Information and Credit Risk

Transcription

1 Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

2 Motivation (1/3) Source: M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

3 Motivation (2/3) Hazard-process approach for Credit-risk. information G can be decomposed: ( ) Ω, F, G = (G t) t 0, P, the total G = F H Market risk: given by the natural movements of interest rates, etc. It can be hedged with F-adapted financial instruments. Default risk: risk that is directly associated with the default. It can be hedged with H-adapted financial instruments. What is H = (H t) t 0? H = ( H t := I {τ t}, t 0 ) is the default indicator process. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

4 Motivation (3/3) Objective Our approach aims to give a qualitative description of the information on τ before the default, thus making τ a little bit less inaccessible. In our approach the information will be carried by β = (β t, t 0) a Brownian bridge between 0 and 0 on the stochastic interval [0, τ]. Thus, in our market model we will deal with: F: the reference filtration. F β : the information on the default event generated by the information process β. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

5 Agenda 1 The Information Process β Introduction and First Properties Conditional Expectation Application: Pricing a Credit Default Swap Classification of τ in F β 2 The Enlarged Filtration G := F F β First Properties Conditional Expectation in G Classification of τ in G: two examples 3 Conclusion, Acknowledgments and References M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

6 Introduction (Ω, F, P) complete probability space, N P the collection of the P-null sets. W = (W t, t 0) is a standard BM. τ : Ω (0, + ) random variable. F (t) := P {τ t}. Assumption τ is independent of W. Definition The process β = (β t, t 0) is called Information process : β t := W t t τ t W τ t (1) M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

7 First Properties (1/2) Let F 0 = ( Ft 0 := σ {β s, 0 s t} ) be the natural filtration of the t 0 process β and let: ) F β = (F t β := Ft+ 0 N P t 0 Proposition τ is an ( Ft+ 0 ) -stopping time, P-a.s. t 0 ( For all t > ) 0, {β t = 0} = {τ t}, P-a.s. τ is an F 0 t N P -stopping time t 0 β is an ( Ft+ 0 ) t 0 -Markov process F t β = Ft 0 N P M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

8 First Properties (2/2) Lemma The process β is an F β -semi-martingale and the process b = (b t, t 0) given by ˆt τ [ ] βs b t := β t + E τ s F s β ds 0 is an F β -Brownian motion stopped at τ. ( Sketch of the proof. The process B = B t := β t + ) t τ β s 0 τ s ds, t 0 is an F β σ {τ}-brownian motion stopped at τ. Then apply standard results from filtering theory. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

9 Conditional Expectation Theorem Let t > 0, g : R + R a Borel function such that E [ g (τ) ] < +. Then, P-almost surely [ ] E g (τ) Ft β = g (τ) I {τ t} + + t g (r) ϕ t (r, β t ) F (dr) + I {τ>t} (2) t ϕ t (r, β t ) F (dr) where ϕ t (r, x), r > t > 0, x R denotes the density of ( ) βt r t(r t) N 0, r M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

10 Application: Pricing a Credit Default Swap (1/2) A Credit Default Swap (CDS) is a financial contract between a buyer and a seller: The buyer wants to insure the risk of default. Protection leg: I {t<τ T } δ (τ) The seller is paid by the buyer to provide such insurance. Fee leg: I {τ>t} k [(τ T ) t] The price S t (k, δ, T ) of the CDS is equal to: S t (k, δ, T ) = E [I {t<τ T } δ (τ) F ] t E [I {τ>t} k [(τ T ) t] F ] t where F ( ) = Ft t 0 is the (yet unspecified) market filtration. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

11 Application: Pricing a Credit Default Swap (2/2) Lemma If the market filtration is F β, for t [u, T ] we have ˆT ˆT S t (k, δ, T ) = I {τ>t} δ(r)dψ t (r) k Ψ t (r)dr { } Where Ψ t (r) := P τ > r Ft β. t t Lemma If the market filtration is H, for t [u, T ] we have ˆT ˆT S t (k, δ, T ) = I {τ>t} δ(r)dg(r) k G(r)dr t t Where G(r) := P {τ > r} M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

12 Classification of τ in F β Theorem Suppose F (t) admits a continuous density with respect to the Lebesgue measure: df (t) = f (t)dt. Then τ is a totally inaccessible stopping time with respect to F β and the compensator K = (K t, t 0) of H = (H t, t 0) is given by K t = ˆτ t 0 f (r) + r ϕ r (v, 0) f (v)dv dl r (3) where l = (l t, t 0) is the local time at 0 of the process β at time t. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

13 First Properties (1/2) Let F = (F t ) t 0 be a filtration on (Ω, F, P) satisfying the usual condition. F t (dr) := P {τ dr F t } Assumption F + σ {τ} is independent of W. Definition The filtration G = ( ) G t := F t Ft β is called enlarged filtration. t 0. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

14 First Properties (2/2) Lemma The process β satisfies the following P {β t+h Γ G t } = P {β t+h Γ F t σ {β t }}, P a.s. for all t, h 0 and Γ B (R). Lemma If E [ τ] < + the process β is a G-semi-martingale and the process b G = ( b G t, t 0 ) given by ˆ bt G := β t + t τ is a G-Brownian motion stopped at τ. 0 E [ ] βs τ s G s ds M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

15 Conditional Expectation in G Theorem Let t > 0, g : R + R a Borel function such that E [ g (τ) ] < +. Then, P-almost surely E [g (τ) G t ] = g (τ) I {τ t} + + t g (r) ϕ t (r, β t ) F t (dr) + I {τ>t} (4) t ϕ t (r, β t ) F t (dr) where ϕ t (r, x), r > t > 0, x R denotes the density of ( ) βt r t(r t) N 0, r M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

16 Classification of τ in G Azéma super-martingale: Z = (Z t := P {τ > t F t }, t 0). If the random time τ avoids F-stopping times (i.e. P {τ = ρ} = 0, ρ F-stopping time), the F-dual predictable projection A = (A t, t 0) of the single-jump process H = (H t, t 0) is continuous and the canonical decomposition of the semi-martingale Z is given by Z t = M t A t where M = (M t = E [A F t ], t 0). Moreover the process N = (N t, t 0) given by N t := H t ˆt τ 0 1 Z s da s is a martingale with respect to G = (G t = F t H t ) t 0. In particular τ is a G -totally inaccessible stopping time. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

17 First Example (1/2) Let τ and η two independent exponentially distributed random variable with parameters λ and µ respectively. Let θ := τ + η and define F 1 = ( Ft 1 ) to be the minimal filtration making θ a stopping t 0 time (τ is not an F-stopping time). G 1 := F 1 F β. Lemma ( ) The compensator K (1) = K (1) t, t 0 of the G 1 -sub-martingale H 1 = ( H t := I {θ t}, t 0 ) is continuous. Thus τ is a totally inaccessible stopping time w.r.t. G 1. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

18 First Example (2/2) Sketch of the proof. The compensator ( is obtained ) through the direct computation of the process K (1) = K (1) t, t 0. Remark The conditional density of τ given F 1, P {τ dr F t } = f t (r) dr, on the set {θ > t}, is: f t (r) = (λ µ) [ λe λr I {r t} + λe µt e r(µ λ) I {0<r<t} ] λe µt µe λt I {θ>t}. Note that the condition f t (r) = f r (r), r t is not satisfied (no immersion in the standard progressive enlargement). M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

19 Second Example (1/9) Let B = (B t, t 0) be a standard Brownian motion independent of W. Let F 2 = ( Ft 2 ) be the natural and completed filtration of B. t 0 Define G 2 := F 2 F β. τ := sup {B t = 0} 0 t 1 Lemma τ is P-a.s. the first hitting time of 0 of the continuous 2-dimensional process X = ( X t := [B t, β t ], t [0, 1] ). Thus τ is a predictable stopping time. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

20 Second Example (2/9) Proof. Let X = (X t, t [0, 1]) be the two-dimensional process given by: [ ] βt X t :=, 0 t 1 B t and let σ be the first hitting time of 0 of X : σ := inf {X t = 0} t [0,1] Since σ is the first entry time in the closed set {0} of the continuous process X we have that σ is a predictable stopping time. Our aim is to show that τ = σ, P a. s. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

21 Second Example (3/9) By definition of τ we have that and so Thus, it must be shown that and, since τ = sup {X t = 0} t [0,1] σ τ, P a. s. P {σ < τ} = 0 P {σ < τ} = E [P {σ < τ τ}] ˆ = P {σ < τ τ = r} F (dr) [0,1] it suffices to show that, conditionally on τ = r, P {σ < τ τ = r} = 0 M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

22 Second example (4/9) Knowing τ = r we have that 1 (B t, 0 t r) is a Brownian bridge of length r between 0 and 0 (see Mansuy and Yor [6]). 2 (β t, 0 t r) is a Brownian bridge of length r between 0 and 0 (from the definition of β). 3 β is independent of τ, B (because it depends only on W which is, by hypothesis, independent of τ, B). This means that, conditionally on τ = r, the process (X t, 0 t r) is a 2-dimensional Brownian bridge from 0 to 0. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

23 Second example (5/9) Let now X r = (Xt r, 0 t r) be a 2-dimensional Brownian bridge from 0 to 0 on the time interval [0, r]. Let σ r be the first hitting time of 0 of the process X r : σ r := inf {X t r = 0} t [0,r] Knowing τ = r, the two processes (X t, 0 t r) and (X r t, 0 t r) have the same finite dimensional distributions and P {σ < τ τ = r} = P {σ r < r} M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

24 Second example (6/9) If we prove that then we have that σ r = r, P a. s. (5) P {σ < τ} = E [P {σ < τ τ}] = ˆ P {σ < τ τ = r} F (dr) = ˆ [0,1] [0,1] P {σ r < r} F (dr) = 0 which implies the thesis. Thus it remains to prove (5). M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

25 Second example (7/9) ) On (Ω, F, F r = (Ft r ) t 0, P r let X r = (Xt r, 0 t r) be a 2-dimensional Brownian bridge from 0 to 0 on the time interval [0, r]. Then, there exists a 2-dimensional (F r, P r )-Brownian motion b = (b t, t 0) such that dx r t = db t X r t r t dt and, for every T (0, r) we can define the process Z r = (Zt r, 0 t T ) by ( Zt r := exp ξt r,t 1 ξ r,t ) 2 t where ξ r,t t := t T ˆ 0 X r s r s db s, 0 t T M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

26 Second example (8/9) By Girsanov theorem, the process Z r is a (F r, P r )-martingale, and the process X r = (X r t, 0 t T ) is a (F r, Q r )-Brownian motion, where Q r F r t := Z r t P r F r t, 0 t T Note that Q r is equivalent to P r only for t [0, T ]. However, for a 2-dimensional Brownian motion, the one-point sets are polar, i.e. Q r {T x T } = 0, x R 2 (see, for example Jeanblanc, Yor and Chesney [3]) where T x := inf {t > 0 : X t = x}. Thus Q r {σ r T } = 0, T (0, r) M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

27 Second example (9/9) Since Q r is equivalent to P r, from the previous relation it follows that P r {σ r T } = 0, T (0, r) P r {σ r > T } = 1, T (0, r) On the other hand, by definition of Brownian bridge P r {X r r = 0} = 1 and so Now the proof is concluded. P r {σ r = r} = 1 M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

28 Conclusion Explicit formulas can be obtained and they appear to be an intuitive generalization of some simple models already present in literature. We can obtain pricing formulas (Credit Default Swap, Zero-Coupon Bond). The information process β t can substitute the default indicator process H t = I {τ t} in many models providing interesting insight on the role of information on financial markets. Under appropriate condition we can compute explicitly the compensator of the default indicator process in the enlarged filtration. We have the first example (at the best of our knowledge) of random time which is predictable in the enlarged filtration G = F F β despite its weak properties in F and F β. M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

29 Acknowledgments Work supported in part by the European Community s FP 7 Programme under contract PITN-GA , Marie Curie ITN Controlled Systems and jointly supervised by prof. Rainer Buckdahn (UBO) and prof. Hans-Juergen Engelbert (FSU). M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29

30 M. L. B. Information on a default time: a new enlargement of filtrations. Working paper, Bielecki, Jeanblanc, Rutkowski. Hedging of Basket of Credit Derivatives in Credit Default Swap Market. Journal of Credit Risk, 3:91-132, M. Jeanblanc, M. Yor, M. Chesney. Mathematical Method for Financial Markets. Springer, N. El Karoui, M. Jeanblanc, Y. Jiao. What happen after the default: the conditional density approach. Stochastic Processes and their Applications, 120, pp R. Mansuy, M. Yor. Random times and Enlargement of Filtrations in Brownian Settings. Lecture Notes in Mathematics, Springer, M. L. Bedini (Université de Bretagne Occidentale, Brest Information - Friedrich and Schiller CreditUniversität, Risk Jena) Jena, March / 29