Introduction Multiple defaults risk model Optimal investment problem Backward system of BSDEs Conclusion Optimal investment under multiple defaults: a BSDE-decomposition approach en PHAM ∗ Huyˆ ∗ LPMA-University Paris Diderot, and Institut Universitaire de France Based on joint works with: Ying Jiao (LPMA-University Paris Diderot), Idris Kharroubi (University Paris Dauphine) Workshop “New advances in Backward SDEs for financial engineering applications” Tamerza, October 25-28, 2010 Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Optimal investment problem Backward system of BSDEs Conclusion The basic problem • Assets portfolio subject to multiple defaults risk ◮ In addition to the default-free assets model (e.g. diffusion model with Brownian W ), introduce jumps at random times modelled by a marked point process ( τ i , ζ i ) i ↔ random measure µ ( dt , de ). • Optimal investment by classical stochastic control methods: ◮ (Quadratic) BSDEs with jumps: this relies on martingale representation in the global filtration generated by W and µ . Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Optimal investment problem Backward system of BSDEs Conclusion Optimal investment problem with defaults revisited • Approach by using: - Point of view of global filtration as progressive enlargement of filtration of the default-free filtration - Decomposition in the default-free filtration ◮ Backward system of BSDEs in Brownian filtration ◮ Get rid of the jump terms and overcome the technical difficulties in BSDEs with jumps ◮ Existence and uniqueness results in a general formulation under weaker conditions Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Multiple defaults times and marks On a probability space (Ω , G , P ): • Reference filtration F = ( F t ) t ≥ 0 : default-free information Progressive information provided, when they occur, by: • a family of n random times τ = ( τ 1 , . . . , τ n ) associated to a family of n random marks ζ = ( ζ 1 , . . . , ζ n ). ◮ τ i default time of name i ∈ I n = { 1 , . . . , n } . ◮ The mark ζ i , valued in E Borel set of R p , represents a jump size at τ i , which cannot be predicted from the reference filtration, e.g. the loss given default. Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Progressive enlargement of filtrations The global market information is defined by: F ∨ D 1 ∨ . . . ∨ D n , = G where D i is the default filtration generated by the observation of τ i and ζ i when they occur, i.e. D i = ( D i D i t = σ { 1 τ i ≤ s , s ≤ t } ∨ σ { ζ i 1 τ i ≤ s , s ≤ t } . t ) t ≥ 0 , → G = F ∨ F µ , where F µ is the filtration generated by the jump random measure µ ( dt , de ) associated to ( τ i , ζ i ). Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Successive defaults For simplicity of presentation, we assume that τ 1 ≤ . . . ≤ τ n Remarks. • This means that we do not distinguish specific credit names, and only observe the ordered defaults: relevant for classical portfolio derivatives, e.g. basket default swaps. • The general multiple random times case for ( τ 1 , . . . , τ n ) can be derived from the ordered case by considering the filtration generated by the corresponding ranked times (ˆ τ 1 , . . . , ˆ τ n ) and the index marks ι i , i = 1 , . . . , n so that (ˆ τ 1 , . . . , ˆ τ n ) = ( τ ι 1 , . . . , τ ι n ). Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Notations • For any k = 0 , . . . , n : Ω k � � = τ k ≤ t < τ k +1 , t Ω k � � = τ k < t ≤ τ k +1 , t − with the convention Ω 0 t = { t < τ 1 } , Ω n t = { τ n ≤ t } . → Scenario of k defaults before time t , the other names having not yet defaulted. Ω k t : k -default scenario at time t , (Ω k t ) k =0 ,..., n partition of Ω. • For k = 0 , . . . , n , τ k = ( τ 1 , . . . , τ k ) , ζ k = ( ζ 1 , . . . , ζ k ) , with the convention τ 0 = ∅ , ζ 0 = ∅ . Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Decomposition of G -adapted and predictable processes Lemma Any G -adapted process Y is represented as: n � t Y k = 1 Ω k t ( τ k , ζ k ) , (1) Y t k =0 where Y k t is F t ⊗ B ( R k + ) ⊗ B ( E k ) -measurable. Remarks. • A similar decomposition result holds for G -predictable t − , and Y k ∈ P F ( R k processes: Ω k t ↔ Ω k + , E k )-measurable in (1) . • Extension of Jeulin-Yor result (case of single random time without mark). • We identify Y with the n + 1-tuple ( Y 0 , . . . , Y n ). Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion • Portfolio of N assets with G -adapted value process S : n � t S k = 1 Ω k t ( τ k , ζ k ) , S t k =0 where S k ( θ k , e k ), θ k = ( θ 1 , . . . , θ k ) ∈ R k + , e k = ( e 1 , . . . , e k ) ∈ E k , indexed F -adapted process valued in R N + , represents the assets value given the past default events τ k = θ k and marks at default ζ k = e k . Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Change of regimes with jumps at defaults • Dynamics of S = S k between τ k = θ k and τ k +1 = θ k +1 : dS k S k t ( θ k , e k ) ∗ ( b k t ( θ k , e k ) dt + σ k t ( θ k , e k ) = t ( θ k , e k ) dW t ) , where W is a m -dimensional ( P , F )-Brownian motion, m ≥ N . • Jumps at τ k +1 = θ k +1 : S k +1 S k 1 N + γ k � � k +1 ( θ k , e k ) ∗ θ k +1 ( θ k +1 , e k +1 ) = θ k +1 ( θ k , e k , e k +1 ) , θ − Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Credit derivative • A credit derivative of maturity T is represented by a G T -measurable random variable H T : n � T H k H T = 1 Ω k T ( τ k , ζ k ) , k =0 where H k T ( ., . ) is F T ⊗ B ( R k + ) ⊗ B ( E k )-measurable, and represents the option payoff in the k -default scenario. Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Exogenous counterparty default • One default time τ ( n = 1) inducing jumps in the price process S of N -assets portfolio: S 0 t 1 t <τ + S 1 = t ( τ, ζ )1 t ≥ τ , S t where S 0 is the price process before default, governed by dS 0 S 0 t ∗ ( b 0 t dt + σ 0 = t dW t ) t and S 1 ( θ, e ), ( θ, e ) ∈ R + × E , is the indexed price process after default at time θ and with mark e : dS 1 S 1 t ( θ, e ) ∗ ( b 1 t ( θ, e ) dt + σ 1 t ≥ θ, t ( θ, e ) = t ( θ, e ) dW t ) , S 1 S 0 θ ( θ, e ) = θ ∗ ( 1 N + γ θ ( e )) . Huyˆ en PHAM Multiple defaults risk and BSDEs
Introduction Multiple defaults risk model Modelling of multiple defaults events Optimal investment problem Assets and credit derivatives model Backward system of BSDEs Examples Conclusion Multilateral counterparty risk • Assets family (e.g. portfolio of defaultable bonds) in which each underlying name is subject to its own default but also to the defaults of the other names (contagion effect). ◮ number of defaults n = N number of assets S = ( P 1 , . . . , P n ) ◮ τ i default time of name P i , and ζ i its (random) recovery rate ( P i is not traded anymore after τ i ) ◮ τ i induces jump on P j , j � = i . Huyˆ en PHAM Multiple defaults risk and BSDEs
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