Density models for credit risk Nicole El Karoui, Ecole - PowerPoint PPT Presentation

Density models for credit risk Nicole El Karoui, Ecole Polytechnique, France e d’´ Monique Jeanblanc, Universit´ Evry; Institut Europlace de Finance Ying Jiao, Universit´ e Paris VII Workshop on stochastic calculus and finance, July 2009, Hong-Kong Financial support from La Fondation du Risque and F´ ed´ eration Bancaire Fran¸ caise 1

Density Hypothesis Density Hypothesis Let (Ω , A , F , P ) be a filtered probability space. A strictly positive and finite random variable τ (the default time) is given. Our goals are • to show how the information contained in the reference filtration F can be used to obtain information on the law of τ , • to investigate the links between martingales in the different filtrations that will appear. 2

Density Hypothesis We assume the following density hypothesis: there exists a non-atomic non-negative measure η such that, for any time t , there exists an F t ⊗ B ( R + ) -measurable function ( ω, θ ) → α t ( ω, θ ) which satisfies P ( τ ∈ dθ |F t ) = α t ( θ ) η ( dθ ) , P − a.s. The conditional distribution of τ is characterized by the survival probability defined by � ∞ G t ( θ ) := P ( τ > θ |F t ) = α t ( u ) η ( du ) θ Let � ∞ G t := G t ( t ) = P ( τ > t |F t ) = α t ( u ) η ( du ) t Observe that the set A t := { G t > 0 } contains a.s. the event { τ > t } . 3

Density Hypothesis The family α t ( . ) is called the conditional density of τ w.r.t. η given F t . Note that • G t ( θ ) = E ( G θ |F t ) for any θ ≥ t � ∞ • the law of τ is P ( τ > θ ) = α 0 ( u ) η ( du ) θ � ∞ • for any t , α t ( u ) η ( du ) = 1 0 • For an integrable F T ⊗ σ ( τ ) r.v. Y T ( τ ), one has, for t ≤ T : � ∞ E ( Y T ( τ ) |F t ) = E ( Y T ( u ) α T ( u ) η ( du ) |F t ) 0 • The default time τ avoids F -stopping times, i.e., P ( τ = ϑ ) = 0 for every F -stopping time ϑ . � ∞ G t ( θ ) := P ( τ > θ |F t ) = α t ( u ) η ( du ) 4 θ

Density Hypothesis The family α t ( . ) is called the conditional density of τ w.r.t. η given F t . Note that • G t ( θ ) = E ( G θ |F t ) for any θ ≥ t � ∞ • taking, w.l.g., α 0 ( u ) = 1, the law of τ is P ( τ > θ ) = η ( du ) θ � ∞ • for any t , α t ( u ) η ( du ) = 1 0 • For an integrable F T ⊗ σ ( τ ) r.v. Y T ( τ ), one has, for t ≤ T : � ∞ E ( Y T ( τ ) |F t ) = E ( Y T ( u ) α T ( u ) η ( du ) |F t ) 0 • The default time τ avoids F -stopping times, i.e., P ( τ = ϑ ) = 0 for every F -stopping time ϑ . � ∞ G t ( θ ) := P ( τ > θ |F t ) = α t ( u ) η ( du ) 5 θ

Density Hypothesis By using the density, we adopt an additive point of view to represent the conditional probability of τ � ∞ G t ( θ ) = α t ( u ) η ( du ) θ In the default framework, the “intensity” point of view is often preferred, and one uses a multiplicative representation as � θ G t ( θ ) = exp( − λ t ( u ) η ( du )) 0 where λ t ( u ) = − ∂ u ln G t ( u ) is the “forward intensity”. 6

Computation of conditional expectations Computation of conditional expectations Let D = ( D t ) t ≥ 0 be the smallest right-continuous filtration such that τ is a D -stopping time, and let G = F ∨ D . Any G t -measurable r.v. H G t may be represented as H G t = H F t 1 { τ>t } + H t ( τ )1 1 { τ ≤ t } where H F t is an F t -measurable random variable and H t ( τ ) is F t ⊗ σ ( τ ) measurable t = E [ H G t 1 { τ>t } |F t ] H F a.s. on A t ; = 0 if not . G t 7

Computation of conditional expectations Computation of conditional expectations Let D = ( D t ) t ≥ 0 be the smallest right-continuous filtration such that τ is a D -stopping time, and let G = F ∨ D . Any G t -measurable r.v. H G t may be represented as H G t = H F t 1 { τ>t } + H t ( τ )1 1 { τ ≤ t } where H F t is an F t -measurable random variable and H t ( τ ) is F t ⊗ σ ( τ ) measurable t = E [ H G t 1 { τ>t } |F t ] H F a.s. on A t ; = 0 if not . G t 8

Computation of conditional expectations Immersion property In the particular case where α t ( θ ) = α θ ( θ ) , ∀ θ ≤ t one has � t � t G t = 1 − α t ( θ ) η ( dθ ) = 1 − α T ( θ ) η ( dθ ) = P ( τ > t |F T ) a.s. 0 0 for any T ≥ t and P ( τ > t |F t ) = P ( τ > t |F ∞ ). This last equality is equivalent to the immersion property (i.e. F martingales are G -martingales). Conversely, if immersion property holds, then P ( τ > t |F t ) = P ( τ > t |F ∞ ) hence, the process G is decreasing and the conditional survival functions G t ( θ ) are constant in time on [ θ, ∞ ), i.e., G t ( θ ) = G θ ( θ ) for t > θ . � ∞ G t := P ( τ > t |F t ) = α t ( u ) η ( du ) 9 t

Dynamic point of view and density process Dynamic point of view and density process Regular Version of Martingales One of the major difficulties is to prove the existence of a universal c` adl` ag martingale version of this family of densities. 10

Dynamic point of view and density process F -decompositions of the survival process G • The Doob-Meyer decomposition of the super-martingale G is given by � t G t = 1 + M F t − α u ( u ) η ( du ) 0 where M F is the c` adl` ag square-integrable martingale defined as � t � ∞ M F � � t = − α t ( u ) − α u ( u ) η ( du ) = E [ α u ( u ) η ( du ) |F t ] − 1 . 0 0 • Let ζ F := inf { t : G t − = 0 } . We define λ F G t − for any t ≤ ζ F and t := α t ( t ) let λ F t = λ F t ∧ ζ F for any t > ζ F . The multiplicative decomposition of G is given by � t � t 0 λ F 0 λ F t e − s η ( ds ) s η ( ds ) dM F G t = L F dL F where t = e t . 11

Dynamic point of view and density process F -decompositions of the survival process G • The Doob-Meyer decomposition of the super-martingale G is given by � t G t = 1 + M F t − α u ( u ) η ( du ) 0 where M F is the c` adl` ag square-integrable martingale defined as � t � ∞ M F � � t = − α t ( u ) − α u ( u ) η ( du ) = E [ α u ( u ) η ( du ) |F t ] − 1 . 0 0 • Let ζ F := inf { t : G t − = 0 } . We define λ F G t − for any t ≤ ζ F and t := α t ( t ) let λ F t = λ F t ∧ ζ F for any t > ζ F . The multiplicative decomposition of G is given by � t � t 0 λ F 0 λ F t e − s η ( ds ) s η ( ds ) dM F G t = L F dL F where t = e t . 12

Dynamic point of view and density process � t Proof: 1) First notice that ( 0 α u ( u ) η ( du ) , t ≥ 0) is an F -adapted continuous increasing process. By the martingale property of ( α t ( θ ) , t ≥ 0), for any fixed t , � ∞ � ∞ G t = α t ( u ) η ( du ) = E [ α u ( u ) η ( du ) |F t ] , a.s.. t t � t From the properties of the density, 1 − G t = 0 α t ( u ) η ( du ) and � t � ∞ M F t := − ( α t ( u ) − α u ( u )) η ( du ) = E [ α u ( u ) η ( du ) |F t ] − 1 . 0 0 2) By definition of L F t and 1), we have � t � t � t 0 λ F 0 λ F 0 λ F dL F s η ( ds ) dG t + e s η ( ds ) λ F s η ( ds ) dM F t = e t G t η ( dt ) = e t , which implies the result. △ 13

Dynamic point of view and density process Relationship with the G -intensity Definition: Let τ be a G -stopping time. The G -compensator Λ G of τ is 1 { τ ≤ t } − Λ G the G -predictable increasing process such that (1 t , t ≥ 0) is a G -martingale. The G -compensator is stopped at τ , i.e., Λ G t = Λ G t ∧ τ . Λ G coincides, on the set { τ ≥ t } , with an F -predictable process Λ F , i.e. Λ G t 1 { τ ≥ t } = Λ F t 1 { τ ≥ t } . • the G -compensator Λ G of τ admits a density given by α t ( t ) λ G 1 { τ>t } λ F t = 1 t = 1 1 { τ>t } . G t − In particular, τ is a totally inaccessible G -stopping time. • For any t < ζ F and T ≥ t , we have α t ( T ) = E [ λ G T |F t ]. 14

Dynamic point of view and density process Relationship with the G -intensity Definition: Let τ be a G -stopping time. The G -compensator Λ G of τ is 1 { τ ≤ t } − Λ G the G -predictable increasing process such that (1 t , t ≥ 0) is a G -martingale. The G -compensator is stopped at τ , i.e., Λ G t = Λ G t ∧ τ . Λ G coincides, on the set { τ ≥ t } , with an F -predictable process Λ F , i.e. Λ G t 1 { τ ≥ t } = Λ F t 1 { τ ≥ t } . • the G -compensator Λ G of τ is absolutely continuous w.r.t. η with α t ( t ) λ G 1 { τ>t } λ F t = 1 t = 1 1 { τ>t } . G t − In particular, τ is a totally inaccessible G -stopping time. • For any t < ζ F and T ≥ t , we have α t ( T ) = E [ λ G T |F t ]. 15

Dynamic point of view and density process Relationship with the G -intensity Definition: Let τ be a G -stopping time. The G -compensator Λ G of τ is 1 { τ ≤ t } − Λ G the G -predictable increasing process such that (1 t , t ≥ 0) is a G -martingale. The G -compensator is stopped at τ , i.e., Λ G t = Λ G t ∧ τ . Λ G coincides, on the set { τ ≥ t } , with an F -predictable process Λ F , i.e. Λ G t 1 { τ ≥ t } = Λ F t 1 { τ ≥ t } . • the G -compensator Λ G of τ is absolutely continuous w.r.t. η with α t ( t ) λ G 1 { τ>t } λ F t = 1 t = 1 1 { τ>t } . G t − In particular, τ is a totally inaccessible G -stopping time. • For any t < ζ F and T ≥ t , we have α t ( T ) = E [ λ G T |F t ]. 16