Modeling Information for Credit Risk - uni-jena.de · M. L. Bedini (ITN - UBO, Brest) Modeling...

Modeling Information for Credit Risk

M. L. Bedini

ITN - UBO, Brest

Jena, June 2010

M. L. Bedini (ITN - UBO, Brest) Modeling Information for Credit Risk Jena, June 2010 1 / 22

Introduction and Motivation

(Ω, F , P) complete probability space. B = (Bt , t ≥ 0) BM, F = FB .

τ ∼ Eλ r.v. independent of B,(Ht , Iτ≤t, t ≥ 0

), H = FH .

CEV Model with Jump to Default (Campi, Polbennikov, Sbuelz ’09)

dSt = Stµ (St) dt + σSρt dBt − St−dHt , t ≥ 0, S0 > 0, ρ < 1

dSt = Stµ(St

)dt + σSρ

t dBt , t ≥ 0, S0 > 0, ρ < 1

Constant Elasticity of Variance (CEV)

Default time for S : η , inf

t > 0 : St = 0

, P η < +∞ = 1.

Default time for the company: θ , τ ∧ η. P θ < +∞ = 1.

St = St for t ∈ [0, θ]. St = 0, ∀t ≥ θ.

θ is predictable on η ≤ τ and totally inaccessible on η > τ w.r.t.F ∨H.



Objective

Our approach aims to give a qualitative description of the information onτ before the default, thus making τ “a little bit less inaccessible”.

Information-based Approach (Brody, Hughston, Macrina ’07)

Let T , σ > 0, HT ∼ B(1, p),(βT

t , 0 ≤ t ≤ T)

Brownian bridge.

ξt = σtHT + βTt

The market filtration Fξ is generated by the information process.

In our approach the information will be carried by(βt , βτ

t , t ≥ 0)

a

Brownian bridge between 0 and 0 on the stochastic interval [0, τ ].



In the CEV Model with Jump to Default and in the Hazard ProcessApproach to Credit-Risk we deal with two filtrations:

F: the reference filtration that models the common knowledge offinancial agents on the market (ex: Brownian filtration). τ is not anF-stopping time.

H: the information on the default event generated by the default

indicator process(Ht , Iτ≤t, t ≥ 0

).

However, before the default we have some more information on τ (thereare periods in which we are“quite confident” the default will not occur).Thus, in our market model we will deal with:

F: the reference filtration.

Fβ : the information on the default event generated by the informationprocess (βt , t ≥ 0).


Agenda

1 Introduction and Motivation

2 Properties of the Information process

3 Enlargement of Filtration

4 Application to a CEV Model with Jump to Default

5 Conclusion and Further Development


Properties

(Ω,F ,P) complete probability space, NP the collection of the P-null sets.W = Wtt≥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 = βtt≥0 is called Information process :

βt , Wt −t

τ ∨ tWτ∨t (1)


Properties

oF

β

=

(oF

β

t , σ βs , 0 ≤ s ≤ t)

t≥0

Fβ =

(Fβ

t ,⋂ε>0

σ βs , 0 ≤ s ≤ t + ε ∨ NP

)t≥0

Proposition

τ is an Fβ-stopping time but, in general, it is not anoF

β

-stopping time.

For all t > 0, βt = 0 = τ ≤ t, P-a.s.

β is an Fβ-Markov process.


Properties

Theorem

Let t > 0, g : R+ → R a Borel function such that E [|g (τ) |] < +∞.Then, P-almost surely on τ > t

E[g (τ) Iτ>t|F

βt

]=

´ +∞t g (r) ϕt (r , βt) dF (r)´ +∞

t ϕt (r , βt) dF (r)Iτ>t (2)

where ϕt (r , x) , r > t > 0, x ∈ R denotes the density of

βrt ∼ N

(0,

t(r − t)

r

)


Properties

Define the Brownian bridge b = bs0≤s≤1 between 0 and 0 on the timeinterval [0, 1] by

bs ,1√τβsτ

The local time L(a) = Ls(a)0≤s≤1 at a ∈ R of the process b is welldefined. Then

(βt , 0 ≤ t ≤ τ) =(√

τb tτ, 0 ≤ t ≤ τ

)The local time lx = lxt t≥0 of the information process β at level x is thendefined as

lxt ,√

τL tτ

(x√τ

)If x = 0 we will write lt instead of l0t .


Properties

Theorem

Suppose F (t) admits a continuous density with respect to the Lebesguemeasure: dF (t) = f (t)dt. Then τ is a totally inaccessible stopping timewith respect to Fβ and the compensator K = Ktt≥0 of

(Iτ≤t, t ≥ 0

)is

given by

Kt =

τ∧tˆ

0

f (r)´ +∞r ϕr (v , 0) f (v)dv

dlr (3)

where lt is the local time at 0 of the process β at time t.


Enlargement of Filtration

Let F = (Ft)t≥0 be a filtration on (Ω, F , P) satisfying the usual

condition. Ft(dr) , P τ ∈ dr |Ft

Assumption

F+∞ ∨ σ τ is independent of W .

Definition

The filtration G =(Gt , Ft ∨ Fβ

t

)t≥0

is called enlarged filtration.



Lemma

The process β satisfies the following

P βt+h ∈ Γ|Gt = P βt+h ∈ Γ|Ft ∨ σ βt , P− a.s.

for all t, h ≥ 0 and Γ ∈ B (R).

Lemma

Denoting by St,v ,´ +∞t ϕt (r , βt) Fv (dr)

P

τ ∈ du|Fv ∨ Fβt

I τ > t = ϕt (u, βt) Fv (du) S−1

t,v I τ > t , P−a.s.



Lemma

Let T > 0 be a constant and Y be an FT -measurable r.v. Let g be abounded Borel function s.t. E [|g (τ,Y ) |] < +∞. Then

E [g (τ,Y ) |Gt ] Iτ>t = E

[´ +∞t g(r ,Y )ϕt(r , βt)FT (dr)´ +∞

t ϕt(r , βt)FT (dr)|Gt

]Iτ>t,P−a.s.


Application: Pricing a Credit Default Swap (1/2)

A Credit Default Swap (CDS) is a financial contract between a buyer anda seller:

The buyer wants to insure the risk of default. Protection leg :

It<τ≤Tδ (τ)

The seller is paid by the buyer to provide such insurance. Fee leg :

Iτ>tk [(τ ∧ T )− t]

The price St (k, δ, T ) of the CDS is equal to:

St (k, δ, T ) = E[It<τ≤Tδ (τ) |Ft

]− E

[Iτ>tk [(τ ∧ T )− t] |Ft

]where F =

(Ft

)t≥0

is the (yet unspecified) market filtration.


Application: Pricing a Credit Default Swap (2/2)

Lemma

If the market filtration is G, for t ∈ [s,T ] we have

St (k, δ, T ) = Iτ>t

− T

t

δ(r)dΨt(r)− k

T

t

Ψt(r)dr

Where Ψt(r) , P τ > r |Gt.

Lemma

If the market filtration is H =(Ht , σ t ∧ τ

)t≥0

, for t ∈ [s,T ] we have

St (k, δ, T ) = Iτ>t

− T

t

δ(r)dG (r)− k

T

t

G (r)dr

Where G (r) , P τ > rM. L. Bedini (ITN - UBO, Brest) Modeling Information for Credit Risk Jena, June 2010 15 / 22

Application to a CEV Model with Jump to Default

In an arbitrage-free incomplete market we observe the price process of theshare of a company:

dSt = Stµ (St) dt + σSρt dBt − St−dHt , t ≥ 0, S0 > 0, ρ < 1

dSt = Stµ(St

)dt + σSρ

t dBt

τ ∈ L0 (Ω, F , P), Ht , Iτ≤t.

Default time for S : η , inf

t > 0 : St = 0

.

Default time: θ , τ ∧ η.

This model is used to describe the positive link between default and equityvolatility.



Note that St = St for t ∈ [0, θ]. Furthermore we have thatP θ < +∞ = 1 and St = 0, ∀t ≥ θ.Let (Wt , t ≥ 0) be a BM.

Assumption

τ , B and W are mutually independent.

F = FB is the filtration generated by the BM (Bt , t ≥ 0).

Fβ is the filtration generated by the information processβt = Wt − t

τ∨t Wτ∨t .

G = F ∨ Fβ is the market filtration.

θ is G-predictable on η ≤ τ and G-totally inaccessible on η > τ.



In the following we will assume for convenience that:

τ is exponentially distributed with parameter λ > 0

µ(x) = 12ρσ2x2ρ−1, x > 0

So that, up to the predictable stopping time η > 0

St =(S1−ρ

0 + (1− ρ) σBt

)1/(1−ρ)

Then we have that

P η ∈ du|Ft = F ηt (du) , Iη≤tδη(du) + Iη>th (u − t,Bt) du

where

h (u, x) =1

u3/2

(x +

S1−ρ0

(1− ρ)σ

)ϕ

(1√u

(x +

S1−ρ0

(1− ρ)σ

))Iu>0



The G-compensator(KG

t , t ≥ 0)

of Iθ≤t is given by:

KGt =

tˆ

0

((E[K+∞|Fβ

r−

]− Kr

)I[η,+∞)(dr) + Iη>rdKr

), t ≥ 0



If the default protection is equal to δ > 0, the market price St(k, δ, T ) of aCDS is equal to

St(k, δ, T ) = Iθ>tδ [Φt(T ) (1− F ηt (T )) + F η

t (T )]+

+Iθ>tk

t −T

t

uˆ

t

rdΦt(r) + u (1− Φt(u))

F ηt (du)

+

−Iθ>tk

T

t

rdΦt(r) + T (1− Φt(T ))

(1− F ηt (T ))

(where Φt(r) = P

τ ≤ r |Fβt

).


Conclusion and Further Development

Modeling the information regarding a default time τ with a Brownianbridge on the stochastic interval [0, τ ], allows to reconcile theInformation-based approach to Credit-Risk with the reduced-formmodels.

Explicit formulas can be obtained and they appear to be an intuitivegeneralization of some simple models already present in literature. Wecan obtain pricing formulas (Credit Default Swap, Zero-CouponBond).

The information process βt can substitute the default indicatorprocess Iτ≤t in many models providing interesting insight on therole of information on financial markets.

Under opportune condition we can compute explicitly thecompensator of the default indicator process in the enlarged filtration.


Bielecki, Jeanblanc, Rutkowski. Hedging of Basket of CreditDerivatives in Credit Default Swap Market. Journal of Credit Risk,3:91-132, 2007.

D. C. Brody, L. P. Hughston & A. Macrina. Beyond hazard rates: anew framework for credit-risk modeling. Advances in MathematicalFinance, Festschrift volume in honor of Dilip Madan. Birkhauser,Basel, 2007.

L. Campi, S. Polbennikov, A. Sbuelz. Systematic Equity-Based CreditRisk: A CEV Model with Jump to Default. Journal of EconomicDynamics and Control, 33:93-108, 2009.

N. El Karoui, M. Jeanblanc, Y. Jiao. What happen after the default:the conditional density approach. Preprint, 2009.


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