Equity-credit modeling under affine jump-diffusion models with jump-to-default

Transcription

1 Equiy-credi modeling under affine jump-diffusion models wih jump-o-defaul Tsz Kin Chung Deparmen of Mahemaics, Hong Kong Universiy of Science and Technology Yue Kuen Kwok Deparmen of Mahemaics, Hong Kong Universiy of Science and Technology 1

2 EQUITY-CREDIT MODELING UNDER AFFINE JUMP-DIFFUSION MODELS WITH JUMP-TO-DEFAULT ABSTRACT This aricle reviews he sochasic models for pricing credi-sensiive financial derivaives using he join equiy-credi modeling approach. The modeling of credi risk is embedded ino a sochasic asse dynamics model by adding he jump-o-defaul feaure. We discuss he class of sochasic affine jump-diffusion models wih jump-o-defaul and apply he models o price defaulable European opions and credi defaul swaps. Numerical implemenaion of he equiy-credi models is also considered. The impac on he pricing behavior of derivaive producs wih he added jump-o-defaul feaure is examined. 1 Inroducion The affine jump-diffusion (AJD models have been widely used in coninuous ime modeling of sochasic evoluion of asse prices, bond yields and credi spreads. Some of he well known examples include he sochasic volailiy (SV model of Heson (1993, sochasic volailiy jump-diffusion models (SVJ of Baes (1996 and Bakshi e al. (1997, and sochasic volailiy coheren jump model (SVCJ of Duffie e al. (2000. The AJD models possess flexibiliy o capure he dynamics of marke prices in various asse classes, while also admi nice analyical racabiliy. The affine erm srucure models, which fall ino he family of AJD models, have been frequenly used o sudy he dynamics of bond yields and credi spreads (Duffie and Singleon, A number of sudies have addressed he imporance of including jump dynamics o valuaion and hedging of derivaives. In he modeling of equiy derivaives, Bakshi e al. (1997 illusrae ha he sochasic volailiy model augmened wih he jump-diffusion feaure produces a parsimonious fi o sock opion prices for boh shor-erm and long-erm mauriies. Empirical 2

3 sudies repored by Baes (1996, Pan (2002 and Erakar (2004 show ha he inclusion of jumps in he modeling of sock price is necessary o reconcile he ime series behavior of he underlying wih he cross-secional paern of opion prices. In paricular, Erakar (2004 concludes from his empirical sudies ha simulaneous jumps in sock price and reurn variance are imporan in caering for differen volailiy regimes. While he AJD models have been successfully applied in valuaion of boh equiy and credi derivaives, he join modeling of equiy and credi derivaives have no been fully addressed in he lieraure. Recenly, a growing lieraure has highlighed such an ineracion beween equiy risk (sock reurn and is variance and credi risk (firm defaul risk. While he risk neural disribuion of sock reurn is fully conveyed by raded opion prices of differen srikes and mauriies, he informaion of he arrival rae of defaul can be exraced from he bond yield spreads or credi defaul swap spreads. Wih he growing liquidiy of he credi defaul swap (CDS markes, he CDS spreads provide more reliable and updaed informaion abou he credi risk of firms. Achyara and Johnson (2007 find ha he CDS marke conains forward looking informaion on equiy reurn, in paricular during imes of negaive credi oulooks. For equiy opions, Cremers e al. (2008, Zhang e al. (2009 and Cao e al. (2010 show ha he ou-of-he-money pu opions, which depic he negaive ail of he underlying risk neural disribuion, are closely linked o yield spreads and CDS spreads of he reference firm. Several innovaive equiy-credi models have been proposed in he lieraure. Carr and Linesky (2006 propose an equiy-credi hybrid model in which he sock price is sen o a cemeery sae upon he arrival of defaul of he reference company. Carr and Wu (2009 inroduce anoher equiy-credi hybrid model which incorporaes jump-o-defaul in which he equiy price drops o zero given he defaul arrival. Carr and Madan (2010 consider a local volailiy model enhanced by jump-o-defaul. Mendoza-Arriaga e al. (2010 and Bayrakar and Yang (2011, respecively, propose a flexible modeling framework o unify he valuaion of equiy and credi derivaives using he ime-changed Markov process and muliscale sochasic volailiy. Cheridio and Wugaler (2011 propose a general framework under affine models wih possibiliy of defaul for he simulaneous modeling of equiy, governmen bonds, corporae bonds and derivaives. In his aricle, we propose an equiy-credi model under he general affine jump-diffusion framework in he presence of jump-o-defaul (JD-AJD model. We illusrae how o use he proposed JD-AJD model for pricing defaulable European opions and credi defaul swaps. The aricle is organized as follows. In he nex secion, we presen he mahemaical framework of he affine jump-diffusion wih jump-o-defaul. The reduced form approach is adoped, where he defaul process is modeled as a Cox process wih sochasic inensiy. We illusrae how o apply he Fourier ransform echnique o derive he join characerisic funcion of he sock price disribuion in he JD-AJD model. In Secion 3, we consider pricing of defaulable 3

4 European coningen claims and credi defauls swaps using he JD-AJD model. We manage o obain closed form pricing formulas of hese wo credi-sensiive derivaive producs, a demonsraion for nice analyical racabiliy of he proposed equiy-credi model. In Secion 4, we discuss he pracical implemenaion of he jump-o-defaul feaure o several popular opion pricing models. We also consider numerical valuaion of defaulable European opions using various numerical approaches, like he Fas Fourier ransform (FFT echniques and Mone Carlo simulaion. The impac of various jump parameers on he pricing behavior of defaulable European opions is examined. Conclusive remarks are presened in he las secion. 2 Affine jump diffusion wih jump-o-defaul This secion summarizes he mahemaical framework of he AJD model (Duffie e al., Consider he filered probabiliy space (Ω, G, {G }, Q, he AJD process of he vecor sochasic sae variable X is defined in some sae space D R n as follows: dx = µ (X d + σ (X dw + dz, (1a where Q is some appropriae equivalen maringale measure adoped for pricing coningen claims, G = σ {X s s < } is he naural filraion generaed by he vecor sae variable X, W is an G -sandard Brownian moion in R n, µ : D R n is he drif vecor, σ : D R n n is he diffusion marix, and Z is a pure jump process whose jumps have a fixed probabiliy disribuion ν on R n and arrive wih inensiy {λ (X : 0}, λ : D [0,. The ineres rae process is specified as {r (X : 0}, r : D [0,. Under he AJD model, he parameer funcions µ (X, σ (X, λ (X and he risk-free ineres rae r (X are specified as follows: µ (X = K 0 + K 1 X, for K 0 R n and K 1 R n n ; { σ (X σ (X T } ij = {H 0 } ij + {H 1 } ij X, for H 0 R n n and H 1 R n n n ; λ (X = l 0 + l 1 X, for l 0 R and l 1 R n ; r (X = r 0 + r 1 X, for r 0 R and r 1 R n. Wihou loss of generaliy, we le he firs componen of X be he logarihm of he sock price S. Under he AJD framework, sochasic ineres rae and sochasic volailiy (expressed as some linear combinaion of he vecor sae variable X can be incorporaed. Also, he correlaion srucures beween he differen facors can be inroduced by specifying he diffusion marix σ (X. I is worh noing ha he drif vecor µ (X is deermined by requiring he 4

5 sock price process o be a maringale under he equivalen maringale measure Q. Specificaion of he defaul process Nex, we exend he AJD framework by incoporaing he jump-o-defaul feaure of he sock price process. Upon he arrival of defaul, he sock price jumps o some consan level called he cemeery sae (Carr and Linesky, In principle, he sae can be a prior known level or a level ha is arbirarily close o zero. Following he reduced form framework, we assume he defaul process o be generaed by a Cox process ha is defined in he same sae space D. Formally, we define he firs jump ime τ d = inf 0 : 0 h (X s ds e as he random ime of defaul arrival. Here, e is he sandard exponenial random variable and he inensiy process of defaul arrival (hazard rae is assumed o be a funcion of he sae variable as represened by {h (X : 0}, h : D [0,, and adaped o G. The filraion H = σ ( 1 {τd <s} s conains he informaion of wheher here has been a defaul by ime. We define F = H G o be he informaion se ha conains he knowledge of he evoluion of he vecor sae variable X and hisory of defaul by ime. In order o have he risk adjused ineres rae o be affine, he hazard rae process h (X is specified as h (X = h 0 + h 1 X, for h 0 R and h 1 R n. (1b The above specificaion falls ino he affine erm srucure framework commonly used in he modeling of ineres rae and credi derivaives (Lando, 1998; Duffie and Singleon, Transform analysis The discouned expecaion of a coningen claim ha pays F (X T when here is no defaul prior expiraion is given by E Q = 1 {τd >}E Q r (X s ds R (X s ds F (X T 1 {τd >T } F F (X T G, where R (X s = r (X s + h (X s is he risk adjused discoun rae a ime s and he payoff F (X T is G T -measurable requiring no knowledge of he defaul process. The price funcion of 5

6 he coningen claim condiional on X = X a ime is defined by V (X, = E Q R (X s ds F (X T. (2 By he Feynman-Kac represenaion formula, we deduce ha V (X, saisfies he following parial inegro-differenial equaion (PIDE: V (X, + LV (X, = 0. (3a Here, L is he infiniesimal generaor as defined by LV (X, = µ (X V X (X, + 1 [σ 2 r (X σ (X T V XX (X, + λ (X [V (X + z, V (X, dν (z R (X V (X,. R n (3b The soluion o he above PIDE subjec o an exponenially affine form of he erminal condiion is saed in Theorem 1. Theorem 1. When he erminal payoff funcion is exponenially affine, where F (X T = exp (u X T, for u C n, he condiional expecaion F 0 (u, X, ; T = E Q R (X s ds e u X T G (4 has he soluion of he form F 0 (u, X, ; T = exp (α (τ + β (τ X, τ = T, where α (τ and β (τ saisfy he following sysem of complex-valued ordinary differenial equaions (ODEs: α (τ = R 0 + K 0 β (τ β (τt H 0 β (τ + l 0 [Λ (β (τ 1, τ > 0 β (τ = R 1 + K T 1 β (τ β (τt H 1 β (τ + l 1 [Λ (β (τ 1, τ > 0, (5 wih he iniial condiions: α (0 = 0, β (0 = u. Here, β (τ T H 1 β (τ is a vecor whose k h componen is given by β i {H 1 } ijk β j and Λ(c is he jump ransform as defined by i j 6

7 Λ(c = exp (c z dν (z for some c C n. R n The proof of Theorem 1 can be found in Duffie e al. (2000. Defaulable bond wih fixed recovery We consider a defaulable zero-coupon bond which pays one dollar when here is no defaul prior o mauriy, oherwise a recovery paymen R p is paid a mauriy. Hence, he erminal payoff can be formulaed as 1 {τd >T } + R p 1 {τd <T }. The non-defaul componen of he zerocoupon bond is given by B 0 (, T ; X = E Q = 1 {τd >}E Q r (X s ds 1 {τd >T } R (X s ds F G = 1 {τd >} exp (α (T + β (T X, (6 by virue of he ransform analysis saed in Theorem 1. Here, α (τ and β (τ saisfy he same sysem of ODEs as depiced in eq. (5, wih he corresponding iniial condiions specified as α (0 = 0, β (0 = (0, 0,..., 0 T. Similarly, he risk neural discouned expecaion of he recovery paymen R p coningen upon defaul is given by B R (, T ; X = R p E Q { ( = R p E Q T r (X s ds 1 {τd >}E Q 1 {τd <T } r (X s ds G F } R (X s ds G. (7a The firs erm can be visualized as he risk-free discoun facor as defined by ( B f (, T ; X = E Q = exp T r (X s ds G ( α (T + β (T X, (7b where α (τ and β (τ saisfy a similar sysem of ODEs as depiced in eq. (5, excep ha he risk-free rae (r 0, r 1 replace he role of he risk adjused discoun rae (R 0, R 1. The corresponding iniial condiions are specified as: α (0 = 0, β (0 = (0, 0,..., 0 T. Adding he 7

8 wo componens yields he defaulable bond price wih fixed recovery as follows: B (, T ; X = B 0 (, T ; X + B R (, T ; X = 1 {τd >} (1 R p exp [α (T + β (T X + R p exp [ α (T + β (T X. (8 2.2 Join characerisic funcion The pre-defaul discouned characerisic funcion, which provides informaion on he evoluion of he sock price dynamics prior o defaul, is defined by he following condiional expecaion: Ψ (ω, T ; X, = E Q where ω = (ω 1, ω 2,..., ω n T = 1 {τd >}E Q r (X s ds R (X s ds exp (iω X T 1 {τd >T } F exp (iω X T G = 1 {τd >} exp (α (T + β (T X, (9 R n. Again, α (τ and β (τ saisfy he same sysem of ODEs as depiced on eq. (5, while he corresponding iniial condiions are specified as α (0 = 0, β (0 = (iω 1, iω 2,..., iω n T. The marginal characerisic funcion of he firs componen of X, which is he logarihm of he sock price, can be obained by seing ω = (ω, 0,..., 0 T. For noaional convenience, we wrie he firs componen of X as x in our subsequen discussion. 3 Pricing of defaulable European opions and credi defaul swaps In his secion, we illusrae he pricing of defaulable European coningen claims and credi defaul swaps under he JD-AJD model. Provided ha analyic soluion o he associaed sysem of ODEs is available, we are able o obain closed form pricing formulas of hese credisensiive derivaives. Forunaely, nice analyic racabiliy of he Ricai sysem of ODEs is feasible for a wide range of sochasic sock price dynamics models, which include he SV model (Heson, 1993, SVJ model (Baes, 1996; Bakshi e al., 1997, SVCJ model (Duffie e al., 2000; Erakar, 2004, and Carr-Wu s model (Carr and Wu, Moreover, he exponenial affine srucure is preserved in he characerisic funcion of he sock price disribuion, which proves o be useful in he compuaion of risk sensiiviy of derivaives. 8

9 3.1 Defaulable European coningen claims Consider a defaulable European coningen claim which pays P (X T when no defaul occurs before mauriy and zero payoff upon defaul (zero recovery. Given ha he payoff depends only on he erminal sock price S = exp (x, he ime- value of he coningen claim is given by P (X, = E Q = 1 {τd >}E Q r (X s ds R (X s ds P (x T 1 {τd >T } F P (x T G, (10 where R (X s = r (X s + h (X s is he risk-adjused discoun rae a ime s. Le P (ω denoe he Fourier ransform of he erminal payoff wih respec o x T, where P (ω = e iωx T P (x T dx T, he erminal payoff can be expressed in he following represenaion as a generalized Fourier ransform inegral: P (x T = 1 2π iε+ iε e iωx T P (ω dω. Here, he parameer ε = Im ω denoes he imaginary par of ω which falls ino some regulariy srip, ε (a, b, such ha he generalized Fourier ransform exiss (Lord and Kahl, By virue of Fubini s heorem, we obain he following inegral represenaion of he ime- value of he coningen claim where P (X, = 1 2π iε+ iε Ψ ( ω P (ω, T dω, (11 Ψ (ω = 1 {τd >}E Q R (X s ds exp (iωx T G. This is precisely he pre-defaul discouned firs-componen marginal characerisic funcion [see eq. (9. If here is a fixed recovery paymen R p o be paid on he mauriy dae upon earlier defaul, he presen value of his recovery paymen is given by { P R (X, = R p E Q r (X s ds G 1 {τd >}E Q ( T } R (X s ds G. (12 9

10 Recall ha he jump ransform is defined by Λ(c = exp(c z dν(z for some c C n. R n I is worh noing ha differen erminal payoff funcions may impose differen resricions on he regulariy srip ε (a, b, inside which he generalized Fourier ransform exiss. Indeed, one has o choose a paricular regulariy srip such ha boh he jump ransform and he Fourier ransform of he erminal payoff exis. Taking he double exponenial jump model as an example, he jump ransform exiss for η 1 < Im ω < η 2, where η 1 > 1 and η 2 > 0 are he parameers defining he sizes of he upward and downward jumps. Usually, here exis cerain resricions on he regulariy srip wih respec o some sandard opion payoffs. In case when he regulariy srip does no conform wih he ransformed opion erminal payoff funcion, one may use he pu-call pariy relaion or oher relaion derived from an appropriae replicaion porfolio o compue he desired opion price. European call opion Consider a call opion which pays (S T K + a mauriy when here is no defaul prior o he mauriy dae T and zero oherwise, so he erminal payoff funcion is given by (S T K + 1 {τd >T } = (e x T K + 1 {τd >T }, where x T = ln S T. The Fourier ransformed of he above erminal payoff funcion is C (ω = e iωx T (e x T K + ln K Keiω dx T = ω 2 iω, for ε = Im ω (1, ε max. The upper bound of ε, as denoed by ε max, can be deermined by he non-explosive momen condiion Ψ ( iε < (Carr and Madan, The call opion price has he following Fourier inegral represenaion C (X, = 1 2π = K π iε+ iε 0 Re Ψ ( ω [ Keiω ln K dω ω 2 iω { e i(ζ+iε ln K Ψ ( (ζ + iε i (ζ + iε (ζ + iε 2 } dζ, ω = ζ + iε. (13a I is worh noing ha along he conour ω = a + ib for b (1, ε max, here is no singulariy in he inegrand and one can perform numerical inegraion wihou much difficuly. 10

11 I can be shown by replacing k = ln K, ζ = v and ε = α + 1 ha he expression in eq. (13a is equivalen o he pricing formulaion in Carr and Madan (1999, where C (X, = e αk π 0 { } e ivk Ψ (v i (α + 1 Re dv. (v iα [v i (α + 1 (13b In oher words, one would obain he same analyic expression of he Fourier inegral represenaion no maer one considers he ransform wih respec o he log-sock price or log-srike price. Lord and Kahl (2007 propose wo approaches o compue he above Fourier inegral. The firs approach is he direc numerical inegraion using an adapive numerical quadraure, such as he Gauss-Kronrod quadraure (for example, he quadgk subrouine in Malab. adapive quadraure can achieve high order of accuracy by choosing appropriaely differen opimal damping facors α for opions a differen srikes. The second approach is o employ he Fas Fourier ransform (FFT echnique o inver he Fourier inegral o obain opion prices on a uniform grid of log-srikes. Since opion prices a discree srikes are obained on a uniform grid of log-srikes, one needs o perform inerpolaion o obain he opion price a an arbirary srike price (Carr and Madan, For shor-mauriy opions, he inerpolaion errors can be quie subsanial. The European pu opion Suppose a pu opion pays a mauriy he following payoff: (K S T + 1 {τd >T } = (K e x T + 1 {τd >T } when here is no defaul prior o mauriy, and a recovery paymen R P 1 {τd <T } o be paid a mauriy when defaul occurs during he conracual period. The ransformed payoff of he non-defaul componen of he pu opion is given by P 0 (ω = e iωx T (K e x T + ln K Keiω dx T = ω 2 iω, for ε = Im ω ( ε max, 0. Inside he regulariy srip where he above Fourier ransform is well defined, he non-defaul componen has he following inegral represenaion P 0 (X, = 1 2π = K π iε+ iε 0 Re Ψ ( ω ( K 0e iω ln K dω ω 2 iω { e i(ζ+iε ln K Ψ ( (ζ + iε i (ζ + iε (ζ + iε 2 } dζ, ω = ζ + iε. (14 I is ineresing o find ha he Fourier ransform of he erminal payoff funcion for he pu opion and he call opion counerpar boh have he same inegral represenaion, hough subjec o differen consrains on he regulariy srip. Noe ha Im ω (1, ε max for he call 11

12 opion and Im ω ( ε max, 0 for he pu opion. As shown earlier, he recovery paymen can be obained similar o P R (X, in eq. (12. The defaulable European pu opion price is hen given by P (X, = P 0 (X, + P R (X,. (15 Remark For he Meron jump-diffusion model, here is no addiional resricion on he regulariy srip. For he Kou double exponenial jump model, he jump ransform exiss only for Im ω ( η 1, η 2, where η 1 > 1 and η 2 > 0. In his case, i is more convenien o implemen he pu opion formula [which requires ε = Im ω (, 0 and obain he call opion price using he pu-call pariy relaion (o be discussed nex. Pu-call pariy relaion under jump-o-defaul In he presence of jump-o-defaul, a porfolio of a long call and a shor pu has he erminal payoff (S T K + 1 {τd >T } [ (K S T + 1 {τd >T } + K1 {τd <T } = (S T K 1 {τd >T } K1 {τd <T }. Hence, he difference of defaulable European call and pu prices is given by C (X, P (X, = 1 {τd >}E Q ( E [K Q exp T + 1 {τd >}E Q [K exp R (X s ds r (X s ds G (S K G R (X s ds G = 1 {τd >}S KB f (, T, (16 where B f (, T is defined in eq. (7b. The pu-call pariy relaion in he presence of jump-odefaul is seen o be he same as he sandard relaion. This is consisen wih he model-free propery of he pu-call pariy relaion. 3.2 Credi defaul swap A credi defaul swap (CDS is an over-he-couner (OTC credi proecion conrac in which a proecion buyer pays a sream of fixed premium (CDS spread o a proecion seller and in 12

13 reurn eniles he proecion buyer o receive a coningen paymen upon he occurrence of a pre-defined credi even. The CDS spread is se a iniiaion of he conrac in such a way ha he expeced presen value of he premium leg received by proecion seller equals ha of he proecion leg received by he proecion buyer. We would like o deermine he fair CDS spread, assuming a fixed and known recovery rae of he underlying risky bond. In pracice, he recovery rae is commonly se o be 30 40% for corporae bonds in he US and Japanese markes. Based on he proposed JD-AJD model, he hazard rae of defaul arrival is assumed o be affine [see eqs. (1a,b, where h (X = h 0 + h 1 X, for h 0 R and h 1 R n ; dx = µ (X d + σ (X dw + dz. For simpliciy, we assume he hazard rae o be independen of he ineres rae process so ha he survival probabiliy is given by S (, T = E Q h (X u du G. Hence, he price of a defaulable zero-coupon bond wih zero recovery is given by D (, T = E Q r (X u + h (X u du G = B f (, T S (, T. Boh S(, T and D(, T can be readily obained using he ransform analysis. We now consider he deerminaion of he fair CDS spread c. Consider a CDS conrac wih uni noional o be iniiaed a ime and expire a ime T, he presen value of he premium leg L P (, T can be formulaed as L P (, T = E Q [ T ( c exp s r (X u + h (X u du ds G. On he oher hand, he corresponding presen value of he proecion leg L R (, T is given by L R (, T = (1 w E Q [ T ( h (X s exp s r (X u + h (X u du ds G, where w is he loss given defaul of he risky bond (assumed o be fixed and known. Equaing 13

14 he wo paymen legs yields he CDS spread as follows: ( h (X s exp [ T E Q c = (1 w [ T E Q exp ( s s r (X u + h (X u du ds G. (17 r (X u + h (X u du ds G When he defaul inensiy assumes he consan value λ, we recover he obvious while simple resul: c = (1 w λ. By inerchanging he order of aking expecaion and performing inegraion, he denominaor and numeraor in eq. (17 can be simplified as and E Q [ T E Q [ T ( exp ( h (X s exp s s r (X u + h (X u du ds G = r (X u + h (X u du ds G = respecively. Hence, he CDS spread can be expressed as T T D (, s ds, [ B f (, s S (, s ds, s [ T 1 T [ c = (1 w D (, s ds B f (, s S (, s ds, (18 s which can be compued direcly from he risk adjused discoun facor and survival probabiliy. 4 Numerical valuaion of defaulable opions and impac of jump-o-defaul feaure In he las wo secions, we presened he general mahemaical formulaion of equiy-credi modeling under affine-diffusion models wih he jump-o-defaul feaure and demonsraed he pricing of defaulable European opions and credi defaul swaps using he proposed equiycredi formulaion. In his secion, we discuss he pracical implemenaion of he jump-odefaul feaure o several popular opion pricing models. Our choices of he sochasic sock price dynamics models include Heson s sochasic volailiy model (Heson, 1993, Meron s jump-diffusion model (Meron, 1976, and Kou s double exponenial jump model (Kou, Firs, we briefly review he mahemaical formulaion of each of hese popular sock price dynamics models and show ha hey are all nesed under he sochasic volailiy jump-diffusion model (SVJ. We hen demonsrae how o find he closed form formula of he characerisic funcion of he SVJ model wih a specified se of parameer funcions. Nex, we repor he numerical experimens ha were performed on valuaion of he prices of defaulable European 14

15 opions using various numerical approaches, namely, (i (ii direc numerical inegraion of he Fourier inegral represenaion of he price funcion; Fas Fourier ransform algorihm of invering he Fourier ransform; (iii Mone Carlo simulaion of he erminal sock price and compuaion of he sampled aver- -aged discouned expecaion of he erminal payoff. Besides comparing opion prices wih varying moneyness (raio of srike price o sock price and mauriies under differen sock price dynamics models, i is also insrucive o compare he implied volailiy values and examine he naure of implied volailiy smile paerns under various moneyness condiions. 4.1 Sochasic dynamics of sochasic volailiy and jump-diffusion models wih jump-o-defaul We presen he sochasic differenial equaions ha govern he sock price dynamics under Heson s sochasic volailiy model. Also, we derive he momen generaing funcions of Meron s Gaussian jump-diffusion model and Kou s exponenial jump-diffusion model. The sochasic volailiy (wih no price jumps and jump-diffusion models (wih non-sochasic volailiy are acually nesed by he SVJ models since hey can be recovered from he SVJ model by swiching off he jump componen and sochasic volailiy componen, respecively. We now derive he closed form represenaion of he characerisic funcion of he SVJ model wih jump-o-defaul. Heson s sochasic volailiy model Le x( = ln S( be he logarihm of he sock price S(. Heson s sochasic volailiy model is specified by dx ( = [r( 12 ν ( + h ( d + ν ( dw s, The pre-defaul dynamics of dν ( = κ [θ ν ( d + σ ν ν ( dwν, (19 where he insananeous variance ν ( is assumed o follow a square-roo process wih a meanreversion level θ, mean-reversion speed κ, and volailiy of volailiy σ ν. The correlaion beween he price process and insananeous variance is given by ρ, where E[dW s dw ν = ρ d. The nice analyical racabiliy of he sochasic volailiy model wih jump-o-defaul prevails provided ha he generalizaion of he hazard rae h ( is sae dependen on he insananeous variance ν (. 15

16 As a remark, we consider he degenerae case where he variance ν ( is aken o be consan. This reduces o he sandard Black-Scholes formulaion, excep wih he inclusion of jump-o-defaul. Le σ be he consan volailiy, h be he consan hazard rae and r be he consan ineres rae. The pre-defaul dynamics of he sock price is given by dx ( = The call price formula can be obained as follows: where K is he srike price, and (r σ2 2 + h d + σ dw s. (20 c (S, τ = SN (d 1 Ke (r+hτ N (d 2, τ = T, (21 ( ln S + r + h + σ2 τ K 2 d 1 = σ, d 2 = d 1 σ τ. τ This call price formula is simply he sandard Black-Scholes opion price formula wih he replacemen of he riskfree ineres rae r by he defaul risk adjused discoun rae r + h. Sochasic sock price models wih jumps To incorporae he sock price jumps prior he arrival of jump-o-defaul, we may modify he sock price dynamics as dx ( = [r( 12 ν ( λm + h( d + ν ( dw s + q dn (, (22 where he jump in sock price is modeled by he couning process dn ( wih inensiy λ and jump size q. Here, he compensaor m = E Q (q 1 is he expeced jump size which renders he discouned sock price process o be a maringale under he equivalen maringale measure Q. The wo popular disribuion specificaions of he jump size q are he Gaussian disribuion (Meron model and he double exponenial disribuion (Kou model. Gaussian jump disribuion (Meron model The Gaussian jump disribuion is a normal disribuion wih mean µ j and volailiy σ j so ha q N (µ j, σ j. The momen generaing funcion of he Gaussian jump disribuion is given by M M (ζ = E [ Q e ζq ( = exp µ j ζ + σ2 j 2 ζ2. (23 16

17 The corresponding jump compensaor is given by ( m = exp µ j + σ2 j 1. 2 Double exponenial jump disribuion (Kou model The double exponenial jump has he asymmeric densiy funcion: f (q = pη 1 e η 1q 1 {q>0} + (1 p η 2 e η 2q 1 {q<0}, where η 1 > 1, η 2 > 0 and 0 < p < 1. The probabiliy of an upward jump and a downward jump are given by p and 1 p, respecively. Empirically, we usually have p < 1/2 such ha he jump is asymmeric wih bias o downward jumps. The momen generaing funcion of he double exponenial jump disribuion is given by M K (ζ = E Q [ e ζq = p η 1 η 1 ζ + (1 p η 2 η 2 + ζ, (24 which is well defined for η 1 < ζ < η 2. The corresponding jump compensaor is given by m = p η 1 η (1 p η 2 η In hese earlier versions of jump-diffusion models, he volailiy is assumed o be consan. One may combine all he feaures of sochasic volailiy, price jumps and jump-o-defaul in he generalized SVJ model wih jump-o-defaul. We demonsrae how o find he closed form analyic represenaion of he characerisic funcion of he SVJ model wih jump-o-defaul, whose dynamics is specified as follows: dx = µ (X d + σ (X dw + dz, wih he sae variables X = (x, ν T. The parameer funcions are defined by µ (X = σ (X = dz = ( r 1ν 2 λm + h, κ (θ ν ( ν 0 ρσ ν ν σ ν 1 ρ 2 ν ( q dn. 0, 17

18 4.2 Characerisic funcions We would like o demonsrae how o derive he closed form represenaion of he characerisic funcion of he SVJ model wih jump-o-defaul. According o he governing equaion of he AJD process as depiced in eqs. 1(a,b, we se he coefficien funcions o be ( ( r λm + h µ (X = + X κθ 0 κ ( σ (X σ (X T ν ρσ ν ν =. ρσ ν ν σνν 2 Taking he ineres rae r and hazard rae h o be consan, he characerisic funcion wih u = (u 0, u 1, u 2 T is given by Ψ (u, X, ; T = E Q R (X s ds e u X T G = exp [α (τ + β 1 (τ x + β 2 (τ ν, τ = T. (25 The ime dependen coefficien funcions, α (τ and β (τ = (β 1 (τ, β 2 (τ T, are obained by solving he following sysem of ODEs dα (τ dτ dβ 1 (τ dτ dβ 2 (τ dτ = (r λm + h β 1 (τ + κθβ 2 (τ + λ [Λ (β (τ 1 h r, = 0, = 1 2 σ2 νβ 2 2 (τ + ρσ ν β 1 (τ β 2 (τ κβ 2 (τ β2 1 (τ 1 2 β 1 (τ, (26 wih iniial condiions: α (0 = u 0 and β (0 = (u 1, u 2 T. Since here are only jumps in sock price, he jump ransform Λ (β (τ = exp (β (τ z dν (z = f (q exp (β 1 (τ q dq R n can be explicily compued as: (i Λ (β (τ = M M (β 1 (τ under he Gaussian jump-diffusion model; (ii Λ (β (τ = M K (β 1 (τ under he double exponenial jump model. I is seen ha β 1 (τ admis a rivial soluion of β 1 (τ = u 1. As a resul, he ODE governing β 2 (τ can be recased as a Riccai equaion dβ 2 (τ dτ = b 0 + b 1 β 2 (τ + b 2 β 2 2 (τ, 18

19 where b 0 = 1 2 u 1 (u 1 1, b 1 = ρσ ν u 1 κ, b 2 = 1 2 σ2 ν, wih iniial condiion: β 2 (0 = u 2. The sysem of ODEs has he following explici soluion: α (τ = u 0 + {(r λm + h u 1 + λ [Λ (u 1 1 h r} τ [ + κθ r τ 1 ( 1 g exp ( τd ln, b 2 1 g β 1 (τ = u 1, β 2 (τ = r 1 g exp ( τd 1 g exp ( τd, (27 where r ± = 1 ( b 1 ± d, g = r u 2, g = r + g, d = b 2 1 4b 0 b 2. 2b 2 r + u 2 r Remark When he ODEs governing α (τ and β j (τ, j = 1, 2, have coupled or non-linear erms, we may no have closed form soluion. One hen has o resor o numerical mehod such as he fourh-order Runga-Kua mehod o solve he sysem of ODEs. Neverheless, he characerisic funcion perserves he simple exponenial expression wih argumen which is a linear combinaion of he soluion of he sysem of ODEs. 4.3 Numerical valuaion of defaulable European opions Firs, we presen numerical calculaions on pricing of defaulable European opions under he JD-AJD model using differen approaches of evaluaing he Fourier inegral of he opion price funcion. We also examine he impac of he jump-o-defaul feaure on he values of he defaulable European opions and he corresponding implied volailiy smile paerns. In our numerical calculaions, he model parameers are specified as: ρ = 0.3, κ = 5, θ = 0.12, σ ν = 0.2, ν 0 = 0.09, λ = 0.5, h = The model parameers are chosen based on he following assumpions. The sochasic volailiy has a mean-reversion speed wih a half-life of 0.2 year under a moderaely upward sloping erm srucure. The hazard rae of 0.02 implies an CDS spread of around bps. For he sock price jump, we assume he arrival inensiy o be 0.5 and he jump sizes are assumed o be eiher he Meron jump or double exponenial jump. The values of he jump parameers are aken o be: Meron jump: µ j = 0.12, σ j = 0.15; Double exponenial jump: p = 0.25, η 1 = 8 and η 2 = 6. The above parameer values are chosen o be similar o hose in Broadie and Kaya (2006. Also, we assume consan ineres rae o be 2% and he spo sock price o be 100 (unless 19

20 oherwise specified. Tables 1 and 2 presen he comparison of numerical esimaes of defaulable European call opion prices a varying mauriies: 3-monh ( = 0.25, 6-monh ( = 0.5, 1-year ( = 1 and 2-year ( = 2 wih he Meron jump and double exponenial jump, respecively. For he adapive numerical inegraion approach, we use he Gauss-Kronrod quadraure (wih relaive olerance of 10 8 o compue he Fourier inegral. In he Mone Carlo simulaion calculaions, we apply he Euler scheme for he numerical simulaion of he log sock price and sochasic volailiy processes, where a reflecing boundary is imposed for he laer. In order o achieve high accuracy, he simulaion is repeaed 1,000,000 imes and he ime sep is kep a As shown in Tables 1 and 2, he numerical opion prices obained from valuaion of he exac characerisic funcion using adapive numerical inegraion are very close o he Mone Carlo esimaes (sandard deviaion of he Mone Carlo simulaion is also repored alongside. Since he jump-o-defaul feaure leads o all-or-nohing in he erminal payoff, he Mone Carlo esimaes are seen o be less accurae, in paricular for long-erm opions. As an alernaive numerical approach, we use he FFT echnique o inver he Fourier inegral and use linear inerpolaion o obain he opion value a he desired srike price. We follow Carr and Madan (1999 o choose he number of grids for he discree Fourier ransform o be N F F T = 4000 and he grid size o be ω = This implies a runcaion a α = N F F T ω = 1, 000. For he FFT implemenaion, he esimaes are consisen wih he esimaes using numerical inegraion quadraure up o 4 decimal places. To analyse he impac of he FFT parameers on numerical accuracy of he opion value calculaions, we vary he number of grids in he Fourier domain (N F F T as 250, 1000, 4000 and 16000, and he grid size ( ω as 0.05, 0.1, 0.25 and 0.5. The oal number of operaions for he FFT algorihm is N F F T log 2 (N F F T, so he same number of grids in he Fourier domain implies he same operaions coun. For each level of compuaional budge, he differen se of FFT parameers produce differen degrees of runcaion error (conrolled by α, discreizaion 2π error (conrolled by ω, and inerpolaion error (conrolled by k =. Table 3 N F F T ω repors he opion price esimaes using he same model parameers as in Table 2 bu wih differen FFT parameers. I is ineresing o noe ha accuracy of he esimaes depends on he rade-off in minimizing he differen sources of errors. Suppose we would like o achieve higher accuracy in compuing he Fourier inegral by is discree approximaion and se he grid size ω o be small (say, ω = 0.05, hen a large number of grids in he Fourier domain is needed o avoid he runcaion error. However, such a small grid size in he Fourier domain inroduces a wide dispersion of he log-srike grids, and leads o larger inerpolaion error. Table 3 shows ha a significan numerical error appears wih ω = Conversely, if we aemp o minimize he inerpolaion error by seing he spacing ω o be large (say, ω = 0.5, he opion price esimae suffers from he discreizaion error in approximaing 20

21 he Fourier inegral. Forunaely, one can use beer numerical inegraion scheme o minimize he impac of discreizaion error. Indeed, when a higher order inegraion scheme is employed (say, Simpson s rule, he discreizaion is negligible even when he spacing is se a 0.5 or higher. Our resuls are consisen wih he observaion in Carr and Madan (1999. As a conclusion, i is desirable o have a coarse grid size in he Fourier domain in order o minimize he runcaion and inerpolaion errors, and adop a higher order quadraure o approximae he Fourier inegral. Implied volailiy smile paerns wih jump-o-defaul I is more convenien o use pu opions o examine he impac of jump-o-defaul. Empirical evidence shows ha he ou-of-he-money (OTM pu opions are more closely linked o he yield spreads and CDS spreads of he underlying firm. Also, he corresponding skew in he implied volailiy smile is srongly correlaed wih he defaul risk of he firm. Figure 1 shows he implied volailiy smile paerns of defaulable European pu opions under differen jumpo-defaul (JD models a varying mauriies of 3 monhs, 6 monhs, 1 year and 2 years. We consider he following jump-o-defaul models: (i Black-Scholes model wih JD (BS-JD model; (ii Heson model wih JD (SV-JD model; and (iii SVJ model wih JD (SVJ-JD model. We also show he fla Black-Scholes implied volailiy as benchmark for he implied volailiy smile generaed by hese models. A higher implied volailiy indicaes a higher pu opion price, so a higher premium o buy he downside proecion. I can be seen ha he presence of jump-o-defaul significanly increases he implied volailiy of he deep OTM pu opions and produces a srongly skewed implied volailiy smile. The jump-o-defaul feaure adds o he implied volailiy smile wih magniude varying from 10 o 30 volailiy poins as he moneyness moves from 0.8 and 0.6. This is consisen wih he empirical evidence ha he implied volailiy of a deep OTM pu opion is srongly correlaed wih he defaul risk of he underlying firm. In paricular, i is imporan o noe ha for shor-erm opions, he implied volailiy generaed by differen JD models converge as he moneyness goes deep ou-of-he-money, excep for he SVJ-JD model which produces a slighly differen paern. Wih he possibiliy of jump-o-defaul, he price of a shor-erm deep OTM opion is primarily deermined by defaul risk while he diffusion sock price dynamics is of less imporance. Given he possibiliy of jump-o-defaul, he skew in he implied volailiy is more persisence for long-erm opions, a feaure ha canno be produced by pure sochasic volailiy models as heir asympoic implied volailiy smile becomes fla as ime-o-mauriy increases since he variance revers o is long-run mean. I is also worh o noe ha he jump-o-defaul feaure inroduces a noable shif in he implied volailiy smile for he long-erm opions (by 10 o 15 volailiy poins for he BS-JD model. This suggess ha he volailiy risk premium 21

22 (ne amoun ha he implied volailiy value exceeds he hisorical volailiy value embedded in he long-erm opions is largely aribued o he defaul risk of he underlying firm. 5 Conclusion The join modeling of equiy risk and credi exposure is imporan in any sae-of-he-ar opion pricing models of credi-sensiive equiy derivaives. Our proposed equiy-credi models aemp o perform pricing of equiy and credi derivaives under a unified framework. We have demonsraed he robusness of adding he jump-o-defaul feaure in he popular affine jump-diffusion models for pricing defaulable European claims and credi defaul swaps. By assuming he hazard rae o be affine, analyic racabiliy in ypical affine jump-diffusion models is mainained even wih he inclusion of he jump-o-defaul feaure. Once he analyic formula is available for he characerisic funcion of he join equiy-credi price dynamics, numerical valuaion of he derivaive prices can be performed easily using a sandard numerical inegraion quadraure or Fas Fourier ransform algorihm. Our numerical experimens showed ha accuracy of he Fas Fourier ransform algorihm may deeriorae for shor-mauriy opions. Also, volailiy skew effecs may be significan for shor-mauriy deep-ou-of-he-money pus under he join modeling of equiy and credi risks. This may be aribued o he observaion ha he price of a shor-erm deep-ou-of-he-money pu may be more sensiive o defaul risk. As fuure research works, one may consider pricing of credi-sensiive exoic equiy derivaives, like variance swap producs wih defaul cap feaure. ACKNOWLEDGEMENT This work was suppored by he Hong Kong Research Grans Council under Projec of he General Research Funds. REFERENCES Acharya, V.V. and Johnson, T.C. Insider Trading in Credi Derivaives. Journal of Finance Economics 84 (2007: Bakshi, G., Cao, C. and Chen, Z. Empirical Performance of Alernaive Opion Pricing Models. Journal of Finance 52(5 (1997:

23 Baes, D.S. Jumps and Sochasic Volailiy: Exchange Rae Processes Implici in Deusche Mark Opions. Review of Financial Sudies 9(1 (1996: Broadie, M. and Kaya, O. Exac Simulaion of Sochasic Volailiy and Oher Affine Jump Diffusion Processes. Operaions Research 54(2, (2006: Bayrakar, E. and Yang, B.A. Unified Framework for Pricing Credi and Equiy Derivaives. To appear in Mahemaical Finance (2011. Cao, C., Yu, F. and Zhong, Z. The Informaion Conen of Opion-Implied Volailiy for Credi Defaul Swap Valuaion. Journal of Financial Markes 13(3 (2010: Carr, P. and Linesky, V. A Jump o Defaul exended CEV models: An applicaion of Bessel Processes. Finance and Sochasics 10 (2006: Carr, P. and Wu, L. Sock Opions and Credi Defaul Swaps: A Join Framework for Valuaion and Esimaion. Journal of Financial Economerics (2009: Carr, P. and Madan, D.B. Opion Valuaion using he Fas Fourier Transform. Journal of Compuaional Finance 2(4 (1999: Carr, P. and Madan, D.B. Local Volailiy Enhanced by a Jump o Defaul. SIAM Journal of Financial Mahemaics 1 (2010: Cheridio, P. and Wugaler, A. Pricing and Hedging in Affine Models wih Possibiliy of Defaul. Working paper of Princeon Universiy (2011. Cremers, M., Driessen, J., Maenhou, P. and Weinbaum, D. Individual Sock-Opion Prices and Credi Spreads. Journal of Banking and Finance 32 (2008: Duffie, D. and Singleon, K. Modeling Term Srucures of Defaulable Bonds. Financial Sudies 12 (1999: Review of Duffie, D., Pan, J. and Singleon, K. Transform Analysis and Asse Pricing for Affine Jump- Diffusions. Economerica 68(6 (2000: Erakar, B. Do Sock Prices and Volailiy Jump? Reconciling Evidence from Spo and Opion Prices. Journal of Finance 59 (3 (2004: Heson, S.A. Closed-form Soluion for Opions wih Sochasic Volailiy wih Applicaion o Bond and Currency Opions. Review of Financial Sudies 6(2 (1993: Kou, S.G. A Jump-Diffusion Model for Opion Pricing. Managemen Science (48 (2002: Lando, D. On Cox Processes and Credi Risky Securiies. Review of Derivaives Research 2 (1998:

24 Lord, R. and Kahl, C. Opimal Fourier Inversion in Semi-analyical Opion Pricing. Journal of Compuaional Finance 10(4 (2007: Meron, R.C. Opion Pricing when Underlying Sock Reurns are Disconinuous. Journal of Financial Economics 3 (1976: Mendoza-Arriaga, R., Carr, P. and Linesky, V. Time-changed Markov Processes in Unified Credi-Equiy Modeling. Mahemaical Finance 20(4 (2010: Pan, J. The Jump-Risk Premia Implici in Opions: Evidence from an Inegraed Time-Series Sudy. Journal of Financial Economics 63 (2002: Zhang, B.Y., Zhou, H. and Zhu, H. Explaining Credi Defaul Swap Spreads wih he Equiy Volailiy and Jump Risks of Individual Firms. Review of Financial Sudies 22(12 (2009:

25 Time-o-Mauriy = 0.25 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.015 Time-o-Mauriy = 0.5 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.022 Time-o-Mauriy = 1 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.032 Time-o-Mauriy = 2 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.047 Table 1: Comparison of numerical esimaes of European call opion prices under he Meron jump model using various numerical approaches. 25

26 Time-o-Mauriy = 0.25 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.016 Time-o-Mauriy = 0.5 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.022 Time-o-Mauriy = 1 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.032 Time-o-Mauriy = 2 Spo price G-K quadraure FFT Mone Carlo (sandard deviaion ( ( ( ( (0.049 Table 2: Comparison of numerical esimaes of European call opion prices under he double exponenial jump model using various approaches. 26

27 Time-o-Mauriy = 0.25 ω 0.5 N F F T G-K quadraure Time-o-Mauriy = 0.5 ω 0.5 N F F T G-K quadraure Time-o-Mauriy = 1.0 ω 0.5 N F F T G-K quadraure Time-o-Mauriy = 2.0 ω 0.5 N F F T G-K quadraure Table 3: Comparison of numerical esimaes of European call opion prices wih double exponenial jump using he FFT algorihm wih varying values of N F F T (number of grids and ω (grid size. The numerical esimaes of opion prices obained from he Gauss-Kronrod inegraion quadraure are used as benchmark values for comparison. The runcaion in he Fourier domain is given by α = N F F T ω. The corresponding spacing in he log-srike deermined by he relaion: k ω = 2π/N F F T. 27

28 Implied Volailiy (% monh pu opions BS BS + JD SV + JD SVJ + JD Implied Volailiy (% monh pu opions BS BS + JD SV + JD SVJ + JD Moneyness (K/S Moneyness (K/S Implied Volailiy (% year pu opions BS BS + JD SV + JD SVJ + JD Implied Volailiy (% year pu opions BS BS + JD SV + JD SVJ + JD Moneyness (K/S Moneyness (K/S Figure 1: Implied volailiy smile paerns of defaulable European pu opions under various jump-o-defaul models a varying mauriies. The implied volailiy values are obained by invering he Black-Scholes formula from he model opion prices. 28