DEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK

Transcription

1 DEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Séphane Crépey Déparemen de Mahémaiques Universié d Évry Val d Essonne 9125 Évry Cedex, France Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne 9125 Évry Cedex, France and Europlace Insiue of Finance Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, NSW 252, Ausralia and Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology -661 Warszawa, Poland Sepember 16, 28 Noe o he Reader: This is an updaed version of he paper forhcoming under he same ile in he journal Mahemaical Finance, mean for consisency wih he laes developmens of he companion paper [4]. The research of T.R. Bielecki was suppored by NSF Gran and Moody s Corporaion gran The research of S. Crépey was suppored by Io33. The research of M. Jeanblanc was suppored by Io33 and Moody s Corporaion gran The research of M. Rukowski was suppored by he 27 Faculy Research Gran PS12918.

2 2 Defaulable Opions in a Markovian Inensiy Model 1 Inroducion In Bielecki e al. [4], we sudied he valuaion and hedging of defaulable game opions in a very general reduced-form model of credi risk. Given a filered probabiliy space (Ω, G, P, used o model he primary marke, i was assumed in [4] ha G = H F, where he filraion H carries he informaion abou he defaul and he reference filraion F represens all oher informaion available o raders. The main echnique employed in [4] was he effecive reducion of he informaion flow from he full filraion G o he reference filraion F. Working under a risk-neural probabiliy measure Q and under suiable condiions on he F opional projecion of he defaul indicaor process H = 1 {τd }, we derived convenien pricing formulae wih respec o he reference filraion F. In addiion, we proved ha, under suiable inegrabiliy and regulariy condiions embedded in he sanding assumpion ha a relaed doubly refleced BSDE admis a soluion under Q, he saeprocess of his soluion corresponds o he minimal (superhedging price wih a (G, Q sigma (or local, under suiable assumpions maringale 1 cos. This resul is acually ineresing even beyond he scope of credi risk, as i provides a general connecion beween, on he one hand, arbirage prices of an opion (a game opion, including American and European opions as special cases, defaulable or no (he laer case corresponding o τ d =, and, on he oher hand, a suiable noion of hedging wih a sigma (or local maringale cos, in a general, possibly incomplee, marke. In he special case of a complee marke, he cos of he relaed hedging sraegies vanishes, and he hedging sraegies are super-hedges in he usual sense. For an efficien pracical implemenaion, a (dynamic pricing model should possess a suiable Markovian propery. For his reason, we propose in his paper a generic Markovian pre-defaul inensiy model of credi risk, which encompasses as a special case he jump-diffusion model sudied in deail in [5] (cf. Subsecion 4.3 of he presen paper. As a prerequisie, we recall in Theorem 2.1 (a varian of he main resuls from [4]. As compared wih [4], we work in his paper under he slighly sronger assumpion ha he doubly refleced BSDE (E associaed wih a defaulable game opion has a soluion (Θ, M, K where K is a coninuous process and M belongs o H 2. Though unnecessary from he sricly mahemaical poin of view (see [4], he laer requiremens are imporan in view of pracical use of our previous resuls, like showing ha (E is well-posed, esablishing he connecion wih a PDE formulaion of he problem in Markovian seings, devising appropriae numerical approximaion schemes, ec. I should be made clear ha in our previous work [4], we simply posulaed ha a primary marke arbirage price process X is given and i saisfies all our assumpions. We did no address he issues of proving exisence and/or building such marke models. In order o fill his gap, we develop in Subsecion 3.1 a generic mehod of consrucing such an arbirage price process X (see Proposiion 3.1. In paricular, we provide in Lemma 3.1 (see also Corollary 3.1 a general condiion which should be imposed on a pre-defaul primary marke model in order o make he model arbirage-free. Under a raher generic specificaion of he infiniesimal generaor of a driving Markov facor process, we subsequenly develop in Subsecions 3.2 o 3.4 he variaional inequaliy approach (cf. (34 o pricing and hedging of a defaulable game opion. Le us sress again ha puing he previous heoreical resuls in a Markovian framework is a necessary sep owards any implemenaion. The generic Markovian model considered in his paper is also ineresing as a concree example of he oherwise absrac maerial presened in [4] or in Secion 2 (see Theorems 2.1 and 2.2 of his paper. Finally, in Secion 4, we illusrae our sudy by considering converible bonds. We specify o his case he general variaional inequaliy (34 and we emphasize he crucial role of he freedom o choose he mos convenien driver (i.e., he parameer process F in equaion (E. 1 Sigma maringales are a relevan generalizaion of local maringales, see, for insance, [7, 18, 24].

3 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski General Se-Up For a finie horizon dae T >, we assume ha he primary marke is composed of he saving accoun and of d risky asses wih price processes defined on a filered probabiliy space (Ω, G, P, where P denoes he saisical probabiliy measure. We posulae ha (cf. [3]: he discoun facor process β, ha is, he inverse of he savings accoun, is a G-adaped, finie variaion, coninuous, posiive and bounded process; he prices of risky asses are G-semimaringales wih càdlàg sample pahs. The primary risky asses, wih R d -valued price process X, are assumed o pay dividends, whose cumulaive value process, denoed by D, is modeled as a G-adaped, càdlàg and R d -valued process of finie variaion. Given he price process X, we define he cumulaive price X of he asse as X = X + β 1 β u dd u. (1 [,] We assume ha β X is locally bounded and ha he primary marke model is free of arbirage opporuniies (hough presumably incomplee, in he sense ha here exiss a risk-neural measure Q M, where M denoes he se of probabiliy measures Q equivalen o P for which β X is a (G, Q local maringale. Noe ha relaxing he assumpion of local boundness on β X, he only saemen ha would change in his paper is he previous one, namely he characerizaion of arbirages prices. This characerizaion would hen be in erms of (G, Q sigma maringales raher han in erms of (G, Q local maringales. Since we wan o avoid he noion of sigma maringales in his paper, o keep i more user s friendly, we prefer o work under his harmless assumpion on β X. We refer he ineresed reader o [4] for he mos general resuls under minimal assumpions. In his paper, similarly as in [3, 4], we work wih he noion of a vecor (as opposed o componenwise sochasic inegral (see Cherny and Shiryaev [7]. By convenion, we denoe by he inegral over (, ]; oherwise, we explicily specify he domain of inegraion as a subscrip of. Also noe ha in wha follows we in fac deal wih righ-coninuous and compleed versions of all relevan filraions, so ha all he filraions under consideraion saisfy he so-called usual condiions. 2 Valuaion and Hedging of Defaulable Opions in he Hazard Process Se-Up: A User s Guide In [4], we derived general hedging resuls for a game opion under fairly general assumpions in he so-called hazard process se-up. In he same framework, and a he cos of slighly sronger assumpions (see Remark A.1(i in he Appendix, we shall now derive varians of hese resuls ha are required in pracical applicaions of he general heory. In his secion, we work under a risk-neural measure Q which is fixed hroughou. So all he measure-dependen noions like (local maringale, compensaor, ec., implicily refer o he probabiliy measure Q. 2.1 Hazard Process Se-Up Given a [, + ]-valued G sopping ime τ d represening he defaul ime of a reference eniy, we assume ha G = H F, where he filraion H is generaed by he defaul indicaor process H = 1 {τd } and F is some reference filraion. We assume ha he process G given by G = Q(τ d > F for R + is (sricly posiive, coninuous and non-increasing. Hence he F hazard process Γ = ln(g of τ d is well defined, coninuous and non-decreasing on R +. The G sopping

4 4 Defaulable Opions in a Markovian Inensiy Model ime τ d is hen an F pseudo-sopping ime ([23], see also [4], which means in paricular ha any F local maringale sopped a τ d is a G local maringale (cf. [23, Theorem 4]. I is also posulaed hroughou Secion 2 ha he defaul ime τ d avoids F sopping imes, ha is, Q(τ d = τ = for any F sopping ime τ. Under he coninuiy assumpion on Γ, his would for insance (bu no only be he case under he hypohesis (no made in his paper ha any F-maringale is coninuous, see Mansuy and Yor [22, p.25]. The sanding assumpion ha τ d avoids F sopping imes implies, in paricular, ha an càdlàg process Y canno jump a τ d, ha is, Y τd := Y τd Y τd =, almos surely. We shall someimes assume, in addiion, ha he F-adaped processes β and Γ are absoluely coninuous wih respec o he Lebesgue measure, specifically: β = exp( r u du for an F-adaped, bounded from below shor-erm ineres rae process r, Γ = γ u du, for a non-negaive F-adaped process γ, called he F inensiy process of τ d. A se-up saisfying he laer assumpions will be referred o as a defaul inensiy se-up. We now recall he concep of a (dividend paying defaulable game opion (see [2, 19, 3, 4] wih incepion dae and mauriy dae T. For any [, T ], le FT (resp. G T denoe he se of [, T ]-valued F (resp. G-sopping imes; given a furher τ FT, le Ḡ T sand for {τ G T ; τ τ d τ τ d }. The sopping ime τ FT in he following definiion is used o model he resricion ha he issuer of a game opion may be prevened from calling he opion during some random ime inerval [, τ (see [3]. Le Ḡ T sand for {τ G T ; τ τ d τ τ d }. Definiion 2.1 A defaulable game opion wih lifing ime of he call proecion τ FT, is a game opion wih he ex-dividend cumulaive discouned cash flows β π(; τ p, τ c given by he formula, for any [, T ] and (τ p, τ c GT Ḡ T, β π(; τ p, τ c = τ β u dd u + 1 {τd >τ}β τ (1 {τ=τp<t }L τp + 1 {τ<τp}u τc + 1 {τ=t } ξ, (2 where τ = τ p τ c and where he dividend process D = (D [,T ] equals D = (1 H u dc u + R u dh u [,] for some coupon process C = (C [,T ], which is a F-predicable càdlàg process wih bounded variaion, and some real-valued, F-predicable, locally bounded recovery process R = (R [,T ], he pu paymen L = (L [,T ] and he call paymen U = (U [,T ] are F-adaped, real-valued, càdlàg processes, he inequaliy L U holds for every [τ d τ, τ d T, and he paymen a mauriy ξ is a real-valued, F T -measurable random variable. Noe ha π(; τ p, τ c is a G τ τd -measurable random variable. We furher assume ha he cumulaive discouned payoff is bounded from below. Specifically, here exiss a consan c such ha β L := β u dd u + 1 {τd >}β (1 {<T } L + 1 {=T } ξ c, [, T ]. [,] In order o ge an upper bound for his payoff, we shall someimes assume ha here exiss a consan c such ha β Û := β u dd u + 1 {τd >}β (1 {<T } U + 1 {=T } ξ c, [, T ]. (3 [,]

5 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 5 The class of defaulable game opions covers as special cases defaulable American opions (case τ = T. I can be shown ha he laer class includes defaulable European opions as a special case (sub-case where he maximum of β L is aained a T, see [3]. One defines likewise for [, T ], defaulable European opions as conracs wih cash flows φ( given by, β φ( = T β u dd u + 1 {τd >T }β T ξ. (4 Defaulable European opions can equivalenly be redefined as he sub-case of defaulable American opions for which he maximum of β L is aained a T (see [3]. All opions considered in his paper are poenially defaulable, so he expressions opion and defaulable opion will be used inerchangeably in he sequel. We are now in he posiion o inroduce he concep of hedging of a game opion. Recall ha X (resp. X is he price process (resp. cumulaive price process of primary raded asses, cf. (1. Definiion 2.2 By a (self-financing primary rading sraegy we mean a pair (V, ζ such ha: V is a G -measurable real-valued random variable represening he iniial wealh, ζ is an R 1 d -valued, β X-inegrable process represening holdings in primary risky asses. The wealh process V of (V, ζ saisfies, for [, T ], wih an iniial condiion V. d(β V = ζ d(β X (5 Definiion 2.3 (i An issuer hedge wih cos process ρ for he game opion wih ex-dividend cumulaive discouned cash flows βπ (cf. (2 is represened by a quadruple (V, ζ, ρ, τ c such ha: (V, ζ is a primary sraegy wih he wealh process V given by (5, a cos process ρ is a real-valued G-semimaringale wih ρ =, a (fixed call ime τ c belongs o Ḡ T, he following inequaliy is valid, for every pu ime τ p GT, β τ V τ + τ β u dρ u β π(; τ p, τ c, a.s. (6 (ii A holder hedge wih cos process ρ for he game opion is a quadruple (V, ζ, ρ, τ p such ha: (V, ζ is a primary sraegy wih he wealh process V given by (5, a cos process ρ is a real-valued G-semimaringale wih ρ =, a (fixed pu ime τ p belongs o GT, he following inequaliy is valid, for every call ime τ c Ḡ T, β τ V τ + τ β u dρ u β π(; τ p, τ c, a.s. (7 Issuer or holder hedges a no cos (ha is, wih ρ = are hus in effec issuer or holder superhedges. Remarks 2.1 (i The process ρ can also be inerpreed as he (running financing cos, ha is, he amoun of cash added o (if dρ or wihdrawn from (if dρ he hedging porfolio in order o ge a (no self-financing any longer superhedge. (ii Regarding he admissibiliy issues (see, e.g., Delbaen and Schachermayer [12], noe ha he l.h.s. of (6 (discouned wealh process wih financing coss included is bounded from below for any issuer hedge (V, ζ, ρ, τ c. Likewise, in he case of a bounded payoff π (ha is, assuming (3, he l.h.s. of (7 (discouned wealh process wih financing coss included is bounded from below for any holder hedge (V, ζ, ρ, τ p.

6 6 Defaulable Opions in a Markovian Inensiy Model Le us now consider he special case of a defaulable European opion wih cash-flows φ (cf. (4. Definiion 2.4 (i An issuer hedge wih cos ρ (a real-valued G-semimaringale wih ρ = for a defaulable European opion is a primary sraegy (V, ζ wih wealh process V such ha T β T V T + β u dρ u β φ(, a.s. If he inequaliy may be replaced by equaliy hen we deal wih an issuer replicaing sraegy wih cos ρ. (ii A holder hedge wih cos ρ (a real-valued G-semimaringale wih ρ = for a defaulable European opion is a primary sraegy (V, ζ wih wealh process V such ha T β T V T + β u dρ u β φ(, a.s. If he inequaliy may be replaced by equaliy hen we deal wih a holder replicaing sraegy wih cos ρ. 2.2 Valuaion and Hedging Resuls We will now sudy valuaion and hedging of a game opion under suiable inegrabiliy and regulariy condiions. These condiions are implicily embedded in he sanding assumpion ha a relaed doubly refleced BSDE (E, saed under he risk-neural measure Q, admis a soluion. Assuming ha (E has a soluion (which will hold under mild condiions, cf. he discussion following Definiion 2.5, we shall deduce explici hedging sraegies wih minimal iniial wealh and a (G, Q local maringale cos for a game opion. Le α = β exp( Γ denoe he credi-risk adjused discoun facor; noe ha he process α is bounded. We define he F-adaped processes D and F of finie variaion by seing, for [, T ], ( D = dcu + R u dγ u, F := α 1 α u d D u. (8 [,] Noe ha from he poin of view of he financial inerpreaion, he process D represens he dividend process (including he recovery paymen as seen in an equivalen ficiious defaul-free world (see [4]. Le F be a given F-adaped, finie variaion process, such ha F F be bounded from below; we shall refer o he process F as he driver in wha follows. We inroduce he r.v. χ and he processes L, Ū defined as wih U = U + F F. [,] χ = ξ + F T F T, L = L + F F, Ū = 1 {< τ} + 1 { τ} U We consider he following doubly refleced BSDE (E wih daa F, χ, L, U, τ (see Definiion 2.5: Θ = χ + (F T F T (F u + Θ u db u + K T K (M T M, [, T ], L Θ Ū, [, T ], T (Θ u L u dk u + = T (Ūu Θ u dku =, where B = ln α. Noe ha since d(α F = α (df F db, we have ha T T F T F F u db u = αu 1 d(α u F u (E

7 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 7 and likewise for Θ. Hence he firs line in (E can be rewrien as T T Θ = χ Θ u db u + αu 1 d(α u F u + K T K (M T M, [, T ], (9 Finally, le us consider he special case of a defaul inensiy se-up, so B = ln α = µ u du wih µ = r + γ, and of a driver F of he form F = α 1 α u f u du, (1 for some F-adaped ime-inegrable process f over [, T ] such ha he process F F is bounded from below. Then (9 can be rewrien as Θ = χ + T (f u µ u Θ u du + K T K (M T M, [, T ]. (11 In his case he relaed driver erms in (E are hus given as inegrals wih respec o he Lebesgue measure, which is he sandard form in he BSDE lieraure (see e.g. [1, 16, 11, 14]. Le H 2 denoe he se of F-maringales wih inegrable quadraic variaion over [, T ] vanishing a ime. Definiion 2.5 By a soluion o (E, we mean a riple (Θ, M, K such ha: he sae process Θ is a real-valued, F-adaped, càdlàg process, M lies in H 2, K is an F-adaped finie variaion coninuous process vanishing a ime, all condiions in (E are saisfied, where in he hird line K + and K denoe he Jordan componens of K (i.e., he decomposiion K = K + K where he non-decreasing coninuous processes K ± vanish a ime and define muually singular measures, and where he convenion ha ± = is made in he hird line. For various specificaions of he presen se-up and ses of echnical assumpions ensuring he exisence and uniqueness of a soluion o (E, we refer he reader o [1, 16, 5, 11]. Basically, for he daa in suiable spaces of square-inegrable processes and random variables, and allowing for jumps of L and U (a oally inaccessible F sopping imes, see Subsecion 4.4, he exisence of a soluion o (E (supplemened by suiable inegrabiliy condiions on he daa and he soluion is essenially equivalen o he so-called Mokobodski condiion, namely, he exisence of a quasimaringale Y such ha L Y U on [, T ] (see Crépey and Maoussi [1], Hamadène and Hassani [16, Theorem 4.1], and previous works in his direcion, saring wih [11]. Recall ha a quasimaringale Y can equivalenly be defined as a difference of wo non-negaive supermaringales, or in erms of a bound on some condiionally expeced variaions of Y on arbirary pariions of [, T ], or as a special semimaringale wih predicable finie variaion componen of inegrable variaion (Proer [24, Chaper III, secion 4]. In paricular, any square inegrable Iô-Lévy process S (Iô-Lévy process wih square inegrable special semimaringale decomposiion componens, or S l for any such process S and consan l, is a quasimaringale (see Crépey and Maoussi [1]. Hence he Mokobodski condiion is saisfied, and he exisence of a soluion for (E holds, whenever L and/or U is given by S or S l for such an Iô-Lévy process S, as i is he case in many pracical applicaions (see [1, 5]. Noe ha equaion (E (including he definiion of he barriers L and U in (E and is consequences are implicily parameerized by he choice of a driver F in (E. In fac (Θ, M, K solves equaion (E for some driver F if and only if ( Θ, M, K, where Θ = Θ + F, solves equaion (E for F = (and accordingly modified barriers. Ye, as we shall see in a concree example in Secion 4 (see, in paricular, Subsecion 4.4, he freedom of choosing he mos convenien driver is imporan in financial applicaions. So a paricular form of F may be seleced in order o deal wih he mos

8 8 Defaulable Opions in a Markovian Inensiy Model racable BSDE, namely, he BSDE wih he simples form of reflecing barriers, which are he mos difficul poin o ackle wih, from he poin of view of solving he BSDE (see Secion 4 and [1, 5]. In he case of Markovian models (see laer secions and [8, 5], his freedom will allow us o deal wih he relaed variaional inequaliies of he mos racable srucure. Given a soluion (Θ, M, K o (E, we define he Q pre-defaul price, price, cumulaive value, and opimal sopping imes, respecively, by, for [, T ], Π = Θ + F F, Π = 1 Π {<τd }, Π = Π + β 1 [,] β u dd u (12 { τp ( = inf u [, T ] ; Π } { u L u T, τc ( = inf u [ τ, T ] ; Π } u U u T. (13 Le N d = H Γ τd sand for he compensaed jump-o-defaul process. Recall ha he process N d is a G-maringale, under our sanding assumpion ha he F hazard process Γ of τ d is nondecreasing and coninuous. The following saemen follows by applicaion of he main resuls of [4] (see he Appendix. The noion of an arbirage price of a game opion referred o in poin (i below is a suiable exension o game opions (Definiion 2.6 in Kallsen and Kühn [19], see also [3] of he No Free Lunch wih Vanishing Risk (NFLVR condiion of Delbaen and Schachermayer [12, 7]. Theorem 2.1 Assume ha he BSDE (E has a soluion (Θ, M, K. Then : (i Π is an arbirage price process for he game opion; (ii The process m given by he formula, for [, T ], m = β Π + β u dk u, (14 is a G-maringale (sopped a τ d, such ha [, τ d ] dm = 1 { τd }β ( dm + ( R Π dn d ; (15 (iii Given an arbirary R 1 d -valued, predicable and locally bounded process ζ, le he process ρ = ρ(ζ be defined by ρ = and, for [, T ], β dρ = dm ζ d(β X. (16 Then (Π, ζ, ρ(ζ, τ c ( is an issuer hedge wih G local maringale cos, and ( Π, ζ, ρ(ζ, τ p ( (cf. (13 is a holder hedge wih G local maringale cos. (iv Π is he minimal iniial wealh of an issuer hedge wih G local maringale cos and, under condiion (3, Π is he minimal iniial wealh of a holder hedge wih G local maringale cos. Remarks 2.2 (i In view of (14 he process m can equivalenly be redefined as he G-local maringale componen of he discouned cumulaive Q-value process β Π. The processes m and β Π are easily seen o coincide on he random inerval [, τc ( τp ( τ d T ] and hus boh can be inerpreed on his inerval as he discouned cumulaive Q-value of a defaulable game opion. (ii One defines he hedging error process (also known as he racking error or he profi and loss process e = e(ζ relaive o he ex-dividend Q-price process Π, as seen from he perspecive of an opion s issuer, by, for [, T ]: Using (14 here comes, for [, T ], β e = β Π + β e = τd ζ u d(β u Xu β Π. (17 β u dk u β u dρ u. (18 Therefore β dρ can also be inerpreed as he G-local maringale componen of he G-special semimaringale βe.

9 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 9 In he case of a European opion, we consider he BSDE (E wih L replaced by L such ha α L = (c + 1, where c is a lower bound on α T χ. Noe ha under mild echnical assumpions his equaion has a soluion (Θ, m, K = (see [4, 1], so ha (E effecively reduces o a sandard BSDE wih no process K involved. We hen define he Q pre-defaul price, price and cumulaive value as, respecively Φ = Θ + F F, Φ = 1 Φ {<τd }, Φ = Φ + β 1 [,] β u dd u (19 Theorem 2.2 In he case of a European opion, assume ha he BSDE (E wih L replaced by L and wih τ = T has a soluion (Θ, M, K =. Then: (i Φ is an arbirage price process for he opion; (ii The discouned cumulaive Q-value process m = β Φ is a G-maringale (sopped a τ d, such ha dm = 1 { τd }β ( dm + ( R Φ dn d ; (2 (iii Given an arbirary R 1 d -valued, predicable and locally bounded process ζ, le he process ρ = ρ(ζ be defined by by ρ = and (16 wih mβ Φ herein. Then (Φ, ζ, ρ(ζ is an issuer replicaing sraegy wih G local maringale cos, and ( Φ, ζ, ρ(ζ is a holder replicaing sraegy wih G local maringale cos. (iv Φ is he minimal iniial wealh of an issuer replicaing sraegy (or hedge wih G local maringale cos and, for bounded R and ξ, Φ is he minimal iniial wealh of a holder replicaing sraegy (or hedge wih G local maringale cos. So, in he European case, he process m exacly corresponds o he discouned cumulaive Q-value process of he opion, and β dρ = βe = β Φ + ζ d(β X β Φ. Remarks 2.3 In Theorem 2.1 or 2.2: (i The special case ρ = corresponds o a paricular form of a model compleeness (aainabiliy of defaulable European claims, cf. Theorem 2.2; see also [5] in which he issuer (or he holder of he opion is able and wishes o hedge all risks embedded in he opion. The case where ρ corresponds o eiher model incompleeness or he siuaion of a complee model in which he issuer (or he holder is able o hedge, bu she prefers no o hedge all he risks embedded in he opion; for insance, she may be willing o ake some bes in specific risk direcions. (ii In cases where ρ may be aken equal o, he minimaliy saemens in pars (i of hese heorems may be used o prove uniqueness of he relaed arbirage prices (see [4, 5]. 3 Markovian Se-Up 3.1 Marke Model Facory In he previous secions we ook a primary marke model saisfying all assumpions as given. The goal of his secion is o presen a generic consrucion of an arbirage-free primary marke model in a defaul inensiy se-up. To his end, we assume ha we are given a sochasic basis (Ω, F, Q endowed wih he following processes: an F-adaped, bounded from below and ime-inegrable process r, which is inended o represen shor-erm ineres rae, an F-adaped, non-negaive and ime-inegrable process γ, which represens he defaul inensiy, an R d -valued càdlàg F-semimaringale X, which is aimed o model he pre-defaul prices of primary asses, as well as he associaed coupon process C and recovery process R, such ha:

10 1 Defaulable Opions in a Markovian Inensiy Model C is an R d -valued, F-predicable process of inegrable variaion, 2 R is an R d -valued, F-predicable and bounded from below process. Relevan ways o consruc such primary daa (Ω, F, Q, r, γ, X, C, R will be given laer in his secion (see Remark 3.1(i. In his subsecion, all he measure-dependen noions implicily refer o he probabiliy measure Q. Given hese primary daa, he consrucion of he primary marke model goes as follows. Firs, we define he discoun facor β = e R rudu. Nex, he so-called canonical consrucion yields a convenien mehod of defining a random ime τ d on an enlarged probabiliy space (Ω, G, Q, such ha (see, e.g., [6]: τ d is a G sopping ime wih respec o G = H F, where H is he filraion generaed by he defaul indicaor process H = 1 {τd }, he process γ is he F inensiy process of τ d, wih relaed F hazard process Γ = γ d, and hus τ d is an F pseudo-sopping ime (cf. Secion 2.1, τ d avoids F sopping imes (under he canonical consrucion, his propery can be shown by condiioning wih respec o F T. Finally, since X is inended o model he pre-defaul prices of primary asses, we se X = 1 X {<τd }. Le us observe ha X τd is a G-semimaringale (since τ d is an F pseudo-sopping ime, cf. Secion 2.1, and hus X is an R d -valued, G-semimaringale on [, T ], which is null on [, T ] [τ d,. The las feaure reflecs he fac ha any value a τ d is embedded in he recovery par of he dividend process D for X, given as D = (1 H u dc u + R u dh u (21 We furher define, for [, T ], he pre-defaul cumulaive dividend process [,] D = dc u + R u dγ u (22 he credi-risk adjused discoun facor α = exp( µ u du wih µ = r + γ, and he pre-defaul cumulaive price X X = X + α 1 Finally, we define he cumulaive price X by seing, for [, T ], β X = 1 {<τd }β X + [,] β u dd u = 1 {<τd }β ( X α 1 α u d D u. (23 α u d D u + β u dd u. (24 [,] The following resul is he analog, relaive o he primary marke, of ideniy (15 for a game opion. The proof is deferred o Appendix A.2. Lemma 3.1 One has, for [, T ] : d(β X = 1 { τd }β ( α 1 d(α X + ( R X dn d. (25 In paricular, if α X is an F local maringale, hen β X is a G local maringale. The siuaion considered in he following Corollary is common in applicaions (see, e.g., Proposiion As opposed o he case of a game opion, we do no assume he variaion of C o be bounded, in order o cover ypical examples, see e.g. [5].

11 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 11 Corollary 3.1 In he special case of a ime-differeniable primary coupon process C = c d for an F-adaped ime-inegrable coupon rae process c, and of a pre-defaul primary price process X wih ime-differeniable predicable finie variaion componen x d for some F-adaped ime-inegrable process x, hen β X is a G local maringale under he following arbirage drif condiion: Proof. Given (23, we have x = µ X c γ R, [, T ]. (26 d(α X = d(α X + α d D = α (d X µ X d + α d D. In view of (22 and (26, his implies ha α X is an F local maringale, hence β X is a G local maringale, by Lemma 3.1. The following proposiion furnishes a generic consrucive procedure for building arbirage price processes for a primary marke model. Proposiion 3.1 Le us be given a sochasic basis (Ω, F, Q, an F-adaped bounded from below and ime-inegrable process r, and an F-adaped, non-negaive and ime-inegrable process γ. Le a random ime τ d and filraions H and G be defined by he canonical consrucion. In addiion, le us be given an R d -valued, càdlàg F special semimaringale X and an R d -valued, primary dividend process D as in (21, such ha α X is an F local maringale. Then he discoun facor β = e R rudu and he primary marke risky price process X = 1 X {<τd } define a primary marke wih arbirage price process X, for any saisical probabiliy P Q. Proof. Mos of he proposiion follows by consrucion of he model. The only poin ha requires a jusificaion is ha X is an arbirage price process. Bu his resuls from Lemma 3.1, which ells us ha β X is a (G, Q local maringale. Henceforh a primary marke arbirage price process X consruced in his way will be called a (Ω, F, Q canonical inensiy marke model. 3.2 Markovian BSDE The marke model consruced above is oo general o be suiable for pracical purposes. In paricular, for compuaional purposes, i is necessary o impose some Markovian srucure on a marke model. For a given valuaion problem a hand, his will be achieved, by producing a pre-defaul facor model Z in which he relaed BSDE (E is in fac a Markovian BSDE (see [8, 14]. If he primary marke is also Markovian in some sense wih respec o Z, i will furher be possible o provide and analyze explici and compuable hedging sraegies. Le us hus be given a (game opion wih daa C, R, L, U, ξ, τ, in a (Ω, F, Q canonical inensiy marke model, and for a driver F of he form (1, for some F-adaped, ime-inegrable process f over [, T ] (which will someimes also be called he driver in he sequel, such ha he process F F is bounded from below. We are hus in he special case where he firs line of (E can equivalenly be rewrien as (11. Definiion 3.1 We say ha he BSDE (E is a Markovian BSDE, if: he inpu daa µ = r + γ, f, χ, L and U of (E (wih he firs line of (E represened by (11 are given by Borel-measurable funcions of an (Ω, F, Q-Markov process Z aking values in a finiedimensional Borel sae space E (wih firs componen given by ime, so 3 r = r(z, γ = γ(z f = f(z, χ = χ(z T, L = L(Z, U = U(Z ; (27 3 The relaed funcions are denoed by he same symbols as he corresponding processes.

12 12 Defaulable Opions in a Markovian Inensiy Model τ is he firs ime of enry (capped a T by he process Z ino a given closed subse of E. In paricular, he sysem made of he specificaion of forward dynamics for Z and of he BSDE (E consiues a decoupled Markovian FBSDE in (Z, Θ, M, K. This equaion is decoupled, in he sense ha he forward componen of he sysem serves as an inpu for he backward componen (i.e., Z is an inpu o (E, cf. (27, bu no he oher way round. Of course, he possibiliy of finding such a process Z and he naure of Z obviously depend on he driver f in (E, so he following developmens are, once again, parameerized by he choice of he process f in (1 (see Secion 4 for a concree example, Subsecion 4.4 in paricular. From he poin of view of he financial inerpreaion, he componens of Z are observable facors. The firs componen of Z (indexed by is simply ime Z =. The remaining componens of Z are inimaely, hough non-rivially, conneced o he pre-defaul price process X as follows: Mos componens of X will usually be given by some componens of Z. Noe, however, ha ypically here will be some primary risky asses in X, inroduced for hedging purposes, which are no represened in Z. In paricular one exra asse is ypically required for hedging defaul risk (if wished, which is no (explicily, a leas presen in he filraion F of Z; The componens of Z ha are no included in X (if any are o be undersood as simple facors ha may be required o Markovianize he payoffs of a game opion (e.g., facors accouning for pah dependence in he opion s payoff and/or non-raded facors such as sochasic volailiy in he dynamics of he asses underlying he opion; There exiss a well-defined and consrucive mapping from a collecion of meaningful and direcly observable economic variables o Z. Noe ha, due o he naure of he model, he observabiliy of he facor process Z in he mahemaical sense of F-adapedness is no sufficien in pracice. For a model o be implemenable, a consrucive mapping from a collecion of meaningful and direcly observable economic variables o Z is really needed. Oherwise, he model would be in fac useless. 3.3 Jump Diffusion Seing wih Regimes Under a raher generic specificaion for he Markov facor process Z, we shall now derive he associaed Markovian BSDE, as well as he relaed obsacles problem, ha is, a coupled sysem of parial inegro-differenial variaional inequaliies. To his end, given an ineger p and a finie se I wih l elemens, we define he following linear operaor A acing on regular funcions 4 Θ = Θ(, x, y, for (, x, y E = [, T ] R p I : AΘ(, x, y = 1 2 p i,j=1 a ij (, x, y 2 x ix j Θ(, x, y (28 p ( + b i (, x, y g(, x, y u i (, x, y, x h(, x, y, dx xi Θ(, x, y i=1 R p ( + g(, x, y Θ(, x + u(, x, y, x, y Θ(, x, y h(, x, y, dx R p + y I λ(, x, y, y (Θ(, x, y Θ(, x, y, where: he a(, x, y are p-dimensional covariance marices, wih a(, x, y = σ(, x, yσ(, x, y T, for some p-dimensional dispersion marices σ(, x, y; he b(, x, y are p-dimensional drif vecor coefficiens; 4 We use he same noaion Θ for he sae-process of a soluion o a BSDE and for a generic funcion devoed o represen he soluion of a relaed PIDE, cf. (34.

13 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 13 he jump inensiy funcion g(, x, y is non-negaive, he h(, x, y, are probabiliy measures on R p, and u(, x, y, x is he jump size funcion,; he inensiy marix funcion [λ(, x, y, y ] y,y I is such ha λ(, x, y, y whenever y y, and λ(, x, y, y = y I\{y} λ(, x, y, y. Under appropriae echnical condiions (see [8] for a deailed se of explici condiions, or see Theorems 4.1 and 5.4 in Chaper 4 of Ehier and Kurz [15] for absrac condiions regarding he exisence and uniqueness of a soluion o he relaed maringale problem wih generaor A, here exiss a sochasic basis (Ω, F, Q on [, T ], endowed wih a p-dimensional Brownian moion W, an ineger-valued random measure J (see Jacod and Shiryaev [18, Definiion II.1.13 p.68], and an (Ω, F, Q-Markov càdlàg process Z = (, X, Y, such ha: The Q-compensaed maringale measure ν of he ineger-valued random measure ν on I which couns he ransiions ν (y of Y o sae y beween ime and ime, is given by d ν (y = dν (y 1 {Y y}λ(, X, Y, y d. (29 Hence Y admis he following special semimaringale canonical represenaion: dy = y I λ(, X, Y, y(y Y d + y I (y Y d ν (y, [, T ] ; (3 The Q-compensaed maringale (random measure J of J is given by J(d, dx = J(d, dx g(, X, Y h(, X, Y, dxd, and he R p -valued process X saisfies, for [, T ], dx = b(, X, Y d + σ(, X, Y dw + u(, X, Y, x J(dx, d. R p (31 In paricular, we hen have he following varian of he Iô formula (see, e.g., Jacod [17, Theorem 3.89 p.19], in which denoes he row-gradien of Θ(, x, y wih respec o x: dθ(z = ( + AΘ(Z d + Θ(Z σ(z dw ( + Θ(, X + u(z, x, Y Θ(Z J(dx, d R p + y I (Θ(, X, y Θ(Z d ν (y, which holds for a sufficienly regular funcion Θ. Remarks 3.1 (i If we suppose ha he inensiy marix of Y does no depend on, x, hen Y is a coninuous ime Markov chain wih finie sae space I. Alernaively, if we ake g(, x, y, x = x, and we suppose ha he coefficiens σ, b, u, g and h do no depend on, x, y, hen X is a Lévy-Poisson process. In general, we deal wih a Y-modulaed Lévy-like componen X and an X -modulaed Markov chain-like componen Y. For simpliciy we do no consider he infinie aciviy case, ha is, he case when he jump measure of X is unbounded. Noe however ha our approach could be exended o he infinie aciviy wihou major changes if wished. Z hus defines a raher general class of Markov processes o be used as facor processes in financial modeling. (ii From he poin of view of inerpreaion, process Y represens regimes ha modulae he dynamics of he risk-neural pricing process. In order o make he calibraion of such a risk-neural pre-defaul model possible, various regimes y I should correspond o non-overlapping (vecorvalued ses of model parameers. Remarks 3.2 For l = 1, ha is, in he case when he regime indicaor process is consan, he one-dimensional process ν in (29 is rivially null and plays no role whasoever, so ha we may and do redefine l as (see Secion 4.3 for a concree example.

14 14 Defaulable Opions in a Markovian Inensiy Model 3.4 Variaional Inequaliy Approach Given such a facor process Z and suiable Borel funcions r and γ, he relaed sochasic basis (Ω, F, Q and processes r = r(z, γ = γ(z can be used as saring poins in he consrucion of a canonical inensiy model wih respec o (Ω, F, Q (cf. Proposiion 3.1. Denoe by P he F-predicable σ-algebra on Ω [, T ] and by B(R p he Borel σ-algebra on R p. Given a furher primary dividend process D, process X may hus be defined as X = X α 1 α u d D u (cf. (23, where, consisenly wih arbirage requiremens (cf. Lemma 3.1, he pre-defaul cumulaive price X is defined over [, T ] by X = and α 1 d(α X = Z dw + Ỹ d ν + Ṽ (x J(dx, d (32 R p for R d p -valued and R d l F-predicable processes Z and Ỹ, and for a d-valued vecor Ṽ of P B(Rp - measurable random funcions. In order for he (fully decoupled and explici Markovian SDE (32 o define a well-posed problem, we impose he following inegrabiliy condiions, denoing I = {y 1,..., y l } (wih he convenion ha l = when here are no regimes, or, equivalenly, only one regime, in he model, cf. Remark 3.2: d i=1 p j=1 E Q( T ( Z i,j d i=1 E Q( T 2 d F + d ( l i=1 j=1 E T i,j Q (Ỹ 2 λ(z, y j d F <, a.s. (Ṽ i R p 2 (xg(z h(z, dx d F <, a.s. Hence well-posedness of he (fully decoupled and explici Markovian SDE (32 follows. Remarks 3.3 The sanding example in which (32 holds along wih all he required condiions is given by he ypical siuaion where X = X(Z for a Borel-measurable funcion X wih (a leas he same regulariy as he funcion Θ o be inroduced in Theorem 3.1, see furher developmens following Theorem 3.1. Having inroduced a furher Markovian BSDE (E, a soluion (Θ, M, K o (E is hen ypically sough for wih M in he form M = Z u dw u + Y u d ν u + V u (x J(dx, du (33 R p for F-predicable processes Z, Y and a P B(R p -measurable random funcion V : Ω [, T ] R p R such ha: p j=1 Q( E T (Zj 2 d F + ( l j=1 E T Q (Y j 2 λ(, X, Y, y j d F <, a.s. ( T E Q V 2 R p (xg(, X, Y h(, X, Y, dx d F <, a.s. We are hus led o look for soluions (Θ, M, K o (E wih M in he form (33, where Z, Y and V are pars of a soluion. We hus ge a Markovian BSDE in he unknowns (Θ, Z, Y, V, K. In paricular, i is shown in [1] ha, under mild regulariy condiions, (E has a unique soluion (Θ, M, K wih M of he form (33, in suiable Hilber spaces. In he Markovian case, [8] esablishes he relaion beween his soluion and he unique soluion in some sense (viscosiy soluion wih polynomial growh in he x variable, Θ(, x, y, o an associaed PIDE obsacle problem. In he simples case where τ = (ha is, no call proecion, he associaed obsacles problem is given by he following sysem of l coupled parial inegro-differenial variaional inequaliies in spacedimension p : max (min ( Θ(, x, y AΘ(, x, y f(, x, y + µ(, x, yθ(, x, y, Θ(, x, y L(, x, y, Θ(, x, y U(, x, y =, < T, (x, y R p I, wih erminal condiion Θ(T, x, y = χ(x, y. So, (34

15 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 15 Theorem 3.1 Under mild condiions, one has, for [, T ] : Θ = Θ(Z. Moreover, in regular cases, we also have in some sense (see [9], for [, T ] : Z = Θσ(Z Y = Θ(, X Θ(Z V ( = Θ (, X + u(z,, Y Θ(Z (35 where Θ(, x sands for ( Θ(, x, y 1,, Θ(, x, y l in he second line. Le us furher assume ha he primary pre-defaul price X saisfies likewise X = X(Z for a funcion X wih he same regulariy as Θ, and ha, consisenly wih (32: α 1 d(α X = Xσ(Z dw + (36 [ X(, X X(Z [ ] d ν + X (, X + u(z, x, Y X(Z ] J(dx, d R p We hus ge a pre-defaul cumulaive primary marke Q-value dynamics for X of he form (32, wih processes Z, Ỹ and random measure Ṽ herein given by, for [, T ] : Z = Xσ(Z Ỹ = X(, X X(Z (37 Ṽ ( = X (, X + u(z,, Y X(Z where X(, ( x sands for X(, x, y 1,, X(, x, y l in he second line. Given (15, (25 and (32, he cos ρ = ρ(ζ relaive o he sraegy ζ (cf. (16 can in urn be expressed in erms of he pricing funcions Θ and X and he relaed dela funcions. Theorem 3.2 In he Markovian se-up, he dynamics (16 for he cos process ρ relaive o he sraegy ζ (and hus he relaed profi and loss process e, cf. (18 may be rewrien as (( dρ = 1 { τd }β Z ζ Z dw + (Y ζ Ỹ d ν ( ( (R + V (x ζ Ṽ (x J(dx, d + Π ( ζ R X (38 dn d R p or more specifically, in regular cases where ideniies (35 and (37 apply: dρ = 1 { τd }β (( Θσ(Z ζ Xσ(Z dw ( [ + [Θ(, X Θ(Z ] ζ X(, X X(Z ] d ν ( + [Θ (, X + u(z, x, Y Θ(Z ] R p [ ζ X (, X + u(z, x, Y X(Z ] J(dx, d ( (R + Π ( ζ R X dn d (39 Remarks 3.4 (i Provided relaed marices are lef-inverible over [, T τ d ] (which means ha he primary marke is sufficienly rich, i is hus possible o hedge compleely he source risks W, ν and H, or any finie subse of heir componens. Noe ha he sraegy consising (under he relaed lef-inveribiliy assumpion o hedge a given se of componens of W, ν and/or H on one hand, on he oher hand creaes some risk via he dependence wr ζ of he remaining erms in (38.

16 16 Defaulable Opions in a Markovian Inensiy Model (ii Of course a perfec hedge (ρ = is hopelesss unless here are no jumps (or only a finie number of jump sizes in X. In he conex of incomplee markes he choice of a hedging sraegy is up o one s opimaliy crierion relaive o he hedging cos (16 (38. For insance, a rader may wish o minimize he (objecive, P variance of T β dρ. Ye he relaed sraegy ζ va is hardly accessible in pracice (in paricular i ypically depends on he objecive model drif, a quaniy nooriously difficul o esimae on financial daa. As a proxy o his sraegy, raders commonly use he sraegy ζ va which locally minimizes he risk-neural variance of he error. In view of (16, he relaed sraegy is given as he soluion of he linear regression problem of βdm agains d(β X, so, formally: ζ va = Cov (d(β X 1 Cov (d(β X, βdm (4 wih Cov (da, db := 1 h lim h Cov (A +h A, B +h B. In he conex of a specific Markovian model i is hen ofen easy o derive an explici formula for (4. 4 Sudy of a Defaulable Converible Bonds In he concluding par of his paper, we shall apply he resuls of he previous secions o he case of a defaulable converible bond wih he underlying S, one of he primary risky asses. 4.1 Specificaion of he Payoffs To describe he covenans of a ypical converible bond (CB, we need o inroduce some addiional noaion: N: he par (nominal value, S: he price process of he asse underlying he CB, S: he pre-defaul value process of S, c cb : he coninuous coupon rae process; a bounded, F progressively measurable process, T i, c i, i =, 1,..., K (T = c = : discree coupon daes and amouns; he discree coupon daes T,..., T K are deerminisic fixed imes wih T K 1 < T T K ; he discree coupon amouns c i are bounded, F Ti 1 -measurable random variables, for i = 1, 2,..., K, A : he accrued ineres a ime, specifically, A = T i 1 c i, T i T i 1 where i is he ineger saisfying T i 1 < T i ; in view of our assumpions, he process A is an F-adaped, càdlàg process wih finie variaion. R: he recovery process on he CB upon defaul of he issuer; an F-predicable, bounded process, κ : he bond s conversion facor, P C: he pu and call nominal paymens, respecively; by assumpion P N C. For a more deailed descripion of covenans of converible bonds, he ineresed reader is referred o [3]. Le us only menion ha a real-life converible bonds ypically includes a posiive call noice period so ha hey may coninue o live some ime beyond he call ime τ c. A converible bond wih posiive call noice period is hus a conrac ha becomes an American opion upon call. In paricular, a converible bond wih posiive call noice period does no sricly fi he definiion 2.1 of a game opion (precise modeling involves hree sopping imes, see [3].

17 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 17 To circumven his difficuly, we developed in [3] an approach consising in reaing such a produc as a conrac ha pays upon call he value of an American opion. A leas in complee markes or, more generally, whenever arbirage prices of defaulable American opions are unambiguously defined, his inerpreaion seems o be reasonable. Noe ha for pracical purposes one can ofen complee he marke, in he sense ha arbirage prices of defaulable game opions become uniquely defined in he compleed marke (see, for insance, [5]. To sum up, we propose in [3] a recursive approach o he valuaion of a CB wih posiive call noice period. In he firs sep, a CB is valued upon call. Subsequenly, we use he resuling price upon call as he payoff a call of a CB wihou call noice period. In his way, a CB wih posiive call noice period can be priced as a reduced converible bond, which is formally defined as follows. Definiion 4.1 A reduced converible bond is a game opion wih coupon process C, recovery process R cb and payoffs L cb, U cb, ξ cb such ha C = ccb u du + T i ci, R cb = (1 ηκ S R, L cb = P κ S + A, ξ cb = N κ S T + A T, and where (U cb [,T ] is some F-adaped càdlàg process saisfying he following inequaliy: C κ S + A U cb, [, T ]. (41 In he financial inerpreaion, U cb represens he pre-defaul value of he reduced converible bond upon a call a ime. In paricular, a converible bond wihou call noice period is a reduced converible bond wih U cb = C κ S + A for [, T ]. Noe ha under our assumpion ha P N C, we obain by (41: L cb T = P κ S T + A T N κ S T + A T = ξ cb C κ S T + A T U cb T. 4.2 Clean Price In he sequel, we shall focus on reduced converible bonds. Definiion 4.2 For a pre-defaul Q-price Π of a (reduced converible bond, by he clean price of his bond we mean he difference Π A. The noion of he clean price is consisen wih he marke convenion for bonds, which relies on subracing he accrued ineres from he rading (diry price. Marke quoaions for bonds are usually given in erms of clean prices (or bond yields, in order o avoid coupon-relaed disconinuiies in quoaions. Le us se a = c i T i T i 1 for [, T ]. Then A = a u du T i and he inegraion by par formula yields (recall ha B = ln α: α A = α u ( au du A u db u c i T i α Ti c i. (42 Le us fix some risk-neural measure Q. The moivaion for he choice of he driver F defined by (43 in Proposiions will be discussed in Subsecion 4.4.

18 18 Defaulable Opions in a Markovian Inensiy Model Proposiion 4.1 (i Considering a reduced converible bond, le us choose he driver F cb := F + A, (43 where F was defined in (8. Then he daa of he doubly refleced BSDE (E ake he following form: F cb, χ = ξ cb A T = N κ S T, L = L cb A = P κ S, U = U cb A, τ. and he sae-componen Θ of a soluion (Θ, M, K wih m H 2, assumed o exis, o (E, is equal o he clean price of he bond. (ii In he defaul inensiy se-up, F cb is of he form (1 wih f = γr cb + c cb + a µa =: f cb. (44 (iii Assume ha he pre-defaul value process S is coninuous. Then he lower barrier process L is coninuous on [, T ]. Moreover, in he case of a converible bond wih no call noice period, U is coninuous on [, T ], and he upper barrier process Ū is coninuous on [ τ, T ]. Proof. (i We have, by (12, in view of he definiion of F cb. Also, Θ = Π + F F cb = Π A, L = L cb + F ( F + A = L cb A. The oher ideniies can be shown similarly. (ii Using he definiion of f cb and (42 wih db u = µ u du in an inensiy defaul model, we ge: α u f cb u du = = α A + α u γ u Ru cb du + α u c cb ( u du + α u au µ u A u du α u dc u + α u γ u Ru cb du = α (A + F = α F cb. (iii I suffices o noe ha for a converible bond wih no call noice period we have U = U cb A = C κ S, and, for τ, Ū = U. Le us summarize our findings a his poin of his secion. Firs, we have shown ha by solving he doubly refleced BSDE (E wih he driver F = F cb given by (44, we obain he clean price of a reduced converible bond as he sae process Θ of a soluion o (E. Second, he relaed driver erms in (E are hen given as inegrals wih respec o he Lebesgue measure, which is he sandard form in he BSDE lieraure. Third, under mild assumpions, he lower and upper barriers for his choice of he driver F are given as coninuous processes (a leas, in case wih no noice period; see Subsecion 4.4 for more abou his, and hus he sae process process Θ of a soluion o he doubly refleced BSDE (E is coninuous, provided he F-maringale M is coninuous. 4.3 A Simple Model The previous observaions prove useful in he pracical implemenaion of a jump-o-defaul inensiy model wih a Markovian srucure, as described in Bielecki e al. [5] (see also Ayache e al. [2] or Andersen and Buffum [1]. In [5], he filraion F is generaed by a sandard Brownian moion W under Q, and a primary marke model composed of he savings accoun and d = 2 risky asses is sudied:

19 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 19 he firs primary risky asse is he sock of a reference firm wih price process S and defaul ime represened by τ d ; he second primary risky asse is a credi defaul swap (CDS wrien on he reference eniy. We choose as pre-defaul facor process Z he pair (, S, where S represens he pre-defaul sock price process, modeled as he unique srong soluion o he following SDE d S = S ( (r( q( + ηγ(, S d + σ(, S dw (45 wih S given as a real-valued F -measurable random variable, and where: he riskless shor ineres rae r(, he equiy dividend yield q(, and he local defaul inensiy γ(, S are bounded Borel funcions, he fracional recovery on S upon defaul, η, is a non-negaive consan, he local volailiy σ(, S is a posiively bounded Borel funcion, he funcions γ(, SS and σ(, SS are Lipschiz in S. I is furher posulaed ha (cf. Subsecion 4.1: he coupon process C = C( := c cb u du + [,] T i for a bounded Borel coninuous coupon rae c cb and deerminisic discree coupon daes and amouns, wih T = and T K 1 < T T K ; he recovery process R is of he form R(, S for a Borel funcion R. We say ha we deal wih a hard call proecion if he lifing ime of call proecion τ = T for some T T. A sandard sof call proecion corresponds o he lifing ime of call proecion given as τ = inf{ > ; S S} T for some S R +. c i Proposiion 4.2 Le us assume eiher a hard call proecion or a sandard sof call proecion. (i Choosing he driver f = f cb as defined in (44, he relaed BSDE (E is Markovian wih respec o he facor process Z = (, S (so X is reduced o X 1 = S and here are no regimes; hus d = 2, p = 1, l =. (ii The pre-defaul primary price process S saisfies he arbirage drif condiion (26. (iii In case τ = (no call proecion, (34 becomes: ( max min ( LΘ + µθ (γr cb + c cb + a µa, Θ P κs, Θ (U cb A =, < T, Θ(T, S = N κs, (46 wih L + (r q + ηγs S + σ2 S 2 2 S 2 c and a 2 = i T i T i (cf. Subsecion 4.2. The funcion Θ 1 defined by (46 is he clean pricing funcion of he relaed reduced converible bond Proof. (i holds by consrucion of he model. Moreover, (he Markovian form of condiion (26 holds for S since in he presen case µ(, S = r( + γ(, S, c(, S = q(s, R(, S = (1 ηs, so ha µ(, SS c(, S γ(, SR(, S = (r( q( + ηγ(, SS which is he drif coefficien funcion of process S. Thus (ii is saisfied. Finally (iii follows by specificaion of he general pricing equaion (34 o he daa of Proposiion 4.1(i.