = y()e c(t ) ρσ yσ r a [ 1 e (ac)(t ) a c ψ(u)e c(t u) du, 1 e c(t ) c (20-A) where α h is he drif, σ h1 and σ h2 are volailiies, and dw 1 and dw 2 are independen sandard Brownian oions under he risk-neural easure P Again, he no-arirage condiion dicaes ha α h, σ h1, and σ h2, saisfy (see Schönucher [1998): SPRING 2003 THE JOURNAL OF DERIVATIVES 55

6 α h (, T )= (24) By recalling ha s() = h(, ), one can show ha he process followed y s is governed y: ds(t )= 2 2 i=1 σ f (, T ) u σ hi (, T ) σ hi (, u) du σ h1 (, T) σ f (u, v) σ h 1 (u, T ) σ hi (T,T)dW i (T ) i=1 { h(, T ) σ h 1(, u) du dv du i=1 (25) Equaion (25) reveals ha he drif er of s is in general pah-dependen If insead we choose he volailiy srucures of he riskless insananeous forward rae and he insananeous forward spread o e: σ f (, T )=σ r e a(t ), σ h 1(, T )=ρ e (T ), [ 2 σ hi (u, v) σ h i (u, T ) σ h1 (u, v) σ f (u, T ) i=1 2 σ h 2(, T )= 1 ρ 2 e (T ), (26-A) (26-B) (26-C) hen he process of s ecoes Markovian By coparing he resuling expression for ds and ha given in Equaion (1-B), we oain: φ(t )= σ2 s [1 2(T ) e h(, T ) h(, T ) 2 [ 1 ρ σ e a(t ) r a [ 2 σ 2 h i (u, T )2σ hi (u, T )σ f (u, T ) i=1 σ hi (u, T ) dw i (u) du } dt e a(t ) 1 e (T ) σ f (, u) du (27) II CONSTRUCTION OF NUMERICAL ALGORITHMS Now we can exend he Hull and Whie [1994 ehodology o fiing he curren er srucures of defaul-free and defaulale ond prices Two-Facor Algorihs Le x = x(r) and z = z(s) e soe specified funcions of he spo ineres rae and spo credi spread, respecively Assue ha oh x and z follow ean reversion processes: (29-A) (29-B) where dw x dw z = ρ ~ d We would like o consruc a wofacor ree ha fis he curren er srucures of prices of defaul-free and defaulale onds hrough nuerical procedures For exaple, he coined Hull-Whie and Black- Karasinski odel corresponds o x(r) = r and z(s) = ln s We firs riefly review soe of he key seps in he Hull-Whie fiing procedure for he one-facor ree, and follow y generalizaion o he wo-facor ree In he one-facor Hull-Whie ree, we firs uild he rinoial ree for he process x* wih x 0 = 0 ha corresponds o ~ θ() = 0, where: (30-A) Noe ha [x*( ) x*() is norally disriued wih ean M x and variance V x as given y and dx =[ θ() ãxd σx dw x, dz =[ φ() d σz dw z, z dx = ãx d σ x dw x, x (0) = 0 M x = [1 e a~ x * () This is consisen wih he expression given in Equaion (11) y noing ha: [ h(, T )= ln D(, T ) P(, T ) (28) V x = σ2 x 2ã [1 e 2ã (30-B) 56 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003

7 We firs consruc a rinoial ree ha siulaes he process x* The node proailiies are oained y equaing he ean and variance of he discree ree process and he coninuous process as given y Equaion (30-B) A key sep in he Hull-Whie fiing procedure is deerinaion of he ie-varying drif α a ie so ha he laice ree can price defaul-free onds a differen auriies as oserved in he arke a he curren ie Le Y e he curren yield of a defaul-free ond auring a ie ; Q, j e he sae price of node (, j), which is he presen value of a securiy ha pays off $ 1 if node (, j) is reached and zero oherwise; and (, j) indicaes he node a ie a which x = α j x Iniially, we have Q 0, 0 = 1, α 0 = x(y 1 ) Also, we define r, j = x -1 (α j x), where x -1 denoes he inverse funcion of x(r) If we have oained α 1 and Q 1, j for all values of j, he sae price a ie is given y: (31) where pj* i is he node proailiy of oving fro node x j* a ie 1 o node x j a ie, and he suaion is aken over all values of j * for which he nodes x * j are conneced o he node x j Fro he relaion in Equaion (2), we deduce ha: (32) where n (j) is he nuer of nodes on each side of he cenral node a ie The ie-varying drif α can hen e deerined nuerically in a recursive anner When x = r, α can e solved explicily y: ln α = Q,j = j e Y1(1) = ( n ) (j) j= n (j) Q,je j r ( 1)Y 1 (33) Moreover, he value of he ie-dependen drif ~ θ( ) can e esiaed y: θ = α α 1 Q 1,j p j j e r 1,j, n (j) j= n (j) ãα, Q,j e x 1 (αj x), (34) where α ~ is he ean reversion rae of x [see Equaion (29-A) In he coined wo-facor ree ha siulaes he correlaed evoluion of r and s, here are nine ranches eanaing fro he (, j, k) node a =, x j = j x, and z k = k z We use he Hull-Whie approach of consrucing proailiy arices ha give he node proailiies of oving fro a node a o he differen nine nodes a 1 We le β denoe he ie-varying drif for he discree credi spread process Siilarly, we ry o deerine β nuerically y fiing he curren er srucure of defaulale ond prices Le Q,j,k denoe he sae price of he (, j, k) node and ~ Y e he curren yield of a defaulale ond auring a ie A iniiaion, β 0 = z( ~ Y 1 Y 1 ); and he successive β, 1, are oained recursively y he relaionship: e Ỹ1(1) = (35) where n (k) is defined siilarly as n(j) When z(s) = s, β can e solved explicily y he forula: ln β = ( n (j) j= n (j) n (j) j= n (j) (36) Once β is known, he value of φ ( ) can e esiaed y: φ = β β 1 n (k) k= n (k) n ) (k) k= n (k) Q,j,ke (r,jk s) ( 1)Ỹ 1 β, (37) where ~ is he ean reversion rae of z[see Equaion (29-B) When he sae prices are known a he nodes, he value of any European-syle credi derivaive a ie 0 whose value depends on r and s can e oained y: Q,j,k e (r,jz 1 (βk z)), Q M,j,k p sp (r M,j,s M,k, M ), p sp (r 0,s 0, 0 )= j k (38) where p sp (r M,j, s M,k, M ) is he erinal payoff a auriy dae M, and he suaion is aken over all erinal nodes SPRING 2003 THE JOURNAL OF DERIVATIVES 57

8 I is quie copuaionally deanding o calculae he sae prices since his wo-facor algorih requires a search process a each node When x(r) = r and z(s) = s, one can derive explici forulas for he ie drifs α and β wihou calculaing he sae prices Furher, when he erinal payoff of a credi derivaive is independen of he ineres rae, we can develop a one-facor fiing algorih via he use of valuaion in he forward-adjused risk-neural easure This reduced one-facor fiing algorih can e considered an exension of he Gran-Vora [2001 approach o derive an explici analyic expression for he ie drif funcion of he one-facor Hull-Whie odel One-Facor Tree for Join Exended Vasicek Processes The no-arirage fiing procedure developed y Gran and Vora [2001 is indeed he discree analogue of he analyic procedure for deriving he ie-dependen drif θ() The discree analogue of he exended Vasicek process for r as defined in Equaion (1-A) can e represened y: r = r 1 r = M r r δ V r Z, (39) where he ean M r and variance V r are defined uch he sae as in Equaion (30-B), Z is a sandard noral variale, and δ is an adjusen paraeer for he no-arirage fiing procedure Noe ha α is equal o E P [r Deducively, we oain: 1 r = r 0 (1 M r ) (1 M r ) 1 j δ j 1 Vr (1 M r ) 1 j Z j, (40) where he noral variales Z j, j = 0,, 1, are independenly and idenically disriued Fro Equaion (39), one can deduce ha: (42) Since he price a ie 0 of a defaul-free ond auring a ie 1 is given y: so ha γ 2 = we hen have (43-A) (43-B) (44) which gives he ie-varying drif α As in he coninuous odel, we can reduce he wofacor algorih o a one-facor version y choosing he defaul-free ond price as he nueraire Firs, we would like o find he ie-varying drif β of he spo spread process under he risk-neural easure P and he forward easure Q T Noe fro Equaions (1-B) and (8) ha he wo easures are relaed y where he las er σ 2 r 2a(1 e a ) 2 [e 2a (1 e 2a ) 2e a (1 e a )(1 e a ) (1 e 2a ) P( 0, 1 )=E P exp r j E P [r = ln P ( 0, 1 ) E P [r = 1 ln P ( 0, 1 ) P ( 0, ) 1 E QT [s( ) = E P [s( ) X 0,,M, E P [r j γ2 2, γ2 γ2 1, 2 (45) ha δ = E P [r 1 (1 M r )E P [r (41) Le γ 2 e he variance of Σ r j I can e shown X j,,m = j ρ σ P (u, M )e ( u) du (46) represens he drif adjusen required for effecing he change of easure 58 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003

9 Using he ond volailiy σ P (, T) defined in Equaion (6), and assuing consan values for ρ and, we have p sp (r 0,s 0, 0 )=P ( 0, M ) k U M,k p sp (s M,k, M ), (52) X j,,m = ρσ r a (47) The discree analogue of he exended Vasicek process followed y s can e represened y (48) where M s, V s, and ξ are defined in an analogous anner as in Equaion (39) Siilarly, he adjusen paraeer ξ is deerined y fiing he er srucure of defaulale ond prices Soluion of he recursive relaion (48) gives 1 s = s 0 (1 M s ) (1 M s ) 1 j (ξ j X j,j1,m ) 1 [ e a(m ) e am (a)j (49) The discree version of Equaion (7) is seen o e D( 0, )=P( 0, )E Q exp (50) fro which we deduce he forula for E P [s (his is equal o β, he ie-varying drif for he credi spread process): E P [s = 1 ln D( 0, 1 ) P( 0, 1 ) 1 E P [s j 1 a Vs (1 M s ) 1 j Z j 1 e ( j) s = s 1 s = M s s ξ X,1,M V s Z, 1 s j X j,j1, [(1 M s ) j 1 γ2 M s 2, (51) where γ ~ 2, is defined siilarly as γ 2, excep ha r, a, and σ r are replaced y s,, and, respecively Under he forward easure, he ie 0 price of a credi spread derivaive is given y: where p sp (s M,k, M ) is he erinal payoff on auriy dae M, and U M,k is he cuulaive proailiy under he forward easure reaching he (M, k) node fro he saring (0, 0) node Noe ha U 0,0 = 1, s,k = β k s, and β = ξ χ 0,, III NUMERICAL RESULTS We ipleen he wo-facor and one-facor fiing algorihs using curren er srucures of defaul-free and defaulale yield curves so as o check he accuracy and he effeciveness of he nuerical algorihs agains he analyic price forulas In our nuerical experiens, we generae he curren yield of he defaul-free Treasury ond y: Y (T )= e 18T (53) We use he Briys and de Varenne [1997 inereporal defaul odel o generae he yield for a high-raed ond and a low-raed ond The spread curves are oained y suracing he yield of he defaul-free Treasury onds fro he yield curves of he defaulale onds Noe ha he yield curves of he defaul-free Treasury ond and he high-raed ond increase onoonically wih ie o auriy, while he yield curve of he lowraed ond exhiis he ypical hup shape These yield curves (shown in Exhii 1) are used as inpu in all susequen calculaions In Exhiis 2-A and 2-B, we plo he ie-dependen drif funcion φ() in he ean reversion of he spo spread process, oained y fiing he yield curves of he high-raed ond and he low-raed ond, respecively The drif φ() is shown o e an increasing funcion of he spo spread volailiy for oh high-raed and low-raed onds Since he spread approaches zero as ie approaches auriy, he drif goes o zero as ends o zero The drif funcions for oh high-raed and low-raed onds reach soe peak value, hen decline a a higher value of and sele a soe asypoic value a infinie The drif is greaer for he low-raed ond since he corresponding spo spread has a higher value Also, he drif shows less sensiiviy o he spo spread volailiy for he low-raed ond SPRING 2003 THE JOURNAL OF DERIVATIVES 59

10 E XHIBIT 1 Yield Curve Plos 012 low-raed yield Yield / Spread 006 low-raed yield spread high-raed yield 004 Treasury yield 002 high-raed yield spread Tie o Mauriy E XHIBIT 2-A Tie-Dependen Drif Funcion for High-Raed Bond = = 02 φ() 004 = NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003

11 E XHIBIT 2-B Tie-Dependen Drif Funcion for Low-Raed Bond φ() = 03 = 02 = E XHIBIT 3 Coparison of One-Facor and Two-Facor Tree Calculaions Average Relaive Errors in Average Relaive Errors in Nuer of Tie Seps Two-Facor Tree Calculaions One-Facor Tree Calculaions 8 318% 292% % 163% % 103% To es he convergence of he wo-facor and onefacor fiing algorihs, we apply he algorihs o price a credi spread pu wih payoff as given in Equaion (12) We use differen coinaions of paraeer values in he pricing odel, and copare he nuerical resuls wih he resuls oained fro he analyic price forula in Equaion (16) wih varying nuers of ie seps Exhii 3 shows he convergence ehaviors of he wo fiing algorihs in ers of average relaive errors (in percenages) The relaive errors have relaively low values ha decline in a roughly linear anner wih increasing nuers of ie seps The one-facor algorih provides eer accuracy In ers of copuaional coplexiy, he one-facor algorih is O(M 2 ), while he wo-facor algorih is O(M 3 ), where M is he nuer of ie seps This propery of a polynoial order of coplexiy is highly desirale, as nuerical ipleenaion does no ake an unreasonale aoun of copuaion ie We also invesigae he pricing ehavior of he credi spread pu opion wih respec o ie o expiraion and spo spread volailiy In Exhii 4-A, we plo he spread pu opion funcion agains ie o expiraion wih varying spo spread volailiy where he underlying is a high-raed ond The spread pu price increases wih longer ie o expiraion and higher spo spread volailiy When he underlying is a low-raed ond, however, he ie-dependen ehavior of he spread pu price exhiis a huped shape (see Exhii 4-B) This ay e due o a declining credi spread a very long auriies for SPRING 2003 THE JOURNAL OF DERIVATIVES 61

12 E XHIBIT 4-A Price of Credi Spread Pu Agains Tie o Expiraion High-Raed Bond 012 = Opion Price = = Tie o Expiraion E XHIBIT 4-B Price of Credi Spread Pu Agains Tie o Expiraion Low-Raed Bond = Opion Price = = Tie o Expiraion 62 NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003

13 low-raed onds Spread pu prices are always seen o e increasing funcions of volailiy, regardless of he credi qualiy of he underlying ond IV CONCLUSION The credi spread is a coon sae variale in pricing odels for credi spread derivaives By assuing ha he insananeous spo riskless ineres rae and he credi spread follow correlaed sochasic processes, we have developed a no-arirage approach for pricing credi spread derivaives Our pricing odel assues he riskless ineres rae follows a ean-revering process wih ie-dependen drif, and he credi spread (or is logarih) follows a siilar process The ie-dependen drif funcions in he ean reversion processes are deerined y fiing curren er srucures of defaul-free and defaulale ond prices When oh he spo rae and he spo spread follow an exended Vasicek process, he ie-dependen drif funcions can e oained in closed for Also, we ake an expecaion under he risk-adjused forward easure o derive analyic price forulas for credi spread derivaives When he logarih of he spread follows he exended Vasicek process, he Black-Karasinki odel does no exhii he sae level of analyic racailiy, u one can always rely on nuerical fiing algorihs o deerine he drif er and copue he price of he credi spread opion We also show ha he wo-facor exended Vasicek odel is a special case of he Heah-Jarrow- Moron odel corresponding o soe specific choice of he volailiy srucures By following he Hull-Whie ree-fiing algorih, we have developed wo-facor and one-facor fiing algorihs for pricing credi spread derivaives Since our rinoial ree consrucion has recoining properies, a polynoial order of copuaional coplexiy is oserved in our nuerical algorih The polynoial order propery is an overriding advanage over he Heah-Jarrow- Moron (HJM) fraework, since he non-recoining ree in he HJM odel leads o an exponenial order of coplexiy The validiy of our analyic price forulas and he versailiy of he nuerical schees are verified hrough nuerical coparison of he copued resuls and he resuls oained y analyic forulas The algorihs are nuerically accurae even wih relaively few ie seps Assuing ha oh he spo rae and he spo spread follow exended Vasicek processes, we oserve ha he price of he credi spread pu is an increasing funcion of he volailiy of he spread process This resul holds for oh high-raed and low-raed underlying risky onds When spread volailiy is relaively low and ie o expiraion is relaively long, he credi spread pu price funcion ay ecoe a decreasing funcion of ie o expiraion for lower-raed underlying onds This propery conrass wih he ehavior of os oher equiy opion producs, u i is consisen wih he decline of he spread of a longer lower-raed ond under low spread volailiy Our work exends pricing odels for credi derivaives in several respecs Under he assupion of exended Vasicek processes for he underlying sae variales, one can oain analyic price forulas for credi spread derivaives and closed-for represenaions of he ie-dependen drif ers using a no-arirage approach ha fis he curren er srucures of ond prices Our onefacor and wo-facor fiing algorihs for pricing credi spread derivaives exhii only a polynoial order of copuaional coplexiy And we show how o choose volailiy srucures so ha he Heah-Jarrow-Moron odel reduces o he Hull and Whie odel REFERENCES Black, F, and P Karasinski Bond and Opion Pricing when he Shor Raes are Lognoral Financial Analyss Journal, July/Augus 1991, pp Briys, E, and F de Varenne Valuing Risky Fixed Rae De: An Exension Journal of Financial and Quaniaive Analysis, 32 (1997), pp Chacko, G, and S Das Pricing Ineres Rae Derivaives: A General Approach Review of Financial Sudies, 15 (2002), pp Cox, JC, JE Ingersoll, Jr, and SA Ross A Theory of he Ter Srucure of Ineres Raes Econoerica, 53 (1985), pp Das, SR Credi Risk Derivaives The Journal of Derivaives, 2, 3 (Spring 1995), pp 7-23 Das, SR, and RK Sundara A Discree-Tie Approach o Arirage-Free Pricing of Credi Derivaives Manageen Science, 46 (2000), pp Duffie, D, and K Singleon Modeling Ter Srucures of Defaulale Bonds Review of Financial Sudies, 12 (1999), pp SPRING 2003 THE JOURNAL OF DERIVATIVES 63

14 Garcia, J, H van Gindersen, and R Garcia On he Pricing of Credi Spread Opions: A Two-Facor HW-BK Algorih Working paper, Aresia BC and Universiy of California a Berkeley, 2001 Gran, D and G Vora An Analyical Ipleenaion of he Hull and Whie Model The Journal of Derivaives, Winer 2001, pp Heah, D, R Jarrow, and A Moron Bond Pricing and he Ter Srucure of Ineres Raes: A New Mehodology for Coningen Clais Valuaion Econoerica, 60 (1992), pp Hull, J, and A Whie Nuerical Procedures for Ipleening Ter Srucure Models II: Two-Facor Models The Journal of Derivaives, Winer 1994, pp Pricing Ineres-Rae-Derivaive Securiies Review of Financial Sudies, 3, 4 (1990), pp Jarrow, R, D Lando, and SM Turnull A Markov Model for he Ter Srucure of Credi Risk Spreads Review of Financial Sudies, 10 (1997), pp Kijia, M, and K Kooriayashi A Markov Chain Model for Valuing Credi Risk Derivaives The Journal of Derivaives, Fall 1998, pp Longsaff, FA, and ES Schwarz Valuing Credi Derivaives The Journal of Fixed Incoe, June 1995, pp 6-12 Mougeo, N Credi Spread Specificaion and he Pricing of Spread Opions Working paper, Ecole des HEC, 2000 Schönucher, PJ Ter Srucure Modeling of Defaulale Bonds Review of Derivaives Research, 2 (1998), pp A Tree Ipleenaion of a Credi Spread Model for Credi Derivaives Working paper, Bonn Universiy, 1999 Tahani, N Esiaing and Valuing Credi Spread Opions wih GARCH Models Working paper, HEC Monreal, 2000 Vasicek, O An Equiliriu Characerizaion of he Ter Srucure Journal of Financial Econoics, 5 (1977), pp To order reprins of his aricle, please conac Ajani Malik a aalik@iijournalsco or NO-ARBITRAGE APPROACH TO PRICING CREDIT SPREAD DERIVATIVES SPRING 2003