Dynamical modelling of successive defaults N. El Karoui M. - PowerPoint PPT Presentation

Dynamical modelling of successive defaults N. El Karoui M. Jeanblanc Y. Jiao 21 Sep, 2007 cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 1 / 23

Outline Motivation and introduction 1 An illustrative example 2 The density process framework 3 Successive defaults 4 cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 2 / 23

Motivation and introduction Motivation and Introduction Rapid development of credit portfolio products : k th -to-default swap, CDOs A practical question proposed by the practitioners: possibility of a recursive procedure to study the successive defaults? Calculation of conditional expectations E [ Y T |G t ] when ( G t ) t ≥ 0 is some large filtration including default informations Study of the “after-default case” by using the density process approach cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 3 / 23

An illustrative example An illustrative example cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 4 / 23

An illustrative example An illustrative example A simple deterministic model of two credits, denote by τ = min ( τ 1 , τ 2 ) . Observable information : whether the first default occurs. The basic hypothesis is based on a stationary point of view of the practitioners P ( τ i > T | τ > t ) = e − µ i ( t ) · ( T − t ) , ( i = 1 , 2 ) where µ i ( t ) characterizes the individual default and it can be renewed with market information at t . The marginal distributions remain in the exponential family. cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 5 / 23

An illustrative example The joint probability Let P ( τ 1 > t 1 , τ 2 > t 2 ) = P ( τ 1 > t 1 ) P ( τ 2 > t 2 ) ρ ( t 1 , t 2 ) . Consider the survival copula function � C ( u , v ) such that � C ( P ( τ 1 > t 1 ) , P ( τ 2 > t 2 )) = P ( τ 1 > t 1 , τ 2 > t 2 ) , then � � � ln u ln v uv ρ if u , v > 0 ; µ 1 ( 0 ) , , � µ 2 ( 0 ) C ( u , v ) = 0 , if u = 0 or v = 0 . Joint probability : If ρ ( t 1 , t 2 ) ∈ C 1 , 1 , then the joint survival probability is given by � t 1 � t 2 � � µ 1 ( s ∧ t 2 ) ds − µ 2 ( s ∧ t 1 ) ds P ( τ 1 > t 1 , τ 2 > t 2 ) = exp − . 0 0 First observation: The joint probability function depends on all the dynamics of the marginal distributions and the copula can not be cmap chosen independently with marginal distributions. Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 6 / 23

An illustrative example The joint probability Proposition: If ρ ( t 1 , t 2 ) ∈ C 2 and if µ 1 ( t ) , µ 2 ( t ) ∈ C 1 , then P ( τ 1 > t 1 , τ 2 > t 2 ) � t 1 ∧ t 2 � � − µ 1 ( 0 ) t 1 − µ 2 ( 0 ) t 2 + = exp ϕ ( s )( t 1 + t 2 − 2 s ) ds . 0 � � ∂ 2 where ϕ ( t ) = t 1 = t 2 = t ln ρ ( t 1 , t 2 ) . In addition, we have � ∂ t 1 ∂ t 2 � t µ i ( t ) = µ i ( 0 ) − ϕ ( s ) ds . 0 µ 1 and µ 2 follow the same dynamics apart from their initial values due to the symmetric information flow and the stationary property. cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 7 / 23

An illustrative example First default and contagious jumps The distribution of the first default is given by � � � t 0 µ 1 ( s ) + µ 2 ( s ) ds P ( τ > t ) = exp − ; For the surviving credit, it becomes complicated. Let D t = D 1 t ∨ D 2 t where D i t = σ ( 1 1 { τ i ≤ s } , s ≤ t ) ( i = 1 , 2 ) . Then � � � t � � µ i ( 0 ) − E [ 1 1 { τ i > T } |D t ] = 1 1 { τ> t } exp − ϕ ( s ) ds ( T − t ) 0 � � � τ � � µ i ( 0 ) − + 1 1 { τ i > t ,τ j ≤ t } exp − ϕ ( s ) ds ( T − t ) 0 � τ · µ j ( 0 ) − ϕ ( τ )( T − τ ) − 0 ϕ ( s ) ds � τ 0 ϕ ( s ) ds . µ j ( 0 ) − ϕ ( τ )( t − τ ) − We observe the contagious jump phenomenon cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 8 / 23

An illustrative example The second default time Second observation on σ = max ( τ 1 , τ 2 ) : the conditional 1 { σ> T } |D τ distribution E [ 1 t ] can not remain in the exponential family neither on { τ > t } nor on { τ ≤ t } , except when τ 1 and τ 2 are independent and identically distributed. Remark: conditioned on the first default, these exists no longer the stationary property, as expected by some market practitioners! We need to study the successive defaults in an abstract way. cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 9 / 23

The density process framework The General Case — the density process framework cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 10 / 23

The density process framework Before-default case and Minimal assumption Let (Ω , G , G , P ) be a filtered probability space representing the market. The filtration G = ( G t ) t ≥ 0 represents the global information on the market. Let τ be a finite G -stopping time. Consider a subfiltration F of G satisfying the following condition presented by Jeulin and Yor (1978), Jacod (1982). (Minimal Assumption): We say ( F , G , τ ) satisfy the Minimal Assumption (MA) if for any t ≥ 0 and any U ∈ G t , there exists V ∈ F t such that U ∩ { τ > t } = V ∩ { τ > t } . cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 11 / 23

The density process framework Before-default case and Minimal assumption Two examples of ( F , G , τ ) satisfing MA: In the single credit case, τ represents one default time and let D = ( D t ) t ≥ 0 where D t = σ ( 1 1 { τ ≤ s } , s ≤ t ) . F satisfies G = D ∨ F . In the multi-credits case, τ represents the first default time τ = min ( τ 1 , · · · , τ n ) and F satisfies G = F ∨ D 1 · · · ∨ D n . A direct consequence: Assume that ( F , G , τ ) satisfy MA. For any G -measurable random variable Y , if P ( τ > t |F t ) > 0, a.s. then E [ 1 1 { τ> t } Y |F t ] E [ 1 1 { τ> t } Y |G t ] = 1 1 { τ> t } 1 { τ> t } |F t ] . E [ 1 cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 12 / 23

The density process framework After-default case and Density process ( H J Hypothesis, Jacod (82)) For any t , θ ≥ 0, we assume that there exists a family of F -adapted processes, called the density process ( α t ( u ) , t ≥ 0 ) , such that � ∞ S t ( θ ) = P ( τ > θ |F t ) = α t ( u ) du . θ Proposition For any t , u ≥ 0, let Y ( t , u ) be a random variable such that Y ( t , u ) ∈ F t ⊗ B ( R ) . If G = F ∨ D and if α t ( u ) > 0, then for any 0 ≤ t ≤ T , � � � � � F t 1 { τ ≤ t } = E Y ( T , s ) α T ( s ) � E [ Y ( T , τ ) |G t ] 1 s = τ 1 1 { τ ≤ t } . � α t ( s ) cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 13 / 23

The density process framework Density and intensity processes The G -compensator process Λ of τ is a predictable process such � t 0 λ G that ( N t = 1 1 { τ ≤ t } − Λ t , t ≥ 0 ) is a G -martingale. If Λ t = s ds , then λ G is called the G -intensity process. Proposition: Assume that ( F , G , τ ) satisfy minimal assumption. If the survival density α t ( u ) exists, then the G -compensator process Λ of τ is given by α t ( t ) � ∞ d Λ t = 1 1 ] 0 ,τ ] ( t ) α t ( u ) du dt . t cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 14 / 23

The density process framework Density and intensity processes The intensity process can be deduced from the density process. However, the reverse is not true in general. Proposition : Assume that ( F , G , τ ) satisfy minimal assumption. If the G -intensity process ( λ G t , t ≥ 0 ) of τ exists. Then, for any u ≥ t , the density of the conditional survival law of τ is given by � � λ G α t ( u ) = E u |F t . The after-default case requires us to know α t ( u ) for u < t , which can not be obtained from the intensity process. cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 15 / 23

Successive defaults Successive defaults cmap Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 16 / 23

Successive defaults Two ordered default times - the first default Consider τ 1 and τ 2 with associated filtrations D 1 and D 2 . Let F such that G = F ∨ D 1 ∨ D 2 . Let τ = min ( τ 1 , τ 2 ) and σ = max ( τ 1 , τ 2 ) with D ( 1 ) and D ( 2 ) . ( F , G , τ ) satisfy the minimal assumption. Hence the first default can be treated in the same way as for a single credit. Proposition: Assume that the joint density process of ( τ 1 , τ 2 ) exists, i.e. � ∞ � ∞ P ( τ 1 > t 1 , τ 2 > t 2 | F t ) = du 1 du 2 p t ( u 1 , u 2 ) . t 1 t 2 Then the density process ( α τ t ( θ ) , t ≥ 0 ) of τ is given by � ∞ � � α τ t ( θ ) = du p t ( θ, u ) + p t ( u , θ ) . cmap θ Workshop Amamef (Vienna) Dynamical modelling of successive defaults 21 Sep, 2007 17 / 23