Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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410 7 Default Risk: An Enlargement of Filtration Approach
InthecasewhereF is continuous, Γ (t)=
t
0
dF (s)
G(s)
, and if F admits a
derivative f, the derivative of Γ is
γ(t)=
f(t)
G(t)
= lim
h→0
1
h P(τ>t)
P(t<τ≤ t + h)=P(τ ∈ dt|τ>t)/dt .
In this case,
G(t)=e
−Γ (t)
=exp
−
t
0
γ(s)ds
.
It follows that
d
t
D(t, T )=(r(t)+γ(t))
D(t, T )dt − P (t, T)γ(t)δ(t)dt ,
where the notation d
t
D(t, T ) denotes the differential of
D with respect to t.
Inthecasewhereδ =0
D(t, T ) = exp
−
T
t
(r + γ)(s)ds
.
Hence, the spot rate has to be adjusted by means of a spread (equal to γ)in
order to evaluate DZCs.
If γ and δ are constant, the credit spread s(t, T )is,ontheset{t<τ},
s(t, T )=
1
T − t
ln
P (t, T )
D(t, T )
= γ −
1
T − t
ln
1+δ(e
γ(T −t)
− 1)
and goes to γ(1 − δ)whent goes to T.
Remark 7.1.1.1 Note that, if τ is a random time with differentiable
cumulative distribution function F , setting Λ(t)=−ln(1 −F(t)) allows us to
interpret τ as the first jump time of an inhomogeneous Poisson process with
intensity the derivative of Λ. However, at this point, we insist that no hidden
Poisson process occurs in our framework, which only involves the time τ and
quantities related with it.
Exercise 7.1.1.2 Compute the dynamics of D.
7.1.2 Defaultable Zero-coupon with Payment at Hit
Here, a defaultable zero-coupon bond with maturity T consists of:
• The payment of one monetary unit at time T if (and only if) default has
not yet occurred.
• Apaymentofδ(τ) monetary units, where δ is a deterministic function,
made at time τ if (and only if) τ<T.
7.1 A Toy Model 411
Value of a Defaultable Zero-coupon
Obviously, if the default has occurred before time t, the value of the DZC is
null (this was not the case for payment of the rebate at maturity). Therefore,
D(t, T )=1
{t<τ}
D(t, T )where
D(t, T ) is a deterministic function, called the
predefault price. The predefault time-t value
D(t, T ) satisfies
D(t, T )=E(P (t, T ) 1
{T<τ}
+ P (t, τ)δ(τ)1
{τ≤T }
|t<τ)
=
P(T<τ)
P(t<τ)
P (t, T )+
1
P(t<τ)
T
t
P (t, s)δ(s)dF(s) .
Hence,
G(t)
D(t, T )=G(T )P (t, T ) −
T
t
P (t, s)δ(s)dG(s) .
The process t → D(t, T ) admits a discontinuity at time τ and the size of the
jump is −
D(τ,T).
A Particular Case
If F is differentiable,
D(t, T )=P
d
(t, T )+
T
t
P
d
(t, s)δ(s)γ(s)ds
with P
d
(t, s) = exp
−
s
t
[r(u)+γ(u)]du
. The defaultable interest rate is
r + γ and is, as expected, greater than r (the value of a DZC with δ =0is
smaller than the value of a default-free zero-coupon). The dynamics of
D(t, T )
are
d
t
D(t, T )=
(r(t)+γ(t))
D(t, T ) − δ(t)γ(t)
dt .
It is interesting to recall (see Subsection 2.3.5)that,ifX is the price of an
asset which pays a deterministic dividend rate κ(t) in a financial market with
a deterministic interest rate ρ,then
dX(t)=ρ(t)X(t)dt − κ(t)dt .
The dynamics of the predefault price can be interpreted as the price in a
default-free market of an asset paying dividend rate γ(t)δ(t), with interest rate
ρ = r + γ.Thequantityγ(t) in the dividend rate represents the probability
that the dividend δ(t) is paid in the time interval [t, t + dt].
412 7 Default Risk: An Enlargement of Filtration Approach
7.2 Toy Model and Martingales
We now present the results of the previous section in a different form, following
closely Dellacherie ([240], page 122). We denote by (D
t
,t ≥ 0) the right-
continuous increasing process D
t
= 1
{t≥τ}
and by D its natural filtration.
The filtration D is the smallest filtration which makes τ a stopping time (we
work with the usual completed filtration). The σ-algebra D
t
is generated by
the sets {τ ≤ s} for s ≤ t (or by the r.v. τ ∧ t). As a consequence, and
this is a key point, every D
t
-measurable random variable H is of the form
H = h(τ ∧ t)=h(τ )1
{τ≤t}
+ h(t)1
{t<τ}
where h is a Borel function.
We assume in this section that the cumulative distribution function F is
continuous.
7.2.1 Key Lemma
We now give some elementary tools to compute the conditional expectation
w.r.t. D
t
, as presented in Br´emaud [124], Dellacherie [240], and Elliott [313].
Lemma 7.2.1.1 If X is any integrable, G-measurable r.v.
E(X|D
s
)1
{s<τ}
= 1
{s<τ}
E(X1
{s<τ}
)
P(s<τ)
. (7.2.1)
Proof: The r.v. E(X|D
s
)isD
s
-measurable. Therefore, it can be written in
the form E(X|D
s
)=h(τ ∧s)=h(τ)1
{s≥τ}
+ h(s)1
{s<τ}
for some function h.
By multiplying both sides by 1
{s<τ}
, and taking the expectation, we obtain
E[1
{s<τ}
E(X|D
s
)] = E[ E(1
{s<τ}
X|D
s
)] = E[1
{s<τ}
X]
= E(h(s)1
{s<τ}
)=h(s)P(s<τ) .
Hence, h(s)=
E(X1
{s<τ}
)
P(s<τ)
, which is the desired result.
Remark 7.2.1.2 If the cumulative distribution function F is continuous,
then τ is a D-totally inaccessible stopping time. (See Dellacherie and Meyer
[244] IV, 107.)
7.2.2 The Fundamental Martingale
Proposition 7.2.2.1 The process (M
t
,t≥ 0) defined as
M
t
= D
t
−
τ∧t
0
dF (s)
G(s)
= D
t
−
t
0
(1 − D
s
)
dF (s)
G(s)
= D
t
− Γ (t ∧ τ )
is a D-martingale.
7.2 Toy Model and Martingales 413
Proof: Let s<t. Then, an application of (7.2.1) with X = 1
{τ≤t}
yields
E(D
t
− D
s
|D
s
)=1
{s<τ}
E(1
{s<τ≤t}
|D
s
)=1
{s<τ}
F (t) − F (s)
G(s)
, (7.2.2)
On the other hand, the quantity
C := E
t
s
(1 − D
u
)
dF (u)
G(u)
D
s
,
is given by
C =
t
s
dF (u)
G(u)
E
1
{τ>u}
D
s
= 1
{τ>s}
t
s
dF (u)
G(u)
1 −
F (u) − F (s)
G(s)
= 1
{τ>s}
F (t) − F (s)
G(s)
which, from (7.2.2) proves the result.
7.2.3 Hazard Function
From Proposition 7.2.2.1, the Doob-Meyer decomposition of the submartin-
gale D is M
t
+ Γ (t ∧ τ). The predictable increasing process A
t
= Γ (t ∧ τ )
is called the compensator of D (it is the dual predictable projection of the
increasing process D, see Section 5.2).
We now assume that F is differentiable with derivative f, therefore the
process
M
t
= D
t
−
τ∧t
0
γ(s)ds = D
t
−
t
0
γ(s)(1 − D
s
)ds
is a martingale; the deterministic positive function γ(s)=
f(s)
1 − F (s)
is called
the intensity of τ. We can now write the dynamics of the value of a
defaultable zero-coupon bond with recovery δ paid at hit.
Proposition 7.2.3.1 The dynamics of a DZC with recovery δ paid at hit is
d
t
D(t, T )=(r(t)D(t, T ) − δ(t)γ(t)(1 − D
t
)) dt −
D(t, T )dM
t
. (7.2.3)
Proof: From D(t, T )=1
{t<τ}
D(t, T )=(1− D
t
)
D(t, T ) and the dynamics
of
D(t, T ), we obtain, from the integration by parts formula,
d
t
D(t, T )=(1− D
t
)d
D(t, T ) −
D(t, T )dD
t
=(1− D
t
)
(r(t)+γ(t))
D(t, T ) − δ(t)γ(t)
dt −
D(t, T )dD
t
=(r(t)D(t, T) − δ(t)γ(t)(1 − D
t
)) dt −
D(t, T )dM
t
.
414 7 Default Risk: An Enlargement of Filtration Approach
We can also write (7.2.3)as
d
t
D(t, T )=(r(t)D(t, T ) − δ(t)γ(t)(1 − D
t
)) dt − D(t
−
,T)dM
t
.
We have just detailed the additive decomposition of (D
t
,t≥ 0),orof1− D;
here now is the multiplicative decomposition of (1 − D
t
,t≥ 0).
Proposition 7.2.3.2 The process L
t
:=1
{τ>t}
exp(Γ (t)) is a D-martingale.
In particular, for t<T,
E(1
{τ>T}
|D
t
)=1
{τ>t}
exp
−
T
t
γ(s)ds
.
The multiplicative decomposition of the supermartingale 1 − D is
1 − D
t
= L
t
exp (−Γ (t))
Proof: We shall give three different arguments, each of which constitutes a
proof.
a) Since the function γ is deterministic, for t>s
E(L
t
|D
s
)=exp(Γ (t)) E(1
{t<τ}
|D
s
) .
From the equality (7.2.1)
E(1
{t<τ}
|D
s
)=1
{τ>s}
G(t)
G(s)
= 1
{τ>s}
exp (−Γ (t)+Γ (s)) .
Hence,
E(L
t
|D
s
)=1
{τ>s}
exp (Γ (s)) = L
s
.
b) Another method is to apply the integration by parts formula for
bounded variation processes (see Subsection 8.3.4 if needed) to the process
L
t
=(1− D
t
)exp(Γ (t)) .
Then,
dL
t
= −e
Γ (t)
dD
t
+ γ(t)e
Γ (t)
(1 − D
t
)dt
= −e
Γ (t)
dM
t
.
c) A third (more sophisticated) method is to note that L is the exponential
martingale of M (see Subsection 8.4.4), i.e., the solution of the SDE
dL
t
= −L
t−
dM
t
,L
0
=1.
7.2 Toy Model and Martingales 415
Exercise 7.2.3.3 In this exercise, F is only assumed continuous on the right,
and G(t
−
) is the left-limit of G at point t. Prove that the process (M
t
,t≥ 0)
defined as
M
t
= D
t
−
τ∧t
0
dF (s)
G(s−)
= D
t
−
t
0
(1 − D
s−
)
dF (s)
G(s−)
is a D-martingale.
7.2.4 Incompleteness of the Toy Model, non Arbitrage Prices
In order to study the completeness of the financial market, we first need to
define the tradeable assets.
If the market consists only of the risk-free zero-coupon bond, the market
is incomplete (one cannot hedge defaultable claims, i.e., the payoff of which
belongs to D
T
), hence, there exists infinitely many e.m.m’s. The discounted
asset prices are constant, hence the set Q of equivalent martingale measures
is the set of probabilities equivalent to the historical one. For every Q ∈Q,
we denote by F
Q
the cumulative distribution function of τ under Q, i.e.,
F
Q
(t)=Q(τ ≤ t) .
Assuming that F
P
= P(τ ≤ t) is continuous, then, since Q ∼ P, F
Q
is also
continuous. The range of prices is defined as the set of prices which do not
induce arbitrage opportunities. In the case of constant interest rate, the range
of prices for a contingent claim H to be delivered at maturity is equal to the
interval
]inf
Q∈Q
e
−rT
E
Q
(H), sup
Q∈Q
e
−rT
E
Q
(H)[
For a DZC with a constant rebate δ paid at maturity, the range of prices is
equal to the set
{E
Q
(e
−rT
(1
{T<τ}
+ δ1
{τ≤T }
)), Q ∈Q}.
This set is exactly the interval ]δe
−rT
,e
−rT
[. Indeed, it is obvious that the
range of prices is included in the interval ]δe
−rT
,e
−rT
[. Now, in the set Q,
one can select a sequence of probabilities Q
n
which converges weakly to the
Dirac measure at point 0 (resp. at point T) (the bounds are obtained as limit
cases: the default appears at time 0
+
, or never).
7.2.5 Predictable Representation Theorem
Every square integrable D-martingale terminates at h(τ ), hence is of the form
E(h(τ)|D
t
). We now prove that this martingale can be written as a stochastic
integral with respect to the fundamental martingale M.
416 7 Default Risk: An Enlargement of Filtration Approach
Proposition 7.2.5.1 The martingale H
t
= E(h(τ )|D
t
) admits the represen-
tation
E(h(τ)|D
t
)=E(h(τ)) −
t∧τ
0
(H
s
−
− h(s)) dM
s
.
Consequently, every square integrable D-martingale (X
t
≥ 0) can be written
as X
t
= X
0
+
t
0
x
s
dM
s
where (x
s
,s≥ 0) is a D-predictable process.
Proof: From Lemma 7.2.1.1
H
t
= h(τ )1
{τ≤t}
+ 1
{t<τ}
E(h(τ)1
{t<τ}
)
P(t<τ)
= h(τ )1
{τ≤t}
+ 1
{t<τ}
(G((t))
−1
E(h(τ)1
{t<τ}
)
=
t
0
h(s)dD
s
+ 1
{t<τ}
(G(t))
−1
∞
t
h(s)f(s)ds .
It remains to apply integration by parts (see equation 9.1.1 ) to obtain
dH
t
=(h(t) − H
t
−
)(dD
t
− (1 − D
t
)
f(t)
G(t)
dt)=(h(t) − H
t
−
)dM
t
7.2.6 Risk-neutral Probability Measures
If DZCs with rebate paid at maturity are traded, their prices are given by the
market, and the equivalent martingale measure Q, chosen by the market, is
such that, on the set {t<τ},
D(t, T )=P (t, T )E
Q
[1
T<τ
+ δ1
t<τ≤T
]
t<τ
.
Therefore, we can characterize the cumulative distribution function of τ under
Q from the market prices of the DZC as follows.
Zero Recovery
If a DZC of maturity T with zero recovery is traded at a price D(t, T )which
belongs to the interval ]0,P(t, T)[ , then, under any risk-neutral probability
Q, the discounted value of D(t, T) is a martingale (for the moment, we do
not know that the market is complete, so we cannot claim that the e.m.m. is
unique), and the following equality holds
D(t, T )=E
Q
(P (t, T )1
{T<τ}
|D
t
)=P (t, T )1
{t<τ}
exp
−
T
t
λ
Q
(s)ds
7.2 Toy Model and Martingales 417
where λ
Q
(s)=
dF
Q
(s)/ds
1 − F
Q
(s)
.ItisobviousthatifD(t, T ) belongs to the range
of viable prices ]0,P(t, T )[, then the function λ
Q
is strictly positive (and the
converse holds true). The process D
t
−
t∧τ
0
λ
Q
(t)dt is a Q-martingale and
λ
Q
is the Q-intensity of τ. Therefore, the value of
T
t
λ
Q
(s)ds is known for
every T as long as there are DZC bonds for each maturity, and the unique
risk-neutral intensity can be obtained from the prices of DZCs as
r(t)+λ
Q
(t)=−
∂
∂T
ln D(t, T )|
T =t
.
Remark 7.2.6.1 It is important to note that there is no relation between the
risk-neutral intensity and the historical one. The risk-neutral intensity can be
greater (resp. smaller) than the historical one. The historical intensity can be
deduced from observations of default time, the risk-neutral one is obtained
from the prices of traded defaultable claims.
Fixed Payment at Maturity
If the prices of DZCs with different maturities and the same δ are known,
then from (7.1.1)
P (0,T) − D(0,T)
P (0,T)(1 − δ)
= F
Q
(T )
where F
Q
(t)=Q(τ ≤ t), so that the law of τ is known under the e.m.m..
However, as noticed in Hull and White [454], extracting default probabilities
from bond prices is in practice, usually complicated. First, the recovery rate is
usually non-zero. Second, most corporate bonds are not zero-coupon bonds.
Payment at Hit
In this case the cumulative function can be obtained using the derivative of
the defaultable zero-coupon price with respect to the maturity. Indeed, writing
for short ∂
T
D for
∂
∂T
D, the partial derivative of the value of the DZC at time
0 with respect to the maturity, and assuming that G =1−F is differentiable,
we obtain
∂
T
D(0,T)=g(T )P (0,T) − G(T )P (0,T)r(T ) − δ(T )g(T )P (0,T) ,
where g(t)=G
(t). Therefore, solving this equation leads to
Q(τ>t)=G(t)=Δ(t)
1+
t
0
∂
T
D(0,s)
1
P (0,s)(1 − δ(s))
(Δ(s))
−1
ds
,
where Δ(t) = exp
t
0
r(u)
1 − δ(u)
du
.
418 7 Default Risk: An Enlargement of Filtration Approach
7.2.7 Partial Information: Duffie and Lando’s Model
Duffie and Lando [273] study the case where τ =inf{t : V
t
≤ m} for V a
diffusion process which satisfies
dV
t
= μ(t, V
t
)dt + σ(t, V
t
)dW
t
.
Here the process W is a Brownian motion. If the information at hand is the
Brownian filtration, and if V is adapted w.r.t. W (i.e., is a strong solution of
the SDE), the time τ is a stopping time w.r.t. this Brownian filtration F,and
hence is predictable and does not admit an intensity, i.e., there is no process
λ such that D
t
−
t
0
(1 − D
s
)λ
s
ds is an F-martingale. We assume now that
the agents do not observe the process V , but they have only the minimal
information D
t
, i.e., they know when the default appears. We assume for
simplicity a null interest rate. In the case where the cumulative distribution
of the hitting time τ admits a derivative f, we have established above that the
value of a defaultable zero-coupon of maturity T is, when the default has not
yet occurred, exp
−
T
t
λ(s)ds
where λ(s)=
f(s)
1−F (s)
. A more general case is
presented in Subsection 7.6.1.
Remark 7.2.7.1 Duffie and Lando showed that the value of the DZC is
exp(−
T
t
λ(s)ds), where
λ(t)=
1
2
σ
2
(t, 0)
∂ϕ
∂x
(t, 0)
and ϕ(t, x) is the conditional density of V
t
when T
0
>t, i.e., the differential
w.r.t. x of
P(V
t
≤ x, T
0
>t)
P(T
0
>t)
,whereT
0
=inf{t ; V
t
=0}.Eveninthecase
where V is a homogeneous diffusion, i.e., dV
t
= μ(V
t
)dt + σ(V
t
)dW
t
,the
equality between Duffie and Lando’s result and ours is not obvious. See Elliott
et al. [315] for a proof based on time reversal properties.
7.3 Default Times with a Given Stochastic Intensity
We now present a case where some additional information is available in the
market, i.e., a stochastic process plays the rˆole of the hazard function. We
construct a default time from this process on an enlarged probability space.
7.3.1 Construction of Default Time with a Given Stochastic
Intensity
Let (Ω,G, F, P) be a filtered probability space. A positive F-adapted process λ
is given. We assume that there exists a r.v. Θ, constructed on Ω, independent
of F
∞
, with the exponential law of parameter 1: P(Θ ≥ t)=e
−t
. We define
7.3 Default Times with a Given Stochastic Intensity 419
the random time τ as the first time when the process (Λ
t
:=
t
0
λ
s
ds, t ≥ 0)
is above the random level Θ, i.e.,
τ =inf{t ≥ 0:Λ
t
≥ Θ}.
In particular, {τ ≥ s} = {Λ
s
≤ Θ}. We assume here that Λ
t
< ∞, ∀t, and
that Λ
∞
= ∞.
We shall refer to this construction as the Cox process model.
Comment 7.3.1.1 The choice of an exponential law for the r.v. Θ has no
real importance. In Wong [849], the time of default is given as
τ =inf{t : Λ
t
≥ Σ}
where Σ a non-negative r.v. independent of F
∞
. Under some regularity
assumptions, this model reduces to the previous one as we discuss now. We
recall that if X is a r.v. with continuous distribution function F ,ther.v.
F (X) is uniformly distributed on [0, 1]. Let Ψ(x)=1−e
−x
be the cumulative
distribution function of the exponential law. If Φ, the cumulative distribution
function of Σ, is strictly increasing, the r.v. Φ(Σ) is uniformly distributed and
Θ = Ψ
−1
(Φ(Σ)) is an exponential r.v. Then,
τ =inf{t : Φ(Λ
t
) ≥ Φ(Σ)} =inf{t : Ψ
−1
[Φ(Λ
t
)] ≥ Θ}
and
F
t
= P(τ ≤ t|F
t
)=P(Λ
t
≥ Σ|F
t
)=1− exp
−Ψ
−1
(Φ(Λ
t
))
.
It remains to write Ψ
−1
(Φ(Λ
t
)) as
t
0
γ
s
ds to recover the previous case. The
case where Φ is not strictly increasing has to be solved carefully, see for
example B´elanger et al. [67].
7.3.2 Conditional Expectation with Respect to F
t
Lemma 7.3.2.1 The conditional distribution function of τ given the σ-
algebra F
t
is, for t ≥ s
P(τ>s|F
t
)=exp(−Λ
s
) .
Proof: The proof follows from the equality {τ>s} = {Λ
s
<Θ}. From the
independence assumption and the F
t
-measurability of Λ
s
for s ≤ t, we obtain
P(τ>s|F
t
)=P(Λ
s
<Θ|F
t
)=exp(−Λ
s
).
In particular, for t ≥ s, P(τ>s|F
t
)=P(τ>s|F
s
), and, letting t →∞,
we obtain
G
s
:=P(τ>s|F
s
)=P(τ>s|F
∞
)=e
−Λ
s
. (7.3.1)
Here, the Az´ema supermartingale P(τ>t|F
t
) is a decreasing process.