CONVERTIBLE BONDS IN A DEFAULTABLE DIFFUSION MODEL

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1 CONVERTIBLE BONDS IN A DEFAULTABLE DIFFUSION MODEL Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 60616, USA Séphane Crépey Déparemen de Mahémaiques Universié d Évry Val d Essonne Évry Cedex, France Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne Évry Cedex, France and Europlace Insiue of Finance Marek Rukowski School of Mahemaics Universiy of New Souh Wales Sydney, NSW 2052, Ausralia and Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology Warszawa, Poland July 11, 2007 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 2007 Faculy Research Gran PS12918.

2 2 Converible Bonds in a Defaulable Diffusion Model 1 Inroducion In [4], working in an absrac se-up, we characerized arbirage prices of generic Converible Securiies CS, such as Converible Bonds CB, and we provided a rigorous decomposiion of a CB ino a bond componen and a game opion componen, in order o give a definie meaning o commonly used erms of CB spread and CB implied volailiy. Moreover, in [5], we showed ha in he hazard process se-up, he heoreical problem of pricing and hedging CS can essenially be reduced o a problem of solving a relaed doubly refleced Backward Sochasic Differenial Equaions R2BSDE for shor in he sequel, see [5]. Finally, in [6], we esablished he formal connecion beween his R2BSDE and relaed variaional inequaliies wih double obsacle in a generic Markovian inensiy model. The relaed mahemaical issues are deal wih in Crépey [13, 14]. In his paper, we sudy CSs, in paricular CBs, in a specific marke se-up. Namely, we consider a primary marke consising of: a savings accoun, a sock underlying he CS, and an associaed credi defaul swap CDS. We model he dynamics of hese hree securiies in erms of Markovian diffusion se-up wih defaul Secion 2. In his model, we give condiions, obained by applicaion of he general resuls of [13, 14], ensuring ha he R2BSDE relaed o a CS has a soluion Proposiion 3.5, and we provide he associaed super-hedging sraegy Theorem 3.2. Moreover, we characerize he pricing funcion of he CS in erms of viscosiy soluions of associaed variaional inequaliies Theorem 3.3, and we prove he convergence of suiable approximaion schemes Theorem 3.4. We hen specify hese resuls o converible bonds and heir sraigh bond and opion componens Secion 4. The above-menioned model appears as he simples equiy-o-credi reduced form model one may hink of he connecion beween equiy and credi in he model being maerialized by he fac ha he defaul inensiy γ depends on he sock level S, and i is hus widely used in he indusry for dealing wih defaulable converible bonds. This was he firs moivaion for he presen sudy. The second moivaion was he fac ha all assumpions ha we posulaed in our previous heoreical works [4, 5, 6] are saisfied wihin his se-up; in his sense, he model is consisen wih our heory of converible securiies. In paricular, we worked in [4, 6] under he assumpion ha he value U cb of a CB upon a call a ime yields, as a funcion of ime, a well-defined process saisfying some naural condiions. In he specific framework of his paper, using uniqueness of arbirage prices Proposiion 2.1 and Theorem 3.1 and a form of coninuous aggregaion propery of he value U cb of a CB upon a call a ime Theorem 4.14ii, we are acually able o prove ha his assumpion is saisfied, and we also give ways o compue U cb Theorems 4.14 and Model In his secion, we inroduce a simple specificaion of he generic Markovian defaul inensiy se-up of [6]. More precisely, we consider a defaulable diffusion model, wih ime and sock-dependen local defaul inensiy and local volailiy see also [2, 1, 18, 19, 28, 11]. 2.1 Canonical Consrucion Le us be given, relaive o a finie horizon dae T > 0, a filered probabiliy space Ω, F, G, Q saisfying he usual condiions, where F is he filraion of a sandard Brownian moion W on [0, T ] under Q. Here Q is devoed o represen a risk-neural probabiliy measure on a financial marke o be defined below. We define wha will laer be inerpreed as he pre-defaul sock price S of he firm underlying a CB, by seing, for [0, T ], d S = S r q + ηγ, S d + σ, S dw, S0 = x R. 1

3 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 3 Assumpion 2.1 i The riskless shor ineres rae r, he equiy dividend yield q and he local defaul inensiy γ, S 0, are bounded Borel-measurable funcions, and η is a non-negaive consan; ii The local volailiy σ, S is a posiively bounded Borel-measurable funcion, so in paricular σ, S σ, for a posiive consan σ; iii The funcions γ, SS and σ, SS are Lipschiz coninuous in S, uniformly in. Noe ha we auhorize negaive values of r and/or q, in order, for insance, o possibly accoun for repo raes in he model. Under hese assumpions he SDE 1 admis a unique srong soluion S. Moreover, he following a priori esimae is available, for any p [2, + see, e.g., [13]: E Q [ sup [0,T ] S p ] C 1 + x p. 2 Remark 2.2 i For x > 0 he soluion of 1 is non-negaive. In his work we find i convenien o define 1 for any iniial condiion x R, even hough only he posiive values will have a financial inerpreaion. This will be useful for he variaional inequaliies approach see Remark ii The fac ha γ may depend on S in his model is crucial, since his dependence acually conveys all he equiy-o-credi informaion in he model. A naural choice for γ is a decreasing e.g., negaive power funcion of S capped when S is close o 0. A possible refinemen is o posiively floor γ. The lower bound on γ hen represens pure credi risk, as opposed o equiy-relaed credi risk. We define he [0, T ] {+ }-valued random defaul ime τ d by he so-called canonical consrucion [8]. Specifically, we se by convenion, inf = { τ d = inf [0, T ]; γu, S } u du ε, 3 0 where ε is a uni exponenial random variable on Ω, F, G, Q independen of W. We se H = 1 τd and M d = H 0 1 H u γu, S u du. Le H be he filraion generaed by he process H and he filraion G be given as F H. Because of our consrucion of τ d, he process γ, S is he F-inensiy of τ d and he process M d is a G-maringale, called he compensaed jump maringale. Moreover, he process Pτ > F = e 0 γu, S udu is coninuous and non-increasing. Finally, he filraion F is immersed in G Hypohesis H holds, in he erminology of [8], in he sense ha all F-maringales are G-maringales. In paricular, he F-Brownian moion W is a G-Brownian moion under Q. Noe also ha F, Q; W has he local maringale predicable represenaion propery, since we assumed ha F is he filraion of he Brownian moion W on [0, T ]. 2.2 Specificaion of he Primary Marke Model We consider a primary marke composed of he savings accoun and wo primary risky asses: he sock S of a reference eniy, ha is, he firm issuing he CS, wih defaul ime represened by τ d ; a CDS conrac B wrien a ime 0 on he reference eniy.

4 4 Converible Bonds in a Defaulable Diffusion Model We denoe he discoun facor process as β, so ha β dynamics of S under Q are ds = S r q d + σ, S dw η dm d = e 0 ru du, and we assume ha he, S 0 = x R. 4 Here η is he fracional loss of he equiy value upon defaul, assumed o be a consan 0 η 1. Observe ha process S, once adjused for ineres raes and dividend yields, is a G, Q local maringale. I is easily seen ha dynamics 1 and 4 are consisen, in he sense ha S is he unique F-predicable process such ha 1 H S = 1 H S for every [0, T ]. Under suiable inegrabiliy assumpions, he Q-dynamics of B are see [7] db = rb d + 1 H ν ν γ, S d + α 1 Σ dw B dm d where: ν is he conraced CDS spread, ν is an F-predicable, bounded proecion paymen process, α = e 0 ru+γu, S udu represens he credi-risk adjused discoun facor, Σ is an F-predicable, bounded process. We define, for every [0, T ], υ = e 0 qudu, Ŝ = υ S, B = B + β 1 [0, τ d ] β uν u dh u ν du υ Ŝ Υ = Diag, X = 1 B, 5 Remark 2.3 In he financial inerpreaion, B denoes, in accordance wih our general convenion for he noaion. in [4, 5, 6], he curren value a ime of a buy-and-hold sraegy in one CDS conrac a ime 0, assuming ha all he CDS paymens are immediaely reinvesed in he savings accoun. As for Ŝ, i represens in his paper he curren value a ime of a buy-and-hold sraegy in one share of S a ime 0, assuming ha all he dividend paymens on S are immediaely reinvesed in he equiy S. These convenions explain he appearance of he facor υ in fron of S above and he vecor Υ in equaion 6 below. Since β X is manifesly a locally bounded process, he arbirage risk-neural pricing measures on our primary marke model are given by probabiliy measures Q Q such ha β X is a G, Q-local maringale see, e.g., [6]. Noe ha W d β X = β Υ Ξ d M d where he G-predicable dispersion marix process Ξ is given by [ Ξ = We work in he sequel under he following σ, S Ŝ ηŝ 1 { τd }α 1 Σ 1 { τd }ν B Assumpion 2.4 Ξ is inverible on [0, τ d T ]. We hen have he following, [0, T ] 6 ], [0, T ]. 7

5 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 5 Proposiion 2.1 For any risk-neural measure Q on he primary marke, we have ha E d Q Q dq G = 1 on [0, τ d T ]. Proof. Given a probabiliy measure Q equivalen o Q on Ω, G T, he Radon-Nikodym densiy Z = E d Q Q dq G is a posiive G, Q-maringale. Therefore, by Kusuoka [27], here exis wo G-predicable processes ϕ and ϕ d such ha dz = Z ϕ dw + ϕ d dm d, [0, T ]. 8 The measure Q is hen risk-neural iff he process β X is a G, Q-local maringale, or equivalenly, if he processes βŝz and β BZ are G, Q-local maringales. These condiions are saisfied if and only if ϕ Ξ 1 { τd }γ, S ϕ d = 0. 9 Unil τ d he unique soluion o 9 is ϕ = ϕ d = 0. We conclude ha Z = 1 on [0, τ d T ]. 3 Converible Securiies We now specify o he above model he noions of converible securiies, converible bonds in paricular, ha were inroduced in a general se-up in [4]. Le 0 respecively T sand for he incepion dae respecively he mauriy dae of a converible securiy CS wih underlying S. For any [0, T ], we wrie FT resp. G T o denoe he se of all F-sopping imes resp. G-sopping imes wih values in [, T ]. Given he ime of lifing of a call proecion of a CS, τ FT 0, le also F T sand for {ϱ F T : ϱ τ}, and Ḡ T sand for {ϱ GT ; ϱ τ d τ τ d }. Le finally τ denoe τ p τ c, for any τ p, τ c GT Ḡ T. Definiion 3.1 A Converible Securiy wih underlying S is a game opion see [4, 5, 6, 26, 25] wih he ex-dividend cumulaive discouned cash flows π; τ p, τ c given by he formula, for any [0, T ] and τ p, τ c GT Ḡ T, τ β π; τ p, τ c = β u dd u + 1 {τd >τ}β τ 1 {τ=τp<t }L τp + 1 {τc<τ p}u τc + 1 {τ=t } ξ, where: he dividend process D = D [0,T ] equals D = 1 H u dc u + [0,] [0,] R u dh u for some coupon process C = C [0,T ], which is a G-adaped càdlàg process wih finie variaion, and some real-valued, G-predicable recovery process R = R [0,T ] ; he pu paymen L is given as a G-adaped, real-valued, càdlàg process on [0, T ], he call paymen U is a G-adaped, real-valued, càdlàg process on [0, T ], such ha L U for [τ d τ, τ d T, 10 he paymen a mauriy ξ is a G T -measurable real random variable. In addiion, he processes R, L and he random variable ξ are assumed o saisfy he following inequaliies, for some posiive consan c: c R c 1 S, [0, T ], c L c 1 S, [0, T ], 11 c ξ c 1 S T.

6 6 Converible Bonds in a Defaulable Diffusion Model Theorem 3.1 If he Q-Dynkin game relaed o he CS admis a value Π, in he sense ha, esssup τp G T essinf τ c Ḡ T E Q π; τp, τ c G = Π 12 = essinf τc Ḡ T esssup τ p G T E Q π; τp, τ c G, [0, T ], and ha Π is a G-semimaringale, hen Π is he unique arbirage ex-dividend price of he CS. Proof. Excep for he uniqueness saemen, his follows by applicaion of he general resuls in [4]. To verify he uniqueness propery we firs noe ha given he esimae 2 on S hence S, he general resuls of [4] also imply ha any arbirage price of a CS is given by he value of he relaed Dynkin Game for some risk-neural measure Q. Now, for any such risk-neural measure Q, we have ha Z = E d Q Q dq G = 1 on [0, τ d T ], by Proposiion 2.1. Furhermore, π; τ p, τ c is a G τd T measurable random variable. Therefore E Q π; τp, τ c G = EQ π; τp, τ c G, 13 for any [0, T ], τ p G T, τ c Ḡ T. In conclusion, he Q-Dynkin Game also has value Π. We now define special cases of CSs, corresponding o American-syle and European-syle CSs, respecively. Formally, Definiion 3.2 A non-callable CS denoed as PB, cf. [4] is a converible securiy wih τ = T, or, equivalenly, Ū =. An Elemenary Securiy ES is a non-callable CS wih bounded variaion dividend process D over [0, T ], bounded paymen a mauriy ξ, and such ha β u dd u + 1 {τd >}β L β u dd u + 1 {τd >T }β T ξ for [0, T. 14 [0,] [0,T ] By Definiion 3.2, PBs and ESs are special cases of CSs. Noe ha given Theorem 3.1, a PB resp. an ES can be redefined in a more sandard way as a financial produc wih ex-dividend cumulaive discouned cash flows π; τ p resp. φ given as, for [0, T ] and τ p GT, β π; τ p = τp β u dd u + 1 {τd >τ p}β τp 1{τp<T }L τp + 1 {τp=t }ξ resp. β φ = T β u dd u + 1 {τd >T }β T ξ for every [0, T ]. Reurning o he case of a general CS, we furher posulae in he Markovian se-up of his paper, ha Assumpion 3.3 he coupon process C = C := [0,] cudu + 0 T i ci, for a bounded Borel-measurable coninuous ime coupon rae funcion c, and deerminisic discree imes and coupons T i and c i, respecively; for reasons ha will become clear in Secion 4.7, we ake he enor of he discree coupons as T 0 = 0 < T 1 < < T K 1 < T K, wih T T K ; he recovery process R is of he form R, S, for a Borel-measurable funcion R; ξ = ξs T, L = L, S, U = U, S, for a Borel-measurable funcion ξ and Borel-measurable funcions L, U such ha for any, S, we have L, S U, S, LT, S ξs UT, S. Definiion 3.4 We define he accrued ineres a ime by A = T i 1 c i, T i T i 1

7 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 7 c where i is he ineger saisfying T i 1 < T i, and we le ρ = i T i T i, so ha ouside discree 1 coupon daes da = ρd. We also se γ = γ, S, µ, S = r + γ, S, µ = µ, S 15 so ha α = e 0 µudu and for [0, T ] wih he convenion ha A 0 = 0: α A = dαa u = α u ρu µ u A u du α Ti c i. 16 [0,] 0 0 T i To a CS wih daa funcions C, R, ξ, L, U, and lifing ime of call proecion τ FT 0, we associae he Borel-measurable funcions f, S, θ for θ real, gs, l, S and h, S defined by and gs = ξs A T, l, S = L, S A, h, S = U, S A, f, S, θ = γ, SR, S + Γ, S µ, Sθ, where Γ, S = c + ρ µ, SA 17 In he case of a non-callable CS, he process U is irrelevan, and in his case we se h, S = +. Moreover, we noe ha in he case of an ES, which is a special case of non-callable CS, he process L plays no role, and herefore we se l, S =. Finally, we define he processes and random variables associaed o a CS parameerized by θ R, regarding f as f θ = f, S, θ, g = g S T, l = l, S, h = h, S. In order o ensure sabiliy of soluions o he relaed BSDEs see below, and, incidenally, o ensure well definedness of he previous processes, we work henceforh under he following Assumpion 3.5 The funcions r, γ, g, l, h, R, c are coninuous. 3.1 Doubly Refleced BSDEs Approach We define: H 2 he se of real-valued, F-predicable processes Θ such ha E Q T 0 Θ2 d <, S 2 he se of real-valued, F-adaped, coninuous processes Θ such ha E Q sup [0,T ] Θ 2 <, A 2 he space of finie variaion coninuous processes k wih coninuous and non decreasing Jordan componens k ± S 2 null a ime 0, A 2 i he space of non-decreasing processes in A2. So k = k + k where k ± A 2 i define muually singular measures on R+, for any k A 2. Given a CS wih daa C, R, ξ, L, U, τ, and given he associaed processes and random variables f, g, l, h, we inroduce he following doubly refleced Backward Sochasic Differenial Equaion E wih daa f, g, l, h, τ R2BSDE for shor, see [6, 13], such ha for [0, T ]: dθ = f Θ d + dk z dw, Θ T = g, l Θ h, Θ l dk + = h Θ dk = 0, E where we se h = 1 {< τ} + 1 { τ} h, using he convenion ha 0 ± = 0.

8 8 Converible Bonds in a Defaulable Diffusion Model Definiion 3.6 i By a soluion o E, we mean a riple of processes Θ, z, k S 2 H 2 A 2 ha saisfies all condiions in E for any 0 T. So, in paricular, Θ and k have o be coninuous processes. ii In he case where τ = T, we have k = 0, so ha E reduces o a refleced BSDE wih daa f, g, l and k A 2 i in he soluion. iii In he special case of an ES, one can show ha k = 0 in any soluion Θ, z, k o E, so ha E reduces o an elemenary BSDE wih daa f, g and no process k involved in he soluion, referred o in he sequel as E. In he se-up of his paper he noions of issuer hedge and holder hedge inroduced in [6, 5] ake he following form. Definiion 3.7 By a primary sraegy, we mean a riple V 0, ζ, Q such ha: V 0 is a G 0 -measurable real-valued random variable represening he iniial wealh, ζ is an R 1 2 -valued bi-dimensional row vecor, β X-inegrable process represening holdings number of unis held in primary risky asses, Q is a real-valued, finie variaion process wih Q 0 = 0, represening he financing cos process of he sraegy. The wealh process V of a primary sraegy V 0, ζ, Q is given by wih he iniial condiion V 0. dβ V = ζ dβ X + β dq, [0, T ], Noe ha a primary sraegy is hus no self-financing in he sandard sense, unless Q = 0. In he conex of his paper his is inended o accoun for dividends as follows. Definiion 3.8 Given a CS wih dividend process D: a An issuer hedge is represened by a riple V 0, ζ, τ c such ha: i τ c belongs o Ḡ0 T, ii V 0, ζ, D is a primary sraegy wih relaed wealh process V such ha, for [0, T ], V τc 1 { τc<τ d } 1 { τc=<t }L + 1 {τc<}u τc + 1 {=τc=t }ξ 0, a.s. b An holder hedge is a riple V 0, ζ, τ p such ha: i τ p belongs o GT 0, ii V 0, ζ, D is a primary sraegy wih relaed wealh process V such ha, for [ τ, T ], V τp + 1 { τp<τ d } 1 {τp =τ p<t }L τp + 1 {<τp}u + 1 {τp==t }ξ 0, a.s. Accouning for dividend gains or losses, issuer or holder hedges are hus in effec issuer or holder self-financing superhedges. By applying general resuls of [6], we obain he following super-hedging resul. Theorem 3.2 Le Θ, z, k be a soluion o E, assumed o exis, and le Θ denoe 1 Θ {<τd } wih Θ := Θ + A. Then Θ is he unique arbirage price process of he CS, and for any [0, T ]: i An issuer hedge wih iniial wealh Θ is furnished by { τc = inf u [ τ, T ]; Θ } u = h u T F T,

9 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 9 and ζ u := 1 u τd [ z u, R u Θ u ] Ξ 1 u, u T, 18 where Θ = Θ and d Θ u µ u Θu du = d Θ u µ u Θu du + dk u, u T. Moreover, he corresponding wealh process is bounded from below and Θ is he smalles iniial wealh of an issuer hedge. ii A holder hedge wih iniial wealh Θ is furnished by { τp = inf u [, T ] ; Θ } u = l u T FT and ζ = ζ above. Moreover, Θ is he smalles iniial wealh of a holder hedge. Proof. By applicaion of he general resuls of [6], Θ saisfies all he assumpions for Π in Theorem 3.1; herefore, i is he unique arbirage price process of he CS. Moreover, under Assumpion 2.4, i and ii resul by applicaion of he general resuls of [6] up o a simple change of variables o accoun for a minor discrepancy beween he definiions of Ŝ in [6] and in he presen paper; cf. Remark 2.3. We hus see ha in he presen se-up any CS has a bilaeral hedging price bilaeral in he sense ha his price Θ ensures super-hedging o boh he issuer and he holder of he claim, saring from he iniial wealh Θ for he former and Θ for he laer, which is also he unique arbirage price. Of course, his conclusion hinges on our emporary assumpion ha he relaed BSDE has a soluion. 3.2 Variaional Inequaliies Approach Le D denoe a closed sub-domain of [0, T ] R given by eiher [0, T ] R as a whole, or [0, T ], S] for some S <. Le hen In p D = [0, T R or [0, T, S, p D = D \ In p D 19 sand for he parabolic inerior and he parabolic boundary of D, respecively. Le L denoe he linear operaor L + r q + ηγs S + σ2 S S 2. Finally, le P be he class of funcions Θ on D bounded by C1 + S p for some real C and ineger p ha may depend on Θ 1. In order o esablish he connecion beween he previous BSDEs and he formally relaed obsacles problems see [6, 13, 14], we posulae henceforh he following Assumpion 3.9 r, q, γ and σ are coninuous funcions and he funcions R, g, h, l associaed o a CS are coninuous and of class P or h = +, in he case where τ = T, and l =, in he case of an ES. Given a coninuous boundary condiion b, where b is a coninuous funcion of class P on p D such ha b = g poinwise a T, we inroduce he following obsacles problem VI on D where f was defined in 17: max min LΘ, S f, S, Θ, S, Θ, S l, S, Θ, S h, S = 0, supplemened by he boundary condiion Θ = b on p D. 1 By a sligh abuse of erminology, we shall say ha a funcion ΘS,.. is of class P if i has polynomial growh in S, uniformly in any oher argumens.

10 10 Converible Bonds in a Defaulable Diffusion Model Remark 3.10 Noe ha VI is defined over a domain in space variable S going o, hough only he posiive par of he domain has a financial inerpreaion cf. Remark 2.2i. If we decided o pose problems VI over bounded spaial domains, hen we would need o impose some appropriae non-rivial boundary condiion a he lower space boundary, in order o ge a well-posed problem. We refer he reader o he Appendix for he definiion of viscosiy soluions which is relevan o cope wih he ime-disconinuiies of f a he T i s in case he produc under consideraion pays discree coupons. Building upon Definiion A.1, we inroduce he following definiion of P semi-soluions o VI on D. Definiion 3.11 By a P subsoluion, resp. supersoluion, resp.resp. soluion Θ of VI on D for he boundary condiion b, we mean a viscosiy subsoluion, resp. supersoluion, resp.resp. soluion of VI of class P on In p D, such ha Θ b, resp. Θ b, resp.resp. Θ = b, poinwise on p D. Theorem 3.3 Le Θ, z, k be a soluion o E, assumed o exis. Then: a Cauchy problem: τ = 0. In his case, process Θ, denoed here as Θ, can be wrien as Θ = Θ, S, where he funcion Θ is a P-soluion of VI on [0, T ] R wih erminal condiion g a T ; b Cauchy Dirichle problem: τ = inf{ > 0 ; S S} T for some S > 0. In his case, process Θ, denoed here as Θ, can be wrien on [0, τ] as Θ, S, where he funcion Θ is a P- soluion of VI on [0, T ], S] wih erminal condiion g a T and Dirichle condiion Θ, S a level S where Θ is he funcion defined in a. Proof. This follows by applicaion of he general resuls of Crépey [14, 13]. Noe, in paricular, ha τ depends a.s.-coninuously on he iniial condiion, x of S, under Assumpion 2.1ii see, for insance, Darling Pardoux [17], which is one of he condiions posulaed for b in [13]. We now come o he issues of uniqueness and approximaion of soluions for VI. For his we make he following addiional Assumpion 3.12 The funcions r, q, γ, σ are locally Lipschiz coninuous. We refer he reader o Crépey [14] or Barles and Souganidis [3] for he definiion of sable, monoone and consisen approximaion schemes o VI and for he relaed noion of convergence of he scheme, involved in he following Theorem 3.4 Le Θ, z, k be a soluion o E, assumed o exis, and le he funcions Θ and Θ be defined as in Theorem 3.3. a Cauchy problem: τ = 0. The funcion Θ is he unique P-soluion, he maximal P-subsoluion, and he minimal P-supersoluion, of VI on D = [0, T ] R wih erminal condiion g a T. Le Θ h h>0 denoe a sable, monoone and consisen approximaion scheme for he funcion Θ. Then Θ h Θ locally uniformly on D as h 0 +. b Cauchy Dirichle problem: τ = inf{ > 0 ; S S} T for some S > 0. The funcion Θ is he unique P-soluion, he maximal P-subsoluion, and he minimal P-supersoluion, of VI on D = [0, T ], S] wih erminal condiion g a T and Dirichle condiion Θ, S a S. Le Θ h h>0 denoe a sable, monoone and consisen approximaion scheme for he funcion Θ. Then Θ h Θ locally uniformly on D as h 0 +, provided Θ h Θ= Θ a S. Proof. Noe, in paricular, ha under our assumpions: he funcions r q + ηγ, SS and σ, SS are locally Lipschiz coninuous;

11 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 11 he funcion f admis a modulus of coninuiy in S, in he sense ha for every R > 0 here exiss a nonnegaive funcion η R coninuous and null a 0 such ha: f, S, θ f, S, θ η R S S for any [0, T ] and S, S, θ R wih S, S, θ R. The resuls hen follows by applicaion of he general resuls of Crépey [14]. The previous resuls show he imporance of having a soluion Θ, z, k o E. By applicaion of he general resuls of [13], we have he following Proposiion 3.5 Assume furher ha l, S = λ, S c for a funcion λ of class C 1,2 wih λ, λ, S S λ, S 2 S 2 2λ of class P and for a consan c R { }. Then E admis a unique soluion Θ, z, k. Example 3.13 The sanding example for he funcion λ, S in Proposiion 3.5 is λ, S = S. In ha case, l corresponds o he payoff funcion of a call opion or, more precisely, o he lower payoff funcion of a converible bond, see Secion 4. Remark 3.14 We refer, in paricular, he reader o he las secion of Crépey [14] regarding he fac ha he poenial disconinuiies of f a he T i s which represen a non-sandard feaure from he poin of view of he classic heory of viscosiy soluions as presened, for insance, in he User s Guide [12] are no a real issue in he previous resuls, provided one works wih he suiable Definiion A.1 of viscosiy soluions o our problems. 4 Applicaion o Converible Bonds 4.1 Converible Bonds and Reduced Converible Bonds As we already poined ou, a converible bond is a special case of a converible securiy. To describe he covenans of a ypical converible bond CB, we need o inroduce he following addiional noaion see [4] for a horough descripion and discussion of he associaed converible bonds covenans: N: he par nominal value, η: he fracional loss on he underlying equiy upon defaul 0 η 1, R : he recovery process on he CB upon defaul of he issuer a ime, given by R = R, S for a coninuous bounded funcion R, κ : he conversion facor, R cb = R cb, S = 1 ηκs R : he effecive recovery process, ξ cb = N κs T + A T : he effecive payoff a mauriy, P C : he pu and call nominal paymens, respecively, such ha P N C, δ 0 : he lengh of he call noice period see below, δ = + δ T : he end dae of he call noice period sared a.

12 12 Converible Bonds in a Defaulable Diffusion Model Real-life converible bonds ypically include a posiive call noice period δ so ha if he issuer calls he bond a ime τ c, hen he holder may eiher redeem he bond for C or conver he bond ino κ shares of sock, a any ime u in [τ c, τ δ c ], where τ δ c = τ c + δ T. Accouning for accrued ineres, he effecive call/conversion paymen o he holder a ime u is C κs u + A u. This clause makes CB wih posiive call noice period difficul o price direcly. To handle his, we developed in [4] a wo sep approach o value a CB wih posiive call noice period. In he firs sep, we value he CB upon call as a Reduced Converible Bond RB, see Definiion 4.1 below. In he second sep, we use his price as he payoff a call ime of a CB wih no call noice period. Definiion 4.1 [4] A reduced converible bond RB is a converible securiy wih recovery process R cb and erminal payoffs L cb, U cb, ξ cb such ha and R cb τ d = 1 ηκs τd R τd, L cb = P κs + A, ξ cb = N κs T + A T, U cb = 1 {<τd }Ũ cb, S + 1 { τd } C κs + A, [0, T ] 20 for a funcion Ũ cb, S joinly coninuous in ime and space, excep for negaive lef jumps of c i a he T i s, and such ha Ũ cb, S C κs + A on he even { < τ d } so U cb C κs + A, [0, T ]. So he discouned dividend process of an RB is given by, for every [0, T ], β u ddu cb := β u cudu + β Ti c i + 1 {0 τd }β τd Rτ cb d. 21 [0,] [0, τ d ] 0 T i, T i<τ d Clearly, a CB wih no noice period δ = 0 is an RB, wih Ũ cb, S = C κs + A. More generally, he financial inerpreaion of he process U cb in an RB is ha U cb represens he value of he RB upon a call a ime. In Secion 4.7, we shall prove ha under mild regulariy assumpions in our model, any CB wheher δ is posiive or no can be inerpreed and priced as an RB. 4.2 Decomposiion of a Reduced Converible Bond Embedded Bond We consider an RB wih dividend process D cb given by 21, and an ES wih he same coupon process as he RB and wih R b and ξ b as follows: R b = R, ξ b = N + A T 22 so ha R cb R b = 1 ηκs R + 0, ξ cb ξ b = κs T N + 0. Thus, his ES corresponds o he defaulable bond wih discouned cash flows given by he expression β φ = := T T τd β u dd b u + 1 {τd >T } β T ξ b β u cudu + <T i T,T i<τ d β Ti c i + 1 {<τd T } β τd Rτ b d + 1 {τd >T } β T ξ b

13 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 13 and he associaed funcions f, S, θ = γ, SR b, S + Γ, S µ, Sθ, gs = N. This bond can be seen as he pure bond componen of he RB ha is, he RB sripped of is opional clauses. Therefore, we shall call i he bond embedded ino he RB, or simply he embedded bond. In he sequel, in addiion o he assumpions made so far, we work under he following reinforcemen of Assumpion Assumpion 4.2 The funcions r, q, γ, SS, σ, SS, γ, S R, S and c are coninuously differeniable in ime, and hrice coninuously differeniable in space, wih bounded relaed spaial parial derivaives. Noe ha hese assumpions cover ypical financial applicaions. In paricular, hey are saisfied when R is consan and for well-chosen parameerizaions of σ and γ, which can be enforced a he ime of he calibraion of he model. Theorem 4.1 i In he case of an RB, BSDE E cf. Definiion 3.6iii associaed wih he embedded bond admis a soluion Φ, z. Denoing Φ = Φ + A, he embedded bond admis a unique arbirage price Φ = 1 <τd Φ, [0, T ]. 23 ii Moreover we have Φ = Φ, S, where he funcion Φ, S is bounded, joinly coninuous in ime and space and wice coninuously differeniable in space, and he process Φ, S is an Iô process wih rue maringale componen, such ha wih v H 2. d Φ = u d + v dw 24 := µ Φ γ R b + Γ d + σ, S S S Φ dw Proof. i By sandard resuls see, e.g., [20, 22], BSDE E wih daa γr b + Γ µθ, N admis a soluion Φ, z. Hence by Theorem 3.2 specified o he paricular case of an ES we obain ha he embedded bond admis a unique arbirage price given by 23. ii By E, we have or equivalenly see [6] and hus, using 16: T Φ = E Q γ u Ru b + Γ u µ u Φu du + ξ b A T F, [0, T ], α Φ = E Q T α Φ = E Q T α u γ u R b u + Γ u du + α T ξ b A T F, [0, T ], α u γu R b udu + cu du + <T i T α Ti c i + α T ξ b F, [0, T ]. Se T α Φ0 = E Q α u γ u Ru b + cudu + α T N + AT F, T, 25 α Φi = E Q αti c i F, Ti. 26

14 14 Converible Bonds in a Defaulable Diffusion Model We have Φ T = Φ 0 T and Φ = Φ 0 + j;t i T j T Φ j on [T i 1, T i, or on [T K 1, T in case i = K 1. Le us denoe generically T or T i by T, and Φ 0 or Φ i by Θ, as appropriae according o he problem a hand. Noe ha Θ is bounded. Moreover, given our regulariy assumpions, we have Θ = Θ, S, where Θ is a C 1,2 [0, T R C 0 [0, T ] R-funcion [34, 22]. Therefore, Φ = Φ A is given by Φ, S, for a funcion Φ, S which is joinly coninuous in ime and space on [0, T ] R and wice coninuously differeniable in space on [0, T R. Moreover, given 25, 26 and he above C 1,2 regulariy resuls, we have d Φ 0 = µ Φ0 γ R b + c d + σ, S S Φ0 S, S dw, < T, This yields d Φ, S = d Φ i = µ Φi d + σ, S S S Φi, S dw, i = 1, 2,..., K, < T i T, da = ρ d, T i, i = 0, 1, 2,..., K. µ Φ γ R b + c + ρ d + σ, S S S Φ, S dw = u d + v dw. Moreover, since Φ and u are bounded in 24, we conclude ha v H Embedded Game Exchange Opion We now define he embedded Game Exchange Opion as an RB wih he dividend process D cb D b, paymen a mauriy ξ cb ξ b, pu paymen L cb Φ, call paymen U cb Φ and call proecion lifing ime τ. This means ha he embedded Game Exchange Opion is a zero-coupon CS wih cash flows β ψ; τ p, τ c = 1 {<τd τ} β τd Rτ cb d Rτ b d {τd >τ} β τ 1 {τ=τp<t } L cb τ p Φ τp + 1 {τ=τc<τ p} U cb τ c Φ τc + 1{τ=T } ξ cb ξ b. Noe ha from he poin of view of he financial inerpreaion see [4] for more abou his, he Game Exchange Opion corresponds o an opion o exchange he embedded bond for eiher L cb, U cb or ξ cb as seen from he perspecive of he holder, according o which player decides firs o sop his game and before T or no. We have he following observaion. Proposiion 4.2 Given an RB, he associaed funcions f, S, θ, g = gs, l = l, S and h = h, S, are: f = γr cb + Γ µθ, g = N κs, l = P κs, h = Ũ cb A, for he RB; f = γr cb R b µθ, g = κs N +, l = P κs Φ, h = Ũ cb A Φ, for he embedded Game Exchange Opion. 4.3 Soluion of he Doubly Refleced BSDEs Theorem 4.3 i The funcions f, g, l, h associaed o an RB or o he embedded Game Exchange Opion cf. Proposiion 4.2, saisfy all he general assumpions of Theorems ii The relaed problems E have unique soluions. Proof. i The resul for he RB follows direcly from he definiion of RB. Then, in view of Proposiion 4.2, he resul for he Game Exchange Opion follows from Theorem 4.1ii. ii In he case of he RB, we are in he siuaion of Example 3.13, so he relaed problem E has a unique soluion Π, v, k, by applicaion of Proposiion 3.5. Now, Φ, z denoing he soluion

15 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 15 o he BSDE E exhibied in Theorem 4.1i, i is immediae o check ha Ψ, w, k solves he Game Exchange Opion-relaed problem E iff Φ + Ψ, z + w, k, solves he RB-relaed problem E, whence in urn he resul for he Game Exchange Opion. Given an RB, we denoe by Π and Ψ he sae-processes firs componens Θ of he soluions o he relaed R2BSDEs. Theorem 4.4 i Ψ defined as 1 <τd Ψ is he unique arbirage price of he embedded Game Exchange Opionand Ψ, ζ, τc resp. Ψ, ζ, τp as defined in Theorem 3.2 is an issuer hedge wih iniial value Ψ resp. holder hedge wih iniial value Ψ saring from ime for he embedded Game Exchange Opion; ii Π defined as 1 <τd Π, wih Π := Π + A, is he unique arbirage price of he RB, and Π, ζ, τc resp. Π, ζ, τp as defined in Theorem 3.2 is an issuer hedge wih iniial value Π resp. holder hedge wih iniial value Π saring from ime for he RB. iii Wih Φ and Φ defined as in Theorem 4.1, we have Π = Φ + Ψ, Π = Φ + Ψ. Proof Given Theorem 4.3i, Saemens i and ii follow by applicaion of Theorem 3.2, whereas iii hen follows from he general resuls in [4]. In he foregoing sub-secions, we will give analyical characerizaions of he so-called pre-defaul clean prices pre-defaul prices less accrued ineres; see [6] in erms of viscosiy soluions o he associaed variaional inequaliies. To ge he corresponding pre-defaul prices, i suffices o add o he clean price process he relaed accrued ineres process. Noe ha in he case of he Game Exchange Opion, here are no discree coupons involved, herefore he pre-defaul clean price and he pre-defaul price coincide. 4.4 Variaional Inequaliies for he No-Proecion Clean Prices We firs assume ha τ = 0 no call proecion. By applicaion of Theorems 4.3, 3.3a and 3.4a, we have he following resul. Theorem 4.5 No-Proecion Clean Prices In he case where τ = 0 no call proecion, we define he following problems on D = [0, T ] R: a Defaulable Bond b Game Exchange Opion max min L Ψ + µ Ψ γr cb R b, Ψ P κs Φ L Φ + µ Φ γr b + Γ = 0, < T, 28 ΦT, S = N,, Ψ Ũ cb A Φ = 0, < T, ΨT, S = κs N +, 29 c RB max min L Π + µ Π γr cb + Γ, Π P κs, Π Ũ cb A = 0, < T, ΠT, S = N κs, 30 Then for any of he problems above, here exiss a P-soluion on D, denoed generically as Θ, S, ha deermines he corresponding No Proecion Pre-defaul Clean Price, say Θ, in he sense ha Θ = Θ, S, T. Moreover, we have uniqueness of he P-soluion and any sable, monoone and consisen approximaion scheme for Θ converges locally uniformly o Θ on D as h 0 +.

16 16 Converible Bonds in a Defaulable Diffusion Model Corollary 4.6 A pair of No Proecion Pre-defaul opimal sopping imes τ p, τ c see Theorem 3.2, boh in he case of he Game Exchange Opion embedded in he RB and of he RB iself, is given by where τ p = inf{u [, T ] ; S u E p } T, τ c = inf{u [, T ] ; S u E c } T, E p := {, S [0, T ] ; Π, S = P κs} E c := {, S [0, T ] ; Π, S = Ũ cb, S A } are he No Proecion Pre-defaul Pu or Conversion Region and he No Proecion Pre-defaul Call Region. Proof. This follows immediaely of Theorems 4.5 and 4.4. So, given a ime [0, T ], assuming ha here is no call proecion, and ha he RB is sill alive a ime : an opimal call ime for he issuer of he RB is given by he firs hiing ime of E c by S afer, if any such hiing ime occurs before T τ d ; an opimal pu/conversion ime for he holder of he RB consiss in puing or convering, whichever is bes, a he firs hiing ime of E p by S afer, if any such hiing ime occurs before T τ d. 4.5 Variaional Inequaliies for he Pos-proecion Prices For any τ FT 0, he associaed Pre-defaul Price coincides on [ τ, T ] wih he Pre-defaul Price corresponding o a lifing ime of call proecion ha would be given by τ 0 := 0. This follows from he general resuls in [5], using also he fac ha he BSDEs relaed o he problems wih lifing imes of call proecion τ and τ 0 boh have soluions, by he previous resuls. Thus No Proecion Prices pre-defaul prices for lifing ime of call proecion := τ 0 = 0 can be also be inerpreed as Pos-proecion Pre-defaul Prices for arbirary τ GT 0. Therefore he resuls of Secion 4.4 also apply o Pos-Proecion Clean Prices. So, Theorem 4.7 Pos-proecion Clean Prices Le τ FT 0. Then for an RB, he embedded Bond and he embedded Game Exchange Opion, he corresponding Pos-Proecion Pre-defaul Clean Price process coincides on [ τ, T ] wih process Θ, S, where Θ is he relaed funcion in Theorem 4.5. Corollary 4.8 The pair of No-Proecion Pre-defaul opimal sopping imes τ p, τ c, and he associaed No-Proecion Pre-defaul Call and Pu Regions E c and E p see Corollary 4.6, can also be inerpreed as a pair of Pos-proecion opimal sopping imes and Pos-proecion Call and Pu Regions, respecively. So assuming ha call proecion have already been lifed namely, for τ, and ha he RB is sill alive, we conclude ha: an opimal call ime for he issuer of he RB is given by he firs hiing ime of E c by S afer, if any such hiing ime occurs before T τ d ; an opimal pu/conversion ime for he holder of he RB consiss in puing or convering, whichever is bes, a he firs hiing ime of E p by S afer, if any such hiing ime occurs before T τ d.

17 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski Variaional Inequaliies for he Proecion Prices We finally consider Proecion Clean Prices Θ, namely, by definiion, Pre-Defaul Clean Prices sopped a τ. As usual, o ge he corresponding Proecion Pre-defaul Prices, i suffices o add he relaed accrued ineres process if here are any discree coupons involved. Le Φ, Ψ, Π denoe he No Proecion Clean Prices Funcions defined in Theorem Hard Call Proecions In he case of hard call proecion τ = T for some T T, he proecion clean prices funcions Θ are soluions of analyical problems as in Theorem 4.5a special case wih one obsacle, h = + wih T insead of T, and erminal condiions equal o he corresponding No Proecion Clean Prices Funcions Θ a T. So, Theorem 4.9 Hard Proecion Clean Prices In he case of τ = T for some T T, we define he following problems VI on D = [0, T ] R : a Game Exchange Opion min L Ψ + µ Ψ γr cb R b, Ψ P κs Φ = 0, < T, b RB Ψ T, S = Ψ T, S, 31 min L Π + µ Π γr cb + Γ, Π P κs = 0, < T, Π T, S = Π T, S. 32 Then for any of he problems 31 or 32, here exiss a P-soluion on D, denoed generically as Θ, S, ha deermines he corresponding Hard Proecion Clean Price, say Θ, in he sense ha Θ = Θ, S, T. Moreover, we have uniqueness of he P-soluion, and any sable, monoone and consisen approximaion scheme for Θ converges locally uniformly o Θ on D as h 0 +. Proof. For eiher problem, we know by Theorem 4.7 ha Θ T = Θ T, S T, and by Theorem 4.5 ha he erminal condiion Θ T, is coninuous and in P. Therefore he resul follows by applicaion of Theorems 4.3, 3.3a and 3.4a. Corollary 4.10 A Hard Proecion Pre-defaul opimal sopping ime τp Opion problem, and for he RB problem as well, is given by { τp = inf u [, T ] ; S u E h } T for he Game Exchange where E h = {, S [0, T ] ; Πu, S = P κ S } T is he Hard Proecion Pre-defaul Pu or Conversion Region. So assuming ha he RB is sill alive a some ime < T, an opimal sraegy for he holder of he RB consiss in puing or convering, whichever is bes, a he firs hiing ime of E h if any before τ d T by S.

18 18 Converible Bonds in a Defaulable Diffusion Model Sof Call Proecions Le us now rea he case of sof call proecions. By applicaion of Theorems 4.3, 3.3b and 3.4b, we have in urn he following resul. Theorem 4.11 Sof Proecion Clean Prices Assuming ha τ = inf{ > 0 ; S S} T for some S > 0, we define he following problems VI on D = [0, T ], S] : a Game Exchange Opion min L Ψ + µ Ψ γr cb R b, Ψ P κs Φ = 0, < T, S < S, b RB Ψ, S = Ψ, S, T, 33 ΨT, S = κs N +, S S, min L Π + µ Π γr cb + Γ, Π P κs = 0, < T, Π, S = Π, S, T, ΠT, S = N κs, S S. 34 Then for any of he problems 33 or 34 here exiss a P-soluion on D, denoed generically as Θ, S, ha deermines he corresponding Sof Proecion Clean Price, say Θ, in he sense ha Θ = Θ, S, τ. Moreover, we have uniqueness of he P-soluion, and any sable, monoone and consisen approximaion scheme for Θ converges locally uniformly o Θ on D as h 0 +, provided i converges o Θ= Θ a S. Corollary 4.12 A Sof Proecion Pre-defaul opimal sopping ime τp Opion problem, and for he RB problem as well, is given by { τp = inf u [, τ] ; S } u E s T for he Game Exchange where E s = {, S [0, T ] ; Π, S = P } κs is he Sof Proecion Pre-defaul Pu or Conversion Region. So assuming ha he sock has no reached he level S ye, and ha he RB is sill alive, an opimal sraegy for he holder of he RB consiss in puing or convering, whichever is bes, a he firs hiing ime of E s if any before τ d τ by S. 4.7 Converible Bonds wih posiive Call Noice Period We now consider he case of a Converible Bond wih posiive Call Noice Period. Noe ha beween he call ime and he end of he noice period δ = + δ T, a CB acually becomes a non-callable CS denoed as PB, cf. Definiion 3.2, ha is, a CB wih no call clause formally, we se τ = δ in he relaed BSDE. For a fixed, we call such a bond -PB, and i has effecive pu paymen equal o he effecive call paymen C u, u [, δ ], of he original CB, and effecive paymen a mauriy C δ see [4].

19 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 19 Proposiion 4.13 In he case of he -PB [0, T ], he associaed funcions fu, S, θ, g = gs and l = lu, S are h = + in all hree cases below: fu, S, θ = γu, SR b u, S + Γu, S µu, Sθ, gs = C, lu, S =, for he embedded Bond he -Bond, in he sequel; fu, S, θ = γu, SR cb R b u, S µu, Sθ, gs = C κs Φ δ, S, lu, S = C κs Φ u, S, where Φ is he No Proecion Clean Price Funcion of he -Bond obained by applicaion of Theorem 4.5, see also 35 below, for he embedded Game Exchange Opion he -Game Exchange Opion, in he sequel; fu, S, θ = γu, SR cb u, S + Γu, S µu, Sθ, gs = C κs, lu, S = C κs, for he -PB iself. Noe ha in view of he proof of Theorem 4.14ii below, i is convenien o define he relaed pricing problems on D := [0, δ ] R, raher han merely on [, δ ] R. Theorem 4.14 Variaional Inequaliies for he embedded PBs Given [0, T ], we define he following problems VI on D = [0, δ ] R : a -Bond L Φ + µ Φ γr b + Γ = 0, u < δ, 35 Φ δ, S = C, b -Game Exchange Opion min L Ψ + µ Ψ γr cb R b, Ψ C κs Φ = 0, u < δ, Ψ δ, S = C κs Φ δ, S, 36 c -PB min L Π + µ Π γr cb + Γ, Π C κs = 0, u < δ, Π δ, S = C κs. 37 i For any of he problems VI above, he corresponding Pre-defaul Clean -Price Θ u can be wrien as Θ u, S u, where he funcion Θ is a P-soluion of VI on D. Moreover, we have uniqueness of he P-soluion, and any sable, monoone and consisen approximaion scheme for Θ converges locally uniformly o Θ on D as h 0 +. ii The funcion Û, S := Π, S is joinly coninuous in ime and space. Hence he funcion Ũ, S = Û, S + A is also coninuous wih respec o, S, excep for lef jumps of size c i a he T i s. Proof. Par i follows by applicaion of previous resuls, in view of Proposiion We now prove ii. Le n, S n, S as n. We decompose Π n n, S n = Π n, S n + Π n n, S n Π n, S n. By i, Π n, S n Π, S as n. Moreover, denoing Ĉ = C κ S, F = γr cb + Γ, we have τ p α u Π u = esssup E τp F u Q α v F v dv + α τp Ĉ τp Fu, u δ. δ u So, assuming n sufficienly close o he lef of, and in view of he Markov propery of he process S, we have on he even { S n = S n }, τ p α n Π n n, S n = esssup τp F n E Q α v F v dv + α τp Ĉ τp δ n n τp esssup τp F n E Q α v F v dv + α τp Ĉ τp δ n F n F n = α n Π n, S n.

20 20 Converible Bonds in a Defaulable Diffusion Model Conversely, for any τ p F n, we have τ δ p := τ p δ n F n, 0 τ p τ p n and τp n δ n τp α v F v dv + α τp Ĉ τp α v F v dv α τp Ĉ τp n τp τ p α v F v dv + α τp Ĉ τp α τp Ĉ τp. Therefore E Q τ p τ p F n E Q α v F v dv + α τp Ĉ τp α v F v dv + α τp Ĉ τp n n τ p E Q α v F v dv F n + E Q α τp Ĉ τp α τp Ĉ τp F n τ p a n F H 2 + E Q α τp Ĉ τp α τp Ĉ τp F n, F n for some finie, posiive consan a. We conclude ha Π n n, S n Π n, S n 0 as n. Bu his is also rue, wih he same proof, as n +. Hence Π n n, S n Π n, S n 0 as n. Finally Π n n, S n Π, S as n, as desired. Theorem 4.15 A CB wih posiive noice period δ > 0 can be inerpreed as an RB wih Ũ cb, S = Ũ, S, where Ũ, S is he funcion defined a Theorem 4.14ii, so ha cf. 20 U cb = 1 {τd >}Ũ, S + 1 {τd } C κs + A. 38 Proof. Firs, he -PB relaed refleced BSDE has a soluion, by Theorems 4.3 and 3.3a applied o he -PB. Thus he -PB has a unique arbirage price process Π u = 1 {u<τd } Π u wih Π u = Π u + A u, by Theorem 3.2. So he arbirage price of he CB upon call ime assuming he CB sill alive a ime is well defined, as Π, which is also equal o Ũ, S cf. Theorem 4.14ii. Moreover, by Theorem 4.14ii, he funcion Ũ, S is joinly coninuous in ime and space, excep for negaive lef jumps of c i a he T i s, and Π C κs + A on he even {τ d > }, by he general resuls of [4]. So U cb defined as 38 saisfies all he requiremens in 20. Therefore, all he resuls of Secion 4 are applicable o a CB, since he laer may be inerpreed as an RB in virue of Theorem Numerical Issues 5.1 Pricing Assume ha τ = 0 no call proecion and ha we have already specified all he parameers for one of he problems 28, 29 or 30, including, in he case of 29 or 30, he funcion Ũ cb. Then one can solve he problem numerically see e.g. [2, 29] and i is known ha, under mild condiions cf. Theorem 3.4 and he Theorems of Secion 4, suiable approximaion schemes will converge owards he P-soluion of he problem as he discreizaion sep goes o 0. Solving he PDEs relaed o he embedded bond is sandard, and herefore we shall no commen on his issue here. To have a fully endogenous specificaion of he problem, one can ake Ũ cb, S = Ũ, S as defined in Theorem 4.14ii in 29 or 30, where Ũ, S is numerically compued by solving he relaed obsacle problems, using Theorem 4.14i. We provide below a pracical algorihm for solving, say 30, wih Ũ cb, S = Ũ, S, using, for example, a fully implici finie difference scheme see, for insance, [33] o discreize L :

21 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski Localize problems 37 for he embedded -PBs and problem 30 for he CB. A naural choice, for he -PBs as for he CB, is o localize he problems on he spaial domain, C κ ], wih a Dirichle boundary condiion equal o κs or a Neumann boundary condiion equal o κ a level C κ ; 2. Discreize he localized domain D loc = [0, T ], C κ ], using, say, one ime sep per day beween 0 and T ; 3. Discreize problems 37 for he embedded -PBs on he subdomain [, δ ] of D loc, for in he ime grid one problem per value of in he ime grid; 4. Solve for Π he discreized problems 37 corresponding o he embedded -PBs for in he ime grid one problem per value of in he ime grid; 5. Discreize problem 30 for he CB on D loc and solve he discreized problem, using he numerical approximaion of Ũ, S := Π, S + A as an inpu for Ũ cb, S in 30. Since he problem for he -PB only has o be solved on he ime-srip [, δ ] of D loc, he overall compuaional cos for solving a CB problem 30 wih posiive call noice period is roughly he same as ha required for solving one CB problem wihou call noice period, plus he cos of solving n PB problems ha would be defined on he whole grid, where n is he number of ime mesh poins in he noice period ha is ypically one monh, so n = 30, for a noice period δ = 1 monh and a ime sep of one day. Finally if a call proecion is in force hen we proceed along essenially he same lines, using he resuls in Secion 4.6. On Figure 1, 2 we ploed he price of he Converible Bond, he embedded Bond and he embedded Game Exchange Opion obained in his way as a funcion of he sock level S a ime 0, in he simple case where δ = 0, no call proecion is in force, and here are no dividends no coupons nor recovery, and for he numerical daa of remaining parameers gahered in Table 1. We ploed in each case he curves corresponding o defaul inensiies of he form γ, S = γ 0 S0 S for γ γ1 0 = 0.02 and γ 1 = 1.2 or γ 1 = 0., curves respecively labeled local and implied on each graph. r q η σ S 0 T P N C κ 5% % 100 5y Table 1: Parameer values Noe ha in case α = 1.2, consisenly wih ypical marke daa, he price of he CB as a funcion of S exhibis he so-called ski jump behavior, namely, i is convex for high values of S and collapsing a he low values. This collapse a low levels of S comes from he collapse of he embedded bond componen of a CB collapse of he bond floor. We refer he ineresed reader o [4] for more abou his. Remark 5.1 An alernaive for pricing would be o use numerical mehods for refleced BSDEs [32, 9, 10]. Given he soluion Θ, z, k of a R2BSDE in a Markovian se-up, he ineres of hese mehods is o provide numerical approximaions no only o he sae-process Θ he price of he CS, bu also o z he dela of he CS, cf. 18. In our case, such mehods would reduce o simple exensions o game problems of he well-known simulaion mehods for American opions [30, 35, 31]. Noe however ha hese mehods are no much used in he indusry a his sage. Beyond he fac ha hey are compuaionally inensive, anoher reason is ha hey do no give a confidence inerval, unlike sandard Mone Carlo mehods for European opions. Ye, in order o ake ino accoun non sandard sof call proecion clauses, or, more generally, o cope wih highly pah-dependen feaures, i may be necessary o resor o such simulaion mehods. 2 We hank Abdallah Rahal from he Mahemaics Deparmens a Universiy of Evry, France, and Lebanese Universiy, Lebanon, for numerical implemenaion of he model and, in paricular, for generaing he picure.

22 22 Converible Bonds in a Defaulable Diffusion Model Converible Bond embedded Bond embedded Opion local implied local implied Sock local implied Figure 1: The Ski-Jump Diagram and is Decomposiion 5.2 Calibraion A furher numerical issue is he calibraion of he model, which consiss in fiing some specific parameers of he model, such as he local volailiy σ and he local inensiy γ in our model, o marke prices of liquidly raded asses. Various inpu insrumens can be used in his calibraion process, such as: vanilla opions on he underlying equiy and/or CDS raded on bonds of he issuer see, e.g., [1]. As i can be seen on Figure 1, he price of he embedded game exchange opion enjoys much beer properies han he price of he CB in erms of convexiy wih respec o he sock price, and hus in urn see [4], in erms of monooniciy wih respec o he volailiy. These simple numerical experimens also suppor he inuiive guess ha he embedded bond concenraes mos of he ineres rae and credi risks of a converible bond, whereas he embedded game exchange opion concenraes mos of he volailiy risk noe in his respec ha he embedded game exchange opion always has a coupon process equal o zero. These feaures sugges ha i could be advanageous o use prices of synheic embedded game exchange opions, raher han prices of CBs, for he purpose of calibraion. We refer he reader o he discussion in he las secion of [4] for a more complee discussion of he poenial benefi of our decomposiion of a converible bond in is bond and opion componens regarding his calibraion issue.