Valuation and Hedging of Convertible Securities with Credit Risk

Valuation and Hedging of Convertible Securities with Credit Risk T.R. Bielecki, S. Cr´pey, M. Jeanblanc and M. Rutkowski e AMAMAEF February 2006 1 Based on • T.R. Bielecki, S. Cr´pey, M. Jeanblanc and M. Rutkowski: Arbitrage e Pricing of Convertible Securities with Credit Risk. Working paper. • T.R. Bielecki, S. Cr´pey, M. Jeanblanc and M. Rutkowski: Pricing and e Hedging Convertible Securities in Default Intensity Set-Ups. Work in Preparation. • T.R. Bielecki, S. Cr´pey, M. Jeanblanc and M. Rutkowski: Valuation and e Hedging of Convertible Bonds in a Markovian Model of Credit Risk. Work in Preparation. • S. Cr´pey, M. Jeanblanc and A. Matoussi: About Doubly Reflected e BSDEs Arising in the Study of Convertible Bonds, Work in preparation. 2 Outline 1. Game Options, Convertible Securities, Convertible Bonds 2. Dynkin Games Characterization of Arbitrage Prices for Convertible Securities 3. Decomposition of the Prices 4. Valuation in the Hazard Process Set-Up 5. Doubly Reflected BSDE Approach in the Intensity Set-Up, Hedging and Verification Principles 6. Variational Inequalities Approach in a Markovian Model of Credit Risk 3 Related works • J. Kallsen and C. K¨hn: Convertible Bonds: Financial Derivatives of u Game Type. In Exotic Option Pricing and Advanced L´vy Models. Edited e by Kyprianou, A., Schoutens, W., and Wilmott, P., Wiley, 2005. • Y. Kifer: Game options, Finance Stoch., 2000. • A. Shiryaev, Yu. Kabanov, D. Kramkov, A. Melnikov: Toward the Theory of Pricing of Options of Both European and American Types. I Discrete Time, II Continuous Time, Theory of Probability and Its Applications, 1994. • S. Hamad`ne : Mixed zero-sum differential game and American game e options, Preprint Universit´ du Maine, Submitted, 2004. e • El Karoui, N., Peng, S., and Quenez, M.-C. Backward stochastic differential equations in finance. Mathematical Finance, 1997. 4 • S. Hamad`ne, J.-P. Lepeltier, A. Matoussi: Double barriers reflected e backward SDE’s with continuous coefficients. Pitman Research Notes in Mathematics Series, 1997. • J.-P. Lepeltier and J. San Martin: Backward SDEs with two barriers and continuous coefficient: an existence result, J. Appl. Probab., 2004 • E. Ayache, P. Forsyth, and K. Vetzal: Valuation of Convertible Bonds with Credit Risk. The Journal of Derivatives, Fall 2003. • Andersen, L. and Buffum, D.: Calibration and implementation of convertible bond models. Working paper, Banc of America Securities, 2003. • El Karoui, N., Kapoudjian, E. Pardoux, C., Peng, S., and Quenez, M.-C: Reflected solutions of backward sde’s, and related obstacle problems for pde’s. Ann. Probab, 1997. • J.-P. Lepeltier and M. Xu: Penalization Method for Reflected Backward Stochastic Differential Equations with one r.c.l.l. barrier. Statistics & Probability Letters, 2005. 5 • Harraj.N., Ouknine.Y. et Turpin I: Double barriers Reflected BSDE’s with jumps and viscosity solutions of parabolic Integrodifferential PDE’s. Journal of Applied Mathematics and Stochastic Analysis, 2005 • Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stochast. Stochast. Reports, 1997. • E. Pardoux, F. Pradeilles, Z. Rao: Probabilistic interpretation of systems of semilinear PDEs. Annales de l’Institut Henri Poincare, serie Probabilites-Statistiques 33, 467–490, 1997. • Darling, R. and Pardoux, E: Backward SDE with Random Terminal Time and Applications to Semilinear Elliptic PDE. The Annals of Probability, 1997. • G. Barles and P.E. Souganidis: Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal., 1991. • B. Bouchard, R. Elie: Discrete time approximation of decoupled Forward-Backward SDE with jumps, Work in preparation. 6 1 Game Options, Convertible Securities, Convertible Bonds 7 1.1 Game Options Given a filtered probability space (Ω, G, P), a game option is a contract with cum-dividend cash-flows, as seen from the perspective of the holder, given by J (τp , τc ) = βτp Lτp 1{τp ≤τc } + βτc Uτc 1{τp >τc } , where: t • τp , τc ∈ GT0 are stopping times under the control of the holder and the issuer of the option, and • the payoffs L and U are G-adapted, c`dl`g, R ∪ {+∞}-valued processes, a a such that, for some real constant C, −C ≤ Lt ≤ Ut for t ∈ [t0 , T ), and LT = UT . 8 1.2 Convertible Securities Let an R+ ∪ {∞}-valued G-stopping time τd represent the default time of a reference entity (firm). We assume that the price evolution of the underlying market (including cash and the stock of the reference entity) can be modeled in terms of stochastic processes defined on a (Ω, G, P). We do not specify the underlying market at this stage so that, in particular, it may be incomplete, but we always assume that it is free of arbitrage opportunities. More specifically, we assume that the stock price S of the reference entity is a G-semimartingale with c`dl`g trajectories, and that the discounted, a a dividend-adjusted stock value βSQ is a G-martingale under a martingale measure for the underlying market, where: • the discount factor process (that is, the inverse of the savings account) βt is G-adapted, positive, and bounded on R+ , with c`dl`g sample paths, a a • the process Q, a generic dividend adjustment factor, is a G-adapted and (strictly) positively bounded process on [t0 , ∞), with c`dl`g sample paths. a a 9 Definition. A convertible security is a financial contract formally represented by a game option, with payoffs L and U such that: βt Lt = βt U t = t0 ≤Ti ≤t, Ti <τd t0 ≤Ti ≤t, Ti <τd X X βTi ci + 1{t0 ≤τd ≤t} βτd Rτd + 1{τd >t} βt 1{tt} βt 1{tτ } βτ 1{τ =τp τp } βτp (C ∨ κSτp + Aτp ), u for τp ∈ Gtδ . βTi ci ¯ Proposition. Assume that the t-PCB price Πt is arbitrage-free. Then ¯t ¯ Πt ≥ C ∨ κSt + At on the event {t < τd }. 21 3 Decomposition of the Prices 22 3.1 CB A B C Spread and Implied Volatility of a Convertible Bond Stock Price 8.39 77.80 25.25 Nominal 16.18 90.97 39.86 CB Price 17.42 93.98 41.80 Credit Spread 135 bp 65 bp 65 bp CB IV 30.2% 21.5% 33.9% Table 1: CB data on names of the CAC40 on May 10, 2005 CB A B C CB Expected Life Oct-10 Nov-07 May-09 Option Strike and Expiry 13.0 Dec-09 90.0 Dec-07 40.0 Dec-08 Option IV 30.7% 20.5% 35.6% Table 2: CBs and the closest listed options 23 3.2 Decomposition of the Cash Flows of a Convertible Security Let us be given: – a CS with coupon schedule (Ti , ci )i , – a non-negative, bounded, G-adapted process Rb and a non-negative, bounded, GT -measurable random variable ξ b . Definition. We define the embedded bond of the CS, as the bond with the ex-dividend cash flows βt φt (τd ) = tT } βT ξ b . Lemma. (i) The cash flows of the CS can be decomposed as follows: ˇ βt πt (τd , τp , τc ) = βt φt (τd ) + βt ψt (τd , τp , τc ), where b ˇ βt ψt (τd , τp , τc ) = 1{t<τd ≤τ } βτd (Rτd − Rτd ) + 1{τd >τ } βτ t ∈ [t0 , T ], 24 „ « ` ´ 1{τ =τp τ } βτ 1{τ =τp 0, a pair of -optimal stopping times (τp , τc ) for the Dynkin Game is given by τp = inf u ∈ [t, T ]; Θu ≤ Lu + ∧ T , τc = inf u ∈ [¯ ∨ t, T ]; Θu ≥ Uu − τ ∧T Remark. The R2BSDE can be rewritten in discounted form as follows: − + −dΘt = f (t, Θt )dt − zt dWt − vt dJt + dkt − dkt , ΘT = ξ + − ¯ ¯ Lt ≤ Θt ≤ Ut , (Θt− − Lt− )dkt = (Uu− − Θt− )dkt = 0 with f (t, Θt ) ≡ F (t) − rt Θt . Letting (Θ, Z, V, K + , K − ) be a solution of the 42 R2BSDE in intrinsic form, then Θ, z ≡ α−1 Z, v ≡ α−1 V, k + ≡ α−1 dK + , k − ≡ α−1 dK − is a solution of the R2BSDE in discounted form, and vice-versa. Remark. Since we work under Hypothesis H, for any solution (Θ, Z, V, K + , K − ) of a (R2)BSDE with respect to F, Θ is a F-semimartingale, hence a G-semimartingale. 43 5.2 Connection with Arbitrage Prices Definition. A Clean Price is an Arbitrage Price minus Accrued Interest. Definition. Given a CS in the present set-up, and under the assumptions of Theorem C, we define the Pre-default Clean Q-Prices of the CS, the embedded Bond and the embedded Option as Π = Π − A, Φ = Φ − A and Ψ = Ψ. Definition. Given a CS in the present set-up, we denote by (R2BSDE), (R2BSDE’) and (BSDE), the (R2)BSDEs with data (F, ξ − AT , L − A, U − A, τ ) , (F , ξ − ξ b , L − Φ, U − Φ, τ ) and (G, ξ b − AT ) , ¯ ¯ respectively, with Ft , Ft , Gt ≡ b b γt Rt + µt − (rt + γt )At , γt (Rt − Rt ), γt Rt + µt − (rt + γt )At and we denote by (r2bsde), (r2bsde’) and (bsde), the analog (R2)BSDEs in discounted form. 44 t ¯t Lemma. For any pricing time t and stopping times (τp , τc ) ∈ FT × FT : P ατ Aτ − αt At = t αu (µu − (ru + γu )Au ) du − t 0, a pair of -optimal stopping times (τp , τc ) t ¯t for the related Dynkin games on GT × GT is given by τp = inf τc = inf or, equivalently τp = inf τc = inf u ∈ [t, T ] ; Ψu ≤ Lu − Φu + ∧T ∧T . u ∈ [t, T ] ; Πu ≤ Lu + ∧T ∧T u ∈ [¯ ∨ t, T ] ; Πu ≥ Uu − τ u ∈ [¯ ∨ t, T ] ; Ψu ≥ Uu − Φu − τ (iii) The same conclusions hold true for Πt = Φt + Ψt and Ψt = Ψt , if (Ψ, Z, V , K + , K − ) is a solution of (R2BSDE’). (iv) Finally, if both equations (R2BSDE) and (R2BSDE’) have solutions (Π, Z, V, K + , K − ) and (Ψ, Z, V , K + , K − ), then Πt + At = Φt + Ψt . 46 5.3 Connection with Hedging Definition. 1. Let (Bt )t∈[t0 ,T ] and (Ot )t∈[t0 ,T ] denote the market price processes of two auxiliary hedging instruments B and O (beyond S) adjusted for their respective dividends. We assume that B and O are arbitrage price processes, so that βt Bt and βt Ot have to be G-martingales under Q, like βt St , where St is a notation for St Qt . Specifically, we assume the following dynamics:     βt St Wt      βt Bt  = Diag(βt St− , βt Bt− , βt Ot− ) Σt d  Jt  d    βt Ot Mt where dMt = dHt − (1 − Ht )γt dt with Ht ≡ 1τd ≤t , and where the 47 time-dependent volatility matrix   Σt ≡  σt  σt σt νt νt νt  −ηt   −ηt −ηt  is such that det Σt = 0, t0 ≤ t ≤ T. 2. A self-financing portfolio built on cash, S, B and O, with cumulative discounted dividend process βX, means a portfolio with discounted wealth process βY such that d(βt Xt ) + d(βt Yt ) = ζt d(βt St ) + ζt d(βt Bt ) + ζt d(βt Ot ) , t0 ≤ t ≤ T ζt = ζt = ζt = 0 , τd < t ≤ T for some G-predictable process ζ ≡ (ζ, ζ , ζ ). We shall denote such a portfolio by P ρ , where ρ ≡ (Y0 , ζ). 3. Let us denote by (βt Xt )t∈[t0 ,T ] the process of cumulative discounted dividends of a CS to be hedged. A perfect hedge for the CS at t0 , from the ∗ perspective of the issuer of the CS, means a pair (ρ, τc ), such that: 48 (i) P ρ is a self-financing portfolio built on cash, S, B and O, with cumulative discounted dividends process βX, ∗ ¯ (ii) τc ∈ G t0 , and ∗ ∗ ∗ ∗ ∗ ∗ (iii) Yt∧τc ∧τd ≥ 1t∧τc <τd 1t≤τc 0, a perfect hedge with initial value Π0 + is ¯0 furnished by τc as in Theorem D(ii) (for t = t0 , so τc ∈ FT ), and   zt   −1 −1   , t0 ≤ t ≤ T ζt ≡ 1t≤τd Σt Diag(βt St− , βt Bt− , βt Ot− )  vt  Rt − Πt + − where Π0 = Π0 and d(αt Πt ) = d(αt Πt ) + dKt − dKt , t0 ≤ t ≤ T . Proof. Use (r2bsde) and the definition of M . 49 Remark. (i) In the special case of a defaultable Bond, the infimum is a minimum in Theorem E (there exists a perfect hedge with initial wealth Π0 for the CS). (ii) In the case where F = FW (Brownian filtration), similar results can be proven with one auxiliary hedging instrument beyond S instead of two above. Moreover, in this case, the infimum is always a minimum in Theorem E (for any CS). (iii) The fact that t0 is the inception date of the instrument to be hedged is irrelevant in the previous results. 50 6 Variational Inequalities Approach in a Markovian Model of Credit Risk 51 6.1 Specification of the Model We consider a CB priced in the following risk-neutral local default intensity and volatility model (SMM, cf. Ayache et al 2003, Andersen & Buffum 2003): dSt = St− ((r(t) − q(t))dt + σ(t, St )dWt − ηdMt ) , t > t0 ; St0 = S0 , where: – the riskless short interest rate r(u) in the economy and the dividend yield q(u) are assumed to be bounded Borel functions of time; – W denotes a standard Brownian motion under the underlying market risk-neutral pricing measure (the underlying market is supposed to be free of arbitrage); – M is the compensated jump process, such that dMt = dHt − (1 − Ht )γt dt with Ht ≡ 1τd ≤t . 52 6.2 Regularity assumptions Assumption D. In the SMM, we assume that: ¯ ¯ 1. General assumptions regarding the CB: At ≡ A(t), Rt ≡ R(t, St ) and ¯ ¯ Xt ≡ X(t, St ) . 2. Assumptions regarding the underlying market: (a) The short rate r, the dividend yield q, the local intensity γ, the ¯ ¯ recovery rate R and the default claim X are bounded Borel functions, and the local volatility σ is a positively bounded Borel function; ¯ ¯ (b) γ(t, S), Sγ(t, S), Sσ(t, S) and γ(t, S)R(t, S)X(t, S) are once continuously differentiable in time, and three times continuously differentiable in space, with bounded spatial partial derivatives. 3. Specific assumptions regarding Utcb : 1t<τd Utcb ≡ 1t<τd U cb (t, St ) where the function U cb (t, S) is jointly continuous in time and space, except for negative left jumps of −ci at the Ti s. 53 6.3 Solution of the BSDE Lemma. (i) We have Φt = Φ(t, St ), where Φ(t, S) is jointly continuous in time and space, and Φ(t, St ) is a strict Itˆ process, such that: o b e e e b e dΦ(t, St ) = σ(t, St )St ∂S Φ(t, St )dWt e b e e e ¯ e ¯ e = ((r + γ(t, St ))Φ(t, St ) + (r + γ(t, St ))A(t) − γ(t, St )R(t, St )X(t, St ) − µt )dt (ii) We have ¯ 0 ≤ Lcb − At = P ∨ κSt ≡ L(St ) ≤ U (t, St ) = U cb (t, St ) − A(t) t ¯ Lcb − Φ = P ∨ κSt − Φ(t, St ) ≡ L (t, St ) ≤ U (t, St ) ≡ U cb (t, St ) − Φ(t, St ) t L=L +Φ, U ≡U +Φ, where the functions L(t, S), U (t, S), L (S) and U (t, S). Theorem E. (R2BSDE) and (R2BSDE’) (note that we are in a special case without jumps, F ≡ FW ) admit unique solutions (Ψ, Z, K + , K − ) and (Π, Z, K + , K − ). Proof. Hamad`ne Lepeltier Matoussi 1997 and Lepeltier San Martin 2004 e 54 6.4 Viscosity Solutions of the Variational Inequalities with time discontinuous forcing terms Note that due to the discontinuities in time that may arise at the coupon dates Ti in the drivers of our BSDEs, we should a priori use the general definition of viscosity solutions for discontinuous operators. However, the very simple nature of the discontinuities involved implies that the ”usual” definition is sufficient for our purposes. Call S-solutions, viscosity solutions with polynomial growth in space, namely bounded by C(1 + S p ) for suitable real C and integers p. 55 6.5 Post-Protection Pre-default Prices Theorem F. Assuming that τ = t0 (no call protection) , let L denote the linear ¯ operator σ2 S 2 2 ∂ 2 − (r + γ)Id L ≡ ∂t + (r − q + ηγ)S∂S + 2 S We define the following analytical problems (P) on D = [t0 , T ] × R+ , where T = T or T i , as appropriate according to the problem at hand: 1. Zero-Coupon Defaultable Bonds −LΦ0 + (r + γ)Φ0 − γRb = 0, t < T ¯ Φ0 (T, S) = N + A(T ) and for 1 ≤ i ≤ K −LΦi + (r + γ)Φi = 0, t < Ti Φi (Ti , S) = ci 56 (3) 2. Defaultable Bond −LΦ + (r + γ)Φ − γRb + µ − (r + γ)A = 0, t < T ¯ Φ(T, S) = N 3. Game Exchange Option (4) “ “ “ ”” b b b b b ¯ b max min −LΨ + (r + γ)Ψ − γ(R − R ), Ψ − P ∨ κS − Φ , “ ”” b e b b ¯ Ψ − U cb − A − Φ = 0, t < T ; Ψ(T, S) = (κS − N )+ (5) 4. CB Then for any of the problems (P) above, the corresponding Pre-default Clean Price Θ can be written as Θ(t, St ), where the function Θ is the unique S-solution of (P) on D. Moreover, in the case of non-coupon bearing 57 “ “ “ ” ” cb b b b b ¯ max min −LΠ + (r + γ)Π − γ R + µ − (r + γ)A , Π − P ∨ κS , ” cb b e b ¯ Π − (U − A) = 0, t < T ; Π(T, S) = N ∨ κS (6) instruments (Zero-Coupon Defaultable Bonds and Game Exchange Option), Θ is also the maximal S-subsolution of (P), as well as its minimal S-supersolution. Proof. Existence by BSDEs, Uniqueness adapted from Barles, Buckdahn & Pardoux 1997 ∗ ∗ A pair of Post-Protection Pre-default optimal stopping times (τp , τc ) in the Dynkin game associated to the game Q-exchange option embedded in the CB, and in the Dynkin game associated to the CB as well, is given by ∗ τp = inf ∗ τc = inf ¯ u ∈ [t, T ] , Π(u, Su ) = P ∨ κSu ∧ T u ∈ [t, T ] ; Π(u, Su ) = π(u, Su ) − A(u) ∧ T 58 6.6 Protection Pre-default Prices ¯ Theorem F’. Assuming that τ = inf{t > t0 ; St ≥ S} ∧ T (standard soft call ¯ protection),we define the following analytical problems (P) on ¯ ¯ D ≡ D = [t0 , T ] × (0, S],where Φ, Ψ, Π denote the Post-Protection Pre-default Prices Functions. 1. Game Exchange Option “ ` ´” ¯ ¯ ¯ b ¯ ¯ ¯ min −LΨ + (r + γ)Ψ − γ Ro , Ψ − P ∨ κS − Φ = 0 ; t < T , S < S ¯ ¯ ¯ Ψ(T, S) = (κS − N )+ , S ≤ S 2. CB “ “ ” ” cb ¯ ¯ b ¯ ¯ min −LΠ + (r + γ)Π − γ R + µ − (r + γ)A , Π − P ∨ κS = 0 , t < T ¯ ¯ b ¯ Π(t, S) = Π(t, S) , t ≤ T ¯ ¯ ¯ Π(T, S) = N ∨ κS , S ≤ S (8) ¯ ¯ b ¯ Ψ(t, S) = Ψ(t, S) , t ≤ T (7) Then for either problem, the corresponding Pre-protection Pre-default Clean 59 ¯ ¯ ¯ Price Θ can be written as Θ(t, St ), where the function Θ is the unique ¯ S-solution of (P) on D. Moreover, in the case of the Game Exchange Option, Θ is also the maximal S-subsolution of (P), as well as its minimal S-supersolution. ∗ A Pre-protection Pre-default optimal stopping time τp for the game Q-exchange option problem, and for the CB problem as well, is given by ∗ τp = inf ¯ ¯ u ∈ [t, T ] ; Π(u, Su ) = P ∨ κSu ∧ T ¯ Similarly, one can show that in the case of hard call protection τ = T < T, the ¯ ¯ Pre-protection Pre-default Clean Prices Functions Θ are solutions of analytical ¯ problems as in Theorem F with T = T , and terminal conditions equal to the ¯ corresponding Pre-default Clean Prices Functions Θ at T . 60 6.7 Variational Inequalities for the embedded PCBs and Reality Check of the Model Recall that if the CB is called at time t, the CB actually becomes a PCB on the time interval [t, tδ ], called the t-PCB, with effective put payment equal to ¯ the effective call payment (C ∨ κSu ) + Au of the original CB, and with market ¯u price process denoted by (Πt )u∈[t,tδ ] . Theorem G. In the SMM, let us be given a CB satisfying all the hypotheses in Assumption D (which includes Assumption A), except maybe for the assumptions regarding the process U cb — that is, except maybe for Assumptions A and D(3). We assume that the embedded PCBs price ¯ processes Πt are arbitrage-free, with associated pricing measures Qt , t0 ≤ t ≤ T. Suppose further that Qt ≡ Q, t ∈ [t0 , T ) . Then Πt ≡ Πt (u, Su ), u and the function Πt (t, S) is continuous with respect to (t, S), except for negative jumps of −ci at the Ti . So Assumption D is satisfied, with U cb (t, S) ≡ Πt (t, S) . 61 6.8 Numerical Solution of the Variational Inequalities Solving the Bond-related PDEs is of course standard. Therefore it is enough to be able to solve the Option-related Variational Inequalities (VIs) , which also allows one to get the result for the CBs, since “Bond+Option=CB.” So we shall concentrate on the Option Problems, which, at least from a theoretical point of view, are (slightly) simpler, as the related operator is continuous . As a consequence, we know by Theorem F that the solutions of the corresponding VIs are not only the unique S-solutions, but also the maximal S-subsolutions, and the minimal S-supersolutions. Otherwise said, we have a semi-continuous comparison principle for these problems. Consequently, Theorem H. (i) Let (Ψh )h>0 denote a stable, monotonous and consistent approximation scheme for the No Protection Pre-default Clean Price function Ψ. Then Ψh → Ψ locally uniformly on D as h → 0+ , provided Ψh → Ψ at T . (ii) Let (Ψh )h>0 denote a stable, monotonous and consistent approximation ¯ scheme for the Soft Protection Pre-default Clean Price Function Ψ. Then ¯ ¯ ¯ Ψh → Ψ locally uniformly on D as h → 0+ , provided Ψh → Ψ(≡ Ψ) at T and 62 ¯ at S. (iii) Let (Ψh )h>0 denote a stable, monotonous and consistent approximation ¯ scheme for the Hard Protection Pre-default Clean Price Function Ψ. Then ¯ ¯ ¯ ¯ Ψh → Ψ locally uniformly on D as h → 0+ , provided Ψh → Ψ(≡ Ψ) at T . Proof. Barles & Souganidis 1991 Here is a practical algorithm for pricing the CB with U cb (t, S) ≡ Πt (t, S), using for example a fully implicit finite difference scheme to discretize L : 1. Localize the VIs for the embedded t-PCBs and for the CB. In the case of the PCBs, we know in advance that the price is equal to κS for κS larger ¯ than C. So a natural choice, obviously in the special case of the PCBs, ¯ but also for the CB, is to localize the VIs on the spatial domain [0, C/κ]; ¯ 2. Discretize the domain D = [t0 , T ] × [0, C/κ], using, say, one time step per day market day between t0 and T ; 3. Discretize the VIs for the embedded PCBs on D t = D ∩ [t, tδ ], for t in the time grid (one problem per value of t in the time grid); 63 4. Solve for Πt the discretized VIs corresponding to the embedded PCBs for t in the time grid (one problem per value of t in the time grid); 5. Discretize the VI for the CB on D and solve the discretized problem, using the numerical approximation of Πt (t, S) ≡ Πt (t, S) + A(t) as an input for U cb (t, S). Since the problem for the t-PCB only has to be solved on the subdomain D t of D, the overall computational cost for pricing a CB with non-zero call notice period is roughly the same as that required for solving one CB problem without call notice period, plus n PCB problems that would be defined on the whole grid, where n is the number of days in the notice period — typically one month, that is n = 20 market days. Finally if a call protection is in force then we proceed along essentially the same lines using Theorem F’, using the Post-Protection Prices numerically determined as above as terminal-boundary conditions . 64 Conclusions & Open Problems 65 Conclusions We derived some results regarding defaultable convertible securities, in particular, • the characterization of arbitrage prices of a CS , • the decomposition of an arbitrage price of a CS (under Q, say) in terms of an arbitrage price of the embedded defaultable bond under Q and an arbitrage price of the embedded Q-game option under Q, • the interpretation of market data, the CB spread and CB implied volatility, in terms of well-established pricing theory, • links between pricing and hedging of a CS and doubly reflected BSDEs in Default Intensity Set-Ups, • links between arbitrage pricing of a CB and variational inequalities in the Standard Market Model, • conditions ensuring convergence of deterministic approximation schemes for the variational inequalities. 66 Open Problems • Relationships between various martingale measures, Q, Q and Qt , should be further investigated. • The issue of hedging strategies in the SMM for a defaultable CB (see Ayache et al. 2003) should be addressed, using for instance equity and a CDS as hedging instruments. Purely dynamic hedging schemes should be compared with partly static hedge involving shorting the embedded bond. • Switching Regime Models should be studied. • Alternative (Monte Carlo or BSDE related) numerical schemes should be considered and benchmarked. 67