A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES. ERHAN BAYRAKTAR University of Michigan. BO YANG Morgan Stanley

Transcription

1 Mahemaical Finance, Vol. 1, No. 3 July 11, A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES ERHAN BAYRAKTAR Universiy of Michigan BO YANG Morgan Sanley We propose a model which can be joinly calibraed o he corporae bond erm srucure and equiy opion volailiy surface of he same company. Our purpose is o obain explici bond and equiy opion pricing formulas ha can be calibraed o find a risk neural model ha maches a se of observed marke prices. This risk neural model can hen be used o price more exoic, illiquid, or over-he-couner derivaives. We observe ha our model maches he equiy opion implied volailiy surface well since we properly accoun for he defaul risk in he implied volailiy surface. We demonsrae he imporance of accouning for he defaul risk and sochasic ineres rae in equiy opion pricing by comparing our resuls o Fouque e al., which only accouns for sochasic volailiy. KEY WORDS: defaulable bond, defaulable sock, equiy opions, sochasic ineres rae, implied volailiy, muliscale perurbaion mehod. 1. INTRODUCTION Our purpose is o build an inensiy-based modeling framework ha can be joinly calibraed o corporae bond prices and sock opions, and can be used o price more exoic derivaives. The same company has socks, sock opions, bonds, and several oher derivaives. When his company defauls, he payoffs of all of hese insrumens are affeced; herefore, heir prices all conain informaion abou he defaul risk of he company. In our framework, we use he Vasicek model for he ineres rae, and use doubly sochasic Poisson process o model defaul. We assume ha he bonds have recovery of marke value and ha socks become valueless a he ime of defaul. Using he muliscale modeling approach of Fouque e al. 3, we obain explici bond pricing equaion wih hree free parameers which we calibrae o he corporae bond erm srucure. On he oher hand, sock opion pricing formula conains seven parameers, hree of which are common wih he bond opion pricing formula. The common parameers are muliplied The firs auhor is suppored in par by he Naional Science Foundaion under an applied mahemaics research gran and a Career gran, DMS-9657 and DMS , respecively, and in par by he Susan M. Smih Professorship. The auhors hank he wo anonymous referees and he anonymous AE for heir consrucive commens, which helped us improve our paper. Any opinions, findings, and conclusions or recommendaions expressed in his maerial are hose of he auhors and do no necessarily reflec hose of he Naional Science Foundaion or Morgan Sanley. Manuscrip received December 7; final revision received June 9. Address correspondence o Erhan Bayrakar, Deparmen of Mahemaics, Universiy of Michigan, Ann Arbor, MI 4819, USA; erhan@umich.edu DOI: /j x C 1 Wiley Periodicals, Inc. 493

2 494 E. BAYRAKTAR AND B. YANG wih he loss rae in he bond pricing formula. The parameers ha remain unknown afer our calibraion o he corporae yield curve are deermined by calibraing hem o he sock opion implied volailiy surface. The calibraion resuls reveal ha our hybrid model is able o produce implied volailiy surfaces ha mach he daa closely. We compare he implied volailiy surfaces ha our model produces o hose of Fouque e al. 3. We see ha even for longer mauriies our model has a prominen skew: compare Figures A.1 and A. in Appendix A. Even when we ignore he sochasic volailiy effecs, our model fis he implied volailiy of he Ford Moor Company well and performs beer han he model of Fouque e al. 3; see Figure A.3 in Appendix A. Observe ha when we ignore he sochasic volailiy our model has one less parameer o calibrae han ha of Fouque e al. 3. This poins o he imporance of accouning for he defaul risk for companies wih low raings. Our model has hree building blocks: 1 We model he defaul even using he muliscale sochasic inensiy model of Papageorgiou and Sircar 8. We also model he ineres rae using an Ornsein Uhlenbeck process Vasicek model. As i was demonsraed in Papageorgiou and Sircar 8, hese modeling assumpions are effecive in capuring he corporae yield curve; We assume he sock price process jumps o zero when he company defauls. This sock price model was considered in Bayrakar 8. Our model specificaion for he sock price differs from he jump o defaul models for he sock price considered by Carr and Linesky 6 and Linesky 6, which ake he defaul inensiy o be funcions of he sock price; 3 We also accoun for he sochasic volailiy in he modeling of he socks because even he index opions when here is no risk of defaul possess implied volailiy skew. We model he volailiy using he fas scale sochasic volailiy model of Fouque, Papanicolaou, and Sircar. We demonsrae on index opions when here is no risk of defaul ha we mach he performance of he wo imescale volailiy model of Fouque e al. 3 see Secion 4.4. The laer model exends Fouque, Papanicolaou, and Sircar by including a slow facor in he volailiy o ge a beer fi o longer mauriy opions. We see from Secion 4.4 ha when one assumes he ineres rae o be sochasic, he calibraion performance of he sochasic volailiy model wih only he fas facor is as good as he wo-scale sochasic volailiy model. This is why we choose he volailiy o be driven by only he fas facor. Even hough ineres rae is sochasic we are able o obain explici asympoic pricing formulas for sock opions. Thanks o hese explici pricing formulas, he inverse problem ha we face in calibraing our model o he corporae bond and sock daa can be solved wih considerable ease. Our modeling framework can be hough of as a hybrid of he models of Fouque, Papanicolaou, and Sircar, which only considers pricing opions in a sochasic volailiy model wih consan ineres rae, and Papageorgiou and Sircar 8, which only considers a framework for pricing derivaives on bonds. Neiher of hese models has he means o ransfer defaul informaion from bond marke o equiy markes and vice versa, which we are se o do in his paper. We should noe ha our model also akes he reasury yield curve, hisorical sock prices, and hisorical spo rae daa o esimae some of is parameers see Secion 4. Our model exends Bayrakar 8 by aking he ineres rae process o be sochasic, which leads o a richer heory and more calibraion parameers, and herefore, beer fi o daa: i When he ineres rae is deerminisic he corporae bond pricing formula urns ou o be very crude and does no fi he bond erm srucure well compare.57 in Bayrakar 8 and 4.1; ii Wih deerminisic ineres raes he bond pricing and he sock opion pricing formulas share only one common erm, he average inensiy of defaul his parameer is muliplied by he loss rae in he bond pricing equaion, under

3 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 495 our loss assumpions. Therefore, he effec of he defaul risk is no accouned for in he implied volailiy surface as much as i should be. And our calibraion analysis demonsraes ha his has a significan impac. When he volailiy is aken o be a consan, boh our new model and he model in Bayrakar 8 have hree free parameers. The model in Bayrakar 8 produces a below par fi o he implied volailiy surface see, e.g., figure 5 in ha paper, whereas our model produces an excellen fi see Secion 4.3 and Figure A.1. The oher defaulable sock models are hose of Carr and Linesky 6, Linesky 6, and Carr and Wu 6, which assume ha he ineres rae is deerminisic. Carr and Linesky 6 and Linesky 6 ake he volailiy and he inensiy o be funcions of he sock price and obain a one-dimensional diffusion for he predefaul sock price evoluion. Using he fac ha he resolvens of paricular Markov processes can be compued explicily, hey obain pricing formulas for sock opion prices. On he oher hand, Carr and Wu 6 uses a Cox-Ingersoll-Ross CIR sochasic volailiy model. This paper also models he inensiy o be a funcion of he volailiy and anoher endogenous CIR facor. The opion prices in his framework are compued numerically using he inverse Fourier ransform. We, on he oher hand, use asympoic expansions o provide explici pricing formulas for sock opions in a framework ha combines a he Vasicek ineres rae model, b fas-mean revering sochasic volailiy model, c defaulable sock price model, and d muliscale sochasic inensiy model. Our calibraion exercise differs from ha of Carr and Wu 6 because hey perform a ime series analysis o obain he parameers of he underlying facors from he sock opion prices and credi defaul swap [CDS] spread ime series, whereas we calibrae our pricing parameers o he daily implied volailiy surface and bond erm srucure daa. Our purpose is o find a risk neural model ha maches a se of observed marke prices. This risk neural model can hen be used o price more exoic, illiquid, or over-he-couner derivaives. For furher discussion of his calibraion mehodology, we refer o Con and Tankov 4 see chaper 13, Fouque, Papanicolaou, and Sircar, Fouque e al. 3, and Papageorgiou and Sircar 8. The res of he paper is organized as follows: In Secion, we inroduce our modeling framework. We also describe how he CDS spread can be compued in our framework. In Secion 3, we inroduce he asympoic expansion mehod. We obain explici asympoic prices for bonds and equiy opions in Secion 3.3. In Secion 4, we describe he calibraion of our parameers and discuss our empirical resuls. Finally, figures show our calibraion resuls.. A FRAMEWORK FOR PRICING EQUITY AND CREDIT DERIVATIVES.1. The Model Le,H, P be a complee probabiliy space supporing i correlaed sandard Brownian moions W = W, W 1, W, W 3, W 4,, wih.1 E [ W, W i ] = ρi, E [ W i, W j ] = ρij, i, j {1,, 3, 4},, where ρ i,ρ ij 1, 1 are such ha he correlaion marix is posiive definie, and ii a Poisson process N independen of W. Le us inroduce he Cox process ime-changed

4 496 E. BAYRAKTAR AND B. YANG Poisson process Ñ N λ s ds,, where. λ = f Y, Z, dy = 1 ɛ m Y d + ν ɛ dw, Y = y, dz = δcz d + δgz dw 3, Z = z, in which ɛ, δ are small posiive consans and f is a sricly posiive, bounded, smooh funcion. We also assume ha he funcions c and g saisfy Lipschiz coninuiy and growh condiions so ha he diffusion process for Z has a unique srong soluion. We model he ime of defaul as.3 τ = inf{ :Ñ = 1}. We also ake ineres rae o be sochasic and model i as an Ornsein Uhlenbeck process.4 dr = α βr d + η dw 1, r = r, for posiive consans α, β,andη. We model he sock price as he soluion of he sochasic differenial equaion τ.5 d X = X r d + σ dw d Ñ λ u du, X = x, where he volailiy is sochasic and is defined hrough σ = σ 1 Ỹ ; dỹ = ɛ m Ỹ ν Ỹ d + ν.6 dw 4 ɛ ɛ, Ỹ = ỹ. Here, is a smooh, bounded funcion of one variable which represens he marke price of volailiy risk. The funcion σ is also a bounded, smooh funcion. Noe ha he discouned sock price is a maringale under he measure P, and a he ime of defaul, he sock price jumps down o zero. The pre-bankrupcy sock price coincides wih he soluion of.7 dx = r + λ X d + σ X dw, X = x. I will be useful o keep rack of differen flows of informaion. Le F ={F, } be he naural filraion of W. Denoe he defaul indicaor process by I = 1 {τ },, and le I ={I, } be he filraion generaed by I. Finally, le G ={G, } be an enlargemen of F such ha G = F I,. Because we will ake ɛ and δ o be small posiive consans, he processes Y and Ỹ are fas mean revering, and Z evolves on a slower ime scale. See Fouque e al. 3 for an exposiion and moivaion of muliscale modeling in he conex of sochasic volailiy models. We noe ha our specificaion of he inensiy of defaul coincides wih ha of Papageorgiou and Sircar 8, who considered only a framework for pricing credi derivaives. Our sock price specificaion is similar o ha of Linesky 6 and Carr and Linesky 6 who considered a framework for only pricing equiy opions on defaulable socks. Our volailiy specificaion, on he oher hand, is in he spiri of Fouque, Papanicolaou, and Sircar.

5 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 497 Bayrakar 8 considered a similar modeling framework o he one considered here, bu he ineres rae was aken o be deerminisic. In his paper, by exending his modeling framework o incorporae sochasic ineres raes, we are able o consisenly price credi and equiy derivaives and produce more realisic yield curves and implied volailiy surfaces... Equiy and Credi Derivaives In our framework, we will price European opions and bonds of he same company in a consisen way. 1. The price of a European call opion wih mauriy T and srike price K is given by.8 [ C; T, K = E exp [ = 1 {τ>} E exp ] r s ds X T K + 1 {τ>t} G ] r s + λ s ds X T K + F, in which he equaliy follows from Lemma 5.1. of Bielecki and Rukowski. This lemma, les us wrie a condiional expecaion wih respec o G in erms of condiional expecaions wih respec o F.. Also, see Linesky 6 and Carr and Linesky 6 for a similar compuaion. On he oher hand, he price of a pu opion wih he same mauriy and srike price is.9 Pu; T = E [ exp [ + E exp [ = 1 {τ>} E exp [ + KE exp ] r s ds K X T + 1 {τ>t} G ] r s ds K1 G {τ T} ] r s + λ s ds K X T + F ] r s ds F [ KE exp ] r s + λ s ds F.. Consider a defaulable bond wih mauriy T and par value of 1 dollar. We assume he recovery of he marke value, inroduced by Duffie and Singleon In his model, if he issuer company defauls prior o mauriy, he holder of he bond

6 498 E. BAYRAKTAR AND B. YANG recovers a consan fracion 1 l of he pre-defaul value, wih l [, 1]. The price of such a bond is.1 B c ; T [ = E exp [ = E exp τ r s ds 1 {τ>t} + exp r s ds ] r s + l λ s ds F, ] 1 {τ T} 1 lb c τ ; T G on {τ >}, see Duffie and Singleon 1999 and Schönbucher In Secion 3, we will obain explici pricing formulas for equiy opions and bonds. These formulas will be calibraed o he observed prices. Once our model is calibraed we can hen deermine he prices of more exoic derivaives. As an example, below we will show how a CDS conrac can be priced in our framework. Consider a CDS wrien on B c, which is an insurance agains losses incurred upon defaul from holding a corporae bond. The proecion buyer pays a fixed premium, he so-called CDS spread, o he proecion seller. The premium is paid on fixed daes T = T 1,...,T M, wih T M being he mauriy of he CDS conrac. We denoe he CDS spread a ime by c ds ; T. Our purpose is o deermine a fair value for he CDS spread so ha wha he proecion buyer buyer is expeced o pay, he value of he premium leg of he conrac, is equal o wha he proecion seller is expeced o pay, he value of he proecion leg of he conrac. For a more deailed descripion of he CDS conrac, see Bielecki and Rukowski or Schönbucher 3. The presen value of he premium leg of he conrac is.11 Premium; T = c ds ; T E [ M = 1 {τ>} c ds ; T m=1 exp M m=1 m [ E exp r s ds m 1 {τ>tm } ] G ] r s + λ s ds F, in which we assumed ha < T 1. The presen value of he proecion leg of he conrac under our assumpion of recovery of marke value is assuming l [, 1.1 Proecion; T = 1 {τ>} E l = 1 {τ>} 1 l = 1 {τ>} l 1 l [ τ exp r s ds B c ; T M E ] 1 {τ TM }lb c τ ; T M G [ exp [ B c ; T M E exp M M r s ds 1 {τ>tm } ] G ] r s + λ s ds F,

7 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 499 in which he second equaliy follows from.1. Now, he CDS spread can be deermined, by seing Proecion; T = Premium; T and using equaions.11 and.1, as.13 c ds l ; T = 1 {τ>} 1 l [ B c ; T M E exp M m=1 [ E exp Noe ha when l = 1 [M.14 Proecion; T = 1 {τ>} E m M ] r s + λ s ds F ] when l [, 1. r s + λ s ds F λ u du exp M ] r s + λ s ds. Observe ha compuing c ds ; T requires he value of l an unobserved quaniy and he value of B c ; T M. This value may or may no be available from he bond price daa. If B c ; T M is no quoed, hen one has o consruc he yield curve o obain his value. Moreover, o compue c ds ; T we also need E[exp i r s + λ s ds F ], i {1,...,M}, which are no available. Because he loss rae may no be equal o 1 hese values canno be recovered direcly from he bond prices. In he nex secion, we will develop approximae pricing formulas for equiy opions and defaulable bonds. We will hen calibrae hese formulas o he opion and bond daa in Secion 4, and obain he value of he loss rae and oher model parameers. If we le B c, T; l denoe he approximaion for he price a ime of a defaulable bond ha maures a ime T, and has loss rae l see 4.1, hen he model-implied CDS spread wih mauriy T M can be obained as.15 cmodel ds, T M = l B c, T M ; l B c, T M ;1. 1 l M B c, T m ;1 Usually, o deermine he CDS spread, one assumes ha he bond has a recovery of face value. We, on he oher hand, use he recovery of marke value assumpion on he bond o deermine he value of he CDS spread. This is because we would like o firs calibrae our model o he bond prices, for which we made a recovery of marke value assumpion. Also, he simpliciy of he CDS spread formula under he recovery of marke value assumpion jusifies our choice. m=1 3. EXPLICIT PRICING FORMULAS FOR CREDIT AND EQUITY DERIVATIVES 3.1. Pricing Equaion 3.1 Le P ɛ,δ denoe P ɛ,δ, X, r, Y, Ỹ, Z = E [ exp ] r s + lλ s ds hx T F. When l = 1andhX T = X T K +, P ɛ,δ is he price of a call opion on a defaulable sock. On he oher hand, when hx T = 1, P ɛ,δ becomes he price of a defaulable bond.

8 5 E. BAYRAKTAR AND B. YANG Using he Feynman Kac formula, we can characerize P ɛ,δ as he soluion of 3. L ɛ,δ P ɛ,δ, x, r, y, ỹ, z =, P ɛ,δ T, x, r, y, ỹ, z = hx, where he parial differenial operaor L ɛ,δ is defined as 3.3 L ɛ,δ 1 ɛ L + 1 ɛ L 1 + L + δm 1 + δm + δ ɛ M 3, in which L ν + m y y y + ν + m ỹ ỹ ỹ + ρ 4vṽ y ỹ, L 1 ρ σ ỹν x x y + ρ 1ην y + ρ 4σ ỹ ν x x ỹ + ρ 14 η ν ỹ ỹ ν ỹ, L + 1 σ ỹx + r + f y, zx + α βr x x + σ ỹηρ 1 x x + 1 η r + lfy, z, M 1 σ ỹρ 3 gzx x z + ηρ 13gz z, M 3 ρ 3 ν gz y z + ρ 34 ν gz ỹ z. M cz z + 1 g z z, 3.. Asympoic Expansion We consruc an asympoic expansion for P ɛ,δ as ɛ, δ. Firs, we consider an expansion of P ɛ,δ in powers of δ 3.4 P ɛ,δ = P ɛ + δ P ɛ 1 + δ Pɛ +. By insering 3.4 ino 3. and comparing he δ and δ erms, we obain ha P ɛ saisfies 1 ɛ L L 1 + L P ɛ ɛ =, P ɛ T, x, r, y, ỹ, z = hx,

9 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 51 and ha P1 ɛ saisfies 1 ɛ L + 1 L 1 + L P M ɛ1 = ɛ M 3 P ɛ ɛ, P1 ɛ T, x, y, ỹ, z, r =. Nex, we expand he soluions of 3.5 and 3.6 in powers of ɛ 3.7 P ɛ = P + ɛ P 1, + ɛ P, + ɛ 3/ P 3, +, 3.8 P ɛ 1 = P,1 + ɛ P 1,1 + ɛ P,1 + ɛ 3/ P 3,1 +. Insering he expansion for P ɛ ino 3.5 and maching he 1/ɛ erms gives L P =. We choose P no o depend on y and ỹ because he oher soluions have exponenial growh a infiniy see, e.g., Fouque e al. 3. Similarly, by maching he 1/ ɛ erms in 3.5 we obain ha L P 1, + L 1 P =. Because L 1 akes derivaives only wih respec o y and ỹ, we observe ha L P 1, =. We choose P 1, no o depend on y and ỹ. Now equaing he order-one erms in he expansion of 3.5 and using he fac ha L 1 P 1, =, we ge ha 3.9 L P, + L P =, which is a Poisson equaion for P, see, e.g., Fouque, Papanicolaou, and Sircar. The solvabiliy condiion for his equaion requires ha 3.1 L P =, where denoes he averaging wih respec o he invarian disribuion of Y, Ỹ, whose densiy is given by 3.11 { [ y, ỹ = 1 πν ν exp 1 y m ]} ỹ m y mỹ m + ρ 1 ρ4 4. ν ν ν ν Le us denoe 3.1 σ 1 σ ỹ, σ σ ỹ, λz = f y, z. To demonsrae he effec of averaging on L, le us wrie 3.13 L := + 1 σ x x + r + λzx + α βr x Togeher wih he erminal condiion + σ 1 ηρ 1 x x + 1 η r + l λz 3.14 P T, x, r; z = hx,

10 5 E. BAYRAKTAR AND B. YANG equaion 3.1 defines he leading order erm P. On he oher hand from 3.9, we can also deduce ha 3.15 P, = L 1 L L P. Maching he ɛ order erms in he expansion of 3.5 yields 3.16 L P 3, + L 1 P, + L P 1, =, which is a Poisson equaion for P 3,. The solvabiliy condiion for his equaion requires ha 3.17 L P 1, = L 1 P, = L 1 L 1 L L P, which along wih he erminal condiion 3.18 P 1, T, x, r; z =, compleely idenifies he funcion P 1,. To obain he second equaliy in 3.17, we used Nex, we will express he righ-hand side of 3.17 more explicily. To his end, le ψ, κ, and φ be he soluions of he Poisson equaions 3.19 L ψỹ = σ ỹ σ 1, L κỹ = σ ỹ σ, and L φy, z = f y, z λz, respecively. Firs observe ha 3. L L P = 1 σ ỹ σ Now, along wih 3.19, we can wrie x P x + σ ỹ σ 1 ηρ 1 x P x + l f y, z λz x P x P. 3.1 L 1 L L P = 1 P κỹx x + ψỹηρ 1x P + l φy, z x P x x P. Applying he differenial operaor L 1 o he las expression yields 3. L1 L 1 L L P = l ρ ν σφ y zx P x + l ρ 1ην φ y z + ρ 4 ν 1 σκ ỹ x x + ρ 14 η ν 1 x P x + σψỹ ηρ 1 x x κ ỹ x 3 P x + ψ ỹ ηρ 1 ν 1 κ ỹ x P x + ψ ỹ ηρ 1 x P x x 3 P x. x P x P x P x

11 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 53 Finally, we inser he expression for P1 ɛ in 3.8 ino 3.6 and collec he erms wih hesamepowersofɛ. Arguing as before, we obain ha P,1 is independen of y and ỹ and saisfies: 3.3 L P,1 = M 1 P, P,1 T, x, r; z = Explici Pricing Formula We approximae P ɛ,δ defined in 3.1 by 3.4 P ɛ,δ = P + ɛ P 1, + δ P,1. Because he Vasicek ineres rae process is unbounded, which implies ha he poenial erm in L or he discouning erm in 3.1 is unbounded, he argumens of Fouque e al. 3 canno be direcly used. However, as in Coon e al. 4 and Papageorgiou and Sircar 8, one can wrie 3.5 in which 3.6 and 3.7 P ɛ,δ, X, r, Y, Ỹ, Z = B, TE T [exp lλ s ds hx T =: B, TF ɛ,δ, X, r, Y, Ỹ, Z, dp T exp dp = [ B, T = E exp B, T r s ds, ] r s ds F. Now, he analysis of Fouque e al. 3 can be used o approximae F ɛ,δ, x, r, y, ỹ, z. As a resul of his analysis for each, x, r, y, ỹ, z, here exiss a consan C such ha P ɛ,δ P ɛ,δ C ɛ + δwhenh is smooh, and P ɛ,δ P ɛ,δ C ɛ logɛ + δ + ɛδ when h is a pu or a call pay-off. In wha follows, we will obain P, P 1,,andP,1 explicily. Our firs objecive is o develop a closed-form expression for P, he soluion of 3.1 and PROPOSITION 3.1. The leading order erm P in 3.4 is given by P, x, r; z = B c, r; z, T, l 1 hexpu exp u m,t 3.8 du, πv,t where F v,t ] 3.9 B c, r; z, T, l exp l λzt + at bt r, in which he funcions as and bs are defined as η as = β α η s + β β α 3.3 exp βs 1 η exp βs 1 3 β 4β3

12 54 E. BAYRAKTAR AND B. YANG and bs = 1 exp βs/β. On he oher hand, 3.31 and 3.3 v,t = σ + ηρ 1 σ 1 + η ηρ1 σ 1 T + β β β η ηρ1 σ 1 exp βt + 3η β3 β β 3 + η, β 3 exp βt m,t = logx + λ T at + bt r 1 v,t. Proof. By applying he Feynman-Kac heorem o 3.1 and 3.14 we have ha [ ] 3.33 P, x, r; z = E exp r s + l λz ds hs T S = x, r = r, where he dynamics of S is given by 3.34 ds = r + λzs d + σ S d W, in which W is a Wiener process whose correlaion wih W 1 is ρ 1 = σ 1 σ ρ 1. Le us define 3.35 in which P, x, r = E [ exp r s ds h S T S = x, r = r ], 3.36 d S = r S d + σ S d W. Then 3.37 P, x, r; z = e l λzt P, x exp λzt, r. Now, by following Geman, Karoui, and Roche 1995, we change he probabiliy measure P o he forward measure P T hrough he Radon Nikodym derivaive 3.6. We can obain he following represenaion of P using he T-forward measure 3.38 P, S, r = B, TE T [ h S T F ] = B, TE T [hf T F ], in which 3.39 F S B, T, which is a P T maringale. Noe ha an explici expression for B, T is available since r is a Vasicek model, and i is given in erms of he funcions a and b 3.4 B, T = expat bt r.

13 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 55 By applying Iô s formula o 3.39, we observe ha he dynamics of F are 3.41 df = F σ1 d W + bt η d W 1, in which W 1 is a P T Brownian moion whose correlaion wih he W which is sill a Brownian moion under P T is ρ 1.GivenX and B, T, he random variable log F T is normally disribued wih variance 3.4 and mean 3.43 v,t = σ T + η b T s ds + η ρ 1 σ bt s ds = m,t = log F 1 σ + η ρ 1 σ + η β β η ρ1 σ T + β η η ρ1 σ exp βt + 3η β3 β β 3 Now he resul immediaely follows. + η, β 3 exp βt σ + b T sη + ρ 1 σ bt sη S ds = log 1 B, T v,t. An immediae corollary of he las proposiion is he following: COROLLARY 3.. i When l = 1, hx = x K +, hen 3.8 becomes 3.44 C, x, r; z = xnd 1 KB c, r; z, T, 1Nd, in which N is he sandard normal cumulaive disribuion funcion and 3.45 x log KB c d 1, =, r; z, T, 1 ± 1 v,t. v,t ii When l = 1, and hx = K x +, hen 3.8 becomes 3.46 Pu, x, r; z = x + xnd 1 KB c, r; z, T, 1Nd + KB c, r; z, T,. iii When hx = 1, hen 3.8 coincides wih 3.3 in Papageorgiou and Sircar 8. PROPOSITION 3.3. The correcion erm ɛ P 1, is given by 3.47 ɛ P1, = T V1 ɛ P zx x + Vɛ x x P x x + lv3 ɛ z x P x α P α + V ɛ 4 x 3 P x α + Vɛ 5 x P η x + Vɛ 6 x P x α,

14 56 E. BAYRAKTAR AND B. YANG in which 3.48 V1 ɛz ɛ l ρ ν σφ y z ν 1 κ ỹ, V ɛ ɛρ4 ν σκỹ, V ɛ 3 z = ɛρ 1 ην φ y z, V ɛ 4 = ɛ 1 ρ 14η ν κỹ ρ 4 ν σψỹ ηρ 1 + ρ 14 η ν ψỹ σ 1 ρ 1 V ɛ 5 = ɛρ 14 η ν ψỹ ρ 1, V ɛ 6 = ɛ ρ 4 ν σψỹ ηρ 1 + ρ 14 η ν ψỹ σ 1 ρ 1 ν ψỹ ηρ 1. Observe ha Vi ɛ, i {1,...,6} may be funcions of he iniial value of he slow facor Z. They do no depend on iniial values of Y, Ỹ; he effec of he fas scale facors are averaged ou in he approximaion formula. The V parameers depend on he fas facors hrough heir mean reversion level, volailiy, and heir correlaion wih he oher sae variables. These parameers also do no depend on r direcly, bu P and P 1, are funcions of his variable. Noe ha if we ake he volailiy, σ, o depend on a slow facor besides he fas facor Ỹ, say Z, hen he parameers V ɛ, Vɛ 4, Vɛ 5, Vɛ 6 will be funcions of he iniial value of Z. On he oher hand, V1 ɛ will depend on boh z and z,andvɛ 3 will only depend on z. One should noe ha he above expressions for hese parameers will sill look he same. Proof. Recall ha P 1, is he soluion of 3.17 and 3.18 and ha he righ-hand side of 3.17 is given by 3.. The resul is a simple algebraic exercise given he following four observaions: 1 x n n commues wih L x n. T x n n P x n solves 3.49 L u = x n n x n P, ut, x, r; z =. 3 By differeniaing 3.1 and 3.14 wih respec o α, weseeha P α, also solves 3.5 L u = P, ut, x, r; z =. 4 Using 1 and above and he equaion we obain by differeniaing 3.1 wih respec o η, we can show ha 1/η σ 1 ρ 1 x P x α P η solves 3.51 L u = P, ut, x, r; z =. REMARK 3.4. By differeniaing 3.1 wih respec o r, we obain 3.5 L P = x x P + β P + P.

15 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 57 Using observaion in he proof of Proposiion 3.3, we see ha x P x P + P solves 1 T β 3.53 L u = P, ut, x, r; z =. Now, i follows from observaion 3 in he proof of Proposiion 3.3 ha P α = 1 T x P β x P + P Using his ideniy, we can express 3.47 only in erms of he Greeks. Nex, we obain an explici expression for P,1, he soluion of 3.3. We need some preparaion firs. By differeniaing 3.1 wih respec o z,weseeha P z solves L u = λ zx P 3.55 x + l λ zp, ut, x, r; z =. As a resul see Observaion in he proof of Proposiion 3.3 P z = T λ z x P 3.56 x lp, from which i follows ha M 1 P can be represened as M 1 P = T λ z σ 1 ρ 3 gz x P x + 1 lx P 3.57 x + ηρ 13 gz x P x l P. PROPOSITION 3.5. The correcion erm δ P,1 is given by 3.58 δ P,1 = V1 δ zt in which + T x P x + 1 lx P x x P x lx P x + lp + V δ z 1 [ x P β α x l P α T x P x l P ], 3.59 V δ 1 z = δ λ z σ 1 ρ 3 gz, V δ z = δ λ zηρ 13 gz. Proof. We consruc he soluion from he following observaions and superposiion since L is linear: 1 We firs observe ha T x n n P x n solves 3.6 L u = T x n n x n P, ut, x, r; z =.

16 58 E. BAYRAKTAR AND B. YANG Nex, we apply L on T P and obain L T P = P + T x P x + β P P, as a resul of which we see ha 3.6 solves 1 β [ P T α x P x P + T P ] 3.63 L u = T P, ut, x, r; z =. 4. CALIBRATION OF THE MODEL In his secion, we will calibrae he loss rae l and he parameers { λz, V1 ɛ z, Vɛ, Vɛ 3 z, Vɛ 4, Vɛ 5, Vɛ 6, Vδ 1 z, Vδ z}, which appear in he expressions 3.8, 3.47, and 3.58 on a daily basis see, e.g., Fouque e al. 3 and Papageorgiou and Sircar 8 for similar calibraion exercises carried ou only for he opion daa or only for he bond daa. We demonsrae his calibraion on Ford Moor Company. Noe ha here are some common parameers beween equiy opions and corporae bonds. Therefore, our model will be calibraed simulaneously o boh of hese daa ses. We will also calibrae he parameers of he ineres rae and sock models o he yield curve daa, hisorical spo rae daa, and hisorical sock price daa. We look a how our model-implied volailiy maches he real opion-implied volailiy. We compare our resuls agains hose of Fouque e al. 3. We see ha even when we make he unrealisic assumpion of consan volailiy, our model is able o produce a very good fi. Finally, in he conex of index opions when λ =, using SPX 5 index opions daa, we show he imporance of accouning for sochasic ineres raes by comparing our model o ha of Fouque, Papanicolaou, and Sircar and Fouque e al Daa Descripion The daily closing sock price daa is obained from finance.yahoo.com. The sock opion daa is from OpionMerics under WRDS daabase, which is he same daabase used in Carr and Wu 6. For index opions, SPX 5 in our case, we use he daa from heir Volailiy Surface file. The file conains informaion on sandardized opions, boh calls and pus, wih expiraions of 3, 6, 91, 1, 15, 18, 73, 365, 547, and 73 calender days. Implied volailiies here are inerpolaed daa using a mehodology based on kernel smoohing algorihm. The inerpolaed implied volailiies are very close o real daa because here are a grea number of opions each day for SPX 5 wih differen mauriies and srikes. The calibraion resuls for index opions are presened in Figure A.4 and only he daa se on June 8, 7 is used.

17 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 59 On Sepember 15, 6 Friday Ford announced ha i would no be paying dividends see, e.g., hp://money.cnn.com/6/9/15/news/companies/ ford/index.hm. Therefore, call opions on Ford afer ha dae do no have early exercise premium saring from Sepember 18, 6. We use Ford s implied volailiy surface on April 4, 7 and June 8, 7 o creae Figures A.3 and A.4, respecively. We excluded he observaions wih zero rading volume or wih mauriy less han 9 days. We find ha he resuls given by using inerpolaed-implied volailiies in he Volailiy Surface File and daa-implied volailiies differ. This may be due o he fac ha here are a limied number of opion prices available for individual companies; ha is, here may no be enough daa poins for he implied volailiies o be accuraely inerpolaed. Therefore, we use he Opion Price file, which conains he hisorical opion price informaion, of he OpionMerics daabase For boh days April 4, 7 and June 8, 7, we use U.S. governmen Treasury yield daa wih mauriies: 1 monh, 3 monhs, 6 monhs, 1 year, years, 3 years, 5 years, 7 years, 1 years, years. This daa se is available a: domesic-finance/deb-managemen/ineres-rae/yield.shml. Corporae bond daa is obained from Bloomberg. Number of available bond quoes and bond mauriies vary. Typically, here are around 15 daa poins, for example, on June 8, we have he following mauriies:.678, 1., , , 1.483, ,.3889,.68, 3.194, 3.694, 3.397, 3.647, 4.17, 4.386, , and The Parameer Esimaion The following parameers can be direcly esimaed from he spo-rae and sock price hisorical daa: 1. The parameers of he ineres rae model {α, β, η} are obained by a leas-square fiing o he Treasury yield curve as in Papageorgiou and Sircar 8.. ρ 1 = σ 1 σ ρ 1, he effecive correlaion beween risk-free spo rae r we use he 1- monh reasury bonds as a proxy for r and sock price in 3.34 is esimaed from hisorical risk-free spo rae and sock price daa. 3. σ, he effecive sock price volailiy in 3.34 is esimaed from he hisorical sock price daa. V6 ɛ, Vδ 1 Now, we deail he calibraion mehod for l, λz and V1 ɛ z, Vɛ, Vɛ 3 z, Vɛ 4, Vɛ 5, z, Vδ z. We will minimize he in-sample quadraic pricing error using nonlinear leas squares o calibrae hese parameers on a daily basis. This way we find a risk neural model ha maches a se of observed marke prices. This risk neural model can hen be used o price more exoic, illiquid, or over-he-couner derivaives. This pracice is commonly employed; and for furher discussion of his calibraion mehodology we refer o Con and Tankov 4 see chaper 13, and he references herein. Our calibraion is carried ou in wo seps in andem: Sep 1. Esimaion of l λz and {lv3 ɛ z, lvδ z} from he corporae bond price daa. The approximae price formula in 3.4 for a defaulable bond is 4.1 B c = B c + ɛ B c 1, + δb c,1,

18 51 E. BAYRAKTAR AND B. YANG in which B c is given by 3.9 and 4. ɛ B c 1, = lv ɛ 3 z Bc α, δb c,1 = lv δ z 1 β [ Bc α ] T + B c + T Bc. We obain {l λz, lv ɛ 3 z, lvδ z} from leas-squares fiing, ha is, by minimizing 4.3 n i=1 B c obs, S i Bmodel c, Si ; l λ, lv3 ɛ z, lvδ z, where Bobs c, S i is he observed marke price of a bond ha maures a ime S i and Bmodel c, S i; l λ, lv3 ɛ z, lvδ z is he corresponding model price obained from 4.1. Here, n is he number of bonds ha are raded a ime. For a fixed value of l λz i follows from 4.1 ha {lv3 ɛ z, lvδ z} can be deermined as he leas-squares soluion of B c [ α, S 1 1, Bc β α + S 1 B c + S 1 ] Bc lv ɛ. 3 z. lv B c [ α, S 1 n, Bc β α + S n B c + S n ] δz Bc Bobs c, S 1 B c, S 1; l λ =.. Bobs c, S n B c, S n; l λ Now, we vary l λz [, M 1 ] and choose he poin {l λ, lv ɛ 3 z, lvδ }z ha minimizes 4.3. Here, we ake M 1 = 1 guided by he resuls of Papageorgiou and Sircar 8. Sep. Esimaion of {l, V1 ɛ z, Vɛ, Vɛ 4, Vɛ 5, Vɛ 6, Vδ 1 z} from he equiy opion daa. These parameers are calibraed from he sock opions daa by a leas-squares fi o he observed implied volailiy. We choose he parameers o minimize 4.4 n I obs, T i, K i I model, T i, K i ; model parameers i=1 n P obs, T i, K i P model, T i, K i ; model parameers, vega T i, K i i=1 in which I obs, T i, K i and I model, T i, K i ; model parameers are observed Black Scholes implied volailiy and model Black Scholes implied volailiy, respecively. The

19 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 511 righ-hand side of 4.4 is from Con and Tankov 4, page 439. Here, P obs, T i, K i is he marke price of a European opion a pu or a call ha maures a ime T i and wih srike price K i and P model, T i, K i ; model parameers is he corresponding model price which is obained from 3.4. As in Con and Tankov 4, vegat i, K i ishemarke implied Black Scholes vega. Le P, T i, K i ; λz be eiher of 3.44 and 3.46 wih K = K i and T = T i.leus inroduce he Greeks, 4.5 g 1 = T x P x, g = T x x P, g x x 3 = x P α x P, g 4 = x 3 P x α, g 5 = x P η x, g 6 = x P α x, g 7 = g 8 = 1 β [ x P α x P T + α P T x x x P x x P P ], x + P T x P x, in which each erm can be explicily evaluaed see Appendix B. Now from 3.4 and he resuls of Secion 3.3 wih l = 1, we can wrie 4.6 P model, T i, K i = P, T i, K i ; λz + V ɛ 1 zg 1T i, K i ; λz + V ɛ g T i, K i ; λz + V3 ɛ zg 3T i, K i ; λz + V4 ɛ g 4T i, K i ; λz + V5 ɛ g 5T i, K i ; λz + V6 ɛ g 6T i, K i ; λz + V1 δ zg 7T i, K i ; λz + V δ zg 8T i, K i ; λz. Firs, le us fix he value of l. Then, from Sep 1, we can infer he values of { λz, V3 ɛ z, Vδ z}. Now he fiing problem in 4.4 is a linear leas-squares problem for {V1 ɛ z, Vɛ, Vɛ 4, Vɛ 5, Vɛ 6, Vδ 1 z}. Nex, we vary l [, 1] and choose {l, V1 ɛ z, Vɛ, Vɛ 4, Vɛ 5, Vɛ 6, Vδ z} so ha 4.4 is minimized Fiing Ford s Implied Volailiy We will compare how well our model fis he implied volailiy agains he model of Fouque e al. 3, which does no accoun for he defaul risk and for he randomness of he ineres raes. Alhough, we only calibrae seven parameers hence we refer o our model as he seven-parameer model o he opion prices see he second sep of he esimaion in Secion 4., we have many more parameers han he model of Fouque e al. 3, which only has four parameers we refer o his model as he four-parameer model. Therefore, for a fair comparison, we also consider a model in which he volailiy is a consan. In his case, as we shall see below, here are only hree parameers o calibrae o he opion prices, herefore we call i he hree-parameer model. Consan Volailiy Model. In his case, we ake σ 1 = σ = σ in he expression for P in Corollary 3.. The expression for δ P,1 remains he same as before. However, ɛ P 1, simplifies o

20 51 E. BAYRAKTAR AND B. YANG 4.7 ɛ P1, = T V1 ɛ P zx x + Vɛ 3 z x P α x + P. α This model has only hree parameers, l, V ɛ 1 z, Vδ 1 z ha need o be calibraed o he opions prices, as opposed o he four-parameer model of Fouque e al. 3. As can be seen from Figure A.3, as expeced, our seven-parameer model ouperforms he four-parameer model of Fouque e al. 3 and fis he implied volailiy daa well. Bu, wha is surprising is ha he hree-parameer model, which does no accoun for he volailiy bu accouns for he defaul risk and sochasic ineres rae, has almos he same performance as he seven-parameer model. The seven-parameer model has a very rich implied volailiy surface srucure, he surface has more curvaure han ha of he four-parameer model of Fouque e al. 3, whose volailiy surface is more fla; see Figures A.1 and A.. The parameers o draw hese figures are obained by calibraing he models o he daa implied volailiy surface on June 8, 7. The seven-parameer model has a recognizable skew even for longer mauriies and has a much sharper skew for shorer mauriies Fiing he Implied Volailiy of he Index Opions The purpose of his secion is o show he imporance of accouning for sochasic ineres raes in fiing he implied volailiy surface. Ineres rae changes should, indeed, be accouned for in pricing long mauriy opions. When we price index opions, we se λ = and our approximaion in 3.4 simplifies o 4.8 P ɛ,δ P + ɛ P 1,, in which P is given by Corollary 3. afer seiing λz =, and 4.9 ɛ P1, = T V1 ɛ P x x + Vɛ x x P x x + V ɛ 4 x 3 P x α + Vɛ 5 x P η x + Vɛ 6 x P α x. Noe ha he difference of 4.8 wih he model of Fouque e al. 3 is ha he laer allows for a slow evolving volailiy facor o beer mach he implied volailiy a he longer mauriies. This was an improvemen on he model of Fouque, Papanicolaou, and Sircar, which only has a fas scale componen in he volailiy model. We, on he oher hand, by accouning for sochasic ineres raes, capure he same performance by using only a fas scale volailiy model. From Figure A.4, we see ha boh 4.8 and Fouque e al. 3 ouperform he model of Fouque e al., especially a he longer mauriies T = 9 monhs, 1 year, 1.5 years, and years, and ha heir performances are very similar. This observaion emphasizes he imporance of accouning for sochasic ineres raes for long mauriy conracs.

21 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 513 APPENDIX A: FIGURES implied volailiy moneyness ime o mauriy FIGURE A.1. Implied volailiy surface corresponding o 4.6, he sevenparameer model. Here, α =.63,β =.134,η =.1, r =.476 σ =.576, λz =.7, V1 ɛ z, Vɛ, Vɛ 3 z, Vɛ 4, Vɛ 5, Vɛ 6, Vδ 1 z, Vδ z =.996,.14,.9,.14,.6514,.334,.1837,.1. implied volailiy moneyness ime o mauriy FIGURE A.. Implied volailiy surface corresponding o he four-parameer model of Fouque e al. 3. Here, r =.46, average volailiy =.546, and he parameers in 4.3 of Fouque e al. 3 are chosen o be V ɛ, Vɛ 3 z, Vδ z, Vδ 1 z =.164,.1718,.6,.63. Noe ha he parameers here and Figure A.1 are boh obained by calibraing he models o he daa implied volailiy surface of Ford Moor Company on June 8, 7. 1

22 514 E. BAYRAKTAR AND B. YANG.55 mauriy = 17 days.9 mauriy = 45 days implied volailiy implied volailiy srikes srikes.44 mauriy = 7 days.44 mauriy = 168 days.4.4 implied volailiy implied volailiy srikes srikes.6 mauriy = 85 days.6 mauriy = 643 days implied volailiy implied volailiy srikes srikes FIGURE A.3. Implied volailiy fi o he Ford call opion daa wih mauriies of [17, 45, 7, 168, 85, 643] calender days on April 4, 7. Model is calibraed across all mauriies bu we ploed he implied volailiies for each mauriy, separaely. Here, sock price x = 8.4, hisorical volailiy σ =.387, 1-monh reasury rae r =.516, esimaed correlaion beween risk-free spo rae 1-monh reasury and sock price ρ 1 =.37. Also α =.37,β =.87,η =.1 which are obained wih a leas-square fiing o he Treasury yield curve on he April 4. o, empy circles = observed daa; x = sochasic vol + sochasic hazard rae + sochasic ineres rae = he seven-parameer model; small full circle = consan vol + sochasic hazard rae + sochasic ineres rae = he hree-parameer model; * = he model of Fouque e al. 3 which has consan ineres rae + sochasic vol slow and fas scales = he four-parameer model.

23 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 515 implied volailiy implied volailiy implied volailiy implied volailiy mauriy = 3 days srikes..15 mauriy = 91 days srikes srikes..15 mauriy = 15 days mauriy = 73 days srikes implied volailiy implied volailiy implied volailiy implied volailiy..15 mauriy = 6 days srikes..15 mauriy = 1 days srikes..15 mauriy = 18 days srikes..15 mauriy = 365 days srikes implied volailiy..15 mauriy = 547 days srikes implied volailiy mauriy = 73 days srikes FIGURE A.4. The fi o he implied volailiy surface of SPX on June 8, 7 wih mauriies [3, 6, 91, 1, 15, 18, 73, 365, 547, 73] calender days. Recall from Secion 4.1 ha we use sandardized opions from he OpionMerics. Models are calibraed across all mauriies, bu we plo he implied volailiy fis separaely. The parameers are: sock price x = , dividend rae =.194, hisorical volailiy σ =.114, 1-monh reasury rae r =.476, esimaed correlaion beween risk-free spo rae 1-monh reasury and sock price ρ 1 =.454. Also, α =.78,β =.1173,η =.41, which are obained from a leas-square fiing o he Treasury yield curve. o, empy circles = observed daa; x = implied volailiy of 4.8, = implied volailiy of Fouque e al. 3; small full circle = implied volailiy of Fouque, Papanicolaou, and Sircar.

24 516 E. BAYRAKTAR AND B. YANG APPENDIX B: EXPLICIT FORMULAE FOR THE GREEKS IN 4.5 When hx = x K +, we can explicily express he Greeks in 4.5 in erms of f x = 1 π exp x / as x C = xfd 1 x, x x C = xfd 1 v,t x x 1 d 1, v,t v,t x C T α x C = K B c exp βt 1, T + Nd β β f d, v,t x C = xfd 1d 1 T exp βt 1 +, α x v,t β β x C = xfd 1 T exp βt 1 +, x α v,t β β x C 1 exp βt x C = K B c, T Nd f d, β v,t x [ C = xfd 1 1 η η T + exp βt 1 x η v,t β β3 η exp βt 1 β3 + 1 log x v 3/ K B,T c,t v,t ρ1 σ + η ρ1 σ T + + 4η β β β β 3 η β 3 exp βt ρ1 σ β + 3η β 3 ]. exp βt REFERENCES BAYRAKTAR, E. 8: Pricing Opions on Defaulable Socks, Appl. Mah. Finance 153, BIELECKI, T. R.,andM. RUTKOWSKI : Credi Risk: Modeling, Valuaion and Hedging, New York: Springer. CARR, P., andv. LINETSKY 6: A Jump o Defaul Exended CEV Model: An Applicaion of Bessel Processes, Finance Soch. 1, CARR, P., andl. WU 6: Sock Opions and Credi Defaul Swaps: A Join Framework for Valuaion and Esimaion, Technical Repor. Available a hp://papers.ssrn.com/sol3/ papers.cfm?absrac_id=7485. CONT, R., andp. TANKOV 4: Financial Modeling wih Jump Processes, Boca Raon, FL: Chapman & Hall. COTTON, P., J.-P. FOUQUE, G. PAPANICOLAOU, andr. SIRCAR 4: Sochasic Volailiy Correcions for Ineres Rae Derivaives, Mah. Finance 14, 173.

25 A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES 517 DUFFIE, D., andk. SINGLETON 1999: Modeling Term Srucure of Defaulable Bonds, Rev. Financ. Sud. 14, FOUQUE, J.-P., G. PAPANICOLAOU, and K. R. SIRCAR : Derivaives in Financial Markes wih Sochasic Volailiy, New York: Cambridge Universiy Press. FOUQUE, J. P., G. PAPANICOLAOU, R. SIRCAR, and K. SOLNA 3: Muliscale Sochasic Volailiy Asympoics, SIAM J. Muliscale Model. Simul. 1, 4. GEMAN, H., N. E. KAROUI, andj. C. ROCHET 1995: Changes of Numéraire, Changes of Probabiliy Measures and Opion Pricing, J. Appl. Probab. 3, LINETSKY, V. 6: Pricing Equiy Derivaives Subjec o Bankrupcy, Mah. Finance 16, PAPAGEORGIOU, E., andr. SIRCAR 8: Muliscale Inensiy Based Models for Single Name Credi Derivaives, Appl. Mah. Finance 151, SCHÖNBUCHER, P. J. 1998: Term Srucure of Defaulable Bond Prices, Rev. Deriva. Res. /3, SCHÖNBUCHER, P. J. 3: Credi Derivaives Pricing Models: Model, Pricing and Implemenaion,NewYork:Wiley.