Multiple Defaults and Counterparty Risks by Density Approach

Transcription

1 Multiple Defaults and Counterparty Risks by Density Approach Ying JIAO Université Paris 7 This presentation is based on joint works with N. El Karoui, M. Jeanblanc and H. Pham

2 Introduction Motivation : study the impact of default events on the markets and the roles of different information using the density approach of default Applications : modelling of multiple defaults and optimal investment with counterparty risks Main ideas : we need the whole term structure of the conditional density the intensity hypothesis is not enough! decompose suitably the initial problem in the global filtration using the before-default and after-default analysis each decomposed problem is solved in the reference default free filtration

3 The progressive enlargement of filtration plays an essential role in the credit risk modelling On the market (Ω, A, P): the default information σ(τ t) and the default-free information F = (F t ) t 0. The global information G = (G t ) t 0 G t = F t σ(τ t). Key remark: any G t -measurable random variable Yt G is written in the form Y G t = Y t 1 {t τ} + Y t (τ)1 {t>τ}, where Y t is F t - measurable and Y t (θ) is F t B(R + )- measurable.

4 For single credit name, the pricing of a default sensitive claim consists of computing the conditional expectation w.r.t. G under some risk-neutral probability. Most credit derivatives cease to exist once the default occurs. So classically, the pricing is on the set {t < τ}. The idea (e.g. Bielecki-Jeanblanc-Rutkowski) is to establish an explicit relationship between the G and the F conditional expectations Key Lemma (Dellacherie-Meyer, Jeulin-Yor): for any A-measurable r.v. Y, 1 {τ>t} E[Y G t ] = 1 {τ>t} E[Y1 {τ>t} F t ] P(τ > t F t ) a.s. (1) on the set A = {S t := P(τ > t F t ) > 0}.

5 Default density approach To study the impact of a default on the market and the multiple defaults, we are interested in what happens after a default, i.e. on {t τ}. Adopting the conditional density of default is suitable for the after-default studies. Similar to the Jacod s hypothesis in the initial enlargement of filtration. Density Hypothesis For any t 0, there exists a family of F t B(R + ) r.v. α t (θ) s.t. for any bounded Borel function f on R +, E[f(τ) F t ] = 0 f(θ)α t (θ)dθ a.s.

6 After-default pricing Let Y T (θ) be a bounded F T B(R + ) r.v., T,θ 0. Then for any t T, on A θ = {α t (θ) > 0}, E [ Y(T,θ)α T (θ) ] F t 1 {τ t} E[Y T (τ) G t ] = 1 {τ t}, a.s. α t (θ) θ=τ (2) The after-default density α t (θ), t θ is needed. Under the H-hypothesis, P(τ > t F t ) = P(τ > t F ), so α t (t) = α T (t), T t. The before-default density implies the whole term structure. In the general case, the after-default density requires more information and the change of probability measure is useful.

7 After-default density without H-hypothesis We begin from a probability measure P where the H-hypothesis holds. Let (N t := 1 {τ t} Λ G t, t 0) be the (G, P)-martingale of pure jump and assume the intensity hypothesis, i.e., Λ G t = t 0 λg s ds = t τ 0 λ s ds where λ is F-adapted. The intensity satisfies λ t = α t (t)/s t. The density of τ under P is α t (θ) = E[α θ (θ) F t ] = E[λ θ S θ F t ] = E[λ θ e Λ θ F t ] t,θ 0. Suppose that F is generated by a Brownian motion W. Then W is also a (G, P)-Brownian motion.

8 Our aim is to find the after-default density under a probability Q where H-hypothesis is not satisfied. By the martingale representation theorem in G (Kusuoka), any positive martingale Q with expectation 1 can be written as the solution of a SDE dq t = Q t (Ψ t dw t + Φ t dn t ), Q 0 = 1 where Ψ and Φ, Φ > 1, are G-predictable processes. Using the decomposed form Ψ t = ψ t 1 {t τ} +ψ t (τ)1 {t>τ} and Φ t = φ t 1 {t τ} +φ t (τ)1 {t>τ}, it follows Q t = q t 1 {t<τ} + q t (τ)1 {t τ} where q t = exp ( t ψ u dw u t 0 ψ 2 udu q t (θ) = q θ (θ) exp ( t ψ u (θ)dw u 1 2 with q θ (θ) = q θ (1 + φ θ ). θ t θ t 0 λ u φ u du ), t 0 ψ u (θ) 2 du ), t θ.

9 The restriction of Q on F is given by t Qt F = E[Q t F t ] = q t S t + q t (u)λ u S u du. 0 Let Q be the probability measure defined by dq = Q t dp on G t. Then α Q t (θ) = α Q t (θ) = 1 Q F t 1 Q F t q t (θ)α θ (θ), t θ E P [α Q θ (θ) F t] = 1 Q F t E P [q θ (1 + φ θ )α θ (θ) F t ], t < θ.

10 Remarks The pricing problem is composed into a before-default one and a after-default one. The initial problem concerns the global information G. The two decomposed pricing formulas are related to the default-free information F. The density process is F-adapted. The filtration F is more familier to work with. This general decomposition methodology is useful to deal with multiple defaults and counterparty risks.

11 Modelling Successive Defaults

12 Introduction The before-default and after-default analysis adapts naturally to study the ordered default events in a recursive manner. For the pricing of credit portfolio derivatives such as basket default swaps and CDOs, it suffices to consider the ordered defaults. Links between the top-down models and bottom-up models of multiple credit modelling.

13 Default information We consider a family of random times (τ 1,,τ n ) on (Ω, A, P), whose increasing-ordered permutation is σ 1 σ 2 σ n. Denote by D i = (Dt i) t 0 the filtration associated with σ i and by D (i) = (D (i) t ) t 0 := D 1 D i. Let G (i) = (G (i) t ) t 0 := F D (i) and define for convenience G (0) = F. Market full information : G (n)

14 Ordered defaults and loss process In the top-down models, one works with the cumulative loss of the underlying portfolio defined by L t := n i=1 1 {τi t}. Loss information: D L = (D L t ) t 0 where D L t = σ(l s, s t) including the loss value and the timing of jumps of the loss process. It holds L t = n i=1 1 {σi t}. The same information flow as the ordered defaults: D (n) t = D L t, t 0.

15 Joint density Joint Density Hypothesis for σ = (σ 1,,σ n ): there exists a family of F t B(R n +)-measurable functions (ω, u) α t (u) where u = (u 1,, u n ) R n +, such that for any bounded Borel function f : R n + R, E[f(σ) F t ] = f(u)α t (u)du, t 0. R n + Remark: α(u) is null outside the set {u 1 u n }. The marginal density of σ (k) := (σ 1,,σ k ) is given by α (k) ( ) t u(k) = α t (u)d(u (>k) ) R n k + where d ( u (>k) ) = duk+1 du n.

16 Pricing of credit portfolio derivatives Payoff of credit portfolio derivatives k th -to-default swap: Y T (σ) = 1 {σk >T } CDO tranche: Y T (σ) = (L T k) + = ( n i=1 1 {σi T } k) + Let T t 0 and Y T (σ) be a positive F T B(R n +)-measurable function on Ω R n +, then E[Y T (σ) G (n) t ] = n i=0 1 {σi t<σ i+1 } ]t, [ d(u (>i)) E[Y T (u)α T (u) F t ] ]t, [ d(u (>i))α t (u) where u (i) = (u 1,, u i ) and σ (i) = (σ 1,,σ i ). The prices depend on the number and the occurrence timing of defaults. u(i) =σ (i)

17 Loss intensity and successive default intensities The loss intensity is the G L -adapted process λ L such that (L t t 0 λl sds, t 0) is a G L -martingale. The G (k) -intensity of σ k is the G (k) -adapted process λ k such that (1 {σk t} t 0 λk sds, t 0) is a G (k) -martingale. The G (k) -intensity of σ k coincides with its G (n) -intensity, it is null outside the set {σ k 1 t < σ k } and is given as λ k t = 1 {σk 1 t<σ k }λ k,f ( ) t σ(k 1) where λ k,f ( ) t u(k 1) is Ft B(R k 1 + )-measurable. The loss intensity is the sum of the intensities of σ k, i.e. n λ L t = λ k t, k=1 a.s..

18 Joint density and successive intensities Under a probability P where H-hypothesis holds between F and G (n), then for any θ R n + such that θ 1 θ n and any 0 t θ n, [ n α t (θ) = E λ i,f ( ) { θi θ θ(i 1) i exp λ i,f ( ) } ] u θ(i 1) du Ft θ i 1 i=1 a.s.. If t > θ n, then α t (θ) = α θn (θ). To obtain the whole term structure of the joint density in the general case, it needs a change of probability measure on G (n) and more information.

19 Bottom-up vs top-down The ordered joint density of σ can be deduced from the non-ordered one of τ = (τ 1,,τ n ).. Denote by β t (θ), t 0, θ = (θ 1,,θ n ) R n +, the joint density of τ. For any θ R n + such that θ 1 θ n, ( ) α t (θ 1,,θ n ) = 1 {θ1 θ n} β t θπ(1),,θ Π(n) where (Π(1),,Π(n)) is a permutation of (1,, n). In particular, if τ is exchangeable, then α t (θ 1,,θ n ) = 1 {θ1 θ n} n! β t (θ 1,,θ n ). Π

20 Joint density models Two important points: compatibility between the joint probability property and the martingale property describe the correlation structure in a dynamic manner An example: how to diffuse a copula model. When t = 0, G 0 (u) = exp( ( n i=1 u2 i ) 1/2 ), which is a c.d.f. of n uni-exponential r.v. linked by a Clayton copula. The conditional probability : ( G t (u) = exp ( n ) ui 2 Mt) i 1/2 A t t 0 where A t = 1 2+X 1/2 s 8 d X X 3/2 s and X s = n i=1 u2 i Mt i s with M i being positive F-martingales such that M0 i = 1 and M i, M j > 0 if i j. t i=1

21 Optimal Investment with Counterparty Risks

22 Introduction The counterparty default risks is often modelled in the literature by the contagion intensity jump (Jarrow-Yu, Crepey-Jeanblanc-Zargari, Herbertsson). We consider a utility maximization problem where the asset value jumps at the default of the counterparty. The aim is to analyze the counterparty risk in optimal investment and measure the impact of default on trading strategy.

23 Asset model Asset value S governed by ds t = S t (µ t dt + σ t dw t γ t dd t ), 0 t T, where µ, σ > 0 and γ are G-predictable processes satisfying T 0 µ t σ t 2 dt + T 0 σ t 2 dt < a.s., D t = 1 τ t and < γ t < 1. Since γ < 1, the asset continues to exist after default. S t = St F1 t<τ + St d(τ)1 t τ, where S F is the price in the market before default: ds F t = St F ( µ F t dt + σt F ) dw t, 0 t T, and S d (θ) is the price after default at τ = θ: ds d t (θ) = Sd t (θ)( µ d t (θ)dt + σd t (θ)dw t), θ < t T, with the initial value S d θ (θ) = SF θ (1 γ θ).

24 Portfolio and wealth process A trading strategy is a G-predictable process π, representing the proportion of wealth invested in the risky asset, with corresponding wealth process: ds t dx t = π t X t, 0 t T. S t By writing π in the form: π t = π F t 1 t τ + π d t (τ)1 t>τ, X t = X F t 1 t<τ + X d t (τ)1 t τ, where X F is the wealth process in the market before default: dxt F = πt F Xt F dst F St F, 0 t T,

25 Portfolio and wealth process after default X d (θ) is the wealth in the market after default at τ = θ: dx d t (θ) = π d t (θ)x d t (θ) dsd t (θ) S d t (θ) X d θ (θ) = X F θ (1 πf θ γ θ). θ < t T We say that a trading strategy π is admissible, denote by π A, if T 0 π t σ t 2 dt <, and π t γ t < 1 a.s. For π A, the wealth process remains strictly positive.

26 The utility maximization problem Given a utility function U strictly increasing, strictly concave and C 1 on (0, ), and satisfying the Inada conditions U (0 + ) =, U ( ) = 0, we consider V 0 = sup J 0 (π) := sup E[U(X T )]. (3) π A π A Incomplete market due to jump default process. Existence and uniqueness of a solution by duality method from general results of Kramkov-Schachermayer Problem (3) recently studied by dynamic programming methods and BSDE in the G-filtration under H-hypothesis by Lim-Quenez. Similar problem by Ankirchner-Blanchet-Scalliet-Eyraud-Loisel

27 Derivation of the decomposition Density approach Explicit solutions with qualitative and quantitative descriptions of the optimal trading strategy, especially w.r.t. the Merton (default-free) strategy. By the law of conditional expectations, and under the density hypothesis, we get J 0 (π) = E [ E[U(X T ) F T ] ] = E [ U(XT F )P[τ > T F T] + E[U(XT d (τ))1 τ T F T ] ] [ T ] = E U(XT F )S T + U(XT d (θ))α T(θ)dθ, 0 where S T = P[τ > T F T ] = T α T(θ)dθ is the conditional survival probability at T.

28 Decomposition theorem Let us introduce the value-function process of the after-default" optimization problem: for (θ, x) [0, T] (0, ), V d θ (x) = ess sup E [ U(X d,x T (θ))α T(θ) ] Fθ. (4) π d (θ) A d (θ) Theorem [ V 0 = sup E U(XT F )S T + π F A F T 0 ] Vθ d (X θ F (1 πf θ γ θ))dθ. (5)

29 Two optimization problems in the complete market. For the after-default problem, Time-inconsistent control problem where the coefficients (µ d (θ), σ d (θ)) depend on the initial time θ density term α T (θ) weighting the utility U We adapt a martingale duality method for solving (4) For the global before-default problem, a dynamic programming type relation (5) backward resolution from the after-default" one, both solved with respect to F The optimal trading strategy is given by ˆπ t = ˆπ F t 1 t τ + ˆπ d t (τ)1 t>τ, 0 t T, where ˆπ F is an optimal control to (4), and ˆπ d (θ) is an optimal control to V d θ (ˆX F θ (1 ˆπF θ γ θ)).

30 Solution to the after-default utility maximization The (local) martingale density of the after-default market: ( t µ d u Z t (θ) = exp (θ) θ σu(θ) d dw u 1 t 2 θ The optimal value process is µd u (θ) σu(θ) d 2 du ( ( Vθ [U d (x) = E I ŷ θ (x) Z )) T(θ) α T (θ) F θ ]. α T (θ) ), θ t T. The corresponding optimal wealth process is equal to: [ ˆX d,x ZT (θ) ( t (θ) = E Z t (θ) I ŷ θ (x) Z T(θ) α T (θ) ) Ft ], θ t T, (6) where I = (U ) 1, and ŷ θ (x) > 0 is F θ B((0, ))- d,x measurable random variable solution to ˆX θ (θ) = x.

31 The global before-default optimization problem In the case of CRRA utility functions: U(x) = xp p for p < 1, p 0, we solve explicitly [ T ] V 0 = sup E U(XT F )S T + Vθ d (X θ F (1 πf θ γ θ))dθ π F A F 0 for which V d θ (x) = U(x)K p θ with K θ = ( [ E α T (θ) ( ZT (θ) α T (θ) ) q Fθ ]) 1 q and q = p 1 p.

32 Theorem (1) V 0 = U(X 0 )Y 0, where Y is strictly positive, and (Y,φ) is the smallest (resp. largest) solution when p > 0 (resp. p < 0), in L b + (F) L2 loc (W) to the BSDE: T T Y t = G T + f(θ, Y θ,φ θ )dθ φ θ dw θ, 0 t T, (7) t t where f(t, Y t,φ t ) = p ess sup πγ t <1 [ (µ F t Y t + σ F t φ t )π 1 p 2 Y t πσ F t 2 + K p t (1 πγ t ) p Remark. We recover Merton value with: G T = 1, K t = 0. p ],

33 (2) The optimal strategy is given by: ˆπ F t [ = arg max (µ F t Y t + σt F φ t )π 1 p πγ t <1 (1 πγ t ) p + K p t p 2 Y t πσ F t 2 ], 0 t T,

34 Numerical illustrations Tests with CRRA utility function U(x) = xp p, p < 1, p 0. At default: γ constant, τ independent of F and τ exp(λ): α t (θ) = α(θ) and S(t) = P(τ > t) = e λt. For γ > 0 (loss at default), we take µ d (θ) = µ F θ T and σ d (θ) = σ F( ) 2 θ T, θ [0, T]. For γ < 0 (gain at default), we take µ d (θ) = µ F( ) 2 θ T and σ d (θ) = σ F (2 θ T ), θ [0, T], µ F = 0.03, σ F = 0.2, T = 1 The after-default optimal strategy: ˆπ d (θ) = µ d (θ) (1 p) σ d (θ) 2. No closed-form solution for the global problem: numerical resolution by Howard algorithm.

35 Optimal stragegy in function of the jump size γ Figure: Optimal strategy vs Merton: p = 0.2, λ = 0.01 and 0.3 respectively. 4 3 Pi Merton Pi F with lambda=0.01 Pi F with lambda= gamma

36 References N. El Karoui, M. Jeanblanc and Y. Jiao (2009), What happens after the default: the conditional density approach, Stochastic Processes and their Applications, 120(7), N. El Karoui, M. Jeanblanc and Y. Jiao (2010), Modelling successive default events, working paper. Y. Jiao (2009), Multiple defaults and contagion risks with global and default-free information, working paper. Y. Jiao and H. Pham (2009), Optimal investment with counterparty risks: a default-density model approach, to appear in Finance and Stochastics.

37 Thank you very much for your attention!