PDE Approach to Credit Derivatives

Transcription

1 PDE Approach to Credit Derivatives Marek Rutkowski School of Mathematics and Statistics University of New South Wales Joint work with T. Bielecki, M. Jeanblanc and K. Yousiph Seminar 26 September, 2007

2 Outline The Model 1 The Model Prices of Traded Assets Change of a Numeraire 2 Martingale Measure 3 Martingale Measure 4 Pricing PDEs 5 Pricing PDEs 6 Case of Two Credit Names Case of m Credit Names

3 REFERENCES

4 References Bielecki, T., Jeanblanc, M. and Rutkowski, M.: PDE approach to valuation and hedging of credit derivatives. Quantitative Finance 5 (2005), Rutkowski, M. and Yousiph, K.: PDE approach to the valuation and hedging of basket credit derivatives. Forthcoming in International Journal of Theoretical and Applied Finance. Bielecki, T., Jeanblanc, M. and Rutkowski, M.: Hedging of credit derivatives in models with totally unexpected default. In: Stochastic Processes and Applications to Mathematical Finance, J. Akahori et al., eds., World Scientific, Singapore, 2006, Bielecki, T., Jeanblanc, M. and Rutkowski, M.: Replication of contingent claims in a reduced-form credit risk model with discontinuous asset prices. Stochastic Models 22 (2006),

5 Default Time Prices of Traded Assets Change of a Numeraire THE MODEL

6 Default Time The Model Default Time Prices of Traded Assets Change of a Numeraire The default time τ is a non-negative random variable on (Ω, G, Q). Note that Q is the statistical probability measure. The filtration generated by the default process H t = 1 {τ t} is denoted by H. We set G = F H, so that G t = F t H t for every t R +, where F = (F t) t R+ is a reference filtration. We define the processes F t and G t as F t = Q{τ t F t} and G t = 1 F t = Q{τ > t F t}.

7 Hazard Process The Model Default Time Prices of Traded Assets Change of a Numeraire The process Γ, given as Γ t = ln(1 F t) = ln G t is the F-hazard process under the statistical probability Q. We shall assume that the F-hazard process is absolutely continuous: Γ t = R t γu du. 0 Hence, the compensated default process M t = H t Z t τ 0 γ u du = H t Z t 0 ξ u du, is a G-martingale under Q, where we denote ξ t = γ t1 {t<τ}.

8 Hypothesis (H) The Model Default Time Prices of Traded Assets Change of a Numeraire Hypothesis (H). We assume throughout that any F-martingale under Q is also a G-martingale under Q. Hypothesis (H) is satisfied if a random time τ is defined through the canonical construction. If the representation theorem holds for the filtration F and a finite family Z i, i n, of F-martingales then, under Hypothesis (H), it holds also for the filtration G and with respect to the G-martingales Z i, i n and M. Remark. Hypothesis (H) is not invariant with respect to an equivalent change of a probability measure, in general.

9 Dynamics of Traded Assets Default Time Prices of Traded Assets Change of a Numeraire Let Y 1, Y 2, Y 3 be semimartingales on (Ω, G, G, Q). We interpret Y i t as the cash price at time t of the ith traded asset in the market model M = (Y 1, Y 2, Y 3 ; Φ), where Φ stands for the class of all self-financing trading strategies. We postulate that the process Y i is governed by the SDE with Y i 0 > 0. dy i t = Y i t `µi dt + σ i dw t + κ i dm t, i = 1, 2, 3, Here W is a one-dimensional Brownian motion and the M is the compensated martingale of the default process H.

10 Assumptions The Model Default Time Prices of Traded Assets Change of a Numeraire We assume that that κ i 1 and κ 1 > 1 so that Yt 1 > 0 for every t R +. This assumptions allows us to take the first asset as a numeraire. Note that the constant coefficient κ 1 > 1 corresponds to a fractional recovery of market value for the first asset. In general, we do not assume that a risk-free security exists. Hence we do not refer to the theory involving the risk-neutral probability associated with the choice of a savings account as a numeraire.

11 Change of a Numeraire Default Time Prices of Traded Assets Change of a Numeraire An equivalent martingale measure e Q is characterized by the property that the relative prices Y i (Y 1 ) 1, i = 1, 2, 3, are e Q-martingales. We will derive the dynamics for the process Y i,1 = Y i (Y 1 ) 1 for i = 2, 3. From Itô s formula, we first obtain «1 d = 1 Yt 1 Yt 1 1 Y 1 t µ 1 + σ ξ t 1 σ 1 dw t + κ κ 1 dm t ««1 + κ 1 dt 1 + κ 1 «.

12 Dynamics of Relative Prices Default Time Prices of Traded Assets Change of a Numeraire Consequently, the Itô s integration by parts formula yields the following dynamics for the processes Y i,1 ( dy i,1 t = Y i,1 t µ i µ 1 σ 1 (σ i σ 1 ) ξ t(κ i κ 1 ) + (σ i σ 1 ) dw t + κ i κ κ 1 dm t ). κ κ 1 «dt

13 Equivalent Martingale Measure Default Time Prices of Traded Assets Change of a Numeraire By assumption, e Q is equivalent to the statistical probability Q on (Ω, G T ) and such that Y i,1, i = 2, 3 are e Q-martingales. Kusuoka (1999) showed that any probability equivalent to Q on (Ω, G T ) is defined by means of its Radon-Nikodým density process η satisfying the SDE dη t = η t `θt dw t + ζ t dm t, η0 = 1, where θ and ζ are G-predictable processes satisfying mild technical conditions (in particular, ζ t > 1 for t [0, T ]). Since M is stopped at τ, we may and do assume that ζ is stopped at τ.

14 Radon-Nikodým Density Default Time Prices of Traded Assets Change of a Numeraire We define e Q by setting d e Q dq = η T = E T (θw )E T (ζm), Q-a.s. Then the processes W c and M b given by, for t [0, T ], Z t cw t = W t θ u du, bm t = M t 0 Z t 0 Z t Z t ξ uζ u du = H t ξ u(1 + ζ u) du = H t bξ u du, 0 0 where b ξ u = ξ u(1 + ζ u), are G-martingales under e Q.

15 Martingale Condition The Model Default Time Prices of Traded Assets Change of a Numeraire Proposition Processes Y i,1, i = 2, 3 are e Q-martingales if and only if drifts in their dynamics, when expressed in terms of c W and b M, vanish. Hence the following equalities hold for i = 2, 3 and every t [0, T ] j Y i,1 t µ 1 µ i + (σ 1 σ i )(θ t σ 1 ) + ξ t(κ 1 κ i ) ζt κ ff 1 = κ 1 Equivalently, we have for i = 2, 3, on the set Y i,1 t 0, µ 1 µ i + (σ 1 σ i )(θ t σ 1 ) + ξ t(κ 1 κ i ) ζt κ κ 1 = 0.

16 Martingale Condition Martingale Measure Example A CASE A: STRICTLY POSITIVE PRIMARY ASSETS

17 Martingale Condition Martingale Measure Example A Case A: standing assumptions: We postulate that κ 1 > 1 so that Y 1 > 0. We assume, in addition, that κ i > 1 for i = 2, 3, so that the price processes Y 2 and Y 3 are strictly positive as well. Martingale condition: From the general theory of arbitrage pricing, it follows that the market model M is complete and arbitrage-free if there exists a unique solution (θ, ζ) such that the process ζ > 1. Since Y i,1 > 0, we search for processes (θ, ζ) such that for i = 2, 3 κ 1 κ i θ t(σ 1 σ i ) + ζ tξ t = µ i µ 1 + σ 1 (σ 1 σ i ) + ξ t(κ 1 κ i ). 1 + κ κ 1 κ 1

18 Martingale Condition The Model Martingale Condition Martingale Measure Example A Since ξ t = γ1 {t τ}, we deal here with four linear equations. For t τ: θ t(σ 1 σ 2 ) + ζ tγ κ 1 κ κ 1 = µ 2 µ 1 + σ 1 (σ 1 σ 2 ) + γ (κ 1 κ 2 )κ κ 1, θ t(σ 1 σ 3 ) + ζ tγ κ 1 κ κ 1 = µ 3 µ 1 + σ 1 (σ 1 σ 3 ) + γ (κ 1 κ 3 )κ κ 1. For t > τ: θ t(σ 1 σ 2 ) = µ 2 µ 1 + σ 1 (σ 1 σ 2 ), θ t(σ 1 σ 3 ) = µ 3 µ 1 + σ 1 (σ 1 σ 3 ). The first (the second, resp.) pair of equations is referred to as the pre-default (post-default, resp.) no-arbitrage condition.

19 Notation The Model Martingale Condition Martingale Measure Example A To solve explicitly these equations, we find it convenient to write a = det A, b = det B, c = det C, where A, B and C are the following matrices:»» σ1 σ A = 2 κ 1 κ 2 σ1 σ, B = 2 µ 1 µ 2 σ 1 σ 3 κ 1 κ 3 σ 1 σ 3 µ 1 µ 3» κ1 κ C = 2 µ 1 µ 2. κ 1 κ 3 µ 1 µ 3,

20 Auxiliary Lemma The Model Martingale Condition Martingale Measure Example A Lemma The pair (θ, ζ) satisfies the following equations θ ta = σ 1 a + c, ζ tξ ta = κ 1 ξ ta (1 + κ 1 )b. In order to ensure the validity of the second equation after the default time τ (i.e., on the set {ξ t = 0}), we need to impose an additional condition, b = 0, or more explicitly, (σ 1 σ 2 )(µ 1 µ 3 ) (σ 1 σ 3 )(µ 1 µ 2 ) = 0. If this holds, then we obtain the following equations θ ta = σ 1 a + c, ζ tξ ta = κ 1 ξ ta.

21 Existence of a Martingale Measure Martingale Condition Martingale Measure Example A Proposition (i) If a 0 and b = 0 then the unique martingale measure e Q has the Radon-Nikodým density of the form d e Q dq = E T (θw )E T (ζm), where the constants θ and ζ are given by and where we write, for t [0, T ], θ = σ 1 + c a, ζ = κ 1 > 1, E t(θw ) = exp θw t 1 2 θ2 t E t(ζm) = `1 + 1 {τ t} ζ) exp ` ζγ(t τ).

22 Existence of a Martingale Measure Martingale Condition Martingale Measure Example A Proposition (ii) If a 0 and b = 0 then the model M = (Y 1, Y 2, Y 3 ; Φ) is arbitrage-free and complete. Moreover, the process (Y 1, Y 2, Y 3, H) has the Markov property under e Q. (iii) If a = 0 and b = 0 then a solution (θ, ζ) exists provided that c = 0 and the uniqueness of a martingale measure e Q fails to hold. In this case, the model M = (Y 1, Y 2, Y 3 ; Φ) is arbitrage-free, but it is not complete. (iv) If b 0 then a martingale measure fails to exist and consequently the model M = (Y 1, Y 2, Y 3 ; Φ) is not arbitrage-free.

23 Martingale Condition Martingale Measure Example A Example A: Extension of the Black-Scholes Model Assume that the asset Y 1 is risk-free, the asset Y 2 Y 1 is default-free, and Y 3 is a defaultable asset with non-zero recovery, so that dyt 1 = ryt 1 dt, dyt 2 = Yt 2 `µ2 dt + σ 2 dw t, dy 3 t = Y 3 t `µ3 dt + σ 3 dw t + κ 3 dm t. We thus have σ 1 = κ 1 = 0, µ 1 = r, σ 2 0, κ 2 = 0, and κ 3 0, κ 3 > 1. Therefore, a = σ 2 κ 3 0, c = κ 3 (r µ 2 ), and the equality b = 0 holds if and only if σ 2 (r µ 3 ) = σ 3 (r µ 2 ).

24 Example A (Continued) Martingale Condition Martingale Measure Example A It is easy to check that θ = r µ 2 σ 2, ζ = 0, and thus under the martingale measure e Q we have (irrespective of whether σ 3 > 0 or σ 3 = 0) dyt 1 = ryt 1 dt, dyt 2 = Yt 2 `r dt + σ2 dw c t, dy 3 t = Y 3 t `r dt + σ3 d c W t + κ 3 dm t. Since ζ = 0 the risk-neutral default intensity bγ coincides here with the statistical default intensity γ. This implies the equality b M = M.

25 Martingale Condition Martingale Measure Example B Stopped Trading CASE B: DEFAULTABLE ASSET WITH ZERO RECOVERY

26 Martingale Condition Martingale Measure Example B Stopped Trading Case B: standing assumptions: We postulate that κ i > 1 for i = 1, 2 and κ 3 = 1. This implies that the price of a defaultable asset Y 3 vanishes after τ, and thus the findings of the preceding section are no longer valid. Martingale condition: Since Y 3 jumps to zero at τ, the first equality in the martingale condition µ 2 µ 1 + (σ 2 σ 1 )(θ t σ 1 ) + ξ t(κ 2 κ 1 ) ζt κ κ 1 = 0 should still be satisfied for every t [0, T ]. The second equality in the martingale condition µ 3 µ 1 + (σ 3 σ 1 )(θ t σ 1 ) + ξ t(κ 3 κ 1 ) ζt κ κ 1 = 0 is required to hold on the set {τ > t} only (i.e. when ξ t = γ).

27 Martingale Condition The Model Martingale Condition Martingale Measure Example B Stopped Trading Lemma Under the present assumptions, the unknown processes θ and ζ in the Radon-Nikodým density of e Q with respect to Q satisfy the following equations µ 2 µ 1 + (σ 2 σ 1 )(θ t σ 1 ) = 0, for t > τ, µ 2 µ 1 + (σ 2 σ 1 )(θ t σ 1 ) + γ(κ 2 κ 1 ) ζt κ κ 1 = 0, for t τ, µ 3 µ 1 + (σ 3 σ 1 )(θ t σ 1 ) + γ( 1 κ 1 ) ζt κ κ 1 = 0, for t τ. This leads to the following result.

28 Existence of a Martingale Measure Martingale Condition Martingale Measure Example B Stopped Trading Proposition The pair (θ, ζ) satisfies the following equations, for t τ, θ ta = σ 1 a + c, ζ tγa = κ 1 γa (1 + κ 1 )b. Moreover, for t > τ, µ 2 µ 1 + (σ 2 σ 1 )(θ t σ 1 ) = 0. Let a 0, σ 1 σ 2 and γ > b/a. Then the unique solution is θ t = 1 {t τ} σ 1 + c + 1 {t>τ} σ 1 µ «1 µ 2, ζ t = κ 1 (1 + κ 1)b > 1. a σ 1 σ 2 γa The model M = (Y 1, Y 2, Y 3 ; Φ) is arbitrage-free, complete, and has the Markov property under the unique martingale measure e Q.

29 Martingale Condition Martingale Measure Example B Stopped Trading Example B: Extension of the Black-Scholes Model Assume that the asset Y 1 is risk-free, the asset Y 2 Y 1 is default-free, and Y 3 is a defaultable asset with zero recovery, so that dyt 1 = ryt 1 dt, dyt 2 = Yt 2 `µ2 dt + σ 2 dw t, dy 3 t = Y 3 t `µ3 dt + σ 3 dw t dm t. This corresponds to the following conditions: σ 1 = κ 1 = 0, µ 1 = r, σ 2 0, κ 2 = 0, κ 3 = 1. Hence a = σ 2 0. Assume, in addition, that γ > b/a = r µ 3 σ 3 σ 2 (r µ 2 ).

30 Example B (Continued) Martingale Condition Martingale Measure Example B Stopped Trading Then we obtain θ = r µ 2 σ 2, ζ = b γa = 1 γ µ 3 r σ «3 (µ 2 r) > 1. σ 2 Consequently, we have under the unique martingale measure e Q dyt 1 = ryt 1 dt, dyt 2 = Yt 2 `r dt + σ2 dw c t, dy 3 t = Y 3 t `r dt + σ3 d c W t d b M t. We do not assume here that b = 0; if this holds then ζ = 0, as in Example A. In Case B, the risk-neutral default intensity bγ and the statistical default intensity γ are different, in general,

31 Case of Stopped Trading Martingale Condition Martingale Measure Example B Stopped Trading Suppose that the recovery payoff at the time of default is exogenously specified in terms of some economic factors related to the prices of traded assets (e.g. credit spreads). The valuation problem for a defaultable claim is reduced to finding its pre-default value, and it is natural to search for a replicating strategy up to default time only. It thus suffices to examine the stopped model in which asset prices and all trading activities are stopped at time τ. In this case, we search for a pair (θ, ζ) of real numbers satisfying θa = σ 1 a + c, ζγa = κ 1 γa (1 + κ 1 )b.

32 Case of Stopped Trading Martingale Condition Martingale Measure Example B Stopped Trading If a 0 then the unique solution (θ, ζ) to the above pair of equations is θ = σ 1 + c a, ζ = κ 1 (1 + κ 1)b γa > 1, where the last inequality holds provided that γ > b/a. As expected, in the stopped model, we obtain the unique martingale measure e Q for any choice of recovery coefficients κ 2 and κ 3. In the case of stopped trading, hedging of a contingent claim after the default time τ is not considered.

33 Pricing PDEs Hedging Example A CASE A: PRICING PDEs AND HEDGING

34 Contingent Claim The Model Pricing PDEs Hedging Example A Let us now discuss the PDE approach in a model in which the prices of all three primary assets are non-vanishing. It is natural to focus on the case when the market model M = (Y 1, Y 2, Y 3 ; Φ) is complete and arbitrage-free. Therefore, we shall work under the assumptions of part (i) in the proposition on the existence of a martingale measure. We are interested in the valuation and hedging of a generic contingent claim with maturity T and the terminal payoff Y = G(Y 1 T, Y 2 T, Y 3 T, H T ). The technique derived for this case can be easily applied to a defaultable claim that is subject to a fairly general recovery scheme.

35 Risk-Neutral Price The Model Pricing PDEs Hedging Example A Let a 0 and b = 0, and let e Q be the unique martingale measure associated with the numeraire Y 1. Then where θ and ζ are explicitly known. d e Q dq = E T (θw )E T (ζm) If Y (YT 1 ) 1 is Q-integrable e then the risk-neutral price of Y equals, for every t [0, T ], π t(y ) = Yt 1 1 E eq`(y T ) 1 Y G t = Y 1 t E eq `(Y 1 T ) 1 G(Y 1 T, Y 2 T, Y 3 T, H T ) Y 1 t, Y 2 t, Y 3 t, H t where the second equality is a consequence of the Markov property of (Y 1, Y 2, Y 3, H) under e Q.

36 Pricing PDEs: Case A The Model Pricing PDEs Hedging Example A Proposition Let the price processes Y i, i = 1, 2, 3 satisfy dyt i = Yt i `µi dt + σ i dw t + κ i dm t with κ i > 1 for i = 1, 2, 3. Assume that a 0 and b = 0. Then the risk-neutral price π t(y ) of the claim Y equals π t(y ) = 1 {t<τ} C(t, Y 1 t, Y 2 t, Y 3 t, 0) + 1 {t τ} C(t, Y 1 t, Y 2 t, Y 3 t, 1) for some function C : [0, T ] R 3 + {0, 1} R. Assume that for h = 0 and h = 1 the function C(, h) : [0, T ] R 3 + R belongs to the class C 1,2 ([0, T ] R 3 +, R).

37 Pricing PDEs: Case A The Model Pricing PDEs Hedging Example A Proposition Then the functions C(, 0) and C(, 1) solve the following PDEs: tc(, 0) + and 3X (α γκ i )y i i C(, 0) i=1 3X σ i σ j y i y j ij C(, 0) αc(, 0) i,j=1 + γˆc(t, y 1 (1 + κ 1 ), y 2 (1 + κ 2 ), y 3 (1 + κ 3 ), 1) C(t, y 1, y 2, y 3, 0) = 0 tc(, 1) + α 3X y i i C(, 1) i=1 3X σ i σ j y i y j ij C(, 1) αc(, 1) = 0 i,j=1 c where α = µ i + σ i, subject to the terminal conditions a C(T, y 1, y 2, y 3, 0) = G(y 1, y 2, y 3, 0), C(T, y 1, y 2, y 3, 1) = G(y 1, y 2, y 3, 1).

38 Comments The Model Pricing PDEs Hedging Example A The valuation problem splits into two pricing PDEs, which are solved recursively. In the first step, we solve the PDE satisfied by the post-default pricing function C(, 1). Next, we substitute this function into the first PDE, and we solve it for the pre-default pricing function C(, 0). The assumption that we deal with only three primary assets and the coefficients are constant can be easily relaxed, but a general result is too heavy to be stated here. Observe that the real-world default intensity γ under Q, rather than the risk-neutral default intensity ˆγ under e Q, enters the valuation PDE.

39 Black and Scholes PDE Pricing PDEs Hedging Example A We consider the set-up of Example A, with a 0 and b = 0. Let Y = G(Y 2 T ) for some function G : R R such that Y (Y 1 T ) 1 is eq-integrable. It is possible to show that π t(y ) = C(t, Y 2 t ). The two valuation PDEs of Proposition A2 reduce to a single PDE tc + (µ 2 σ 2 θ)y 2 2 C σ2 2y C (µ 2 σ 2 θ)c = 0 with θ = (r µ 2 )/σ 2. After simplifications, we obtain the classic Black and Scholes PDE tc + ry 2 2 C σ2 2y C rc = 0.

40 Trading Strategies The Model Pricing PDEs Hedging Example A Recall that φ = (φ 1, φ 2, φ 3 ) is a self-financing strategy if the processes φ 1, φ 2, φ 3 are G-predictable and the wealth process satisfies V t(φ) = φ 1 t Yt 1 + φ 2 t Yt 2 + φ 3 t Yt 3 dv t(φ) = φ 1 t dy 1 t + φ 2 t dy 2 t + φ 3 t dy 3 t. We say that φ replicates a contingent claim Y if V T (φ) = Y. If φ is a replicating strategy for a claim Y then, for t [0, T ], π t(y ) = φ 1 t Y 1 t + φ 2 t Y 2 t + φ 3 t Y 3 t. To find a replicating strategy, we combine the sensitivities of the valuation function C with respect to primary assets with the jump C t = C t C t associated with default event.

41 Hedging with Sensitivities and Jumps Pricing PDEs Hedging Example A Proposition Under the present the assumptions, the claim G(YT 1, YT 2, YT 3, H T ) is replicated by φ = (φ 1, φ 2, φ 3 ), where the components φ i, i = 2, 3, are given in terms of the valuation functions C(, 0) and C(, 1): φ 2 t = φ 3 t = 1 ay 2 t 1 ay 3 t! 3X (κ 3 κ 1 ) σ i Yt i i C σ 1 C (σ 3 σ 1 )( C κ 1 C) i=1! 3X (κ 2 κ 1 ) σ i Yt i i C σ 1 C (σ 2 σ 1 )( C κ 1 C) i=1 and φ 1 equals φ 1 t = (Y 1 t ) 1 C t 3X i=2 φ i tyt i.

42 Pricing PDEs Hedging Example A Example A: Extension of the Black-Scholes Model Assume that the asset Y 1 is risk-free, the asset Y 2 Y 1 is default-free, and Y 3 is a defaultable asset with non-zero recovery, so that dyt 1 = ryt 1 dt, dyt 2 = Yt 2 `µ2 dt + σ 2 dw t, dyt 3 = Yt 3 `µ3 dt + σ 3 dw t + κ 3 dm t with σ 2 0 and κ 3 0, κ 3 > 1. We may assume, without loss of generality, that C does not depend explicitly on the variable y 1. Assume that a = σ 2 κ 3 0 and σ 2 (r µ 3 ) = σ 3 (r µ 2 ). The following result combines and adapts previous results to the present situation.

43 Example A: Pricing PDEs Pricing PDEs Hedging Example A Corollary The arbitrage price of a claim Y = G(YT 2, YT 3, H T ) can be represented as π t(y ) = C(t, Yt 2, Yt 3, H t), where C(t, y 2, y 3, 0) satisfies tc(, 0) + ry 2 2 C(, 0) + y 3 (r κ 3 γ) 3 C(, 0) rc(, 0) + 1 3X σ i σ j y i y j ij C(, 0) + γ`c(t, y 2, y 3 (1 + κ 3 ), 1) C(t, y 2, y 3, 0) = 0 2 i,j=2 with C(T, y 2, y 3, 0) = G(y 2, y 3, 0), and C(t, y 2, y 3, 1) satisfies tc(t, y 2, y 3, 1) + ry 2 2 C(t, y 2, y 3, 1) + ry 3 3 C(t, y 2, y 3, 1) rc(t, y 2, y 3, 1) + 1 3X σ i σ j y i y j ij C(t, y 2, y 3, 1) = 0 2 i,j=2 with C(T, y 2, y 3, 1) = G(y 2, y 3, 1).

44 Example A: Hedging The Model Pricing PDEs Hedging Example A Corollary The replicating strategy for Y equals φ = (φ 1, φ 2, φ 3 ), where! 3X φ 1 t = (Yt 1 ) 1 C t φ i tyt i, φ 2 t = 1 σ 2 κ 3 Y 2 t κ 3 i=2 3X i=2 σ i y i i C(t, Y 2 t, Y 3 t, H t ) σ 3`C(t, Y 2 t, Y 3 t (1 + κ 3 ), 1) C(t, Y 2 t, Y 3 t, 0)!, φ 3 t = 1 `C(t, Y 2 κ 3 Yt 3 t, Yt (1 3 + κ 3 ), 1) C(t, Yt, 2 Yt, 3 0).

45 Example A: Survival Claim Pricing PDEs Hedging Example A By a survival claim we mean a claim of the form Y = 1 {τ>t } X, where an F T -measurable random variable X represents the promised payoff. In other words, a survival claim is a contract with zero recovery in the case of default prior to maturity T. We assume that the promised payoff has the form X = G(Y 2 T, Y 3 T ), where Y i T is the (pre-default) value of the ith asset at time T. It is obvious that the pricing function C(, 1) is now equal to zero, and thus we are only interested in the pre-default pricing function C(, 0).

46 Example A: Survival Claim Pricing PDEs Hedging Example A Corollary The pre-default pricing function C(, 0) of a survival claim of the form Y = 1 {τ>t } G(Y 2 T, Y 3 T ) solves the PDE tc(, 0) + ry 2 2 C(, 0) + y 3 (r κ 3 γ) 3 C(, 0) + 1 3X σ i σ j y i y j ij C(, 0) (r + γ)c(, 0) = 0 2 i,j=2 with C(T, y 2, y 3, 0) = G(y 2, y 3 ). The components φ 2 and φ 3 of a replicating strategy φ are given by the following expressions φ 2 t = 1 κ 3 σ 2 Y 2 t κ 3 3X i=2 σ i Yt i i C(, 0) σ 3 C(, 0), φ 3 C(, 0) t =. κ 3 Yt 3

47 Pricing PDEs Example B CASE B: PRICING PDEs AND HEDGING

48 Pricing PDEs Example B Standing assumptions: We now assume that the prices Y 1 and Y 2 are strictly positive, but κ 3 = 1 so that Y 3 is a defaultable asset with zero recovery. Of course, the price Yt 3 vanishes after default, that is, on the set {t τ}. We assume here that a 0 and σ 1 σ 2, but we no longer postulate that b = 0. We still assume that γ > b/a, however. Let us denote α i = µ i + σ i c a, β i = µ i σ i µ 1 µ 2 σ 1 σ 2.

49 Valuation PDEs: Case B Pricing PDEs Example B Proposition Let the price processes Y i, i = 1, 2, 3, satisfy dyt i = Yt i `µi dt + σ i dw t + κ i dm t with κ i > 1 for i = 1, 2 and κ 3 = 1. Assume that a 0, σ 1 σ 2, γ > b/a. Consider a contingent claim Y with maturity date T and the terminal payoff G(Y 1 T, Y 2 T, Y 3 T, H T ). In addition, we postulate that the pricing functions C(, 0) and C(, 1) belong to the class C 1,2 ([0, T ] R 3 +, R).

50 Pricing PDEs: Case B The Model Pricing PDEs Example B Proposition Then the pre-default pricing function C(t, y 1, y 2, y 3, 0) satisfies the pre-default PDE tc(, 0) + 3X (α i γκ i )y i i C(, 0) i=1 + γ b a α 1 + κ 1 b a 3X σ i σ j y i y j ij C(, 0) i,j=1 ˆC(t, y1 (1 + κ 1 ), y 2 (1 + κ 2 ), 0, 1) C(t, y 1, y 2, y 3, 0) C(, 0) = 0 subject to the terminal condition C(T, y 1, y 2, y 3, 0) = G(y 1, y 2, y 3, 0).

51 Pricing PDEs: Case B The Model Pricing PDEs Example B Proposition The post-default pricing function C(t, y 1, y 2, 1) solves the post-default PDE 2X tc(, 1) + β i y i i C(, 1) + 1 2X σ i σ j y i y j ij C(, 1) β 1 C(, 1) = 0 2 i=1 i,j=1 subject to the terminal condition C(T, y 1, y 2, 1) = G(y 1, y 2, 0, 1). The components of the replicating strategy φ are given by the general formulae.

52 Example B (Continued) Pricing PDEs Example B We assume that the processes Y 1, Y 2, Y 3 satisfy Let us write br = r + bγ, where dyt 1 = ryt 1 dt, dyt 2 = Yt 2 `µ2 dt + σ 2 dw t, dy 3 t = Y 3 t `µ3 dt + σ 3 dw t dm t. bγ = γ(1 + ζ) = γ b a = γ + µ 3 r + σ 3 σ 2 (r µ 2 ) > 0 stands for the default intensity under e Q. The quantity br is interpreted as the credit-risk adjusted short-term rate. Straightforward calculations show that the following corollary is valid.

53 Example B: Pricing PDEs Pricing PDEs Example B Corollary Assume that σ 1 = κ 1 = κ 2 = 0, κ 3 = 1 and Then C(, 0) satisfies the PDE γ > b/a = r µ 3 σ 3 σ 2 (r µ 2 ). tc(t, y 2, y 3, 0) + ry 2 2 C(t, y 2, y 3, 0) + bry 3 3 C(t, y 2, y 3, 0) brc(t, y 2, y 3, 0) + 1 3X σ i σ j y i y j ij C(t, y 2, y 3, 0) + bγc(t, y 2, 1) = 0, 2 i,j=2 with C(T, y 2, y 3, 0) = G(y 2, y 3, 0), and the function C(, 1) solves tc(t, y 2, 1) + ry 2 2 C(t, y 2, 1) σ2 2y C(t, y 2, 1) rc(t, y 2, 1) = 0, with C(T, y 2, 1) = G(y 2, 0, 1).

54 Example B: Survival Claim Pricing PDEs Example B For a survival claim, we have C(, 1) = 0, and thus we obtain following results. Corollary The pre-default pricing function C(, 0) of a survival claim Y = 1 {τ>t } G(Y 2 T, Y 3 T ) solves the following PDE: tc(t, y 2, y 3, 0) + ry 2 2 C(t, y 2, y 3, 0) + bry 3 3 C(t, y 2, y 3, 0) + 1 3X σ i σ j y i y j ij C(t, y 2, y 3, 0) brc(t, y 2, y 3, 0) = 0 2 i,j=2 with the terminal condition C(T, y 2, y 3, 0) = G(y 2, y 3 ).

55 Corollary B2 (Continued) Pricing PDEs Example B Corollary The components φ 2 and φ 3 of the replicating strategy are, for every t < τ, φ 2 t = φ 3 t = 1 σ 2 Y 2 t 3X i=2 1 C(t, Yt, 2 Yt, 3 0). Yt 3 σ i Yt i i C(t, Yt, 2 Yt, 3 0) + σ 3 C(t, Yt, 2 Yt, 3 0), We have φ 3 t Yt 3 = C(t, Yt, 2 Yt, 3 0) for every t [0, T ]. Hence the following relationships holds, for every t < τ, φ 3 t Y 3 t = C(t, Y 2 t, Y 3 t, 0), φ 1 t Y 1 t + φ 2 t Y 2 t = 0. The last equality is a special case of a balance condition introduced in Bielecki et al. (2006) in a semimartingale set-up.

56 Case of Two Credit Names Case of m Credit Names PDE APPROACH TO BASKET CLAIMS

57 Case of Two Credit Names Case of Two Credit Names Case of m Credit Names We first consider a special case of two credit names: Let τ 1 and τ 2 be strictly positive random variables defined on a probability space (Ω, G, Q). We introduce the corresponding jump processes H i t = 1 {τi t} for i = 1, 2, and we denote by H i the filtration generated by the process H i. Finally, we set G = F H 1 H 2, where the filtration F is generated by some Brownian motion W (which is also a G-Brownian motion). We now need at least four traded assets, since we deal with three (possibly independent) sources of uncertainty.

58 Dynamics of Traded Assets Case of Two Credit Names Case of m Credit Names Standing assumptions: For the sake of simplicity, we assume that Yt 1 = 1, so that Y 1 represents the savings account corresponding to the short-term rate r = 0. We postulate that the asset price Y i satisfies, for i = 2, 3, 4, dyt i = Yt `µi i dt + σ i dw t + κ i dmt 1 + ψ i dmt 2 where M i is the Q-martingale associated with the default process H i, that is, Z t Mt i = Ht i γu(1 i Hu) i du. 0 To ensure the Markov property, we assume that γ i u = g i (u, H 1 u, H 2 u ). Defaults cannot occur simultaneously: H 1 t H 2 t = 0.

59 Contingent Claim The Model Case of Two Credit Names Case of m Credit Names Consider a contingent claim of the form Y = G(Y 2 T, Y 3 T, Y 4 T, H 1 T, H 2 T ). Its arbitrage price can be represented as a function π t(y ) = C(t, Y 2 t, Y 3 t, Y 4 t, H 1 t, H 2 t ) or equivalently, as a quadruplet of functions: C(, 1, 1), C(, 0, 1), C(, 1, 0) and C(, 0, 0). The pricing functions satisfy the terminal condition C(T, y 2, y 3, y 4, h 1, h 2 ) = G(y 2, y 3, y 4, h 1, h 2 ). The process C t = C(t, Y 2 t, Y 3 t, Y 4 t, H 1 t, H 2 t ) is a G-martingale under e Q.

60 Pricing PDEs The Model Case of Two Credit Names Case of m Credit Names Let bγ 1 0 and bγ 2 0 be the intensities of τ 1 and τ 2 prior to the first default, bγ 1 2 be the intensity of the default time τ 1 on the event {τ 2 t < τ 1 }, bγ 2 1 be the intensity of the default time τ 2 on the event {τ 1 t < τ 2 }. We obtain the following pricing PDE prior to the first default: 4X tc(, 0, 0) (κ i bγ ψ i bγ 0)y 2 i i C(, 0, 0) i=2 4X σ i σ j y i y j ij C(, 0, 0) i,j=2 + bγ 1 0`C(, 1, 0) C(, 0, 0) + bγ 2 0`C(, 0, 1) C(, 0, 0) = 0.

61 Pricing PDEs (continued) Case of Two Credit Names Case of m Credit Names After the first default, we have 4X tc(, 1, 0) ψ i bγ 1y 2 i i C(, 1, 0) i=2 + bγ 2 1`C(, 1, 1) C(, 1, 0) = 0, 4X σ i σ j y i y j ij C(, 1, 0) i,j=2 4X tc(, 0, 1) κ i bγ 2y 1 i i C(, 0, 1) i=2 + bγ 2`C(, 1 1, 1) C(, 0, 1) = 0, and after the second default 4X σ i σ j y i y j ij C(, 0, 1) i,j=2 tc(, 1, 1) + 1 4X σ i σ j y i y j ij C(, 1, 1) = 0. 2 i,j=2

62 Case of m Credit Names Case of Two Credit Names Case of m Credit Names Standing assumptions: Let the random times τ 1, τ 2,..., τ m, defined on a common probability space (Ω, G, Q), represent the default times of m credit names. Under real-world probability Q, the price processes Y 1, Y 2,..., Y n of primary traded assets are governed by dyt i = Yt i µ i t dt + dx k=1 σ k i (t) dw k t + mx κ l i(t) dmt l where the G-martingales M l, l = 1, 2,..., m are given by Z τl Mt l = Ht l t Z t γu l du = Ht l ξu l du. 0 0 l=1

63 The Markovian Model The Model Case of Two Credit Names Case of m Credit Names The processes µ i, σ i, κ i are given by some functions on R + R n and µ i t = µ i (t, Y 1 t,..., Y n t ), σ i (t) = σ i (t, Y 1 t,..., Y n t ) κ i (t) = κ i (t, Y 1 t,..., Y n t ). The functions above are sufficiently regular, so that the SDE admits a unique strong solution for i = 1, 2,..., n. The pre-default intensities λ l are deterministic functions of asset prices, that is, λ l t = λ l (t, Yt, 1..., Yt ) n for every t R + and l = 1, 2,..., m.

64 Kusuoka s Theorem The Model Case of Two Credit Names Case of m Credit Names Proposition Any probability measure e Q equivalent to Q on (Ω, G T ) is given by the Radon-Nikodým derivative process η satisfying, for t [0, T ], dq e G t = η t = dq dy Z. Y m Z E t θu k dwu k. E t ζu l dmu l k=1 0 where θ 1, θ 2,..., θ d, ζ 1, ζ 2,..., ζ m are some G-predictable processes such that ζ l t > 1 for every t [0, T ]. The processes W fk, k = 1,..., d and M e l, l = 1,..., m are G-martingales under Q e where l=1 0 fw k t = W k t Z t θu k du, 0 Z t M e l t = Mt l ξuζ l u l du. 0

65 Martingale Condition The Model Case of Two Credit Names Case of m Credit Names Assume that the number of primary traded assets is equal to the number of driving orthogonal martingales W 1,..., W d, M 1,..., M m plus one, i.e., n = d + m + 1. In addition, let the price Y 1 be strictly positive. Proposition A probability measure e Q equivalent to Q on (Ω, G T ) is a martingale measure associated with a numeraire Y 1 if and only if the processes θ and ζ satisfy the following equation Y i,1 t µ 1 µ i + for i = 2, 3,..., n. dx mx (σ1 k σi k )(θt k σ1) k + k=1 l=1 ξt(κ l l 1 κ l i) ζl t κ l 1 = κ l 1

66 Pre-default Martingale Condition Case of Two Credit Names Case of m Credit Names Lemma Martingale condition can be represented as follows where: A tx t = b t x t = (θ, λζ) T is an R d+m -valued process with λζ = (λ 1 ζ 1,..., λ m ζ m ), the R n 1 -valued process b t is explicitly known, the (n 1) (m + d) matrix A t given by 2 σ 1 1 σ σ1 d σ d κ 1 1 κ κ 1 1 A t = σ1 1 σn 1... σ1 d σn d κ 1 1 κ1 n... 1+κ 1 1 κ m 1 κm 2 1+κ m 1 κ m 1 κm n 1+κ m

67 Existence of a Martingale Measure Case of Two Credit Names Case of m Credit Names The pre-default intensities λ l t satisfy the equality λ l t = γ l t on the event {τ (1) > t}, that is, prior to occurrence of the first default. Proposition Assume that the pre-default intensities λ l t, l = 1,..., m are strictly positive for every t [0, T ]. Then the martingale measure Q e for the relative prices Y i,1, i = 2, 3,..., m stopped at τ (1) T exists and is unique if and only if A 1 t exists. The Radon-Nikodým derivative of e Q with respect to Q on (Ω, G T ) is given by dq e dy Z. dq = Y m Z E T θu k dwu k. E T ζu l dmu l. k=1 0 l=1 0

68 First-to-Default Claim (FTDC) Case of Two Credit Names Case of m Credit Names Let us denote τ (1) = τ 1 τ 2... τ m = min (τ 1, τ 2,..., τ m). Definition A first-to-default claim with maturity T is a defaultable claim (X, Z, τ (1) ), where X is a constant amount payable at maturity if no default occurs, and Z = (Z 1, Z 2,..., Z l ) is the vector of G-adapted processes, where Z l τ (1) specifies the recovery received at time τ (1) if the lth name is the first defaulted name, that is, on the event {τ l = τ (1) T }. Assumptions: The processes Z l, l = 1, 2,..., m, are given by some real-valued functions on [0, T ] R n, specifically, Zt l = Z l (t, Yt 1,..., Yt n ). X = g(yt 1,..., YT n ) for some function g : R n R.

69 Valuation of an FTDC The Model Case of Two Credit Names Case of m Credit Names Assuming that Y is admissible, that is, Y (Yτ 1 (1) ) 1 is Q-integrable, e we can represent the risk-neutral value of Y on the random interval [0, τ (1) ) as follows π t(y ) = Yt 1 E eq`y (Y 1 τ(1) ) 1 G t. In the Markovian set-up, we can deduce the existence of a function C : [0, T ] R n + R representing the pre-default price of the claim. Lemma There exists a function C : [0, T ] R n + R such that we have for every t [0, T ] π t(y ) = C(t, Yt 1,..., Yt n ) on the event {τ (1) > t}.

70 Pricing PDE for an FTDC Case of Two Credit Names Case of m Credit Names Proposition The function C(t, y 1,..., y n) satisfies the following PDE tc nx i,j=1 k=1 (α 1 + β)c + dx σi k σj k y i y j ij C + mx l=1 nx i=1 α i λ l 1 + ζ l 1 + κ l l C = 0 1 mx l=1 κ l iλ l (1 + ζ l ) y i i C with the terminal condition C(T, y 1,..., y n) = g(y 1,..., y n), where and α i = µ i + dx σi k (θ k σ1), k β = k=1 mx λ l κ l ζl, 1 + κ l 1 l C = Z l (t, y 1 (1 + κ l 1),..., y n(1 + κ l n)) C(t, y 1,..., y n). l=1

71 Replication of an FTDC Case of Two Credit Names Case of m Credit Names Let C t be a candidate for the pre-default arbitrage price of an FTDC (X, Z, τ (1) ). Our goal is to establish the existence of a self-financing trading strategy φ such that nx C t = V t(φ) = φ i tyt i on the interval [0, τ (1) T ]. Equivalently, e C = C(Y 1 ) 1 satisfies dc e Vt(φ) t = d = Y 1 t i=1 nx i=2 φ i t dy i,1 t. In that case, we say that a trading strategy φ replicates an FTDC We will show that an FTDC can be replicated and thus the pre-default risk-neutral value is also the arbitrage price of an FTDC prior to default.

72 Notation The Model Case of Two Credit Names Case of m Credit Names Let P 1 t P 1 t stand for the 1 d vector = ˆ P n i=1 σ1 i Y i t i C σ 1 1C t... P n i=1 σd i Y i t i C σ d 1 C t Let P 2 t the 1 m vector for the 1 m vector P 2 t = h 1 C t κ 1 1 C t 1+κ mc t κ m 1 C t 1+κ m 1 i.

73 Lemma The Model Case of Two Credit Names Case of m Credit Names Lemma The Itô differential of e C t can be represented as follows d e C t = (Y 1 t ) 1 P t d ew t where P t = [P 1 t, P 2 t ] and 2 d ew t = 6 4 dw f t 1. d f W d t d e M 1 t d e M m t

74 Lemma The Model Case of Two Credit Names Case of m Credit Names Lemma The joint dynamics of relative prices Y i,1 t, i = 2,..., n can be represented as follows dy t = Y t A t d ew t where y t is the (n 1) 1 vector y t = Y 2,1 t... Y n,1 t and the diagonal (n 1) (n 1) matrix Y t equals 2 3 Y 2,1 t... 0 Y t = Y n,1 t 3 7 5

75 Replicating Strategy The Model Case of Two Credit Names Case of m Credit Names Proposition Consider a first-to-default claim (X, Z, τ (1) ) with the pricing function C. The claim can be replicated by the self-financing trading strategy φ = (φ 1,..., φ n ) where (φ 2 t,..., φ n t ) = (Yt ) 1 1 P ty 1 A 1 and φ 1 t = (Y 1 t ) 1 C t nx i=2 t t φ i tyt i.

76 Example: Four Assets and Two Defaults Case of Two Credit Names Case of m Credit Names We consider a market model with four primary assets that are driven by two possible sources of default and a one-dimensional Brownian motion. We thus have under the real-world probability Q, for i = 1,..., 4, dyt i = Yt i µ i (t) dt + σi 1 (t) dwt 1 + 2X κ l i(t) dmt l. Note that condition n = m + d + 1 is satisfied and the matrix A t becomes 2 σ1 1 σ 1 κ 1 1 κ1 2 κ κ 1 1 κ2 2 1+κ A t = 6 σ1 1 σ3 1 κ 1 1 κ1 3 κ κ 1 1 κ κ σ1 1 σ4 1 κ 1 1 κ1 4 κ 2 1 κ2 4 1+κ 1 1 l=1 1+κ 2 1

77 Example (continued) The Model Case of Two Credit Names Case of m Credit Names Assuming that the matrix A t is non-singular and λ l t 0 for t [0, T ], we find that the unique martingale measure e Q is given by dq e Z. dq = E T 0 2Y Z θu 1 dwu 1. E T ζu l dmu l l=1 where θ 1, ζ 1 and ζ 2 are given by 2 3 θ 1 4 λ 1 ζ 1 5 = A 1 λ 2 ζ 2 t b t with 2 b t = 6 4 µ 2 µ 1 + σ1(σ σ2) 1 + P 2 l=1 λl (κ l 1 κ l 2) κl 1 1+κ l 1 µ 3 µ 1 + σ1(σ σ3) 1 + P 2 l=1 λl (κ l 1 κ l 3) κl 1 1+κ l 1 µ 4 µ 1 + σ1(σ σ4) 1 + P 2 l=1 λl (κ l 1 κ l 4) κl 1 1+κ l

78 Example (continued) The Model Case of Two Credit Names Case of m Credit Names The dynamics of relative prices Y i,1, i = 2, 3, 4, under e Q are given by dy i,1 t = Y i,1 t (σi 1 σ1) 1 dw f t 1 2X l=1 κ l i κ l κ l 1 d e M l t Consider a first-to-default claim (X, Z, τ (1) ) where Z = (Z 1, Z 2 ). Then P t becomes h P4 i P t = i=1 σ1 i Yt i i C σ1c 1 t 1 C t κ 1 1 C t 1+κ C t κ 2 1 C t 1+κ 2 1 where the function C solves the pre-default pricing PDE The replicating strategy for an FTDC (X, Z, τ (1) ) can be found from the equality (φ 2 t, φ 3 t, φ 4 t ) = (Yt ) 1 1 P ty 1 t A 1 t, combined with the formula φ 1 t = (Y 1 t ) 1 C t 4X i=2 φ i tyt i.

79 Final Remarks The Model Case of Two Credit Names Case of m Credit Names In a single-name case: we distinguished between the case of strictly positive assets and the case of zero recovery for defaultable asset, we examined the pre-default and post-default pricing PDEs, explicit representation for replicating strategies were derived. In a multi-name case: we concentrated on the case of a first-to-default claim, the pricing PDE and the formula for replicating strategy were derived, the method can be extended to kth-to-default claims.