Pricing American Options under the Constant Elasticity of Variance Model and subject to Bankruptcy

Transcription

1 Pricing American Opions under he Consan Elasiciy of Variance Model and subjec o Bankrupcy João Pedro Vidal Nunes ISCTE Business School Complexo INDEG/ISCTE, Av. Prof. Aníbal Beencour, Lisboa, Porugal. Tel: Fax: joao.nunes@isce.p An earlier version of his paper was presened a he 2006 Derivaives Securiies and Risk Managemen Conference (Arlingon, Virginia) and a he 2006 Bachelier Finance Sociey Fourh World Congress (Tokyo), under he ile A General Characerizaion of he Early Exercise Premium. The auhor hanks an anonymous referee, whose correcions have significanly improved his aricle, as well as he helpful programing codes and commens provided by Nico Temme. Of course, all he remaining errors are he auhor s responsibiliy exclusively.

2 Pricing American Opions under he Consan Elasiciy of Variance Model and subjec o Bankrupcy Absrac This paper proposes an alernaive characerizaion of he early exercise premium ha is valid for any Markovian and diffusion underlying price process as well as for any parameerizaion of he exercise boundary. This new represenaion is shown o provide he bes pricing alernaive available in he lieraure for medium- and long-erm American opion conracs, under he Consan Elasiciy of Variance model. Moreover, he proposed pricing mehodology is also exended easily o he valuaion of American opions on defaulable equiy, and possesses appropriae asympoic properies.

3 I. Inroducion This paper proposes a new analyical approximaion for he American opion value ha canbeappliedunderanymarkovian anddiffusion underlying asse price process. In conras wih he previous lieraure, he proposed characerizaion of he American opion can accommodae he risk of defaul aached o he underlying equiy, and is shown o converge o he exac perpeual soluion, being herefore exremely accurae, even for long-erm conracs. The absence of an exac and closed-form pricing soluion for he American pu (or call, bu on a dividend-paying asse) sems from he fac ha he opion price and he early exercise boundary mus be deermined simulaneously as he soluion of he same free boundary problem se up by McKean (1965). Consequenly, he vas lieraure on his subjec, which is reviewed for insance in Barone-Adesi (2005), has proposed only numerical soluion mehods and analyical approximaions. The numerical mehods include he finie difference schemes inroduced by Brennan and Schwarz (1977), and he binomial model of Cox, Ross, and Rubinsein (1979). These mehods are boh simple and convergen, bu hey are also oo ime-consuming and do no provide he comparaive saics aached o an analyical soluion. One of he firs analyical approximaions is due o Barone-Adesi and Whaley (1987), who use he quadraic mehod of MacMillan (1986). Despie is high efficiency and he accuracy improvemens brough by subsequen exensions (see for example, Ju and Zhong (1999)), his mehod is no convergen. Johnson (1983) and Broadie and Deemple (1996) provide lower and upper bounds for American opions, which are based on regression coefficiens ha are esimaed hrough a ime-demanding calibraion o a large se of opions conracs. As argued in Ju ((1998), p. 642), his economeric approach is no convergen and can generae less accurae hedging raios, because he regression coefficiens are opimized only for pricing purposes. More recenly, Sullivan (2000) approximaes he opion value funcion hrough Chebyshev polynomials and employs a Gaussian quadraure inegraion scheme a each discree exercise dae. Alhough he speed and accuracy of he proposed numerical 1

4 approximaion can be enhanced via Richardson exrapolaion, is convergence properies are sill unknown. Geske and Johnson (1984) approximae he American opion price hrough an infinie series of mulivariae normal disribuion funcions. Alhough convergence can be insured by adding more erms, only he firs few erms are considered, and a Richardson exrapolaion scheme is employed in order o reduce he compuaional burden. 1 Anoher fas and accurae convergen mehod is he randomizaion approach of Carr (1998), which also uses Richardson exrapolaion. I mus be noed, however, ha one of he main disadvanages of exrapolaion schemes is he indeerminaion of he sign for he approximaion error. Kim (1990), Jacka (1991), Carr, Jarrow, and Myneni (1992), and Jamshidian (1992) proposed he so-called inegral represenaion mehod o describe he early exercise premium. However, he numerical efficiency of his approach depends on he specificaion adoped for he early exercise boundary. For insance, Ju (1998) derives fas and accurae approximae soluions based on a mulipiece exponenial represenaion of he early exercise boundary. All he sudies menioned are based on he Black and Scholes (1973) model, and mos of hem differ only in he specificaion adoped for he exercise boundary. Kim and Yu (1996) and Deemple and Tian (2002) consiue wo noable excepions: hey exend he inegral represenaion mehod o alernaive diffusion processes. However, and in opposiion o he sandard geomeric Brownian moion case, such an exension does no offer a closedform soluion for he inegral equaion characerizing he early exercise premium, which undermines he compuaional efficiency of his approach. Based on he opimal sopping approach iniiaed by Bensoussan (1984) and Karazas (1988), his paper derives an alernaive characerizaion of he American opion price ha is valid for any coninuous represenaion of he exercise boundary and for any Markovian (and diffusion) price process describing he dynamics of he underlying asse price. The proposed characerizaion possesses a leas hree advanages over he exended inegral represenaion 1 Chung and Shackleon (2007) generalize he Geske-Johnson mehod hrough a wo-poin scheme based no only on he iner-exercise ime dimension, bu also on he ime o mauriy of he opion conrac. 2

5 of Kim and Yu ((1996), equaions 10 or 13): 1) i converges o he perpeual American opion price as he opion mauriy ends o infiniy; 2) is accuracy does no deeriorae as he opion mauriy is lenghened; and 3) i can be adaped easily o he conex of defaulable sock opions pricing models. Alhough knowledge of he firs passage ime densiy of he underlying price process o he exercise boundary is required by he proposed pricing soluion, i is shown ha such opimal sopping ime densiy can be recovered easily from he ransiion densiy funcion. Hence, he proposed characerizaion of he American opion price requires only an efficien valuaion formula for is European counerpar, as well as knowledge of he underlying asse price ransiion densiy funcion. To exemplify he proposed pricing mehodology, several parameerizaions of he early exercise boundary are esed under he usual geomeric Brownian moion assumpion and he Consan Elasiciy of Variance (CEV) model. Special aenion is devoed o his laer framework since i is consisen wih wo well known facs ha have found empirical suppor in he lieraure: he exisence of a negaive correlaion beween sock reurns and realized sock volailiy (leverage effec), as documened, for insance, in Bekaer and Wu (2000); and he inverse relaion beween he implied volailiy and he srike price of an opion conrac (implied volailiy skew), which is observed, for example, by Dennis and Mayhew (2002). Alhough he pricing of European opions under he CEV process has become well esablished since he seminal work of Cox (1975), he same canno be said abou he valuaion of American opions. As noed by Nelson and Ramaswamy ((1990), p. 418), he simple binomial processes approximaion proposed by hese auhors becomes inaccurae as opion mauriy is increased. Alernaively, and as shown in Secion VI, he inegral approach suggesed by Kim and Yu ((1996), subsecion 3.4) and Deemple and Tian ((2002), Proposiion 3) is less efficien han he proposed pricing mehodology, unless such recursive scheme is acceleraed hrough Richardson exrapolaion, in which case is accuracy may deeriorae for medium- and long-erm opions. This paper is inended o fill his gap in he lieraure. Addiionally, he opimal sopping approach presened in his paper is also adaped o he conex of he jump o defaul exended CEV model (JDCEV) proposed by Carr and Linesky (2006). This exension of he lieraure provides analyical pricing soluions for 3

6 American opions on defaulable equiy, which are consisen wih boh he aforemenioned leverage effec, and he posiive relaionship beween defaul indicaors and equiy volailiy ha is documened, for insance, by Campbell and Taksler (2003). This paper proceeds as follows. Based on he opimal sopping approach, Secion II separaes he American opion ino a non-deferrable rebae and a European down-and-ou opion. In Secion III, such represenaion is shown o be equivalen o he usual decomposiion beween a European opion and an early exercise premium. A new analyical characerizaion is offered for he early exercise premium, and is asympoic properies are esed. Secion IV provides an efficien algorihm o recover he firs hiing ime densiy of he underlying price process, which allows he comparison, in Secion VI, of he differen specificaions of he early exercise boundary discussed in Secion V. Secion VII exends he new represenaion of he early exercise premium o he JDCEV model, and Secion VIII concludes. II. Model Seup The valuaion of American opions will be firs explored in he conex of a sochasic ineremporal economy wih coninuous rading on he ime-inerval [,T], forsomefixed ime T >, where uncerainy is represened by a complee probabiliy space (Ω, F, Q). Throughou he paper, Q will denoe he maringale probabiliy measure obained when he numéraire of he economy under analysis is aken o be a money marke accoun B,whose dynamics are governed by he following ordinary differenial equaion: (1) db = rb d, where r denoes he riskless ineres rae, which is assumed o be consan. Alhough he alernaive represenaion of he early exercise premium ha will be proposed in Proposiion 1 requires only ha he underlying asse price process S be a Markovian diffusion, he subsequen empirical analysis will be based on he following one-dimensional diffusion process: (2) ds S =(r q) d + σ (, S) dw Q, 4

7 where q represens he dividend yield for he asse price, σ (, S) corresponds o he insananeous volailiy (per uni of ime) of asse reurns and W Q R is a sandard Brownian moion, iniialized a zero and generaing he augmened, righ coninuous, and complee filraion F = {F : }. Neverheless, equaion (2) encompasses several well known opion pricing models as special cases: for example, i corresponds o he geomeric Brownian moion if σ (, S) =σ is a consan; and i yields he CEV process when (3) σ (, S) =δs β 2 1, for δ, β R. 2 Hereafer, he analysis will focus on he valuaion of an American opion on he asse price S, wih srike price K, and wih mauriy dae T,whoseime- ( T ) value will be denoed by V (S, K, T ; φ), whereφ = 1 foranamericancallorφ =1foranAmericanpu. Since he American opion can be exercised a any ime during is life, i is well known see, for example, Karazas ((1988), Theorem 5.4) ha is price can be represened by he Snell envelope: (4) V 0 (S, K, T ; φ) =supe Q e r[(t τ) ] (φk φs T τ ) + ª F 0, τ T where T is he se of all sopping imes for he filraion F generaed by he underlying price process and aking values in [, ]. 3 Since he underlying asse price is a diffusion and boh ineres raes and dividend yields are assumed o be deerminisic, for each ime [,T] here exiss a criical asse price E below (above) which he American pu (call) price equals is inrinsic value and, herefore, early exercise should occur see, for insance, Carr, Jarrow, and Myneni ((1992), equaions 1.2 and 1.3). Consequenly, he opimal policy should be o exercise he American opion 2 The underlying asse can be hough of as a sock, a sock index, an exchange rae, or a financial fuures conrac, so long as he parameer q is undersood as, respecively, a dividend yield, an average dividend yield, he foreign defaul-free ineres rae, or he domesic risk-free ineres rae. 3 E Q (X F ) denoes he expeced value of he random variable X, condiional on F, and compued under he equivalen maringale measure Q. Similarly, Q (ω F ) will represen he probabiliy of even ω, condiional on F, and compued under he probabiliy measure Q. 5

8 when he underlying asse price firs ouches is criical level. Represening he firs passage ime of he underlying asse price o is moving boundary by (5) τ e := inf { > : S = E } and considering ha he American opion is sill alive a he valuaion dae (i.e., φs 0 > φe 0 ), equaion (4) can hen be resaed as: (6) V 0 (S, K, T ; φ) = E Q e r[(t τ e) ] (φk φs T τ e ) + ª F 0 = E Q e r(τ e ) (φk φe τ e ) 1 {τ e <T } F 0 +e r(t 0) E Q (φk φst ) + 1 {τ e T } F 0, where he firs line of equaion (6) follows from equaion (5), and 1 {A} denoes he indicaor funcion of he se A. Noe ha K E τ e for he American pu, because he exercise rq boundary is limied from above by min ³K, K see, for insance, Huang, Subrahmanyam, and Yu ((1996), foonoe 5). For he American call, K E τ e because he early exercise rq boundary is limied from below by max ³K, K see, for example, Kim and Yu ((1996), p. 67). For φ =1, equaion (6) is equivalen o Kim and Yu ((1996), eq. 7) and decomposes he American pu ino wo componens. The firs one corresponds o he presen value of a non-deferrable (and, in general, also non-consan) rebae (K E τ e ),payableahe opimal sopping ime τ e. The second componen is simply he ime- price of a European down-and-ou pu on he asse S, wih srike price K, mauriy dae a ime T, and (imedependen) barrier levels {E, T }. Assuming a convenien parameric specificaion for he barrier funcion E, i is possible o conver equaion (6) ino a closed-form soluion. Such an approach was pursued, for insance, by Ingersoll (1998) using boh consan and exponenial specificaions, and by Sbuelz (2004), also under a consan barrier formulaion. Unforunaely, he ime pah {E, T } of criical asse prices, which is called he exercise boundary, is no known ex ane and herefore he assumpion of a specific parameric form for he barrier funcion simply ransforms equaion (6) ino a lower bound for he rue American pu opion value. 6

9 III. The Early Exercise Premium Similarly o Kim (1990), Jacka (1991), and Carr, Jarrow, and Myneni (1992), he American opion price can be divided ino wo componens: he corresponding European opion price and an early exercise premium. For his purpose, and because (7) 1 {τ e T } =1 1 {τ e <T }, equaion (6) can be rewrien as: And, since V 0 (S, K, T ; φ) = E Q e r(τ e ) (φk φe τ e ) 1 {τ e <T } F 0 +e r(t 0) E Q (φk φst ) + F 0 e r(t 0) E Q (φk φst ) + 1 {τ e <T } F 0. (8) e r(t ) E Q (φk φst ) + F 0 := v0 (S, K, T ; φ) can be undersood (under a deerminisic ineres rae seing) as he ime- price of he corresponding European opion (wih echnical feaures idenical o hose of he American conrac under analysis), hen (9) V 0 (S, K, T ; φ) = v 0 (S, K, T ; φ) +E Q e r(τ e ) (φk φe τ e ) 1 {τ e<t } F 0 e r(t 0) E Q (φk φst ) + 1 {τ e<t } F 0. The las wo erms on he righ-hand side of equaion (9) correspond o he early exercise premium, for which an analyical soluion will be proposed in he nex proposiion. A. An Alernaive Characerizaion The proposiion presened below provides a new characerizaion for he early exercise premium. 7

10 Proposiion 1 Assuming ha he underlying asse price S follows a Markovian diffusion process and ha he ineres rae r is consan, he ime- valueofanamericanopion V 0 (S, K, T ; φ) on he asse price S, wih srike price K, and wih mauriy dae T can be decomposed ino he corresponding European opion price v 0 (S, K, T ; φ) and he early exercise premium EEP 0 (S, K, T ; φ), i.e., (10) V 0 (S, K, T ; φ) =v 0 (S, K, T ; φ)+eep 0 (S, K, T ; φ), wih (11) EEP 0 (S, K, T ; φ) := Z T e r(u ) [(φk φe u ) v u (E,K,T; φ)] Q (τ e du F 0 ), where Q (τ e du F 0 ) represens he probabiliy densiy funcion of he firs passage ime τ e, as defined by equaion (5), φ = 1 foranamericancallandφ =1for an American pu. Proof. Noing ha he only random variable conained in he second erm on he righ-hand side of equaion (9) is he firs passage ime, hen Z (12) E Q e r(τ e T ) (φk φe τ e ) 1 {τ e <T } F 0 = e r(u 0) (φk φe u ) Q (τ e du F 0 ). Concerning he hird erm on he righ-hand side of equaion (9), i is necessary o consider he join densiy of he wo random variables involved: he firs passage ime τ e and he erminal asse price S T. Hence, (13) Z E Q (φk φst ) + 1 {τ e <T } F 0 = R (φk φs) + Q (S T ds, τ e <T F 0 ), where he inegraion can be resriced o he domain R + if, for example, he geomeric Brownian moion assumpion is imposed. Because he underlying asse price is assumed o be a Markov process, he join densiy conained in equaion (13) is simply he convoluion beween he densiy of he firs passage ime τ e and he ransiion probabiliy densiy funcion of he erminal asse price S T : (14) Q (S T ds, τ e <T F 0 )= Z T Q (S T ds S u = E u ) Q (τ e du F 0 ). 8

11 Therefore, combining equaions (13) and (14), (15) = = E Q (φk φst ) + 1 {τ e <T } F 0 Z T Z (φk φs) + Q (S T ds S u = E u ) Q (τ e du F 0 ) Z T R E Q (φk φst ) + S u = E u Q (τ e du F 0 ). Moreover, considering equaion (8), he expecaion conained in he righ-hand side of equaion (15) can be expressed in erms of a European opion price: Z (16) E Q (φk φst ) + T 1 {τ e<t } F 0 = e r(t u) v u (E,K,T; φ) Q (τ e du F 0 ). Finally, combining equaions (9), (12) and (16), he early exercise represenaion (11) follows. Under he usual geomeric Brownian moion assumpion, equaion (11) yields a closedform soluion o he early exercise premium (modulo o he specificaion of he firs passage ime densiy), because he erm v u (E,K,T; φ) can be compued using he Meron (1973) formulae. The same reasoning applies o he CEV model since, in his case, European opion prices can be compued hrough he analyical soluions provided by Cox (1975) or Schroder (1989). Noe, however, ha he proof of Proposiion 1 relies only on he much weaker assumpion of a Markovian and diffusive asse price. Tha is, he early exercise represenaion (11) is sill valid for oher asse price processes beyond he general class represened by he sochasic differenial equaion (2). The represenaion offered by Proposiion 1 is also amenable o an inuiive inerpreaion. Since he value-maching condiion implies ha (φk φe u )=V u (E,K,T; φ), hen equaion (11) can be rewrien as = EEP 0 (S, K, T ; φ) Z T e r(u ) [V u (E,K,T; φ) v u (E,K,T; φ)] Q (τ e du F 0 ). 9

12 Using equaion (10), oday s early exercise premium can now be easily undersood as he discouned expecaion of he early exercise premium sopped a he firs passage ime: 4 (17) EEP 0 (S, K, T ; φ) =E Q e r(τ e ) EEP τ e (E,K,T; φ) 1 {τ e<t } F 0. Tha is, he discouned and sopped early exercise premium is, as expeced, a maringale under measure Q. 5 Such an inerpreaion is subsanially differen from he one implici in he characerizaion of he American opion already offered by Kim (1990), Jacka (1991), Carr, Jarrow, and Myneni (1992), Kim and Yu (1996), and Deemple and Tian (2002). For all hese auhors, he early exercise premium corresponds o he compensaion ha he opion holder would require (in he sopping region) in order o pospone exercise unil he mauriy dae. Under he geomeric Brownian moion assumpion, and for some early exercise boundary specificaions see, for example, Ju (1998) i is possible o obain closed-form soluions for such early exercise represenaion. However, for more general underlying diffusion price processes, as he ones proposed by Kim and Yu (1996), and Deemple and Tian (2002), i is necessary o solve numerically and recursively a se of value-maching implici inegral equaions, which can be oo ime-consuming for pracical purposes. To improve efficiency, Huang, Subrahmanyam, and Yu (1996) calculae only opion values based on a few poins on an approximaion o he exercise boundary, and hen use Richardson exrapolaion. Such acceleraed recursive scheme is very fas bu no very accurae, especially for medium- and long-erm opions see, for example, Ju ((1998), Tables 1 and 2). Alernaively, he new characerizaion offered by Proposiion 1 can be efficienly applied for any early exercise boundary specificaion, and under any Markovian (and diffusion) 4 I is well known ha he discouned price process of an American opion is a supermaringale under he risk-neural measure. Neverheless, such relaive price process behaves as a maringale during any period of ime in which i is no opimal o exercise he opion. Therefore, he same resul obains unil he firs passage ime o he exercise boundary. 5 Alernaively and as suggesed by an anonymous referee, he righ-hand-side of equaion (11) is simply he expeced value of he cash flow ha arises from liquidaing (a he firs passage ime o he exercise boundary) a saic porfolio ha includes a long posiion on an American opion and a shor posiion on he corresponding European conrac. 10

13 underlying price process, which consiues an innovaion wih respec o he represenaions of he early exercise premium already offered in he lieraure. B. Asympoic Properies Before implemening Proposiion 1 and in order o invesigae is limis, he asympoic properies of he early exercise represenaion (11) are firs explored. Proposiion 2 Under he assumpions of Proposiion 1, he early exercise premium and he American opion value saisfy he following boundary condiions for T : (18) lim r 0 EEP (S, K, T ;1)=0, (19) V T (S, K, T ; φ) =(φk φs T ) +, (20) lim S V (S, K, T ;1)=0, (21) lim S 0 V (S, K, T ; 1) = 0, and (22) lim S E V (S, K, T ; φ) =(φk φe ), where φ = 1 for an American call or φ =1for an American pu. Proof. See Appendix A. Once he general diffusion process (2) is adoped, he usual parabolic parial differenial equaion follows for he price of he American opion. Proposiion 3 Under he diffusion process (2), he American opion value funcion given by Proposiion 1 saisfies, for φs >φe and T, he parial differenial equaion (23) LV (S, K, T ; φ) =0, 11

14 where L is he parabolic operaor (24) L := σ (, S)2 S 2 2 +(r q) S 2 S2 S r +, φ = 1 for an American call and φ =1foranAmericanpu. Proof. See Appendix B. The relevance of Proposiions 2 and 3 emerges from he fac ha he American opion price is, under he sochasic differenial equaion (2), he unique soluion of he iniial value problem represened by he parial differenial equaion (23) and by he boundary condiions (19) hrough (22). Nex proposiion shows ha he American opion represenaion conained in Proposiion 1 converges o he appropriae perpeual limi. This resul conrass wih he characerizaion offered by Carr, Jarrow, and Myneni (1992) or Kim and Yu (1996), and can be relevan for he pricing of long-erm opion conracs. Explici pricing soluions are also given for boh he Meron (1973) and he CEV models, which will be used in he subsequen empirical analysis. The laer resul consiues an innovaion wih respec o he previous lieraure. Proposiion 4 Under he geomeric Brownian moion assumpion, ha is for σ (, S) =σ in equaion (2), he American opion value funcion given by Proposiion 1 converges, in he limi, o he perpeual formulae given by McKean (1965) or Meron (1973), i.e. µ γ(φ) E (25) lim V (S, K, T ; φ) =(φk φe ), T S where φs >φe, E denoes he consan exercise boundary, q r (26) γ (φ) := r q σ2 + φ 2 q σ σ2 r 2, σ 2 φ = 1 for an American call and φ =1foranAmericanpu. Under he CEV model and for r 6= q, he perpeual American opion price is equal o (27) lim T V (S, K, T ; φ) = (φk φe ) µ η(φ) S exp {η (φ)[x (S ) x (E )]} E h i M φ(β 2) η (φ)+( 1) η(φ) α, β 1 2η(φ) ;( 1) η(φ) x (S β 2 ) M φ(β 2) h η (φ)+( 1) η(φ) α, β 1 2η(φ) β 2 ;( 1) η(φ) x (E ) i, 12

15 where 1 {r>q,β<2} φ =1 (28) η (φ) := 1 1 {r>q,β>2} φ = 1, (29) α := r (β 2) (r q), (30) x (S) := 2(r q) δ 2 (β 2) S2 β, and M (a, b; z) λ>0 (31) M λ (a, b; z) := U (a, b; z) λ<0, wih M (a, b; z) and U (a, b; z) represening he confluen hypergeomeric Kummer s funcions. 6 For r = q, (32) lim T V (S, K, T ; φ) =(φk φe ) where r S β (33) ε (S) := 2S1 2 δ β 2, I 1 ε β 2 ;φ(β 2) (S ) 2r E I 1 ε β 2 ;φ(β 2) (E ) 2r, and I ν (z) λ>0 (34) I ν;λ (z) := K ν (z) λ<0, wih I ν (z) and K ν (z) represening he modified Bessel funcions. 7 Proof. See Appendix C. 6 As defined by Abramowiz and Segun ((1972), equaions and ). 7 See, for insance, Abramowiz and Segun ((1972), p. 375). 13

16 IV. The Firs Passage Time Densiy To implemen he new American opion value represenaion offered by Proposiion 1, i is necessary o compue he firs passage ime densiy of he underlying asse price o he moving exercise boundary. Following Buonocore, Nobile, and Ricciardi ((1987), eq. 2.7), a Fore (1943)-ype inegral equaion can be obained for he opimal sopping ime densiy under consideraion. Noably, such non-linear inegral equaion involves only he ransiion densiy funcion of he underlying asse price. This resul, conained in he nex proposiion, is valid for any Markovian underlying diffusion process and for any coninuous represenaion of he exercise boundary. Proposiion 5 Assuming ha he underlying asse price S follows a Markovian diffusion process and considering ha he opimal exercise boundary is a coninuous funcion of ime, he firs passage ime densiy of he underlying asse price o he moving exercise boundary is he implici soluion of he following non-linear inegral equaion: (35) Z u Q (φs u φe u S v = E v ) Q (τ e dv F 0 )=Q (φs u φe u F 0 ), for φs 0 American pu. >φe 0,whereu [,T], andwihφ = 1 foranamericancallorφ =1for an Proof. Assuming ha he exercise boundary is coninuous on [,u] and ha φs 0 >φe 0, while using definiion (5), he disribuion funcion of he opimal sopping ime can be wrien as: 8 Q (τ e u F 0 ) = Q inf (φs v φe v ) 0,φS u φe u v u 0 F +Q inf (φs v φe v ) 0,φS u >φe u v<u 0 F. 8 Noice ha inf 0 v<u [ (S v E v )] = sup 0 v<u (S v E v ). 14

17 Since Q [inf 0 v u (φs v φe v ) 0,φS u φe u F 0 ]=Q (φs u φe u F 0 ) and because he underlying price process is assumed o be Markovian, (36) Q (τ e u F 0 ) = Q (φs u φe u F 0 ) + Z u Q (φs u >φe u S v = E v ) Q (τ e dv F 0 ). Finally, considering ha Q (τ e u F 0 )= R u Q (τ e dv F 0 ), equaion (35) follows immediaely from equaion (36). Proposiions 1 and 5 show ha an explici soluion for he European opion and knowledge of he ransiion densiy funcion of he underlying price process are he only requiremens for he analyical valuaion of he American conrac. Hence, he proposed mehodology can be fruifully applied o many oher Markovian pricing sysems besides he sandard case covered by equaion (2). One of such exensions will be discussed in Secion VII. Proposiion 5 can be specialized easily for he Meron (1973) and he CEV models, which will be used in he numerical analysis o be presened in Secion VI. For σ (, S) =σ, he underlying price process as given by equaion (2) becomes lognormally disribued, and equaion (35) can be resaed as Z u µ µ (37) Φ φ Ez v Eu z Eu Q (τ e dv F 0 )=Φ φ z, u v u 0 wih (38) E z v := ³ ³ S0 ln E v + r q σ2 (v 2 0 ), σ and where Φ ( ) represens he cumulaive densiy funcion of he univariae sandard normal disribuion. Equaion (37) is consisen wih Park and Schuurmann ((1976), Theorem 1) and similar o he inegral equaion used by Longsaff and Schwarz ((1995), eq. A6). For σ (, S) =δs β 2 1, i is well known see, for example, Schroder ((1989), eq. 1) for β<2, or Emanuel and MacBeh ((1982), eq. 7) for β>2 ha (39) Q (S u E u S v = E v )= Q χ2 ( 2 2 β,2κe2 β u ) 2κE 2 β Q χ2 (2+ 2 β 2,2κE2 β v e (2 β)(r q)(u v) ) 15 v e (2 β)(r q)(u v) β<2, 2κE 2 β u β>2

18 wih (40) κ := 2(r q) (2 β) δ 2 [e (2 β)(r q)(u v) 1], and where Q χ 2 (a,b) (x) represens he complemenary disribuion funcion of a non-cenral chi-square law wih a degrees of freedom and non-cenraliy parameer b. Combining equaions (35) and (39), a non-linear inegral equaion follows immediaely for he opimal sopping ime densiy under he CEV model. Excep for such crude criical asse price specificaions as, for example, he consan and exponenial funcional forms used by Ingersoll (1998) under he geomeric Brownian moion assumpion, he opimal sopping ime densiy is no known in closed-form. Following Kuan and Webber (2003), he nex proposiion shows ha such firs passage ime densiy can be efficienly compued, for any exercise boundary specificaion, hrough he sandard pariion mehod proposed by Park and Schuurmann (1976). Proposiion 6 Under he assumpions of Proposiion 5, and dividing he ime-inerval [,T] ino N sub-inervals of (equal) size h := T,hen N (41) EEP 0 (S, K, T ; φ) = NX nh i φk φe 0 + (2i 1)h 2 i=1 v 0 + (2i 1)h 2 o (E,K,T; φ) (2i 1)h r e 2 [Q (τ e = + ih) Q (τ e = +(i 1) h)], where φ = 1 for an American call or φ = 1 for an American pu. Q (τ e = + ih) are obained from he following recurrence relaion: The probabiliies (42) Q (τ e = + ih) h = Q (τ e = +(i 1) h)+ nf φ E 0 +ih; E 0 + (2i 1)h 2 ( Xi 1 i F φ (E 0 +ih; S 0 ) F φ he 0 +ih; E 0 + (2j 1)h 2 j=1 [Q (τ e = + jh) Q (τ e = +(j 1) h)]}, io 1 for i =1,...,N,whereQ (τ e = )=0,andwih (43) F φ (E u ; S 0 ):=Q (φs u φe u F 0 ) 16

19 represening he risk-neural cumulaive densiy funcion, for φ =1, or he complemenary disribuion funcion, for φ = 1, of he underlying price process. Proof. Equaions (41) and (42) are obained via discreizaion of equaions (11) and (35) for he pariion < 1 <...< N = T,where i = + ih (i =1,...,N), and u = i+ i 1 2. V. Specificaion of he Exercise Boundary The pricing soluion offered by Proposiion 1 depends on he specificaion adoped for he exercise boundary {E, T }. Alhough such an opimal exercise policy is no known ex ane (i.e., before he soluion of he pricing problem), is main characerisics have already been esablished in he lieraure: 9 i) The exercise boundary is a coninuous funcion of ime see, for insance, Jacka ((1991), Proposiions and 2.2.5); ii) E is a nondecreasing funcion of ime for he American pu, bu non-increasing for he American call conrac see Jacka ((1991), Proposiion 2.2.2); iii) he exercise boundary is limied by rq E T = φ min ³φK, φ K as saed in Van Moerbeke (1976); and iv) lim E = E,where E represens he (consan) criical asse price for he perpeual American case. As described by Ingersoll ((1998), p. 89), in order o price an American opion, i is necessary o choose a parameric family E of exercise policies E (θ), where each policy is characerized by an n-dimensional vecor of parameers θ R n. Then, he early exercise value (as given by equaion (11)) is expressed as a funcion of θ and maximized wih respec o he parameers. Since he chosen family E may no conain he opimal exercise boundary, he resuling American opion price consiues a lower bound for he rue opion value. Of course, he more general he specificaion adoped for he exercise boundary, he smaller he approximaion error associaed wih he American price esimae should be. However, he parameric families already proposed in he lieraure have been chosen no for heir generaliy bu because hey provide fas analyical pricing soluions. In order o 9 Bunch and Johnson (2000) propose, under he Meron (1973) model, an approximaion for he criicalsock-price funcion which is accurae for small imes o mauriy. 17

20 measure he accuracy improvemen provided by more general families of exercise policies, Secion VI will consider he following parameric specificaions: 1. Consan exercise boundary: (44) E (θ) =θ 1,θ 1 > 0. This is he simples specificaion one can adop and has already been used by Ingersoll (1998) and Sbuelz (2004), under he geomeric Brownian moion assumpion. Alhough i yields a closed-form soluion for equaion (11), such an exercise boundary canno simulaneously saisfy previously saed requiremens (iii) and (iv). 2. Exponenial family: (45) E (θ) =θ 1 e θ 2(T ),θ 1 > 0,φθ 2 < 0. This specificaion, already proposed by Ingersoll (1998) for he geomeric Brownian moion process, also yields an analyical soluion for equaion (11), bu again canno simulaneously saisfy requiremens (iii) and (iv). 3. Exponenial-consan family: (46) E (θ) =θ 1 + e θ 2(T ),φθ 2 < 0. This new parameerizaion corresponds o a simple modificaion of equaion (45) and has never been proposed in he lieraure. Secion VI will show ha i can produce smaller pricing errors han equaion (45) for he same number of parameers. 4. Polynomial family: (47) E (θ) = nx θ i (T ) i 1. i=1 18

21 Because he exercise boundary is assumed o be coninuous and definedonheclosed inerval [,T], he Weiersrass approximaion heorem implies ha E can be uniformly approximaed, for any desired accuracy level, by he polynomial (47). By increasing he degree of he polynomial (and herefore, he number of parameers o be esimaed), his new class of exercise policies allows he pricing error o be arbirarily reduced. Secion VI will reveal ha wih only five parameers i is possible o obain smaller pricing errors han wih many alernaive specificaions already proposed in he lieraure. 5. CJM family: (48) E (θ) =φ min µφk, φ rq K e θ ³ 1 T + E 1 e θ 1 T,θ 1 0. Equaion (48) corresponds o an exponenially weighed average beween he erminal bound and he perpeual limi of he exercise boundary, and fulfills all of requiremens (i) (iv). Such a specificaion was proposed by Carr, Jarrow, and Myneni ((1992), p. 93), bu has never been esed since i does no yield an analyical soluion for he American opion price. The nex secion will show ha, wih only one parameer, he magniude of pricing errors produced by his specificaion is similar o ha associaed wih he bes parameerizaions already available in he lieraure. VI. Numerical Resuls To es he accuracy and efficiency of he pricing soluions proposed in Proposiion 1 and he influence of he exercise boundary specificaion on he early exercise value, all he parameric families described in Secion V will be compared for differen consellaions of he coefficiens conained in equaion (2), and under wo special cases: he geomeric Brownian moion and he CEV processes. For his purpose, he maximizaion of he early exercise value (wih 19

22 respec o he parameers defining he exercise policy) is implemened hrough Powell s mehod, as described in Press, Flannery, Teukolsky, and Veerling ((1994), Secion 10.5). 10 To enhance he efficiency of he proposed valuaion mehod, he parameers defining he exercise policy are firs esimaed by discreizing boh Proposiions 1 and 5 using only N =2 4 ime-seps. Then, and based on such an approximaion for he opimal exercise boundary, he early exercise premium is compued from Proposiion 6 using N =2 8 ime seps. The crude discreizaion adoped in he opimizaion sage should no compromise he accuracy of he pricing formulae proposed because, as noed by Ju ((1998), p. 642) in he conex of he Meron (1973) model, a deailed descripion of he early exercise boundary is no necessary o generae accurae American opion values. Table 1 values shor mauriy American pu opions under differen specificaions of he exercise boundary, and based on he opion parameers conained in Broadie and Deemple ((1996), Table 1), and Ju ((1998), Table 1) for he Black and Scholes (1973) model. Accuracy is measured by he average absolue percenage error (over he 20 conracs considered) of each valuaion approach and wih respec o he exac American opion price. This proxy of he rue American pu value (fourh column) is compued hrough he binomial ree model wih 15, 000 ime seps, as suggesed by Broadie and Deemple ((1996), p. 1222). Efficiency is evaluaed by he oal CPU ime (expressed in seconds) spen o value he whole se of conracs considered. All compuaions were made wih Pascal programs running on an Inel Penium GHz processor under a Linux operaing sysem. Inser Table 1 abou here. The American pu prices produced by he analyical pricing soluions associaed wih he consan and exponenial boundary specificaions (fifh and sixh columns of Table 1), as 10 This mehod requires evaluaions only of he funcion o be maximized and herefore is faser han a conjugae gradien or a quasi-newon algorihm. Neverheless, i is always possible o use a more robus opimizaion mehod, because he derivaives of he firs passage ime densiy can be compued hrough a recurrence relaion similar o equaion (42). Deails are available upon reques. 20

23 given by equaions (44) and (45), respecively, are obained from Ingersoll ((1998), secions 4 and 5). All he oher early exercise boundary approximaions (i.e., from he sevenh o he enh columns of Table 1) are implemened hrough Proposiion 6. For comparison purposes, he las hree columns of Table 1 conain he American pu prices generaed by he full (wih 2, 000 ime seps) 11 and he 10-poin acceleraed recursive mehods of Huang, Subrahmanyam, and Yu (1996), and by he hree-poin mulipiece exponenial funcion mehod proposed by Ju (1998). The choice of he mulipiece exponenial approximaion as a benchmark for he bes pricing mehods already proposed in he lieraure, under he geomeric Brownian moion assumpion, follows from Ju ((1998), Tables 3 and 5): i is faser han he randomizaion mehod of Carr (1998) (for he same accuracy level) and much more accurae, for hedging purposes, han he economeric approach of Broadie and Deemple (1996). The fases approximaions (in erms of CPU ime) are he consan, he exponenial, and he hree-poin mulipiece exponenial specificaions, as well as he acceleraed recursive mehod of Huang, Subrahmanyam, and Yu (1996): hey all possess compuaional imes below 0.2 seconds for he range of all conracs under consideraion. However, he pricing errors generaed by he consan and he exponenial parameerizaions can be significan. For insance, he average mispricing of he consan parameerizaion equals 41 basis poins. Addiionally, and as shown by Ju ((1998), Tables 1 and 2), he accuracy of he 10-poin recursive scheme deerioraes as he opion mauriy increases. Wih he same number of parameers as he already known exponenial approximaion, he new exponenial-consan parameerizaion can yield pricing errors abou hree imes smaller. Even more ineresingly, he CJM approximaion suggesed by Carr, Jarrow, and Myneni (1992) and esed here possesses an accuracy similar o he hree-poin mulipiece exponenial approach. This resul is relevan since he CJM approximaion saisfies all he requiremens described in Secion V for he early exercise boundary specificaion. Table 1 also shows ha he implemenaion of a polynomial approximaion is able o achieve smaller pricing errors han he Ju (1998) approach. The Huang, Subrahmanyam, 11 As suggesed by Deemple and Tian ((2002), p. 924). 21

24 and Yu (1996) full recursive mehod yields an even higher precision level, bu a he expense of a prohibiive compuaional effor. Overall, aking ino consideraion boh accuracy and efficiency, he bes pricing mehodology, under he geomeric Brownian moion assumpion, is sill he mulipiece exponenial approach of Ju (1998). Neverheless, he dispariy of pricing errors conained in Table 1 shows ha he early exercise premium depends largely on he specificaion adoped for he early exercise boundary. Inser Table 2 abou here. Tables 2 and 3 repea he analysis conained in Table 1 for he same parameer values, bu under he CEV model. Table 2 assumes β =3(> 2) and prices American pu conracs wih a ime-o-mauriy of six monhs, while Table 3 considers a square roo process wih β =1(< 2) and American call opions wih a ime-o-mauriy of one year. Parameer δ is compued from equaion (3) by imposing he same insananeous volailiy as in Table 1. The proxy of he exac American opion price (fourh column) is now compued hrough he Crank-Nicolson finie difference mehod wih 15, 000 ime inervals and 10, 000 space seps. Besides he early exercise boundary specificaions described in Secion V, Tables 2 and 3 also conain he full recursive scheme (elevenh column), as suggesed by Deemple and Tian ((2002), Proposiion 3), and a 10-poin acceleraed recursive approach (las column), along he lines of Kim and Yu ((1996), subsecion 3.4). 12 Inser Table 3 abou here. As before, he consan specificaion generaes excessively large (absolue) pricing errors and he new exponenial-consan parameerizaion yields an accuracy higher han he 12 The rinomial approach developed by Boyle and Tian (1999) for he valuaion of barrier and lookback opions under he CEV model (for 0 β<2) can also be used o price American sandard calls and pus. However, he numerical experimens run have shown ha he adoped Crank-Nicolson scheme possesses beer convergence properies. 22

25 exponenial specificaion for American pu conracs (see Table 2). In conras, Table 3 shows ha he exponenial boundary is more accurae for American call conracs han he new formulaion given by equaion (46). Under he CEV model, he CJM approximaion presens an excellen performance even hough he pricing errors are now affeced by he approximaion employed o evaluae he non-cenral chi-square disribuion funcion, 13 as well as by he roo-finding rouine used o exrac he opimal consan exercise boundary E from equaions (27) and (32). In erms of accuracy, he Deemple and Tian (2002) approach consiues he bes pricing mehod for he CEV model. However, his approach is based on he full recursive mehod (wih 2, 000 ime seps) of Huang, Subrahmanyam, and Yu (1996), which is very ime consuming six imes slower han he exac Crank-Nicolson implici finie-difference scheme. The acceleraed recursive scheme of Kim and Yu (1996) is much more efficien bu can also be inaccurae for medium- and long-erm opions. The las column of Table 3 shows a mean absolue percenage error of abou 16 basis poins. On he conrary, Tables 1 hrough 3 show ha he accuracy of he pricing mehodology proposed in Proposiion 1 is no affeced by he ime-o-mauriy of he opion conrac under valuaion. Moreover, for almos all he parameerizaions esed (wih he single excepion of he polynomial specificaion), he compuaional ime of he proposed pricing mehodology corresponds o less han one second per conrac. Inser Table 4 abou here. Under he CEV model, he bes rade-off beween accuracy and efficiency is given by he polynomial approximaions presened in Tables 2 and 3, since heir accuracy can always be improved by increasing heir degree. Table 4 applies differen polynomial specificaions o a 13 Equaion (39) is compued from rouine cumchn, which is conained in he Forran library of Brown, Lovao, and Russell (1997). This rouine is based on Abramowiz and Segun ((1972), eq ), and is found o be more precise han he algorihm offered by Schroder (1989) or he Wiener germ approximaions proposed by Penev and Raykov (1997), especially for large values of he non-cenraliy parameer or of he upper inegraion limi. 23

26 random sample of 1, 250 American pu opions, where all he opion parameers, wih he excepion of β and δ, are exraced from he same uniform disribuions as in Ju ((1998), Table 3). 14 Wih a six-degree polynomial i is possible o obain an average absolue percenage error (compued agains he Crank-Nicolson soluion) of only 1.5 basispoinsanda maximum absolue percenage error of abou 9 basis poins, which corresponds o a higher accuracy han ha associaed wih he 10-poin acceleraed recursive scheme. As expeced, he pricing errors produced by he specificaions described in Secion V are negaive because any approximaion of he opimal exercise policy can yield only a lower bound for he rue American opion price. The only excepion corresponds o he 10-poin recursive mehod, which migh be explained by he non-uniform convergence of he Richardson exrapolaion employed. In summary, he numerical resuls presened in Tables 2, 3 and 4 configure he implemenaion of Proposiion 1 hrough a polynomial specificaion of he early exercise boundary as he bes pricing alernaive, under he CEV model, for medium- and long-erm American opion conracs. VII. Exension o Credi Risk Modeling This secion shows ha he opimal sopping approach proposed in his paper is easily exended o he conex of he Carr and Linesky (2006) model, yielding analyical pricing soluions for American equiy opions under defaul risk. Carr and Linesky (2006) consruc a unified framework for he valuaion of corporae liabiliies, credi derivaives, and equiy derivaives as coningen claims wrien on a defaul- 14 From he uniform disribuion adoped for he insananeous volailiy, he parameer δ is obained from equaion (3). Parameer β is assumed o possess a uniform disribuion beween 0 and 4.0. The scenario β < 0 is ignored because i would imply unrealisic economic properies for he CEV process; namely, bankrupcy would be aainable for sufficienly negaive values of β (which is implausible, for insance, when considering opions on sock indices), and underlying asse price volailiy would explode as he spo price ends o he origin. 24

27 able sock. The price of he defaulable sock is modeled as a ime-inhomogeneous diffusion process solving he sochasic differenial equaion (49) ds S =[r q + λ (, S)] d + σ (, S) dw Q, wih S 0 > 0, and where he ineres rae r and he dividend yield q are now deerminisic funcions of ime, while he insananeous volailiy of equiy reurns σ (, S) and he defaul inensiy λ (, S) can also be sae-dependen. Again, F = {F : } is he filraion generaed by he sandard Wiener process W Q is aken as given. 15 R, and he equivalen maringale measure Q The pricing model proposed by Carr and Linesky (2006) can eiher diffuse or jump o defaul. In he firs case, bankrupcy occurs a he firs passage ime of he sock price o zero: (50) τ 0 := inf { > : S =0}. 1 1 {<τ0 } Alernaively, he sock price can also jump o a cemeery sae whenever he hazard process R λ (u, S) du is greaer or equal o he level drawn from an exponenial random variable Θ independen of W Q and wih uni mean, i.e. a he firs jump ime Z 1 ¾ (51) ζ := inf ½> : λ (u, S) du Θ 1 {<τ 0 } of a doubly-sochasic Poisson process wih inensiy λ (, S). Therefore, he ime of defaul is simply given by (52) ζ = τ 0 ζ, and D = {D : } is he filraion generaed by he defaul indicaor process D = 1 {>ζ}. 15 The inclusion of he hazard rae λ (, S) in he drif of equaion (49) compensaes he sockholders for defaul (wih zero recovery) and insures, under measure Q, an expeced rae of reurn equal o he risk-free ineres rae. Neverheless, such equivalen maringale measure will no be unique because he arbirage-free marke considered by Carr and Linesky (2006) is incomplee in he sense ha he jump o defaul will no be modeled as a sopping ime of F. 25

28 Using he same erminology as in Secion II, he ime- valueofanamericanopionon he sock price S, wih srike price K, and wih mauriy dae T can now be represened by he following Snell envelope: V 0 (S, K, T ; φ) = sup ne Q he i T τ r (53) l dl 0 (φk φs T τ ) + 1 {ζ>t τ} G0 τ T h +E Q e io ζ r l dl 0 (φk) + 1 {ζ T τ} G0, where T is he se of all sopping imes (aking values in [, ]) forheenlargedfilraion G = {G : },wihg = F D. For he American call (φ = 1) here is no recovery if he firm defauls. However, for he American pu (φ =1), he second expecaion on he righ-hand side of equaion (53) corresponds o a recovery paymen equal o he srike K a he defaul ime ζ T τ. Moreover, since he (unknown) early exercise boundary lies beween zero (he bankrupcy boundary) and S 0 (given ha he American pu is assumed o be alive on he valuaion dae), hen he defaul even canno precede he early exercise of he opion conrac, ha is {ζ T τ} = {ζ T }. Therefore, (54) V 0 (S, K, T ; φ) =V 0 (S, K, T ; φ)+v D (S, K, T ; φ), where (55) V 0 (S, K, T ; φ) = sup τ T ne Q he τ r l dl (φk φs τ ) + 1 {ζ>τ} 1 {τ<t} G0 i +e T r l dl E Q (φk φst ) + 1 {ζ>t} 1 {τ T } G 0 o, and (56) V D (S, K, T ; φ) =(φk) + E Q ³ e ζ r l dl 1 {ζ T } G0. For he American pu, he erm V D 0 (S, K, T ;1) is essenially an American-syle defaulconingen claim, which is similar o he floaing leg of a credi defaul swap (CDS), and can be valued hrough Carr and Linesky ((2006), eq. 3.4). Concerning he American opion value condiional on no defaul, ideniy (7) implies ha (57) V 0 (S, K, T ; φ) =v 0 (S, K, T ; φ)+eep 0 (S, K, T ; φ), 26

29 where (58) v 0 (S, K, T ; φ) =e T r l dl E Q (φk φst ) + 1 {ζ>t} G 0 represens he ime- price of he corresponding European opion (condiional on no defaul unil he mauriy dae T ), and he early exercise premium is equal o (59) EEP 0 (S, K, T ; φ) = sup ne Q he i τ r l dl 0 (φk φs τ ) + 1 {ζ>τ} 1 {τ<t} G0 τ T e T r l dl E Q (φk φst ) + 1 {ζ>t} 1 {τ<t} G 0 o. The nex proposiion provides an analyical represenaion of he early exercise premium giveninequaion(59). Proposiion 7 Under he pricing model defined by equaions (49) hrough (52), he ime- value of he early exercise premium for an American opion on he sock price S, wih srike price K, and wih mauriy dae T is equal o (60) = EEP 0 (S, K, T ; φ) Z T e u r l dl 0 (φk φe u ) + vu 0 (E,K,T; φ) SP (,u) Q (τ e du F 0 ), where {E u, u T } ishe(unknown)earlyexerciseboundary,φ =1( 1) foranamerican pu (call), and (61) SP (,u):=e Q h e u λ(l,s)dl 1 {τ 0 >u} i F 0 represens he risk-neural probabiliy of surviving beyond ime u>.thefirs passage ime τ e is defined by equaion (5), and is probabiliy densiy funcion is sill recovered hrough equaion (35). Proof. See Appendix D. Because boh he defaul inensiy and he insananeous sock volailiy have been lef unspecified, Proposiion 7 can be applied o many defaulable sock models already available in he lieraure, such as hose proposed by Madan and Unal (1998) or Linesky (2006). 27

30 Carr and Linesky (2006) ry o accommodae he leverage effec by adoping a CEV specificaion for he insananeous sock volailiy: (62) σ (, S) =a S β, where β <0 is he volailiy elasiciy parameer, and a > 0 is a deerminisic volailiy scale funcion. To be consisen wih he empirical evidence of a posiive relaionship beween defaul probabiliies and equiy volailiy, Carr and Linesky (2006) furher assume ha he defaul inensiy is an increasing affine funcion of he insananeous sock variance: (63) λ (, S) =b + cσ (, S) 2, where c 0, andb 0 is a deerminisic funcion of ime. In summary, equaions (49) o (52), ogeher wih equaions (62) and (63) consiue he jump o defaul exended CEV model (JDCEV) proposed by Carr and Linesky (2006). 16 Compared wih he previous lieraure on defaulable sock models, he JDCEV model offers an exac mach o he erm srucures of CDS spreads and/or a-he-money implied volailiies (hrough he ime-dependen funcions a and b ), even hough i explicily incorporaes he dependency on he curren sock price S of boh λ (, S) and σ (, S). Neverheless, he JDCEV model preserves analyical racabiliy since i offers closed-form soluions for boh European opions and he ransiion densiy funcion of he underlying sock price process. The nex proposiion provides an analyical soluion for he disribuion funcion of he price process S, whichallowshefirs passage ime densiy o be deermined for he JDCEV model hrough he numerical soluion of he non-linear inegral equaion (35). 16 Noe ha he ime-homogeneous version of he JDCEV model wih b = c =0is reduced o a CEV process wih absorpion a poin. In his case, equaions (11) and (60) differ only because he former akes he survival probabiliy o equal 1. However, since Delbaen and Shirakawa ((2002), Theorem 4.2) have shown ha he CEV model admis arbirage opporuniies when i is condiioned o be sricly posiive, Proposiion 7 should also be applied o he CEV process. 28