Derivation of the Price of Bond in the Recovery of Market Value Model

Transcription

1 Derivation of the Price of Bond in the Recovery of Market Value Model By YanFei Gao Department of Mathematics & Statistics, McMaster University Apr. 2 th, 25

2 1 Recovery models For the analysis of reduced-form pricing models, there are some recovery assumptions, such as the zero recovery, recovery of treasury, recovery of par and the recovery of market value. Consider the doubly stochastic model with a stochastic intensity λ t and assume that the recovery rate R t at time t. Assume that the default time τ >, the price at time zero of a defaultable bond with maturity time T is the sum of a recovery term and a survival term. It is given by P T = E Q [e R τ rsds R τ 1 τ T + e R T rsds 1 τ>t ] = E Q [E Q [e R τ rsds R τ 1 τ T F T ]] + E Q [e R T rsds 1 τ>t ] = E Q [ = e R τ (rs+λsds R u λ u du] + E Q [e R T rs+λsds ] E Q [R u λ u e R u rs+λsds ]du + E Q [e R T rs+λsds ] Zero Recovery In the zero recovery model, R t = for all t, so the expression of P T term of Equation (1). (1) is the survival P T = E Q [e R τ rs+λsds ] (2) Recovery of treasury In the recovery of treasury, the recovery rate is in the form R t = RP T, where < R 1. Then P RT T = R E Q [P ut λ u e R u (rs+λs)ds ]du + P T = RE Q [e R T rsds λ u e R u λsds du] + P T = RE Q [e R u rsds (1 e R T λsds )] + P T (3) = RP T + (1 R) P T Recovery of par In the recovery of par model, R t = R, where < R < 1, for all t. P RT T = R E Q [λ u e R u (rs+λs)ds ]du + P T (4) For the recovery of market value, assume that, at each time t, conditional on all information available up to but not including time t, a specified risk-neutral mean fraction L t of market value is lost if default occurs at time t. One can consider the (1-L t ) as the recovery rate R t at time t. The formula of the price at time t of a defaultable bond is derived in the following section. 1

3 2 Derivation of the Price in RMV Model 2.1 Discrete-time Valuation Under risk-neutral probability measure, let h t be the hazard rate for default at time t and let L t be the expected fractional loss in market value if default were to occur at time t, conditional on the information available up to time t (Duffie and Singleton, 1999). Suppose that, X is paid at time T in the event of no default. Then, the initial market value of the defaultable claim to X is P T = E Q [e R T Rtdt X] (5) where E Q denotes risk-neutral, conditional expectation at data. Consider a defaultable case that, X t+τ is paid at maturity date t+τ, and nothing before date t+τ. Let ϕ t be the recovery in units of account in the event of default at time t, and r t be the default-free short rate. The hazard rate h t is also the conditional probability at time t of default between t and t+1 given the information available at time t in the event of no default by t. Therefore, if the asset has not defaulted by time t, its market value P t is the present value of receiving ϕ t+1 in the event of default between t and t+1 plus the present value of receiving V t+1 in the event of no default, that is P t = h t e rt E Q t (ϕ t+1 ) + (1 h t )e rt E Q t (P t+1 ) (6) where E Q t (.) denotes the expectation under Q, conditional on information available to investors at date t (Duffie and Singleton, 1999). By recursively solving Equation () forward over the life of the bond, P t can be expressed equivalently as P t = h t e rt E Q t (ϕ t+1 ) + (1 h t )e rt E Q t (P t+1 ) τ 1 = E Q t [ h t+j e P j j k= r t+k ϕ t+j+1 (1 h t+l 1 )] + E Q t [e P τ 1 τ k= r t+k X t+τ (1 h t+j 1 )] (7) j= l= The equation can be simplified by taking the risk-neutral expected recovery at time s, where s t, in the event of default at time s+1, to be a fraction of the risk-nuetral expected survivalcontingent market value at time s+1. Let R s be the recovery rate at time s. Thus, RMV(Recovery of Market Value): Substituting RMV into Equation () then get E Q s (ϕ s+1 ) = R s E Q s (P s+1 ) (8) P t = h t e rt R t E Q t (P t+1 ) + (1 h t )e rt E Q t (P t+1 ) = E Q t (P t+1 )[h t e rt R t + (1 h t )e rt ] j=1 2

4 By the approximation, we have e x 1 + x, for small x. Then e Yt = h t e rt R t + (1 h t )e rt = e rt [h t R t + 1 h t ] = e rt {h t [R t 1] + 1} e Yt+rt = h t (R t 1) + 1 Y t + r t = h t (R t 1), (By the approximation) Y t = r t + h t (1 R t ) Therefore, Equation (5) can be expressed as P t = E Q t (P t+1 )[h t e rt R t + (1 h t )e rt ] = E Q t (P t+1 )e Yt = E Q t (P t+1 )e Yt Substituting Equation (8) and Y t = r t + h t (1 R t ) into Equation (), then P t = E Q t (e P τ 1 k= Y t+k X t+τ ) = E Q t (e P τ 1 k= r t+k+h t+k (1 R t+k X t+τ ) For the inhomogeneous Poisson process reduced form model, h t = λ t. Therefore, for the discrete-time, 2.2 Continuous-time Valuation P t = E Q t (e P τ 1 k= r t+k+λ t+k (1 R t+k X t+τ ) (9) We fix a probability space (Ω, F, P ) and a family F t : t of σ algebras satisfying the usual conditions (Protter, 199). A predictable short-rate process r is also fixed, so that it is possible at any time t to invest one unit of account in default-free deposits and roll over the proceeds until a later time s for a market value at that time of e R s t rudu. Suppose Z is paid at a stopping time τ, which is denoted by (Z,τ). In order to have the payment can be made based on currently available information, we assume that Z is F t -measurable. Taking an equivalent martingale measure Q relative to the short-rate process r. Thus, the ex dividend price process U of any given (Z,τ) is defined by U t = {E Q t [e R τ t rudu Z], if t < τ, if t τ (1) where E Q t denotes the expectation under the risk-neutral measure Q, given F t (Duffie and Singleton, 1999). 3

5 Suppose (X,T) and (X,T ) are two pairs of claims. The first claim (X,T) means that the issuer has to pay X at date T, and the second claim (X,T ) defines the stopping time T at which the issuer defaults and X is paid to claimholders. Hence, the claim (Z,τ) can be re-written as τ = min(t, T ), andz = X1 {T <T } + X 1 {T T } (11) Suppose the default time T has a risk-neutral default hazard rate process h, which means that the process N t is {, if t < τ N t = (12) 1, if t τ If default occurs at time t, then we will suppose that the claim pays X = R t U t (13) where U t = lim s t U s is the price of the lcaim just before default, and R t is the random variable describing the recovery rate of market value of the claim at default (Duffie and Singleton, 1999). Similarly to the discrete-time model, we guess the continuous-time formula is P t = E Q t [e R T t Y sds X] = E Q t [e R T t (r s+λ sr s)ds X] To confirm the formula above, we set a gain process G, after discounting at the short-rate process r, must be a martingale under Q. G t = e R t rsds P t (1 N t ) + t (14) e R s rudu R s P s dn s (15) The first term is the survival term, and the second term is the discounted payout of the claim upon default. This Equation is similar to the Equation (6). P T =X and G is a Q martingale. Applying Ito s formula (Protter 199) to Equation (15), using Equation (13), we can see that for G to be a Q martingale, it is necessary and sufficient that P t = t (r s + R s λ s )ds + m t (16) for some Q martingale m. Since P jumps at most a countable number of times, we can replace V s in Equation (15) with P s. Given the terminal boundary condition P T = X, this implies Equation (14). (Duffie and Singleton, 1999) 4

6 3 Reference [1] Unifying discrete structural credit risk models and reduced-form models, Cho-Jieh, Chen and Harry, Panjer. (22) [2] Modeling term structures of defaultable bonds, Darrell, Duffie and Kenneth J., Singleton. (1999) [3] Backward-Forward stochastic differential equations, Annals of Applied Probability, 3,