Modelling of successive default events Nicole El Karoui, CMAP, Ecole - PowerPoint PPT Presentation

Modelling of successive default events Nicole El Karoui, CMAP, Ecole Polytechnique e d’´ Monique Jeanblanc, Universit´ Evry, France Ying Jiao, Ecole Sup´ erieure d’Ing´ enieur L´ eonard de Vinci Princeton, Credit Risk, May 2008 Financial support from Fondation du Risque and F´ ed´ eration des Banques Fran¸ caises 1

The aim of this talk is • to give a general framework for multi-defaults modelling • to obtain the dynamics of derivative products in a multiname setting. 2

The basic tool is the conditional law of the default(s) with respect to a reference filtration 3

Single Name Single Name 4

Single Name Notation • G is the global market filtration, • τ is a default time, • H t = 1 1 τ ≤ t is the default processes, • H is the natural filtration of H , with H ⊂ G , • F is a reference filtration, with F ⊂ G . 5

Single Name On the set { τ > t } : ”before-default” ( F , G , τ ) satisfy the minimal assumption (MA) if ∀ t ≥ 0 and Z G ∈ G t , ∃ Z F ∈ F t such that Z G ∩ { τ > t } = Z F ∩ { τ > t } . • If G τ := F ∨ H , then ( F , G τ , τ ) enjoys MA. • Under MA, for any G ∞ -measurable (integrable) r.v. Y G , E [ Y G 1 1 { τ>t } |F t ] 1 { τ>t } E [ Y G |G t ] = 1 1 1 { τ>t } a.s. P ( τ > t |F t ) on the set A := { ω : P ( τ > t |F t )( ω ) > 0 } . 6

Single Name On the set { τ ≤ t } : ”after-default” 1. We assume that G = G τ = F ∨ H 2. J-Hypothesis (Jacod for enlargement of filtration purpose): We assume that there exists a family of F t ⊗ B ( R + ) r.vs α t ( θ ) such that P ( τ ∈ dθ |F t ) = α t ( θ ) dθ dθ ⊗ d P − a.s. and for any θ the process α t ( θ ) , t ≥ 0 is a right-continuous martingale Under the above hypotheses, we can compute G t -conditional expectations on the set { τ ≤ t } A ”weak version” of J-hypothesis consists of the existence of the density only for 0 ≤ t ≤ θ . This is useful for before-default studies, but not sufficient for the after-default ones. 7

Single Name Density and F -martingales We assume J-hypothesis and introduce the c` adl` ag (super) martingales • Conditional survival process ( S t = P ( τ > t |F t ) , t ≥ 0) (Az´ ema super-martingale) • Conditional probability process ( S t ( θ ) = P ( τ > θ |F t ) , t ≥ 0). Note that, for t ≤ θ , one has S t ( θ ) = E [ S θ |F t ], • Family of martingales (for θ ∈ R ): the densities ( α t ( θ ) , t ≥ 0) of S t ( θ ) 8

Single Name Decompositions of Survival process S (super-martingale) • The Doob-Meyer decomposition of S is S t = M F t − A F t with – the F -martingale � t � t M F t = − ( α t ( u ) − α u ( u )) du = S t − S 0 + α u ( u ) du 0 0 � t – the increasing process A F t = 0 α u ( u ) du • The multiplicative decomposition is S t = L F t D F t where � t – The F -martingale L F is given as dL F 0 λ F s ds dM F t = e t � � t � – The decreasing process D F is D F 0 λ F t = exp − s ds where t = α t ( t ) λ F on { S t > 0 } . S t 9

Single Name Links with the classical intensity approach • The G - intensity is the G -adapted process λ G such that � t 0 λ G (1 1 { τ ≤ t } − s ds, t ≥ 0) is a G -martingale. • Under the weak version of J-Hypothesis, α t ( t ) λ G 1 { τ>t } λ F t = 1 t = 1 1 { τ>t } a.s.. S t − • For any θ ≥ t , α t ( θ ) = E [ λ G θ |F t ] a.s.. Note that the intensity approach does not contain enough information to study the after-default case (i.e. for θ < t ). 10

Single Name A characterization of G -martingales Any G t -measurable r.v. X can be written as X = X t 1 1 { τ>t } + X t ( τ )1 1 { τ ≤ t } where X t and X t ( θ ) are F t -measurable. The process M X is a G -martingale if its decomposition as M X := X t 1 1 { τ>t } + X t ( τ )1 1 { τ ≤ t } t satisfies � t • ( X t S t + 0 X s ( s ) α s ( s ) ds, t ≥ 0) is an F -martingale • For any θ , ( X t ( θ ) α t ( θ ) , t ≥ θ ) is an F -martingale Remark: The first condition is equivalent to : � t X t L F 0 ( X s − X s ( s )) L F s λ F t − s ds is an F -martingale 11

Single Name Immersion hypothesis Immersion holds if (and only if) any F -martingale is a G -martingale. Under immersion hypothesis, • α t ( θ ) = α t ∧ θ ( θ ) • S is a non-increasing process • L F is a constant • the process M X := X t 1 1 { τ>t } + X t ( τ )1 1 { τ ≤ t } t is a G -martingale if (a) X t ( θ ) is a F -martingale on [ θ, ∞ ). � t 0 ( X s − X s ( s )) λ F (b) X t − s ds is an F -martingale 12

Single Name Toy Exemple: Cox process model � t 0 λ F Let τ = inf { t : Λ t := s ds ≥ Θ } where Λ is an F -adapted increasing process, Λ 0 = 0, lim t →∞ Λ t = + ∞ Θ is a G -measurable r.v. independent of F ∞ , Θ i ∼ exp(1). F is immersed in G = F ∨ H . The conditional distribution of τ is ⎧ ⎨ E [ e − Λ θ | F t ] , P ( τ > θ | F t ) = for θ > t ⎩ e − Λ θ , P ( τ > θ | F t ) = for θ ≤ t and the density is ⎧ ⎨ E [ λ θ e − Λ θ | F t ] , α t ( θ ) = for θ > t ⎩ λ θ e − Λ θ , = for θ ≤ t 13

Single Name Modelling the density process Two possible solutions: • Model S t ( θ ) := P ( τ > θ |F t ) and then take derivatives w.r.t. θ • Model the density α t ( θ ) as a family of strictly positive martingales � ∞ such that α s ( θ ) dθ = 1 0 Remarks : • for fixed θ , both processes are positive F martingales • reference to the interest models • distinction between θ ≥ t (classical part) and θ < t (non-classical part) 14

Single Name F -martingale representation and HJM framework Model the family of F -martingales S t ( θ ) in the HJM framework Suppose dS t ( θ ) S t ( θ ) = Φ t ( θ ) dM t , t, θ ≥ 0 where M is a continuous multi-dimensional F -martingale, then � � � θ S t ( θ ) = S t ( t ) exp − t λ t ( u ) du where λ t ( θ ) is the forward intensity and �� t � � t 0 Φ s ( θ ) dM s − 1 0 | Φ s ( θ ) | 2 d � M � s * S t ( θ ) = S 0 ( θ ) exp ; 2 � � � t � t � t 0 Φ s ( s ) dM s − 1 0 | Φ s ( s ) | 2 d � M � s 0 λ F * S t = exp − s ds + . 2 � t � t * λ t ( θ ) = λ 0 ( θ ) − 0 ϕ s ( θ ) dM s + 0 ϕ s ( θ )Φ s ( θ ) d � M � s . 15

Single Name For any θ ≥ 0, assume that λ 0 ( θ ) is a family of positive probability densities. Let b ( θ ) be a given family of non-negative F -adapted processes. Define �� t � � t b s ( θ ) dW s − 1 b s ( θ ) 2 ds ϕ t ( θ ) = − b t ( θ ) λ 0 ( θ ) exp 2 0 0 and let � � � t α t ( θ ) = λ t ( θ ) exp − λ t ( v ) dv 0 � θ � t where λ t ( θ ) = λ 0 ( θ ) − 0 ϕ s ( θ ) dW s + 0 ϕ s ( θ )Φ s ( θ ) ds . Then the family ( α t ( θ ) , t ≥ 0) is a density process. 16

Single Name Examples of martingale density process • Compatibility between martingale and probability properties – the r.v. S t ( θ ) is [0 , 1]-valued – for any t , the map θ → S t ( θ ) is non-increasing • A Generalized exponential model: ∀ t, θ ≥ 0, let � � − θM t − 1 2 θ 2 � M � t S t ( θ ) = exp where M is an F -martingale. • Exponential law S 0 ( θ ) = P ( τ > θ ) = exp( − θM 0 ). • Probability condition : M t + 1 2 θ � M � t ≥ 0 17

Single Name Comparison with interest rate modelling � T • Zero-coupon B ( t, T ) = E [ e − t r s ds |F t ]. Short rate r t = − ∂ T | T = t log B ( t, T ). • Defaultable zero-coupon without actualization 1 { τ>t } B τ ( t, T ) . E [1 1 { τ>T } |G t ] = 1 1 { τ>t } E [ S T /S t |F t ] := 1 Intensity λ F t = − ∂ T | T = t log B τ ( t, T ). 18

Single Name A general construction of density process. I. 1. Start with P 0 with immersion hypothesis, and τ with density process ( α 0 t ( θ ) , t ≥ 0) constant in time after θ 2. Let Z G t = Z t 1 1 { τ>t } + Z t ( τ )1 1 { τ ≤ t } a positive ( G , P 0 )-martingale with expectation 1 3. Define d P = Z G T d P 0 on G T . The RN density of P w.r.t. P 0 on F t is � t 0 Z s ( s ) α 0 Z F t = Z t S t + s ( s ) ds . 4. Then the density process of τ under P is θ ( θ ) Z t ( θ ) α 0 α P t ( θ ) = for θ < t Z F t E [ Z θ ( θ ) α 0 θ ( θ ) |F t ] α P t ( θ ) = for θ ≥ t. Z F t 19

Single Name A general construction of density process. II. 1. Start with P 0 with τ independent of F ∞ with density f 2. Let q ∞ ( u ) a family of F ∞ -measurable r.v. such that � ∞ q ∞ ( u ) f ( u ) du = 1 0 3. Define d P = q ∞ ( τ ) d P 0 on G ∞ . 4. Then, setting q t ( u ) = E 0 ( q ∞ ( u ) |F t ), the RN density of P w.r.t. P 0 � ∞ on F t is Z F t = q t ( u ) f ( u ) du and the density process of τ 0 under P is α P q t ( θ ) f ( θ )( Z F t ) − 1 t ( θ ) = 20

Two ordered default times Two ordered default times 21

Two ordered default times Notation Two G -stopping times: τ = τ (1) := min( τ 1 , τ 2 ) σ = τ (2) := max( τ 1 , τ 2 ) . and Before-default and after-default analysis extended naturally to the ordered defaults • Filtrations: H (1) for τ and H (2) for σ respectively. Let G (1) = F ∨ H (1) G (2) = F ∨ H (1) ∨ H (2) = G (1) ∨ H (2) . and • On the set { t < τ } , it suffices to apply directly the previous studies • On the sets { τ ≤ t < σ } and { σ ≤ t } a recursive procedure using G (1) as the reference filtration and G (2) as the global filtration 22