Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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430 7 Default Risk: An Enlargement of Filtration Approach
Remarks 7.5.1.2 (a) If τ is F
∞
-measurable, then equality (7.5.1)isequiv-
alent to: τ is an F-stopping time. Moreover, if F is the Brownian filtration,
then τ is predictable and the Doob-Meyer decomposition of G is G
t
=1−F
t
,
where F is the predictable increasing process.
(b) Though the hypothesis (H) does not necessarily hold true, in general,
it is satisfied when τ is constructed through a Cox process approach (see
Section 7.3).
(c) This hypothesis is quite natural under the historical probability, and
is stable under particular changes of measure. However, Kusuoka provides an
example where (H) holds under the historical probability and does not hold
after a particular change of probability. This counterexample is linked with
dependency between different defaults (see Subsection 7.5.3).
(d) Hypothesis (H) holds in particular if τ is independent of F
∞
.See
Greenfeld’s thesis [406] for a study of derivative claims in that simple setting.
Comment 7.5.1.3 Elliott et al. [315] pay attention to the case when F is
increasing. Obviously, if (H) holds, then F is increasing, however, the reverse
does not hold. The increasing property of F is equivalent to the fact that every
F-martingale, stopped at time τ,isaG-martingale. Nikeghbali and Yor [675]
proved that this is equivalent to E(m
τ
)=m
0
for any bounded F-martingale m
(see Proposition 5.9.4.7). It is worthwhile noting that in Subsection 7.6.1,
the process F is not increasing.
7.5.2 Pricing Contingent Claims
Assume that the default-free market consists of F-adapted prices and that
the default-free interest rate is F-adapted. Whether the defaultable market
is complete or not will be studied in the following section. Let Q
∗
be the
e.m.m. chosen by the market and G
∗
the Q
∗
-survival probability, assumed to
be continuous. From (7.4.1) the discounted price of the defaultable contingent
claim X ∈F
T
is
R
t
E
Q
∗
(X1
{τ>T}
R
T
|G
t
)=1
{t<τ}
(G
∗
t
)
−1
E
Q
∗
(XG
∗
T
R
T
|F
t
) .
If (H) holds under Q
∗
and if G
∗
is differentiable, then G
∗
t
=exp(−
t
0
λ
∗
s
ds)
and
E
Q
∗
(XR
T
1
{τ>T}
|G
t
)=1
{t<τ}
E
Q
∗
X exp
−
T
t
(r
s
+ λ
∗
s
)ds
|F
t
.
(7.5.2)
Particular Case: Defaultable Zero-coupon Bond
The price at time t of a default-free bond paying 1 at maturity t satisfies
7.5 Conditional Survival Probability Approach and Immersion 431
P (t, T )=E
Q
∗
exp
−
T
t
r
s
ds
F
t
,
for any e.m.m. Q
∗
(recall that the market is incomplete as long as no
defaultable asset is traded). Now, if a defaultable zero-coupon bond is traded
in the market with a price - given by the market - equal to D(t, T), we have,
for a particular Q
∗
,
D(t, T )=E
Q
∗
1
{T<τ}
exp
−
T
t
r
s
ds
G
t
= 1
{τ>t}
E
Q
∗
exp
−
T
t
[r
s
+ λ
∗
s
] ds
F
t
.
The value of λ
∗
is obtained from the price of the defaultable zero coupons,
and is related to the conditional law of τ given F
t
as we have explained in
Subsection 7.3.1.
7.5.3 Correlated Defaults: Kusuoka’s Example
Kusuoka [552] assumes that the default times τ
i
,i =1, 2 are independent
exponential random variables, i.e., they have joint density
f(x, y)=λ
1
λ
2
e
−(λ
1
x+λ
2
y)
1
{x≥0,y≥0}
under the historical probability P. The reference filtration F is trivial and the
observation filtration is G = D
1
∨D
2
where D
i
is the natural filtration of the
process D
i
t
= 1
{τ
i
≤t}
. Define a measure Q as
Q|
G
T
= η
T
P|
G
T
where
dη
t
= η
t−
(κ
1
t
dM
1
t
+ κ
2
t
dM
2
t
),η
0
=1,
the (P, G)-martingales (M
i
,i=1, 2) are defined by M
i
t
= D
i
t
−λ
i
(t ∧τ
i
)and
κ
i
t
= 1
{τ
j
≤t}
α
i
λ
i
− 1
, for i = j, i, j =1, 2 .
Using Girsanov’s Theorem (see Section 9.4), the processes
M
i
t
= D
i
t
−
t∧τ
i
0
λ
i
s
ds
where
λ
1
t
= λ
1
1
{τ
2
>t}
+ α
1
1
{τ
2
≤t}
λ
2
t
= λ
2
1
{τ
1
>t}
+ α
2
1
{τ
1
≤t}
432 7 Default Risk: An Enlargement of Filtration Approach
are (Q, G)-martingales. The two default times are no longer independent
under Q. Furthermore, the (H) hypothesis does not hold under Q between
D
2
and D
1
∨ D
2
, in particular
Q(τ
1
>T|D
1
t
∨D
2
t
) = 1
{τ
1
>t}
E
Q
exp
−
T
t
λ
1
u
du
|D
2
t
.
7.5.4 Stochastic Barrier
We assume that the (H)-hypothesis holds under P and that F is strictly
increasing and continuous. Then, there exists a continuous strictly increasing
F-adapted process Γ such that
P(τ>t|F
∞
)=e
−Γ
t
.
Our goal is to show that there exists a random variable Θ, independent of F
∞
,
with exponential law of parameter 1, such that τ =inf{t ≥ 0:Γ
t
≥ Θ}. Let
us set Θ :=Γ
τ
. Then
{t<Θ} = {t<Γ
τ
} = {C
t
<τ},
where C is the right inverse of Γ ,sothatΓ
C
t
= t. Therefore
P(Θ>u|F
∞
)=e
−Γ
C
u
= e
−u
.
We have thus established the required properties, namely, that the exponential
probability law of Θ and its independence of the σ-algebra F
∞
. Furthermore,
τ =inf{t : Γ
t
>Γ
τ
} =inf{t : Γ
t
>Θ}.
Comment 7.5.4.1 This result is extended to a multi-default setting for a
trivial filtration in Norros [677] and Shaked and Shanthikumar [782].
7.5.5 Predictable Representation Theorems
We still assume that the (H) hypothesis holds and that F is absolutely
continuous w.r.t. Lebesgue measure with density f. We recall that
M
t
:=D
t
−
t∧τ
0
λ
s
ds
is a G-martingale where λ
s
= f
s
/G
s
. Kusuoka [552] establishes the following
representation theorem:
Theorem 7.5.5.1 Suppose that F is a Brownian filtration generated by
the Brownian motion W . Then, under the hypothesis (H), every G-square
integrable martingale (H
t
,t ≥ 0) admits a representation as the sum of a
stochastic integral with respect to the Brownian motion W and a stochastic
integral with respect to the discontinuous martingale M :
7.5 Conditional Survival Probability Approach and Immersion 433
H
t
= H
0
+
t
0
φ
s
dW
s
+
t
0
ψ
s
dM
s
where, for any t, E
t
0
φ
2
s
ds
< ∞ and E
t
0
ψ
2
s
λ
s
ds
< ∞.
In the case of a G-martingale of the form E(X1
{T<τ}
|G
t
)whereX is F
T
-
measurable, or for E(h
τ
|G
t
)whereh is predictable, one can be more precise.
Proposition 7.5.5.2 Suppose that hypothesis (H) holds, that G is continuous
and that every F-martingale is continuous.
Then, the martingale H
t
= E(h
τ
|G
t
) , where h is an F-predictable process
such that E(|h
τ
|) < ∞, admits the following decomposition as the sum of a
G-continuous martingale and a G-purely discontinuous martingale:
H
t
= m
h
0
+
t∧τ
0
G
−1
u
dm
h
u
+
]0,t∧τ]
(h
u
− J
u
) dM
u
. (7.5.3)
Here m
h
is the continuous F-martingale
m
h
t
= E
∞
0
h
u
dF
u
|F
t
,
and J
t
= G
−1
t
(m
h
t
−
t
0
h
u
dF
u
). Moreover, J
u
= H
u
on the set {u<τ}.
Proof: From (7.3.3) we know that
H
t
= E(h
τ
|G
t
)=1
{τ≤t}
h
τ
+ 1
{τ>t}
G
−1
t
E
∞
t
h
u
dF
u
|F
t
= 1
{τ≤t}
h
τ
+ 1
{τ>t}
J
t
. (7.5.4)
From the fact that G is a decreasing continuous process and m
h
a continuous
martingale, and using the integration by parts formula, we deduce, after some
easy computations, that
dJ
t
= G
−1
t
dm
h
t
+J
t
G
t
d(G
−1
t
)−h
t
G
−1
t
dF
t
= G
−1
t
dm
h
t
+J
t
G
−1
t
dF
t
−h
t
G
−1
t
dF
t
.
Therefore,
dJ
t
= G
−1
t
dm
h
t
+(J
t
− h
t
)
dF
t
G
t
or,inanintegratedform,
J
t
= m
h
0
+
t
0
G
−1
u
dm
h
u
+
t
0
(J
u
− h
u
)
dF
u
G
u
.
Note that, from (7.5.4), J
u
= H
u
for u<τ. Therefore, on {t<τ},
H
t
= m
h
0
+
t∧τ
0
G
−1
u
dm
h
u
+
t∧τ
0
(J
u
− h
u
)
dF
u
G
u
.
434 7 Default Risk: An Enlargement of Filtration Approach
From (7.5.4), the jump of H at time τ is h
τ
− J
τ
= h
τ
− H
τ−
. Therefore,
(7.5.3) follows. Since hypothesis (H) holds, the processes (m
h
t
,t ≥ 0) and
(
t
0
G
−1
u
dm
h
u
,t ≥ 0) are also G-martingales. Hence, the stopped process
(
t∧τ
0
G
−1
u
dm
h
u
,t≥ 0) is a G-martingale.
7.5.6 Hedging Contingent Claims with DZC
We assume that (H) holds under the risk-neutral probability Q chosen by the
market, and that the process G
t
= Q(τ>t|F
t
) is continuous.
We suppose moreover that:
• The default-free market including the default-free zero-coupon with
constant interest rate and the risky asset S, is complete and arbitrage
free, and
dS
t
= S
t
(rdt + σ
t
dW
t
) .
Here W is a Q-BM.
• A defaultable zero-coupon with maturity T and price D(t, T )istradedon
the market.
• The market which consists of the default-free zero-coupon P (t, T ), the
defaultable zero-coupon D(t, T ) and the risky asset S is arbitrage free (in
particular, D(t, T ) belongs to the range of prices ]0,P(t, T )[ ).
We now make precise the hedging of a defaultable claim and check that any
G
T
-measurable square integrable contingent claim is hedgeable.
The market price of the DZC and the e.m.m. Q are related by
D(t, T )e
−rt
= E
Q
(e
−rT
1
{T<τ}
|G
t
)
= 1
{t<τ}
G
−1
t
E
Q
(e
−rT
G
T
|F
t
)=L
t
m
t
(7.5.5)
where m
t
= E
Q
(e
−rT
G
T
|F
t
)isanF-martingale and L
t
= 1
{t<τ}
G
−1
t
.
We recall that a triple (a, b, c)ofG-predictable processes is a hedging
strategy for the contingent claim Z ∈G
T
if, denoting by
Z
t
= a
t
e
rt
+ b
t
S
t
+ c
t
D(t, T )
the time-t value of this strategy, the self-financing relation
dZ
t
= a
t
re
rt
dt + b
t
dS
t
+ c
t
d
t
D(t, T )
holds and
Z
T
= a
T
e
rT
+ b
T
S
T
+ c
T
D(T,T)=Z.
From the no-arbitrage hypothesis, we obtain
Z
t
e
−rt
= E
Q
(Z
T
e
−rT
|G
t
) .
7.5 Conditional Survival Probability Approach and Immersion 435
Terminal Payoff
In a first step we study the case of a terminal payoff of the particular form
Z = X1
{T<τ}
where X ∈ L
2
(F
T
). We compute E
Q
(X1
{T<τ}
e
−rT
|G
t
), and we
give the hedging strategy for X1
{T<τ}
based on the riskless asset, the risky
asset and the defaultable zero-coupon bond.
Theorem 7.5.6.1 The hedging strategy (a, b, c) for the defaultable contingent
claim X1
{T<τ}
, based on the riskless bond, the asset and the defaultable zero-
coupon satisfies
c
t
D(t, T )=e
rt
E
Q
(Xe
−rT
1
{T<τ}
|G
t
) .
Hence, a
t
e
rt
+ b
t
S
t
=0.
More precisely, let (V
X
t
−v
X
t
S
t
,v
X
t
) be the hedging strategy for the default-
free contingent claim XG
T
,and(V
t
− v
t
S
t
,v
t
) the hedging strategy for the
default-free contingent claim G
T
, i.e.,
e
−rt
V
X
t
= E
Q
(XG
T
e
−rT
|F
t
)=x +
t
0
v
X
s
d(e
−rs
S
s
)
e
−rt
V
t
= E
Q
(G
T
e
−rT
|F
t
)=x +
t
0
v
s
d(e
−rs
S
s
) . (7.5.6)
Then, on {t<τ}
(i) c
t
=
V
X
t
V
t
,
(ii) b
t
= G
−1
t
v
X
t
−
V
X
t
V
t
v
t
,
(iii) a
t
= −G
−1
t
v
X
t
−
V
X
t
V
t
v
t
e
−rt
S
t
.
Obviously, a
t
= b
t
= c
t
=0on {τ ≤ t}.
Proof: Let us reduce attention to the simple case r = 0. The time-t price of
the defaultable claim X1
{T<τ}
is Z
t
which is defined by
Z
t
= E
Q
(X1
{T<τ}
|G
t
)=L
t
V
X
t
where L
t
= 1
{t<τ}
G
−1
t
and V
X
t
= E
Q
(XG
T
|F
t
). The price of the DZC satisfies
D(t, T )=L
t
V
t
. Hence Z
t
= E
Q
(X1
{T<τ}
|G
t
)=
V
X
t
V
t
D(t, T ) .
We now check that there exists a triple (a, b, c) determining a self-financing
portfolio such that a
t
e
rt
+ b
t
S
t
= 0 (in particular, c
t
D(t, T )=Z
t
, hence the
process c satisfies (i)). The self-financing condition reads
b
t
S
t
σdW
t
+ c
t
(L
t−
dV
t
+ V
t
dL
t
)=dZ
t
= L
t−
dV
X
t
+ V
X
t
dL
t
,
where we have used the fact that the martingales L and V (resp. L and V
X
)
are orthogonal. From the choice of c, this equality reduces to
436 7 Default Risk: An Enlargement of Filtration Approach
b
t
S
t
σdW
t
+ c
t
L
t−
dV
t
= L
t−
dV
X
t
,
i.e.,
b
t
+ c
t
L
t
−
v
t
= L
t
−
v
X
t
.
Hence the form of b given in the theorem.
Comments 7.5.6.2 (a) It is worthwhile comparing this result with the
results of Subsection 2.4.5.
(b) See Bielecki et al.[92] for a more complete study of hedging strategies,
using a martingale approach, and trading strategies satisfying a
t
e
rt
+b
t
S
t
=0.
Rebate Part
The representation theorem also provides a hedging strategy for a rebate h
paid at hit. We assume that F is differentiable, with derivative f. We denote by
C
h
t
the price of the default-free contingent claim which consists of a dividend
hf paid between time t and T , i.e.,
C
h
t
e
−rt
= E
Q
T
t
e
−ru
f
u
h
u
du|F
t
,
and by μ
h
the associated hedging strategy:
C
h
t
e
−rt
+
t
0
e
−ru
f
u
h
u
du = C
h
0
+
t
0
μ
h
s
d(e
−rs
S
s
) .
Proposition 7.5.6.3 Let (a, b, c) be the hedging strategy for the rebate part,
i.e., the self-financing strategy such that
e
−rt
(a
t
e
rt
+ b
t
S
t
+ c
t
D(t, T )) = E
Q
(h
τ
1
{τ≤T }
e
−rτ
|G
t
) .
Then, on the set {t<τ}, one has c
t
D(t, T )=e
rt
E
Q
(h
τ
1
{τ≤T }
e
−rτ
|G
t
) − h
t
.
More precisely, the hedging strategy before the default time of the rebate part,
paid at hit, consists of:
(i) c
t
=
1
V
t
(C
h
t
− G
−1
t
h
t
),
(ii) b
t
= G
−1
t
μ
h
t
−
1
V
t
v
t
C
h
t
+
1
V
t
v
t
h
t
,
(iii) a
t
e
rt
= S
t
1
V
t
v
t
h
t
− G
−1
t
μ
h
t
+
1
V
t
v
t
C
h
t
+ h
t
where V and v are defined in (7.5.6).
Proof: We give the proof for the case r =0.
We compute the quantity E
Q
(h
τ
1
{τ≤T }
|G
t
), which corresponds to the
price of the rebate, when the compensation is paid at hit. The representation
theorem 7.5.5.2 states that
7.6 General Case: Without the (H)-Hypothesis 437
E
Q
(h
τ
1
{τ≤T }
|G
t
)=C
h
0
+
t∧τ
0
G
−1
u
μ
h
u
dS
u
+
[0,t∧τ[
(h
u
− J
u−
)dM
u
,
where, on the set {t<τ},
J
t
= E
Q
(h
τ
1
{τ≤T }
|G
t
)=G
−1
t
E
Q
T
t
h
u
dF
u
|F
t
= G
−1
t
C
h
t
.
The value of a DZC given in (7.5.5) can be written D(t, T )=1
{t<τ}
G
−1
t
V
t
.
In particular,
d
t
D(t, T )=−D(t−,T)dM
t
+ L
t
dV
t
= −D(t−,T)dM
t
+ L
t
v
t
dS
t
It follows that
E
Q
h
τ
1
{τ≤T }
|G
t
= C
h
0
+
t∧τ
0
G
−1
u
μ
h
u
dS
u
−
[0,t∧τ[
(h
u
− G
−1
u
C
h
u
)
1
D(u, T )
[d
u
D(u, T ) − v
u
L
u
dS
u
]
which leads to, using that D(t, T )=G
−1
t
V
t
on the set {t<τ},
E
Q
h
τ
1
{τ≤T }
|G
t
= V
h
0
+
t∧τ
0
G
−1
u
v
h
u
− C
h
u
v
h
u
V
u
+
v
u
h
u
V
u
dS
u
−
[0,t∧τ[
(h
u
− G
−1
u
C
h
u
)
1
G
u
V
u
d
u
D(u, T ) .
Comments 7.5.6.4 (a) Under the (H) hypothesis, the Kusuoka represen-
tation theorem and the form of the dynamics of a DZC imply that if the
default-free market is complete, the defaultable market is complete as soon as
a defaultable zero-coupon is traded.
(b) See also B´elanger et al. [67], Jeanblanc and Rutkowski [488]and
Bielecki et al. [89] for an extensive study of hedging strategies and Bielecki
et al. [90]and Section 7.9 for a PDE approach, based on Itˆo’s calculus for
processes with jumps.
7.6 General Case: Without the (H)-Hypothesis
7.6.1 An Example of Partial Observation
As pointed out by Jamshidian [476], “one may wish to apply the general
theory perhaps as an intermediate step, to a subfiltration that is not equal
438 7 Default Risk: An Enlargement of Filtration Approach
to the default-free filtration. In that case, F rarely satisfies hypothesis (H)”.
We present here a simple case of such a situation. Assume that
dV
t
= V
t
(μdt + σdW
t
),V
0
= v
i.e., V
t
= ve
σ(W
t
+νt)
= ve
σX
t
, with ν =(μ − σ
2
/2)/σ and X
t
= W
t
+ νt.We
denote by F
W
t
= σ(W
s
,s ≤ t) the natural filtration of the Brownian motion
(this is also the natural filtration of X).
The default time is assumed to be the first hitting time of α with α<v,
i.e.,
τ =inf{t : V
t
≤ α} =inf{t : X
t
≤ a}
where a = σ
−1
ln(α/v). Here, the reference filtration F is the filtration of the
observations of V at discrete times t
1
, ···t
n
where t
n
≤ t<t
n+1
, i.e.,
F
t
= σ(V
t
1
,...,V
t
n
,t
i
≤ t)
and we compute F
t
= P(τ ≤ t|F
t
). Let us recall that (See Subsection 3.2.2)
P(inf
s≤t
X
s
>z)=Φ(ν, t, z) , (7.6.1)
where
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Φ(ν, t, z)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
N
νt − z
√
t
− e
2νz
N
z + νt
√
t
, for z<0,t>0,
0, for z ≥ 0,t≥ 0,
Φ(ν, 0,z)= 1, for z<0.
We now divide the study into three steps:
On t<t
1
. In that case, for a<0, we obtain
F
t
= P(τ ≤ t)=P
inf
s≤t
X
s
≤ a
=1− Φ(ν, t, a)=N
a − νt
√
t
+ e
2νa
N
a + νt
√
t
.
On t
1
<t<t
2
.
F
t
= P(τ ≤ t|X
t
1
)=1− P(τ>t|X
t
1
)
=1− E
1
{inf
s<t
1
X
s
>a}
P
inf
t
1
≤s<t
X
s
>a|F
W
t
1
|X
t
1
The independence and stationarity of the increments of X yield
P
inf
t
1
≤s<t
X
s
>a|F
W
t
1
= Φ(ν, t − t
1
,a− X
t
1
) .
7.6 General Case: Without the (H)-Hypothesis 439
Hence
F
t
=1− Φ(ν, t − t
1
,a− X
t
1
)P(inf
s<t
1
X
s
>a|X
t
1
) .
From Exercise 3.2.2.1,forX
t
1
>a, we obtain (we omit the parameter ν in
the definition of Φ)
F
t
=1− Φ(t − t
1
,a− X
t
1
)
1 − exp
−
2 a
t
1
(a − X
t
1
)
. (7.6.2)
The case X
t
1
≤ a corresponds to default and, therefore, for X
t
1
≤ a, F
t
=1.
The process F is continuous and increasing in [t
1
,t
2
[.
When t approaches t
1
from above, one has lim
t→t
+
1
Φ(t − t
1
,a− X
t
1
)=1
for X
t
1
>a, hence F
t
+
1
=exp
−
2a
t
1
(a − X
t
1
)
.ForX
t
1
>a,thejumpofF
at t
1
is
ΔF
t
1
=exp
−
2a
t
1
(a − X
t
1
)
− 1+Φ(t
1
,a).
For X
t
1
≤ a, Φ(t − t
1
,a− X
t
1
) = 0 by the definition of Φ(·)and
ΔF
t
1
= Φ(t
1
,a).
General Observation Times t
i
<t<t
i+1
<T, i ≥ 2.
For t
i
<t<t
i+1
,
P(τ>t|X
t
1
,...,X
t
i
)=P
inf
s≤t
i
X
s
>aP
inf
t
i
≤s<t
X
s
>a|F
t
i
|X
t
1
,...,X
t
i
= Φ(t − t
i
,a− X
t
i
)P
inf
s≤t
i
X
s
>a|X
t
1
,...,X
t
i
.
We write K
i
for the second term on the right-hand side
K
i
= P
inf
s≤t
i
X
s
>a|X
t
1
,...,X
t
i
= P
inf
s≤t
i−1
X
s
>aP
inf
t
i−1
≤s<t
i
X
s
>a|F
t
i−1
∨ X
t
i
|X
t
1
,...,X
t
i
.
Obviously,
P
inf
t
i−1
≤s<t
i
X
s
>a|F
t
i−1
∨ X
t
i
= P
inf
t
i−1
≤s<t
i
X
s
>a|X
t
i−1
,X
t
i
=exp
−
2
t
i
− t
i−1
(a − X
t
i−1
)(a − X
t
i
)
.
Therefore,
K
i
= K
i−1
exp
−
2
t
i
− t
i−1
(a − X
t
i−1
)(a − X
t
i
)
. (7.6.3)