Credit Risk Modeling

Vulnerable Swaption In this section, we relax the assumption that Y 1 is the price of a default-free bond. We now let Y 1 and Y 2 to be arbitrary default-free assets, with dynamics dY i t = Y i t ( μi,t dt + σi,t dWt ) , i = 1, 2. (4.36) We still take the defaultable zero-coupon bond with zero recovery and the price process Y 3 t = D(t, T ) to be the third traded asset. We maintain the assumption that the model is arbitrage-free, but we no longer postulate that it is complete. In other words, we postulate the existence an EMM Q, as defined in subsection on the arbitrage-free property, but not the uniqueness of Q. We take the first asset as the numéraire, so that all prices are expressed in units of Y . In particular, Y 1,1 t = 1 for every t ∈ R+, and the relative prices Y 2,1 and Y 3,1 satisfy under Q (cf. Proposition 4.3.1) dY 2,1 t = Y 2,1 t (σ2,t − σ1,t) dŴt, dY 3,1 t = Y 3,1 t− ( (σ3,t − σ1,t) dŴt − dM̂t ) . It is natural to postulate that the driving Brownian noise is two-dimensional. In such a case, we may represent the joint dynamics of relative prices Y 2,1 and Y 3,1 under Q as follows dY 2,1 t = Y 2,1 t (σ2,t − σ1,t) dW 1 t , dY 3,1 t = Y 3,1 t− ( (σ3,t − σ1,t) dW 2 t − dM̂t ) , where W , W 2 are one-dimensional Brownian motions under Q, such that d〈W , W 〉t = ρt dt for a deterministic instantaneous correlation coefficient ρ taking values in [−1, 1]. 152 CHAPTER 4. HEDGING OF DEFAULTABLE CLAIMS We assume from now on that the volatilities σi, i = 1, 2, 3 are deterministic. Let us set αt = 〈ln Ỹ , ln Ỹ 〉t = ∫ t 0 ρu(σ2,u − σ1,u)(σ3,u − σ1,u) du, (4.37) and let Q̂ be an equivalent probability measure on (Ω,GT ) such that the process Ŷt = Y 2,1 t e−αt is a Q̂-martingale. To clarify the financial interpretation of the auxiliary process Ŷ in the present context, we introduce the concept of credit-risk-adjusted forward price relative to the numéraire Y . Definition 4.3.3 Let Y be a GT -measurable claim. An Ft-measurable random variable K is called the time-t credit-risk-adjusted Y -forward price of Y if the pre-default value at time t of a vulnerable forward contract, represented by the claim 1{T<τ}(Y 1 T ) −1(Y −KY 1 T ) = 1{T<τ}(Y (Y 1 T )−1 −K), equals 0. The credit-risk-adjusted Y -forward price of Y is denoted by F̂Y |Y 1(t, T ) and it is also interpreted as an abstract defaultable swap rate. The following auxiliary results are easy to establish, by arguing along the same lines as in Lemmas 4.3.2 and 4.3.3. Lemma 4.3.4 The credit-risk-adjusted Y -forward price of a survival claim Y = (X, 0, τ) equals F̂Y |Y 1(t, T ) = π̃t(X, 0, τ)(D̃(t, T ))−1, where X = X(Y 1 T ) −1 is the price of X in the numéraire Y 1 and π̃t(X, 0, τ) is the pre-default value of a survival claim with the promised payoff X. Proof. It suffices to note that for Y = 1{T<τ}X we have 1{T<τ}(Y (Y 1 T ) −1 −K) = 1{T<τ}X −KD0(T, T ), where X = X(Y 1 T ) −1, and to consider the pre-default values. ¤ Lemma 4.3.5 The credit-risk-adjusted Y -forward price of the asset Y 2 equals F̂Y 2|Y 1(t, T ) = Y 2,1 t e αT−αt = Ŷte , where α, assumed here to be deterministic, is given by formula (4.37). Proof. It suffices to find an Ft-measurable random variable K for which D̃(t, T )Eb Q ( Y 2 T (Y 1 T ) −1 −K ∣Ft ) = 0. From the last equality, we obtain K = F̂Y 2|Y 1(t, T ), where F̂Y 2|Y 1(t, T ) = Eb Q ( Y 2,1 T ∣Ft ) = Y 2,1 t e αT−αt = Ŷt eT . We have used here the facts that Ŷt = Y 2,1 t e −αt is a Q̂-martingale and α is deterministic. ¤ We are in a position to examine a vulnerable option to exchange default-free assets with the payoff C T = 1{T<τ}(Y 1 T ) −1(Y 2 T −KY 1 T ) = 1{T<τ}(Y 2,1 T −K)+. (4.38) The last expression shows that the option can be interpreted as a vulnerable swaption associated with the assets Y 1 and Y . It is useful to observe that C T Y 1 T = 1{T<τ} Y 1 T ( Y 2 T Y 1 T −K )+ , 4.3. MARTINGALE APPROACH 153 so that, when expressed in units of the numéraire Y , the payoff becomes C T = D (T, T )(Y 2,1 T −K)+, where C t = C t (Y 1 t ) −1 and D(t, T ) = D(t, T )(Y 1 t ) −1 stand for the prices relative to the numéraire Y . It is clear that we deal here with a model analogous to the model examined in previous subsections in which, however, all prices are expressed in units of the numéraire asset Y . This observation allows us to directly deduce the valuation formula from Proposition 4.3.2. Proposition 4.3.3 Let us consider the market model (4.36) with a two-dimensional Brownian motion W and deterministic volatilities σi, i = 1, 2, 3. The credit-risk-adjusted Y -forward price of a vulnerable call option, with the terminal payoff given by (4.38), equals F̂Cd|Y 1(t, T ) = F̂tN ( d+(F̂t, t, T ) −KNd−(F̂t, t, T ) ) , where we write F̂t = F̂Y 2|Y 1(t, T ) and d±(z, t, T ) = ln z − ln K ± 12v(t, T ) v(t, T ) with v(t, T ) = ∫ T t (σ2,u − σ1,u) du. The replicating strategy φ in the spot market satisfies, on the event {t < τ}, φt Y 1 t = −φt Y 2 t , φt = D̃(t, T )(Y 1 t )−1N(d+(t, T ))eαT−αt , φt D̃(t, T ) = C̃ t , where d+(t, T ) = d+ ( F̂Y 2|Y 1(t, T ), t, T ) . Proof. The proof is analogous to that of Proposition 4.3.2 and thus it is omitted. ¤ It is worth noting that the payoff (4.38) was judiciously chosen. Suppose instead that the option payoff is not defined by (4.38), but it is given by an apparently simpler expression C T = 1{T<τ}(Y 2 T −KY 1 T ). Since the payoff C T can be represented as follows C T = Ĝ(Y 1 T , Y 2 T , Y 3 T ) = Y 3 T (Y 2 T −KY 1 T ), where Ĝ(y1, y2, y3) = y3(y2 − Ky1), we deal with an option to exchange the second asset for K units of the first asset, but with the payoff expressed in units of the defaultable asset Y . When expressed in relative prices, the payoff becomes C T = 1{T<τ}(Y 2,1 T −K)+. where 1{T<τ} = D(T, T )Y 1 T . It is thus rather clear that it is not longer possible to apply the same method as in the proof of Proposition 4.3.2. 4.3.3 Defaultable Asset with Non-Zero Recovery In this section, we still postulate that Y 1 and Y 2 are default-free assets with price processes dY i t = Y i t ( μi,t dt + σi,t dWt ) , where W is a one-dimensional Brownian motion. We now assume, however, that dY 3 t = Y 3 t−(μ3 dt + σ3 dWt + κ3 dMt) with κ3 > −1 and κ3 6= 0. We assume that Y 3 0 > 0, so that Y 3 t > 0 for every t ∈ R+. We shall briefly describe the same steps as in the case of a defaultable asset with zero recovery. 154 CHAPTER 4. HEDGING OF DEFAULTABLE CLAIMS Arbitrage-Free Property As usual, we need first to impose specific constraints on model coefficients, so that the model is arbitrage-free. In general, an EMM Q exists if there exists a pair (θ, ζ) such that, for i = 2, 3, θt(σi − σ1) + ζtξt κi − κ1 1 + κ1 = μ1 − μi + σ1(σi − σ1) + ξt(κi − κ1) κ1 1 + κ1 . To ensure the existence of a solution (θ, ζ) on the event {τ < t} under the present assumptions, we impose the condition σ1 − μ1 − μ2 σ1 − σ2 = σ1 − μ1 − μ3 σ1 − σ3 , that is, μ1(σ3 − σ2) + μ2(σ1 − σ3) + μ3(σ2 − σ1) = 0. Since κ1 = κ2 = 0, on the event {τ ≥ t}, we have to solve the following equations θt(σ2 − σ1) = μ1 − μ2 + σ1(σ2 − σ1), θt(σ3 − σ1) + ζtγκ3 = μ1 − μ3 + σ1(σ3 − σ1). If, in addition, (σ2 − σ1)κ3 6= 0, we obtain the unique solution θ = σ1 − μ1 − μ2 σ1 − σ2 = σ1 − μ1 − μ3 σ1 − σ3 , ζ = 0 > −1, so that the martingale measure Q exists and is unique. Observe that, since ζ = 0, the default intensity under Q coincides here with the default intensity under the real-life probability Q. It is interesting to note that, in a more general situation when all three assets are defaultable with non-zero recovery, the default intensity under Q coincides with the default intensity under the real-life probability Q if and only if the process Y 1 is continuous. For more details, the interested reader is referred to Bielecki at al. [14] where the general case is studied. 4.3.4 Two Defaultable Assets with Zero Recovery We shall now assume that we have only two assets and both are defaultable assets with zero recovery. This case was recently examined by Carr [44], who studied an imperfect hedging of digital options. Note that here we present results for replication, that is, perfect hedging. We shall briefly outline the analysis of hedging of a survival claim. Under the present assumptions, we have, for i = 1, 2, dY i t = Y i t− ( μi,t dt + σi,t dWt − dMt ) , (4.39) where W is a one-dimensional Brownian motion, so that Y 1 t = 1{t<τ}Ỹ 1 t , Y 2 t = 1{t<τ}Ỹ 2 t , with the pre-default prices governed by the SDEs dỸ i t = Ỹ i t ( (μi,t + γt) dt + σi,t dWt ) . (4.40) The wealth process V associated with the self-financing trading strategy (φ, φ) satisfies, for every t ∈ [0, T ], Vt = Y 1 t ( V 1 0 + ∫ t