Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets

Подождите немного. Документ загружается.

420 7 Default Risk: An Enlargement of Filtration Approach

Remarks 7.3.2.2 (a) For t<s, we obtain P(τ>s|F

)=E(exp (−Λ

) |F

(b) If the process λ is not positive, the process Λ is not increasing and we

obtain, for s<t,

P(τ>s|F

)=P



sup

u≤s

<Θ



=exp



−sup

u≤s



7.3.3 Enlargements of Filtrations

Write as before D

= 1

{τ≤t}

and D

= σ(D

; s ≤ t). Introduce the ﬁltration

= F

∨D

, that is, the enlarged ﬁltration generated by the underlying

ﬁltration F and the process D. (We denote by F the original Filtration and

by G the enlarGed one.) We shall frequently write G = F ∨D. The ﬁltration

G is the smallest one which contains F and such that τ is a stopping time.

It is easy to describe the events which belong to the σ-algebra G

on the

set {τ>t}. Indeed, if G

∈G

,thenG

∩{τ>t} =



∩{τ>t} for some

event



∈F

Therefore, from the monotone class theorem, any G

-measurable random

variable Y

satisﬁes 1

{τ>t}

= 1

{τ>t}

, where y

is an F

-measurable random

variable.

7.3.4 Conditional Expectations with Respect to G

Lemma 7.3.4.1 (Key Lemma) Let Y be an integrable r.v.. Then,

{τ>t}

E(Y |G

)=1

{τ>t}

E(Y 1

{τ>t}

)

E(1

{τ>t}

)

= 1

{τ>t}

E(Y 1

{τ>t}

Proof: From the remarks on G

-measurability, if Y

= E(Y |G

), there exists

an F

-measurable r.v. y

such that

{τ>t}

E(Y |G

)=1

{τ>t}

Taking conditional expectation w.r.t. F

of both members of the above

equality, we obtain y

E(Y 1

{τ>t}

)

E(1

{τ>t}

)

. The last equality in the proposition

follows from E(1

{τ>t}

)=e

−Λ

. 

Corollary 7.3.4.2 If X is an integrable F

-measurable random variable,

then, for t<T

E(X1

{T<τ}

)=1

{τ>t}

E(Xe

−Λ

) . (7.3.2)

7.3 Default Times with a Given Stochastic Intensity 421

Proof: The conditional expectation E(X1

{τ>T}

)isequalto0ontheG

measurable set {τ<t},whereasforanyX ∈ L

E(X1

{τ>T}

)=E(X1

{τ>T}

)=E(Xe

−Λ

The result follows from Lemma 7.3.4.1. 

We now compute the expectation of an F-predictable process evaluated at

time τ , and we give the F-Doob-Meyer decomposition of the increasing process

D. We denote by F

:=P(τ ≤ t|F

) the conditional distribution function.

Lemma 7.3.4.3 (i) If h is an F-predictable (bounded) process, then

E(h

)=E





∞

−Λ





= E





∞





and

E(h

)=e





∞





{τ>t}

+ h

{τ≤t}

. (7.3.3)

In particular

E(h

)=E





∞

−Λ



= E





∞



(ii) The process (D

−



t∧τ

ds, t ≥ 0) is a G-martingale.

Proof: Let B

∈F

and h the elementary F-predictable process deﬁned as

= 1

{t>v}

. Then,

E(h

)=E



{τ>v}



= E



E(1

{τ>v}

∞

)





= E



P(v<τ|F

∞

)





= E



−Λ



It follows that

E(h

)=E





∞

−Λ





= E





∞

−Λ





and (i) is derived from the monotone class theorem. Equality 7.3.3 follows

from the key Lemma 7.3.4.1.

The martingale property (ii) follows from the integration by parts formula.

Indeed, let s<t. Then, on the one hand from the key Lemma

E(D

− D

)=P(s<τ≤ t|G

)=1

{s<τ}

P(s<τ≤ t|F

)

P(s<τ|F

)

= 1

{s<τ}



1 − e



−Λ



422 7 Default Risk: An Enlargement of Filtration Approach

On the other hand, from part (i), for s<t,





t∧τ

s∧τ





= E (Λ

t∧τ

− Λ

s∧τ

)=E(ψ

)

= 1

{s<τ}





∞

−Λ





where ψ

= Λ

t∧u

− Λ

s∧u

= 1

{s<u}

(Λ

t∧u

− Λ

). Consequently,



∞

−Λ

du =



(Λ

− Λ

)λ

−Λ

du +(Λ

− Λ

)



∞

−Λ



−Λ

du − Λ



∞

−Λ

du + Λ

−Λ



−Λ

du − Λ

−Λ

+ Λ

−Λ

= e

−Λ

− e

−Λ

It follows that

E(D

− D

)=E





t∧τ

s∧τ





hence the martingale property of the process D

−



t∧τ

du . 

Comment 7.3.4.4 Property (ii) shows that λ

{t<τ}

is the G-intensity of τ

(see  Section 7.7).

Remark 7.3.4.5 As we said before in Subsection 7.2.4, if no defaultable as-

sets are traded, the defaultable market is incomplete. A change of probability

can aﬀect the law of the r.v. Θ, hence, can aﬀect the value of the intensity.

We shall study this problem in a more general hazard process framework in

the following section.

7.3.5 Conditional Expectations of F

∞

-Measurable Random

Variables

Lemma 7.3.5.1 Let X be an integrable F

∞

-measurable r.v.. Then

E(X|G

)=E(X|F

) . (7.3.4)

Proof: Let X be an integrable F

∞

-measurable r.v.. The σ-algebra G

generated by r.v’s of the form B

h(τ ∧ t)whereB

∈F

and h = 1

[0,a]

Therefore, to prove that E(X|G

)=E(X|F

), it suﬃces to check that

E(B

h(τ ∧ t)X)=E(B

h(τ ∧ t)E(X|F

))

7.3 Default Times with a Given Stochastic Intensity 423

for any B

∈F

and any h = 1

[0,a]

.Fort ≤ a, the equality is obvious. For

t>a, we have from the equality (7.3.1)



{τ≤a}

E(X|F

)



= E



{τ≤a}

E(B

X|F

)



= E



XE(1

{τ≤a}

)



= E



XE(1

{τ≤a}

∞

)



= E(B

{τ≤a}

) .

The result follows. 

Remark 7.3.5.2 The equality (7.3.4)impliesthateveryF-square integrable

martingale is a G-martingale. However, equality (7.3.4) does not apply to any

∞

-measurable r.v.; in particular, since τ is a G-stopping time and not an

F-stopping time, P(τ ≤ t|G

)=1

{τ≤t}

is not equal to P(τ ≤ t|F

7.3.6 Correlated Defaults: Copula Approach

An approach to modelling dependent credit risks is the use of copulas.

If X

,i =1,...,n are random variables with cumulative distribution

function F

,andiftheF

’s are strictly increasing, then, setting U

= F

)

P(X

≤ x

, ∀i)=P(F

) ≤ F

), ∀i)=P(U

≤ F

), ∀i)

hence, the joint law of the X

can be characterized in terms of the joint law

of the vector U. Note that the r.v. U

has a uniform law, and that, a priori

the U

’s are not independent.

Basic Deﬁnitions

Deﬁnition 7.3.6.1 A mapping C deﬁned on [0, 1]

is a copula if it satisﬁes:

(i) C(u

,...,u

) is increasing with respect to each component u

(ii) C(1,...,u

,...,1) = u

, for every i, for every u

∈ [0, 1],

(iii) for every a, b ∈ [0, 1]

with a ≤ b (i.e., a

≤ b

, ∀i)



...



(−1)

+···+i

C(u

1,i

,...,u

n,i

) ≥ 0 ,

where u

j,1

= a

j,2

= b

A copula is the cumulative distribution function of an n-uple (U

, ··· ,U

)

where the U

are r.v’s with uniform law on the interval [0, 1]. The survival

copula is



C(u

,...,u

)=P(U

,...,U

Property (iii) indicates that the probability that the n-tuple belongs to

the rectangle



i=1

] is non-negative.

Theorem 7.3.6.2 (Sklar’s Theorem.) Let F be an n-dimensional cumu-

lative distribution with margins F

. Then there exists a copula C such that

F (x)=C(F

),...,F

)) .

424 7 Default Risk: An Enlargement of Filtration Approach

The copula of the n-dimensional random variable (X

,...,X

) gives

information on the dependence of the marginal one-dimensional random

variables X

. From the deﬁnition, if the marginal distributions are strictly

increasing,

C(u

,...,u

)=F (F

−1

),...,F

−1

))

= P(X

≤ F

−1

),...,X

≤ F

−1

))

= P(F

) ≤ u

,...,F

) ≤ u

) .

A particular copula is the independence copula C(u

,...,u



i=1

where the diﬀerent random variables U

are independent and uniformly

distributed.

One of the copulas used by practioners is the Gaussian copula, given by

C(v

,...,v

)=N



−1

),...,N

−1

)



where N

is the c.d.f for the n-variate central normal distribution with the

linear correlation matrix Σ,andN

−1

is the inverse of the c.d.f. for the

univariate standard normal distribution.

Copula and Threshold

As in Subsection 7.3.1, we deﬁne τ

=inf{t : Λ

(t) ≥ Θ

} where Λ

are F-

adapted, increasing processes and Θ

are random variables, independent of

∞

, with a survival copula



C(u

,...,u

)=P(Φ

(Θ

) >u

, ∀i ≤ n)

where Φ

is the cumulative distribution function of Θ

, assumed to be strictly

increasing. Then, the joint law of default times may be characterized by

P(τ

, ∀i ≤ n)=P(Λ

) <Θ

, ∀i ≤ n)

= P(Φ

(Λ

)) <Φ

(Θ

), ∀i ≤ n)

= E





C(Ψ

),...,Ψ

))



where Ψ

(t)=Φ

(Λ

(t)). We have also, for t

>t,∀i,

P(τ

, ∀i ≤ n|F

)=P(Λ

) <Θ

, ∀i ≤ n|F

)

= E





C(Ψ

),...,Ψ

))|F



In particular, if τ =inf

(τ

)andY ∈F

E(Y 1

{τ

>T }

)=1

{τ>t}

E(Y 1

{τ

>T }

{τ>t}

)

P(τ>t|F

)

= 1

{τ>t}





C(Ψ

(t),...,Ψ

(T ),...Ψ

(t))



C(Ψ

(t),...,Ψ

(t),...Ψ

(t))





7.3 Default Times with a Given Stochastic Intensity 425

Comment 7.3.6.3 The reader is refered to various works, e.g., Bouy´eetal

[115], Coutant et al. [200] and Nelsen [668] for deﬁnitions and properties of

copulas and to Frey and McNeil [358], Embrechts et al. [323]andLi[588]for

ﬁnancial applications.

7.3.7 Correlated Defaults: Jarrow and Yu’s Model

Let us deﬁne τ

=inf{t : Λ

(t) ≥ Θ

},i =1, 2whereΛ

(t)=



(s)ds and

are independent random variables with exponential law of parameter 1. As

in Jarrow and Yu [481], we consider the case where λ

is a constant and

(t)=λ

+(α

− λ

{τ

≤t}

= λ

{t<τ

}

+ α

{τ

≤t}

Assume for simplicity that r = 0. Our aim is to compute the value of a

defaultable zero-coupon with default time τ

, with a rebate δ

paid at maturity:

(t, T )=E(1

{τ

>T }

+ δ

{τ

<T }

), for G

= D

∨D

We compute the joint law of the pair (τ

,τ

) given by the survival probability

G(s, t)=P(τ

>s,τ

>t).

 Case t ≤ s: For t ≤ s<τ

, one has λ

(t)=λ

t. Hence, the following

equality

{τ

>s}∩{τ

>t} = {τ

>s}∩{Λ

(t) <Θ

} = {τ

>s}∩{λ

t<Θ

}

= {λ

s<Θ

}∩{λ

t<Θ

}

leads to

for t ≤ s, P(τ

>s,τ

>t)=e

−λ

. (7.3.5)

 Case t>s

{τ

>s}∩{τ

>t} = {{t>τ

>s}∩{τ

>t}} ∪ {{τ

>t}∩{τ

>t}}

{t>τ

>s}∩{τ

>t} = {t>τ

>s}∩{Λ

(t) <Θ

}

= {t>τ

>s}∩{λ

+ α

(t − τ

) <Θ

The independence between Θ

and Θ

implies that the r.v. τ

= Θ

/λ

independent of Θ

, hence

P(t>τ

>s,τ

>t)=E



{t>τ

>s}

−(λ

+α

(t−τ

))





du 1

{t>u>s}

−(λ

u+α

(t−u))

−λ

−α

+ λ

− α

−α



−s(λ

+λ

−α

)

− e

−t(λ

+λ

−α

)



;

426 7 Default Risk: An Enlargement of Filtration Approach

Setting Δ = λ

+ λ

− α

, it follows that

P(τ

>s,τ

>t)=

−α



−sΔ

− e

−tΔ



+ e

−λ

. (7.3.6)

In particular, for s =0,

P(τ

>t)=



−α

− e

−(λ

+λ



+ Δe

−λ



 The computation of D

reduces to that of

P(τ

>T|G

)=P(τ

>T|F

∨D

)

where F

= D

. From the key Lemma 7.3.4.1,

P(τ

>T|D

∨D

)=1

{t<τ

}

P(τ

>T|D

)

P(τ

>t|D

)

Therefore, using equalities (7.3.5)and(7.3.6)

(t, T )=δ

+ 1

{τ

>t}

(1 − δ

−λ

(T −t)

 The computation of D

follows from

P(τ

>T|D

∨D

)=1

{t<τ

}

P(τ

>T|D

)

P(τ

>t|D

)

and

P(τ

>T|D

)=1

{τ

>t}

P(τ

>t,τ

>T)

P(τ

>t)

+ 1{τ

≤ t}P(τ

>T|τ

) .

Some easy computations lead to

(t, T )=δ

+(1− δ

{τ

>t}



{τ

≤t}

−α

(T −t)

{τ

>t}

(λ

−α

(T −t)

+(λ

− α

−(λ

+λ

)(T −t)

)



7.4 Conditional Survival Probability Approach

We present now a more general model. We deal with two kinds of information:

the information from the asset’s prices, denoted by (F

,t ≥ 0), and the

information from the default time τ , i.e., the knowledge of the time when

the default occurred in the past, if it occurred. More precisely, this latter

information is modelled by the ﬁltration D =(D

,t ≥ 0) generated by the

default process D

= 1

{τ≤t}

7.4 Conditional Survival Probability Approach 427

At the intuitive level, F is generated by the prices of some assets, or by

other economic factors (e.g., long and short interest rates). This ﬁltration can

also be a subﬁltration of that of the prices. The case where F is the trivial

ﬁltration is exactly what we have studied in the toy model. Though in typical

examples F is chosen to be the Brownian ﬁltration, most theoretical results

do not rely on such a speciﬁcation. We denote G

= F

∨D

Special attention is paid here to the hypothesis (H), which postulates the

immersion property of F in G. We establish a representation theorem, in

order to understand the meaning of a complete market in a defaultable world,

and we deduce the hedging strategies for some defaultable claims. The main

part of this section can be found in the surveys of Jeanblanc and Rutkowski

[486, 487].

7.4.1 Conditional Expectations

The conditional law of τ with respect to the information F

is characterized

by P(τ ≤ u|F

). Here, we restrict our attention to the survival conditional

distribution (the Az´ema’s supermartingale)

:=P(τ>t|F

) .

The super-martingale (G

,t ≥ 0) admits a decomposition as Z − A where Z

is an F-martingale and A an F-predictable increasing process. We assume in

this section that G

> 0 for any t and that G is continuous.

It is straightforward to establish that every G

-random variable is equal,

on the set {τ>t},toanF

-measurable random variable.

Lemma 7.4.1.1 (Key Lemma.) Let X be an F

-measurable integrable r.v.

Then, for t<T

E(X1

{T<τ}

)=1

{τ>t}

E(X1

{τ>T}

)

E(1

{τ>t}

)

= 1

{τ>t}

)

−1

E(XG

) .

(7.4.1)

Proof: The proof is exactly the same as that of Corollary 7.3.4.2. Indeed,

{τ>t}

E(X1

{T<τ}

)=1

{τ>t}

where x

is F

-measurable. Taking conditional expectation w.r.t. F

of both

sides, we deduce

E(X1

{τ>T}

)

E(1

{τ>t}

)

= 1

{τ>t}

)

−1

E(XG

) .



428 7 Default Risk: An Enlargement of Filtration Approach

Lemma 7.4.1.2 Let h be an F-predictable process. Then,

E(h

{τ≤T }

)=h

{τ≤t}

− 1

{τ>t}

)

−1







(7.4.2)

In terms of the increasing process A of the Doob-Meyer decomposition of G,

E(h

{τ≤T }

)=h

{τ≤t}

+ 1

{τ>t}

)

−1







Proof: The proof follows the same lines as that of Lemma 7.3.4.3. 

As we shall see, this elementary result will allow us to compute the value

of defaultable claims and of credit derivatives as CDS’s. We are not interested

with the case where h is a G-predictable process, mainly because every G-

predictable process is equal, on the set {t ≤ τ},toanF-predictable process.

Lemma 7.4.1.3 The process M

:=D

−



t∧τ

is a G-martingale.

Proof: The proof is based on the key lemma and Fubini’s theorem. We leave

the details to the reader. 

In other words, the process



t∧τ

is the G-predictable compensator

of D.

Comment 7.4.1.4 The hazard process Γ

:=−ln G

is often introduced. In

the case of the Cox Process model, the hazard process is increasing and is

equal to Λ.

7.5 Conditional Survival Probability Approach and

Immersion

We discuss now the hypothesis on the modelling of default time that we

require, under suitable conditions, to avoid arbitrages in the defaultable

market. We recall that the immersion property, also called (H)-hypothesis

(see Subsection 5.9.1) states that any square integrable F-martingale is

a G-martingale. In the ﬁrst part, we justify that, under some ﬁnancial

conditions, the (H)-hypothesis holds. Then, we present some consequences

of this condition and we establish a representation theorem and give an

important application to hedging.

7.5 Conditional Survival Probability Approach and Immersion 429

7.5.1 (H)-Hypothesis and Arbitrages

If r is the interest rate, we denote, as usual R

=exp(−



ds).

Proposition 7.5.1.1 Let S be the dynamics of a default-free price process,

represented as a semi-martingale on (Ω,G, P) and F

= σ(S

,s ≤ t) its

natural ﬁltration. Assume that the interest rate r is F

-adapted and that there

exists a unique probability Q, equivalent to P on F

, such that the discounted

process (S

, 0 ≤ t ≤ T ) is an F

-martingale under the probability Q.

Assume also that there exists a probability



Q, equivalent to P on G

= F

∨D

such that (S

, 0 ≤ t ≤ T ) is a G-martingale under the probability



Q.Then,

(H) holds under



Q and the restriction of



Q to F

is equal to Q.

Proof: We give a “ﬁnancial proof.” Under our hypothesis, any Q-square

integrable F

-measurable r.v. X can be thought of as the value of a contingent

claim. Since the same claim exists in the larger market, which is assumed to

be arbitrage free, the discounted value of that claim is a (G,

Q)-martingale.

From the uniqueness of the price for a hedgeable claim, for any contingent

claim X ∈F

and any G-e.m.m.

(XR

)=E

(XR

) .

In particular, E

(Z)=E

(Z) for any Z ∈F

(take X = ZR

−1

and

t = 0), hence the restriction of any e.m.m.

Q to the σ-algebra F

equals

Q. Moreover, since every square integrable (F

, Q)-martingale can be written

as E

(Z|F

)=E

(Z|G

), we obtain that every square integrable (F

Q)-

martingale is a (G,



Q)-martingale. 

Some Consequences

We recall that the hypothesis (H), studied in Subsection 5.9.4, reads in this

particular setting:

∀t, P(τ ≤ t|F

∞

)=P(τ ≤ t|F

):=F

. (7.5.1)

In particular, if (H) holds, then F is increasing. Furthermore, if F is

continuous, then the predictable increasing process of the Doob-Meyer

decomposition of G =1− F is equal to F and

−



t∧τ

is a G-martingale.