Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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440 7 Default Risk: An Enlargement of Filtration Approach
Hence,
P(τ ≤ t|F
t
)=1 ifX
t
j
<afor at least one t
j
, t
j
<t,
=1− Φ(t − t
i
,a− X
t
i
)K
i
, otherwise
where
K
i
= k(t
1
,X
t
1
, 0)k(t
2
− t
1
,X
t
1
,X
t
2
) ···k(t
i
− t
i−1
,X
t
i−1
,X
t
i
)
and k(s, x, y)=1− exp
−
2
s
(a − x)(a − y)
.
Comment 7.6.1.1 It is also possible, as in Duffie and Lando [273], to assume
that the observation at time [t]isonlyV
[t]
+ where is a noise, modelled
as a random variable independent of V . Another example, related to Parisian
stopping times is presented in C¸¸ etin et al. [158].
Exercise 7.6.1.2 Prove that the process ζ defined by ζ
t
=
i,t
i
≤t
ΔF
t
i
is
an F-martingale.
Hint: This is a trivial check. For details, see Jeanblanc and Valchev [489].
7.6.2 Two Defaults, Trivial Reference Filtration
We present some results on the case of two default times in the particular case
where the reference filtration F is the trivial filtration. We denote by G the
filtration H
1
∨H
2
and by G(u, v)=P(τ
1
>u,τ
2
>v) the survival probability
of the pair (τ
1
,τ
2
)andbyF
i
(s)=P(τ
i
≤ s)=
s
0
f
i
(u)du the marginal
cumulative distribution functions. We assume that G is twice continuously
differentiable. Our aim is to study the (H) hypothesis between H
1
and G.
Filtration H
i
From Proposition 7.2.2.1, for any i =1, 2, the process
M
i
t
= H
i
t
−
t∧τ
i
0
f
i
(s)
1 − F
i
(s)
ds (7.6.4)
is an H
i
-martingale.
Filtration G
Lemma 7.6.2.1 The H
2
Doob-Meyer decomposition of F
1|2
t
:=P(τ
1
≤ t|H
2
t
)
is
dF
1|2
t
=
G(t, t)
G(0,t)
−
∂
2
G(t, t)
∂
2
G(0,t)
dM
2
t
+
H
2
t
∂
1
h(t, τ
2
) − (1 − H
2
t
)
∂
1
G(t, t)
G(0,t)
dt
where
h(t, v)=1−
∂
2
G(t, v)
∂
2
G(0,v)
.
7.6 General Case: Without the (H)-Hypothesis 441
Proof: Some easy computation enables us to write
F
1|2
t
= H
2
t
P(τ
1
≤ t|τ
2
)+(1− H
2
t
)
P(τ
1
≤ t<τ
2
)
P(τ
2
>t)
= H
2
t
h(t, τ
2
)+(1− H
2
t
)
G(0,t) − G(t, t)
G(0,t)
, (7.6.5)
Introducing the deterministic function ψ(t)=1− G(t, t)/G(0,t), the sub-
martingale F
1|2
t
has the form
F
1|2
t
= H
2
t
h(t, τ
2
)+(1− H
2
t
)ψ(t) (7.6.6)
The function t → ψ(t) and the process t → h(t, τ
2
) are continuous and of
finite variation, hence the integration by parts formula leads to
dF
1|2
t
= h(t, τ
2
)dH
2
t
+ H
2
t
∂
1
h(t, τ
2
)dt +(1− H
2
t
)ψ
(t)dt − ψ(t)dH
2
t
=(h(t, τ
2
) − ψ(t)) dH
2
t
+
H
2
t
∂
1
h(t, τ
2
)+(1− H
2
t
)ψ
(t)
dt
=
G(t, t)
G(0,t)
−
∂
2
G(t, τ
2
)
∂
2
G(0,τ
2
)
dH
2
t
+
H
2
t
∂
1
h(t, τ
2
)+(1− H
2
t
)ψ
(t)
dt .
Now, we note that
T
0
G(t, t)
G(0,t)
−
∂
2
G(t, τ
2
)
∂
2
G(0,τ
2
)
dH
2
t
=
G(τ
2
,τ
2
)
G(0,τ
2
)
−
∂
2
G(τ
2
,τ
2
)
∂
2
G(0,τ
2
)
1
{τ
2
≤t}
=
T
0
G(t, t)
G(0,t)
−
∂
2
G(t, t)
∂
2
G(0,t)
dH
2
t
and substitute it into the expression of dF
1|2
:
dF
1|2
t
=
G(t, t)
G(0,t)
−
∂
2
G(t, t)
∂
2
G(0,t)
dH
2
t
+
H
2
t
∂
1
h(t, τ
2
)+(1− H
2
t
)ψ
(t)
dt .
From
dH
2
t
= dM
2
t
−
1 − H
2
t
∂
2
G(0,t)
G(0,t)
dt ,
where M
2
is an H
2
-martingale, we get the Doob-Meyer decomposition of F
1|2
:
dF
1|2
t
=
G(t, t)
G(0,t)
−
∂
2
G(t, t)
∂
2
G(0,t)
dM
2
t
−
1 − H
2
t
G(t, t)
G(0,t)
−
∂
2
G(t, t)
∂
2
G(0,t)
∂
2
G(0,t)
G(0,t)
dt
+
H
2
t
∂
1
h(t, τ
2
)+(1− H
2
t
)ψ
(t)
dt
and, after the computation of ψ
(t), one obtains
442 7 Default Risk: An Enlargement of Filtration Approach
dF
1|2
t
=
G(t, t)
G(0,t)
−
∂
2
G(t, t)
∂
2
G(0,t)
dM
2
t
+
H
2
t
∂
1
h(t, τ
2
) − (1 − H
2
t
)
∂
1
G(t, t)
G(0,t)
dt .
As a consequence:
Lemma 7.6.2.2 The (H) hypothesis is satisfied for H
1
and G if and only if
G(t, t)
G(0,t)
=
∂
2
G(t, t)
∂
2
G(0,t)
.
Proposition 7.6.2.3 The process
H
1
t
−
t∧τ
1
0
a(s)
1 − F
1|2
(s)
ds ,
where a(t)=H
2
t
∂
1
h(t, τ
2
) − (1 − H
2
t
)
∂
1
G(t,t)
G(0,t)
and h(t, s)=1−
∂
2
G(t,s)
∂
2
G(0,s)
, is a
G-martingale.
Proof: The result follows from Lemma 7.4.1.3 and the form of the Doob-
Meyer decomposition of F
1|2
.
7.6.3 Initial Times
In order that the prices of the default-free assets do not induce arbi-
trage opportunities, one needs to prove that F-martingales remain G-semi-
martingales. We have seen in Proposition 5.9.4.10 that this is the case when
the random time τ is honest. However, in the credit risk setting, default
times are not honest (see for example the Cox model). Hence, we have to
give another condition. We shall assume that the conditions of Proposition
5.9.3.1 are satisfied. This will imply that F-martingales are F ∨ σ(τ)- semi-
martingales, and of course G-semi-martingales.
For any positive random time τ, and for every t,wewriteq
t
(ω, dT)the
regular conditional distribution of τ,and
G
T
t
(ω)=Q(τ>T|F
t
)(ω)=q
t
(ω, ]T,∞[).
For simplicity, we introduce the following (non-standard) definition:
Definition 7.6.3.1 (Initial Times) The positive random time τ is called
an initial time if there exists a probability measure η on B(R
+
) such that
q
t
(ω, dT) η(dT ).
7.6 General Case: Without the (H)-Hypothesis 443
Then, there exists a family of positive F-adapted processes (α
u
t
,t≥ 0) such
that
G
T
t
=
∞
T
α
u
t
η(du) .
From the martingale property of (G
T
t
,t≥ 0), i.e., for every T , for every s ≤ t,
G
T
s
= E(G
T
t
|F
s
), it is immediate to check that for any u ≥ 0, (α
u
t
,t≥ 0) is a
positive F-martingale. Note that Q(τ ∈ du)=α
u
0
η(du), hence α
u
0
=1.
Remark that in this framework, we can write the conditional survival
process G
t
:= G
t
t
as
G
t
= Q(τ>t|F
t
)=
∞
t
α
u
t
η(du)=
∞
0
α
u
u∧t
η(du) −
t
0
α
u
u
η(du)=M
t
−
A
t
where M is an F-martingale (indeed, (α
u
u∧t
)
t
is a stopped F-martingale)
and
A an F-predictable increasing process. The process (G
T
t
,t ≥ 0) being
a martingale, it admits a representation as
G
T
t
= G
T
0
+
t
0
g
T
s
dW
s
where, for any T , the process (g
T
s
,s ≥ 0) is F-predictable. In the case where
η(du)=ϕ(u)du, using the Itˆo-Kunita-Ventzel formula (see Theorem 1.5.3.2),
we obtain:
Lemma 7.6.3.2 The Doob-Meyer decomposition of the conditional survival
process (G
t
,t≥ 0) is
G
t
t
=1+
t
0
g
s
s
dW
s
−
t
0
α
s
s
ϕ(s) ds . (7.6.7)
Lemma 7.6.3.3 The process
M
t
:= H
t
−
t
0
(1 − H
s
)G
−1
s
α
s
s
ϕ(s) ds
is a G-martingale.
Proof: This follows directly from the Doob-Meyer decomposition of G given
in Lemma 7.6.3.2.
Using a method similar to Proposition 5.9.4.10, assuming that G is
continuous, it is possible to prove (see Jeanblanc and Le Cam [482] for details)
thatif X is a square integrable F-martingale
Y
t
= X
t
−
t∧τ
0
d X, G
u
G
u
−
t
t∧τ
d
X, α
θ
u
α
θ
u−
θ=τ
(7.6.8)
444 7 Default Risk: An Enlargement of Filtration Approach
is a G-martingale. Note that, the first integral, which describes the bounded
variation part before τ, is the same as in progressive enlargement of filtration
– even without the honesty hypothesis – (see Proposition 5.9.4.10), and that
the second integral, which describes the bounded variation part after τ,isthe
same as in Proposition 5.9.3.1.
Exercise 7.6.3.4 Prove that, if τ is an initial time with E
Q
(1/α
τ
∞
) < ∞,
there exists a probability
Q equivalent to Q under which τ and F
∞
are
independent.
Hint: Use Exercise 5.9.3.4.
Exercise 7.6.3.5 Let τ be an initial time which avoids F-stopping times.
Prove that the (H)-hypothesis holds if and only if α
u
t
= α
u
t∧u
.
Exercise 7.6.3.6 Let (K
u
t
,t≥ 0) be a family of F-predictable processes
indexed by u ≥ 0 (i.e., for any u ≥ 0,t→ K
u
t
is F-predictable).
Prove that E(K
τ
t
|F
t
)=
∞
0
K
u
t
α
u
t
η(du) .
7.6.4 Explosive Defaults
Let
dX
t
=(θ − k(t)X
t
)dt + σ
X
t
dW
t
where the parameters are chosen so that P(T
0
< ∞)=1whereT
0
is the
first hitting time of 0 for the process X. Andreasen [17] defines the default
time as in the Cox process modelling presented in Subsection 7.3.1, setting
the process (λ
t
,t ≥ 0) equal to 1/X
t
before T
0
and equal to +∞ after time
T
0
. Note that the default time is not a totally inaccessible stopping time
(obviously, P(τ = T
0
) is not null).
The survival probability is, for F
t
= F
X
t
,
P(τ>T|G
t
)=1
{t<τ}
E
exp
−
T
t
λ
s
ds
|F
t
:=1
{t<τ}
L(t, T, X
t
) .
The process
L(t, T, X
t
)exp
−
t
0
λ
s
ds
= L(t, T, X
t
)exp
−
t
0
X
−1
s
ds
is a local-martingale, hence
∂
t
L +(θ − k(t)x)∂
x
L +
1
2
σ
2
x∂
xx
L −
1
x
L =0,
and, taking into account the boundary conditions
L(t, T, 0) = 0,L(T,T,x)=1,L(t, T, ∞)=1
7.7 Intensity Approach 445
one gets
L(t, T, x)=
Γ (β − γ)
Γ (β)
M
γ,β,−
x
2K(t, T )
x
2K(t, T )
γ
where M is the Kummer function (see Appendix A.5.6)and
γ =
−(θ − σ
2
/2) +
(θ − σ
2
/2)
2
+2σ
2
σ
2
,
β =2(γ + θ/σ
2
)
and
K(t, T )=
σ
2
4
T
t
exp
u
t
k(s)ds
du .
Comment 7.6.4.1 Campi et al. [138] work with a model where the default
is the first time when the process X hits the barrier 0, with
dX
t
= X
t−
(μdt + X
β
t
dW
t
− dN
t
) .
Here N is a Poisson process. In other words X
t
= Y
t
1
{t<τ}
where Y is a
CEV process and τ is an exponentially distributed random variable, which is
independent of Y .
7.7 Intensity Approach
7.7.1 Definition
In the intensity approach, the default time τ is a G-stopping time for a
given filtration G. From the Doob-Meyer Theorem, there exists a unique G-
predictable increasing process Λ
G
such that the process M
t
= D
t
− Λ
G
t
is a
G-martingale. This process Λ
G
satisfies Λ
G
t
= Λ
G
t∧τ
. The continuity of Λ
G
is
equivalent to the fact that τ is a G-totally inaccessible stopping time.
In what follows, we assume that Λ
G
is absolutely continuous w.r.t.
Lebesgue measure, i.e., Λ
G
t
=
t
0
λ
G
s
ds. Then, the process λ
G
, called the
G-intensity of τ, vanishes after τ.
Comment 7.7.1.1 Note that, in the Cox Process approach with F-adapted
intensity λ, or in the conditional survival probability approach (see Sec-
tion 7.4)whereλ denotes the F-adapted process given by λ
s
ds = dF
s
/G
s
,we
have λ
G
t
= 1
{τ<t}
λ
t
.
Lemma 7.7.1.2 The process L
t
= 1
{t<τ}
exp
Λ
G
t
is a local martingale.
Proof: From Itˆo calculus (See Subsection 8.3.4)
dL
t
=exp
Λ
G
t
(−dD
t
+(1− D
t−
)λ
G
t
dt)=−exp
Λ
G
t
dM
t
.
446 7 Default Risk: An Enlargement of Filtration Approach
7.7.2 Valuation Formula
Proposition 7.7.2.1 For every integrable r.v. X ∈G
T
:
E(X1
{T<τ}
|G
t
)=1
{τ>t}
V
t
− E(ΔV
τ
1
{τ≤T }
|G
t
)
where V
t
= e
Λ
G
t
E
Xe
−Λ
G
T
|G
t
.
Proof: By application of the integration by parts formula to the product
U
t
= V
t
(1 − D
t
), we obtain
dU
t
= −V
t
dD
t
+(1− D
t
)dV
t
− ΔV
t
ΔD
t
Writing V
t
= e
Λ
G
t
m
t
,wherem
t
= E
Xe
−Λ
G
T
|G
t
is a G-martingale, one gets
dV
t
= e
Λ
G
t
dm
t
+ m
t
λ
G
t
e
Λ
G
t
dt = e
Λ
G
t
dm
t
+ V
t
λ
G
t
dt ,
hence,
dU
t
= −V
t
(dD
t
− λ
G
t
dt)+(1− D
t
)e
Λ
G
t
dm
t
− ΔV
t
ΔD
t
.
By integration, we obtain U
t
= E(ΔV
τ
1
{t<τ≤T }
+ U
T
|G
t
). It remains to note
that U
T
= 1
{T<τ}
X, and the result follows.
Exercise 7.7.2.2 Assume that τ is an exponential r.v. with parameter λ and
that G is the filtration generated by 1
{τ≤t}
.Checkthatλ
G
t
= 1
{t<τ}
λ.Let
Y
t
= E(exp −(T ∧ τ)|G
t
). Compute the jump of Y at time τ and deduce
E(1
{t<τ≤T }
|G
t
) using Proposition 7.7.2.1. Of course, here, the conditional
survival methodology is more powerful.
7.8 Credit Default Swaps
We present briefly Credit Default Swaps (CDS). The reader can consult
Bielecki et al. [94, 97], Brigo and Alphonsi [128] and Jamshidian [477]for
more details. We assume here that the interest rate r is constant.
A credit default swap (CDS) with a constant rate κ and recovery at default
δ is a contract between the buyer and the seller. The buyer (i.e., the buyer of
protection against a reference entity which defaults at time τ), pays a premium
κ to the seller till τ ∧T where T is the maturity of the CDS: the amount κdt is
paid during the time dt. If the default occurs before T , the seller pays, at the
default time, δ(τ) to the buyer. Note that we assume here that the premium is
paid in continuous time, for simplicity; usually, the premium is paid at some
predetermined dates T
i
.
The function δ :[0,T] → R represents the default protection,andκ is the
CDS rate (also termed the spread, premium or annuity of the CDS).
7.8 Credit Default Swaps 447
The price of the CDS is the expectation of the difference of the discounted
payoffs and is given by the formula
S
t
(κ, δ, T ; r)=e
rt
E
Q
e
−rτ
δ(τ)1
{τ≤T }
−
T ∧τ
t∧τ
e
−ru
κdu
G
t
.
Here, G is the information available for the protection buyer. We shall note
in short S
t
(κ) this price. Of course, it vanishes after time τ. Note that this
price does not remain positive: at date t, the buyer is not allowed to cancel
the contract; if S
t
< 0, the buyer has to pay −S
t
(κ) to the seller (or to the
new owner of this CDS) to do so.
At time 0, the price of a CDS is null: the spread κ is chosen so that
κ =
E
Q
e
−rτ
δ(τ)1
{τ≤T }
E
Q
T ∧τ
0
e
−ru
du
.
7.8.1 Dynamics of the CDS’s Price in a single name setting
In this subsection, G = F ∨ H,andG
t
= Q(τ>t|F
t
).
Proposition 7.8.1.1 The price of a CDS equals, for any t ∈ [0,T],
S
t
(κ)=1
{t<τ}
1
e
−rt
G
t
E
Q
T
t
e
−ru
G
u
(δ
u
λ
u
− κ) du
F
t
(7.8.1)
Proof: This relies on a direct and simple application of the key Lemma
7.4.1.1.
The dynamics of the CDS’s price can now be obtained:
Proposition 7.8.1.2 If the immersion property holds, then
dS
t
(κ)=−S
t
−
(κ) dM
t
+(1−H
t
)
rS
t
(κ)+κ −λ
t
δ
t
dt +(1−H
t
)e
rt
G
−1
t
dn
t
(7.8.2)
where n
t
= E
Q
T
0
e
−ru
G
u
(δ
u
λ
u
− κ) du
F
t
.
Proof: Apply Itˆo’s formula.
Comment 7.8.1.3 Note that the risk-neutral dynamics of the CDS’s dis-
counted price do not constitute a martingale. This price is indeed an ex-
dividend price. The cum-dividend price S
cum
(κ) satisfies, for every t ∈ [0,T],
d
t
S
cum
t
(κ)=rS
cum
t
(κ) dt +
δ
t
− S
t−
(κ)
dM
t
+(1− H
t
)G
−1
t
e
rt
dn
t
.
The case where immersion is not satisfied is presented in [97].
448 7 Default Risk: An Enlargement of Filtration Approach
7.8.2 Dynamics of the CDS’s Price in a multi-name setting
An important feature of price’s dynamics is that the price of a CDS depends
strongly of the choice of the observation filtration. We present here the case
where a second firm can default at time τ
2
and where the two default times
are correlated, in the simple case of a trivial filtration F and a null interest
rate. Let us denote by G(t, s)=P(τ
1
>t,τ
2
>s) the survival joined law of
the two defaults, assumed to be regular. Let
S(κ)
t
= 1
{t<τ
1
}
E(δ(τ
1
) − κ(τ ∧ T −t)
+
|G
t
)
be the price of a CDS written on τ
1
, with spread κ and recovery the function
δ,whereG = H
1
∨ H
2
. Then, 1
{t<τ
1
}
S
t
(κ)=1
{t<τ
1
}
S
t
(κ), where an easy
computation leads to
S
t
(κ)=
1
G(t, t)
−
T
t
δ(u)∂
1
G(u, t) du − κ
T
t
G(u, t) du
. (7.8.3)
Hence the dynamics of the pre-default ex-dividend price
S
t
are
d
S
t
(κ)=
λ
1
(t)+
λ
2
(t)
S
t
(κ)+κ
1
−
λ
1
(t)δ(t) −
λ
2
(t)S
t|2
(κ
1
)
dt,
where for i =1, 2 the function
λ
i
(t)=−
∂
i
G(t,t)
G(t,t)
is the (deterministic) pre-
default intensity of τ
i
and S
t|2
(κ) is given by the expression
S
t|2
(κ)=
−1
∂
2
G(t, t)
T
t
δ(u)f(u, t) du + κ
T
t
∂
2
G(u, t) du
.
In the financial interpretation, S
1|2
(t) is the ex-dividend price at time t of a
CDS on the first credit name, under the assumption that the default τ
2
occurs
at time t and the first name has not yet defaulted (recall that simultaneous
defaults are excluded).
Let us now consider the event {τ
2
≤ t<τ
1
}. It is not difficult to show
that in that case the ex-dividend price of a CDS equals
S
t
(κ)=
1
∂
2
G(t, τ
2
)
−
T
t
δ(u)f(u, τ
2
) du − κ
T
t
∂
2
G(u, τ
2
) du
. (7.8.4)
Consequently, on the event {τ
2
≤ t<τ
1
} we obtain
d
S
t
(κ)=
λ
1|2
(t, τ
2
)
S
t
(κ) − δ(t)
+ κ
dt,
where λ
1|2
(t, s)=−
f(t,s)
∂
2
G(t,s)
, evaluated at s = τ
2
, represents the value of the
default intensity process of τ
1
with respect to the filtration H
2
on the event
{τ
2
<t}.
7.9 PDE Approach for Hedging Defaultable Claims 449
7.9 PDE Approach for Hedging Defaultable Claims
We briefly present a PDE approach for defaultable claims. We assume that Y
1
is the price of the savings account, with deterministic interest rate. A default
free asset is supposed to be traded in the market with price dynamics
dY
2
t
= Y
2
t
μ
2,t
dt + σ
2,t
dW
t
. (7.9.1)
A defaultable asset with price dynamics
dY
3
t
= Y
3
t
−
(μ
3,t
− κ
3,t
λ
t
1
{t≤τ}
) dt + σ
3,t
dW
t
+ κ
3,t
dD
t
, (7.9.2)
is also traded in the market. All the processes are assumed to be G-adapted,
where G is the filtration generated by W and D. Here, W is a Brownian
motion, D
t
= 1
{τ≤t}
is the default process (see Section 7.4) and the process
M
t
= D
t
−
t
0
(1 − D
s
)λ
s
ds is assumed to be a G-martingale. The Brownian
motion W is assumed to be a G-Brownian motion, hence, the (H)hypothesis
holds between F
W
and G.
Our aim is to replicate a contingent claim of the form
Y = 1
{T<τ}
g
0
(Y
2
T
,Y
3
T
)+1
{T ≥τ}
g
1
(Y
2
T
,Y
3
T
)=G(Y
2
T
,Y
3
T
,D
T
),
which settles at time T .
7.9.1 Defaultable Asset with Total Default
We first assume that the third asset is subject to total default, i.e., κ
3
= −1,
dY
3
t
= Y
3
t
−
μ
3,t
dt + σ
3,t
dW
t
− dM
t
.
A first step is to find some condition on the coefficients such that the market
is arbitrage free.
Equivalent Martingale Measure
Let Y
i,1
= Y
i
/Y
1
be the relative price of the ith-asset in terms of the first
one. Our goal is now to find a martingale measure Q
1
(if it exists) for the
relative prices Y
2,1
and Y
3,1
. The dynamics of Y
3,1
under P are
dY
3,1
t
= Y
3,1
t
−
μ
3,t
− r(t)
dt + σ
3,t
dW
t
− dM
t
!
.
Let Q
1
be any probability measure equivalent to P on (Ω, G
T
). The asso-
ciated Radon-Nikod´ym density L satisfies (we use Kusuoka’s representation
Theorem 7.5.5.1)
dL
t
= L
t
−
(θ
t
dW
t
+ ζ
t
dM
t
)
for some G-predictable processes θ and ζ.