Density models for credit risk

Density models for credit risk Nicole El Karoui, Ecole Polytechnique, France Monique Jeanblanc, Université d Évry; Institut Europlace de Finance Ying Jiao, Université Paris VII Workshop on stochastic calculus and finance, July 29, Hong-Kong Financial support from La Fondation du Risque and Fédération Bancaire Française 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 canbeusedtoobtaininformationonthelawofτ, 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 )= θ α t (u)η(du) Let G t := G t (t) =P(τ >t F t )= t α t (u)η(du) Observe that the set A t := {G t > } 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)= θ for any t, α t (u)η(du) =1 α (u)η(du) 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 ) The default time τ avoids F-stopping times, i.e., P(τ = ϑ) =for every F-stopping time ϑ. G t (θ) :=P(τ >θ F t )= θ α t (u)η(du) 4

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 (u) = 1, the law of τ is P(τ >θ)= θ for any t, α t (u)η(du) =1 For an integrable F T σ(τ) r.v. Y T (τ), one has, for t T : η(du) E(Y T (τ) F t )=E( Y T (u)α T (u)η(du) F t ) The default time τ avoids F-stopping times, i.e., P(τ = ϑ) =for every F-stopping time ϑ. G t (θ) :=P(τ >θ F t )= θ α t (u)η(du) 5

Density Hypothesis By using the density, we adopt an additive point of view to represent the conditional probability of τ G t (θ) = θ α t (u)η(du) In the default framework, the intensity point of view is often preferred, and one uses a multiplicative representation as G t (θ) =exp( θ λ t (u)η(du)) 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 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 {τ t} where Ht F is an F t -measurable random variable and H t (τ) isf t σ(τ) measurable H F t = E[HG t 1 {τ>t} F t ] G t a.s. on A t ; = if not. 7

Computation of conditional expectations Immersion property In the particular case where α t (θ) =α θ (θ), θ t one has G t =1 α t (θ)η(dθ) =1 α T (θ)η(dθ) =P(τ >t F T ) a.s. 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) fort>θ. G t := P(τ >t F t )= t α t (u)η(du) 9

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 càdlàg martingale version of this family of densities. 1

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 G t =1+Mt F α u (u)η(du) where M F is the càdlàg square-integrable martingale defined as M F t = ( αt (u) α u (u) ) η(du) =E[ α u (u)η(du) F t ] 1. Let ζ F := inf{t : G t =}. We define λ F t := α t(t) G t for any t ζ F and let λ F t = λ F t ζ for any t>ζ F.Themultiplicative decomposition of F G is given by G t = L F t e λf s η(ds) where dl F t = e λf s η(ds) dm F t. 11

Dynamic point of view and density process Proof: 1) First notice that ( α u(u)η(du),t ) is an F-adapted continuous increasing process. By the martingale property of (α t (θ),t ), for any fixed t, G t = t α t (u)η(du) =E[ t α u (u)η(du) F t ], a.s.. From the properties of the density, 1 G t = α t(u)η(du) and M F t := (α t (u) α u (u))η(du) =E[ α u (u)η(du) F t ] 1. 2) By definition of L F t and 1), we have dl F t = e λf s η(ds) dg t + e λf s η(ds) λ F t G t η(dt) =e λf s η(ds) dm F 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 the G-predictable increasing process such that (1 {τ t} Λ G t,t ) 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 λ G t =1 {τ>t} λ F t =1 {τ>t} α t (t) G t. In particular, τ is a totally inaccessible G-stopping time. For any t<ζ F and T t, wehaveα 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 the G-predictable increasing process such that (1 {τ t} Λ G t,t ) 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 λ G t =1 {τ>t} λ F t =1 {τ>t} α t (t) G t. In particular, τ is a totally inaccessible G-stopping time. For any t<ζ F and T t, wehaveα 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 the G-predictable increasing process such that (1 {τ t} Λ G t,t ) 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 λ G t =1 {τ>t} λ F t =1 {τ>t} α t (t) G t. In particular, τ is a totally inaccessible G-stopping time. For any t<ζ F and T t, wehaveα t (T )=E[λ G T F t]. 16

Dynamic point of view and density process Proof: 1) The G-martingale property of (1 {τ t} λg s η(ds),t ) is equivalent to the G-martingale property of This follows from (1 {τ>t} e λg s η(ds) =1 {τ>t} e λf s η(ds),t ) E[1 {τ>t} e λf s η(ds) E[1 {τ>t} e λf s η(ds) F s ] G s ] = 1 {τ>s} G s = 1 {τ>s} E[G t e λf s η(ds) F s ] G s =1 {τ>s} L F s G s, where the last equality follows from the F-local martingale property of L F. Moreover, the continuity of the compensator Λ G implies that τ is totally inaccessible. 17

Dynamic point of view and density process 2) By the martingale property of density, for any T t, α t (T )=E[α T (T ) F t ]. Applying 1), we obtain [ α t (T )=E α T (T ) 1 ] {τ>t} F t = E[λ G T F t ], t <ζ F, G T hence, the value of the density can be partially deduced from the intensity. 18

Dynamic point of view and density process G-martingale characterization Acàdlàg process Y G is a G-martingale if and only if there exist an F-adapted càdlàg process Y and an F t B(R + )-optional process Y t (.) such that Y G t = Y t 1 {τ>t} + Y t (τ)1 {τ t} and that (Y t G t + Y s(s)α s (s)η(ds), t ) is an F-local martingale; (Y t (θ)α t (θ),t θ) isanf-martingale on [θ, ζ θ ). 19

Dynamic point of view and density process Any F-martingale Y F is a G-semimartingale. Moreover, it admits the decomposition Yt F where M Y,G is a G-martingale and A Y,G := A t 1 {τ>t} + A t (τ)1 {τ t} is an optional process with finite variation given by = M Y,G t + A Y,G t A t = d[y F,G] s G s and A t (θ) = θ d[y F,α(θ)] s α s (θ). 2

Dynamic point of view and density process Girsanov theorem Let Zt G = z t 1 {τ>t} + z t (τ)1 {τ t} be a positive G-martingale with Z G =1andletZt F = z t G t + z t(u)α t (u)η(du) be its F projection. Let Q be the probability measure defined on G t by dq = Zt G dp. Then, α Q t (θ) =α t (θ) z t(θ), t [θ, ζ θ ); Zt F and: (i) the Q-conditional survival process is defined on [,ζ F )by G Q t = G t z t Z F t (ii) the (F, Q)-intensity process is λ F,Q t (iii) L F,Q is the (F, Q)-local martingale = λ F t z t (t) z t,η(dt)- a.s.; L F,Q t = L F t z t Z F t exp (λ F,Q s λ F s)η(ds),t [,ζ F ) 21

Dynamic point of view and density process Proof: For any t [,ζ F ), the Q-conditional probability can be calculated by G Q t = Q(τ >t F t )= E[1 {τ>t} Z G t F t ] Z F t = z t G t Z F t and, for any θ t, Q(τ θ F t )= EP [1 {τ θ} Z G t F t ] Z F t = EP [1 {τ θ} z t (τ) F t ] Z F t = θ z t(u)α t (u)η(du). Z F t The density process is then obtained by taking derivatives. Finally, we use λ F,Q t = α Q t (t)/g Q t and L F,Q t = G Q t e λf,q s η(ds). 25

Dynamic point of view and density process Given a density process, it is possible to construct a random variable τ such that P(τ >θ F t )=G t (θ) aswepresentnow: Let (Ω, A, F, P) andτ a random variable with law P(τ >t)=g (t) = η(du), independent of F t, constructed on an extended probability space, and α the given density process. Define dq Gt = Q G t dp Gt with Q G 1 t = 1 t<τ α t (u)η((du)+ 1 τ t α t (τ). G t t Then, Q is a probability which coincides with P on F t and under Q, α Q = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case) 26

Dynamic point of view and density process Given a density process, it is possible to construct a random variable τ such that P(τ >θ F t )=G t (θ) aswepresentnow: Let (Ω, A, F, P) andτ a random variable with law P(τ >θ)=g (θ) = η(du), independent of F θ, constructed on an extended probability space, and α the given density process. Define dq Gt = Q G t dp Gt with Q G 1 t = 1 t<τ α t (u)η((du)+ 1 τ t α t (τ). G t t Then, Q is a probability which coincides with P on F t and under Q, α Q = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case) 27

Dynamic point of view and density process Given a density process, it is possible to construct a random variable τ such that P(τ >θ F t )=G t (θ) aswepresentnow: Let (Ω, A, F, P) andτ a random variable with law P(τ >t)=g (t) = η(du), independent of F t, constructed on an extended probability space, and α the given density process. Define dq Gt = Z G t dp Gt with Zt G 1 = 1 t<τ α t (u)η(du)+ 1 τ t α t (τ). G t t Then, Q is a probability which coincides with P on F t and under Q, α Q = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case) 28

Dynamic point of view and density process Given a density process, it is possible to construct a random variable τ such that P(τ >θ F t )=G t (θ) aswepresentnow: Let (Ω, A, F, P) andτ a random variable with law P(τ >t)=g (t) = η(du), independent of F t, constructed on an extended probability space, and α the given density process. Define dq Gt = Z G t dp Gt with Zt G 1 = 1 t<τ α t (u)η(du)+ 1 τ t α t (τ). G t t Then, Q is a probability which coincides with P on F t and under Q, α Q = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case) 29

Dynamic point of view and density process The change of probability measure generated by the two processes z t =(L F t ) 1, z t (θ) = α θ(θ) α t (θ) provides a model where the immersion property holds true, and where the intensity processes does not change. More generally, the only changes of probability measure for which the immersion property holds with the same intensity process are generated by a process z such that (z t L F t,t ) is an uniformly integrable martingale. 3

Dynamic point of view and density process Assume that immersion property holds under P. 1) Let the Radon-Nikodým density (Zt G,t ) be a pure jump martingale with only one jump at time τ. Then, the (F, P)-martingale (Zt F,t ) is the constant martingale equal to 1. Under Q, the intensity process is λ F,Q t = λ F z t (t) t,η(dt)-a.s., and the immersion property still z t holds. 2) The only changes of probability measure compatible with immersion property have Radon-Nikodým densities that are the product of a pure jump positive martingale with only one jump at time τ, and a positive F-martingale. 31

Modelling of density process Modelling of density process 32

Modelling of density process HJM framework and short rate models To model the family of density processes we make references to the classical interest rate models. We suppose in what follows that F is a Brownian filtration. See D.C. Brody and L. Hugston (21) for related approach 33

Modelling of density process Suppose that for any θ, the bounded martingale (G t (θ),t ) satisfies dg t (θ) =Z t (θ)dw t where (Z t (θ),t ) is an F-predictable process. If the process z t (θ) such that Z t (θ) = θ z t(u)η(du) is bounded by an integrable process, then 1. dα t (θ) = z t (θ)dw t. 2. The martingale part in the Doob-Meyer decomposition of G is given by Mt F =1 Z s(s)dw s. See D.C. Brody and L. Hugston (21) for related approach 34

Modelling of density process Proof: 1) is obvious by definition. 2) is obtained by using previous results and integration by part, in fact, ( s ) Wt F =1 η(du) z s (u)dw s =1 dw s z s (u)η(du). u Observe in addition that Z t () = since G t () = 1 for any t, which implies 2) See D.C. Brody and L. Hugston (21) for related approach 35

Modelling of density process We can also consider (G t (θ),t ) in the classical HJM models where its dynamics is given in multiplicative form. We also deduce the dynamics of the forward rate, in both forward and backward forms. The density can then be calculated as α t (θ) =λ t (θ)g t (θ). See D.C. Brody and L. Hugston (21) for related approach 36

Modelling of density process For any t, θ, let Ψ(t, θ) = Z t(θ) G t (θ) and define ψ(t, θ) by Ψ(t, θ) = θ ψ(t, u)η(du). Recall the forward rate λ t(θ) ofτ given by λ t (θ) = θ ln G t (θ). Then 1. G t (θ) =G (θ)exp ( Ψ(s, θ)dw s 1 2 Ψ(s, θ) 2 ds) ; 2. λ t (θ) =λ (θ) ψ(s, θ)dw s + ψ(s, θ)ψ(s, θ) ds; ( 3. G t =exp λf sη(ds)+ Ψ(s, s)dw s 1 t 2 ds) Ψ(s, s) 2 ; dg t (θ) =Z t (θ)dw t 37

Modelling of density process For any t, θ, let Ψ(t, θ) = Z t(θ) G t (θ) and define ψ(t, θ) by Ψ(t, θ) = θ ψ(t, u)η(du). Recall the forward rate λ t(θ) ofτ given by λ t (θ) = θ ln G t (θ). Then 1. G t (θ) =G (θ)exp ( Ψ(s, θ)dw s 1 2 Ψ(s, θ) 2 ds) ; 2. λ t (θ) =λ (θ) ψ(s, θ)dw s + ψ(s, θ)ψ(s, θ) ds; ( 3. G t =exp λf sη(ds)+ Ψ(s, s)dw ) s 1 t 2 Ψ(s, s) 2 d s ; dg t (θ) =Z t (θ)dw t 38

Modelling of density process Proof: The process G t (θ) is the solution of the equation dg t (θ) G t (θ) =Ψ(t, θ)dw t, t, θ. Hence 1), from which we deduce immediately 2) by differentiation w.r.t. θ. Now, note that by 1), ln G t = λ (s)η(ds)+ Ψ(s, t)dw s 1 2 Ψ(s, t) 2 ds 41

Modelling of density process Woreover, we have by 2) that λ s (s)η(ds) = = + λ (s)η(ds) η(ds) s λ (s)η(ds) η(ds) s ψ(u, s)ψ(u, s) du + 1 2 ( Ψ(u, t) 2 ψ(u, s)dw u (Ψ(u, t) Ψ(u, u))dw u Ψ(u, u) 2 )du. Observe in addition that by definition of the forward rate λ t (θ) and then, we have λ s (s) =λ F s, which implies 3). Finally, 4) is a direct result from 2). As a conditional survival probability, G t (θ) is decreasing on θ, whichis equivalent to that λ t (θ) is positive. This condition is similar as for the zero coupon bond prices. 42

Modelling of density process In the second approach, we can borrow short rate models for the F-intensity λ F of τ, and then obtain the dynamics of conditional probability G t (T )fort t. The monotonicity condition of G t (T )ont is equivalent to the positivity condition on λ F. 43

Modelling of density process Example: The non-negativity of λ is satisfied if for any θ, the process ψ(θ)ψ(θ) is non negative, or if ψ(θ) is non negative; for any θ, the local martingale ζ t (θ) =λ (θ) ψ s(θ)dw s is a Doléans-Dade exponential of some martingale, i.e., is solution of ζ t (θ) =λ (θ)+ ζ s (θ)b s (θ)dw s, that is, if ψ s(θ)dw s = b s(θ)ζ s (θ)dw s. Here the initial condition is a positive constant λ (θ). λ t (θ) =λ (θ) ψ(s, θ)dw s + ψ(s, θ)ψ(s, θ) ds 44

Modelling of density process Hence, we set ( ψ t (θ) = b t (θ)ζ t (θ) = b t (θ)λ (θ)exp b s (θ)dw s 1 2 ) b 2 s(θ)ds where λ is a positive intensity function and b(θ) is a non-positive F-adapted process. Then, the family ( α t (θ) =λ t (θ)exp where θ ) λ t (v) dv, λ t (θ) =λ (θ) satisfies the required assumptions. ψ s (θ) dw s + ψ s (θ)ψ s (θ) ds λ t (θ) =λ (θ) ψ(s, θ)dw s + ψ(s, θ)ψ(s, θ) ds 45

Modelling of density process Other examples 46

Modelling of density process Example: Cox-like construction. Here λ is a non-negative F-adapted process, Λ t = λ sds Θ is a given r.v. independent of F with unit exponential law V is a F -measurable non-negative random variable τ =inf{t :Λ t ΘV }. For any θ and t, G t (θ) =P(τ >θ F t )=P(Λ θ < ΘV F t )=P ( exp Λ θ V e Θ ) F t. Let us denote exp( Λ t /V )=1 ψ sds, with ψ s =(λ s /V )exp s (λ u /V ) du, and define γ t (s) =E (ψ s F t ). Then, α t (s) =γ t (s)/γ (s). 47

Modelling of density process Backward construction of the density Let ϕ(,α) be a family of densities on R +, depending of some parameter and X F a random variable. Then and we can chose ϕ(u, X)du =1 α t (u) =E(α (u) F t )=E(ϕ(u, X) F t ) 48

Modelling of density process Other example Let β(θ) be a family of non negative F-martingales with expectation equal to 1 and set where L β t = β t (θ)η(dθ). dβ t (θ) =β t (θ)σ t (θ)dw t α t (θ) = β t(θ) L β t Then, α is a family of densities under Q with dq = L β t dp and dα t (θ) =α t (θ)(σ t (θ) σ t )(dw t σ t dt) where σ t = 1 β L β t (θ)σ t (θ)η(dθ). Note the similarity with normalized t and non-normalized filter. See also Brody and Hughston. 49

Multidefaults Multidefaults Computation of prices in case of multidefaults is now easy In a first step, one orders the default Computation before the first default are done in the reference filtration Between the first and the second default, one takes as new reference filtration the filtration generated by the first default and the previous reference filtration, as explained previously for the after default computations and we continue till the end 5

Multidefaults Let τ = τ 1 τ 2 andσ = τ 1 τ 2 and assume that G t (θ 1,θ 2 ):=P(τ >θ 1,σ >θ 2 F t )= α t (u, v)dη(u, v). θ 1 θ 2 The F-density of τ is given by α τ F t (θ 1 )= θ 1 α t (θ 1,v)η 2 (dv), a.s.. For any θ 2,t, the G (1) -density of σ is given by α α σ G(1) t t (u, θ 2 )η 1 (du) t (θ 2 )=1 {τ>t} G τ F t (t) +1 {τ t} α t (τ,θ 2 ) α τ F t (τ), a.s.. 51

Multidefaults Begin at the beginning, and go on till you come to the end. Then,... Lewis Carroll, Alice s Adventures in Wonderland. 52

Multidefaults Begin at the beginning, and go on till you come to the end. Then, stop. Lewis Carroll, Alice s Adventures in Wonderland. 53

Multidefaults THANK YOU FOR YOUR ATTENTION 54