We denote by H = (Ht, t ∈ R+) the right-continuous increasing process Ht= 1{t≥τ }, referred to as the default indicator process. Let H stand for the natural filtration of the process H. It is clear that the filtration H is the smallest filtration which makes τ a stopping time. More explicitly, for any t ∈ R+, the σ-field Htis generated by the events {s ≥ τ } for s ≤ t. The key observation is that any Ht-measurable random variable X has the form
X = h(τ )1{t≥τ }+ c1{t<τ }, where h : R+→ R is a Borel measurable function and c is a constant.
Remark 2.1.3 It is worth mentioning that if the cumulative distribution function F is continuous then τ is known to be a totally inaccessible stopping time with respect to H (see, e.g., Dellacherie [57] or Dellacherie and Meyer [60], Page 107). We will not use this important property explicitly, however.
Our next goal is to derive some useful valuation formulae for defaultable bonds with differing recovery schemes.
For the sake of simplicity, we will first assume that a bond is represented by a single payoff at its maturity T . Therefore, it is possible to value a bond as a European contingent claim X maturing at T , by applying the standard risk-neutral valuation formula
πt(X) = B(t) EQ
µ X
B(T )
¯¯
¯ Ht
¶
= B(t, T ) EQ(X | Ht).
For the ease of notation, we will consider, without loss of generality, a defaultable bond with the face value L = 1.
Constant Recovery at Maturity
A defaultable (or corporate) zero-coupon bond (a DZC) with maturity T , unit par value, and recovery value δ paid at maturity, consists of:
• the payment of one monetary unit at time T if default has not occurred before T , i.e., if τ > T ,
• the payment of δ monetary units, made at maturity, if τ ≤ T , where δ ∈ [0, 1] is a constant.
The price at time 0 of the defaultable zero-coupon bond is formally defined as the expectation under Q of the discounted payoff, so that
Dδ(0, T ) = B(0, T ) EQ
¡1{T <τ }+ δ1{τ ≤T }¢ . Consequently,
Dδ(0, T ) = B(0, T ) − (1 − δ)B(0, T )F (T ).
The value of the defaultable zero-coupon bond is thus equal to the value of the default-free zero-coupon bond minus the discounted value of the expected loss computed under the risk-neutral probability. Of course, for δ = 1 we recover, as expected, the price of a default-free zero-coupon bond.
Obviously, the price defined above is not a hedging price, since the payoff at maturity of the defaultable bond cannot be replicated by trading in primary assets; recall that only default-free zero-coupon bonds are traded in the present setup. Therefore, we deal with an incomplete market model and the risk-neutral pricing formula for the defaultable zero-coupon bond is thus postulated, rather than derived from replication.
The value of the bond at any date t ∈ [0, T ] depends whether or not default has happened before this time.
On the one hand, if default has occurred before or at time t, the constant payment of δ will be made at maturity date T and thus the price of the DZC is obviously δB(t, T ).
On the other hand, if default has not yet occurred before or at time t, the date of its occurrence is uncertain. It is thus natural in this situation to define the ex-dividend price Dδ(t, T ) at time t ∈ [0, T [ of the DZC maturing at T as the conditional expectation under Q of the discounted payoff
B(t, T )¡
1{T <τ }+ δ1{τ ≤T }
¢ (2.1)
given the information, which is available at time t, that is, given the no-default event {τ > t}.
In view of specification (2.1) of the bond’s payoff, we thus obtain Dδ(t, T ) = 1{t≥τ }δB(t, T ) + 1{t<τ }Deδ(t, T ), where the pre-default value eDδ(t, T ), t ∈ [0, T ], is defined as
Deδ(t, T ) = EQ
¡B(t, T ) (1{T <τ }+ δ1{τ ≤T })¯
¯ t < τ¢ . To compute eDδ(t, T ), we observe that
Deδ(t, T ) = B(t, T )
³
1 − (1 − δ)Q(τ ≤ T¯
¯ t < τ)´
= B(t, T ) µ
1 − (1 − δ)Q(t < τ ≤ T ) Q(t < τ )
¶
= B(t, T ) µ
1 − (1 − δ)G(t) − G(T ) G(t)
¶
, (2.2)
where we denote G(t) = 1 − F (t). Let us define, for every t ∈ [0, T ], Bγ(t, T ) = B(t, T )G(T )
G(t) = exp³
− Z T
t
(r(u) + γ(u)) du´ . Then pre-default value of the bond can be represented as follows
Deδ(t, T ) = Bγ(t, T ) + δ¡
B(t, T ) − Bγ(t, T )¢ .
In particular, for δ = 0, that is, for the case of the bond with zero recovery, we obtain the equality De0(t, T ) = Bγ(t, T ), and thus the price D0(t, T ) satisfies
D0(t, T ) = 1{t<τ }De0(t, T ) = 1{t<τ }Bγ(t, T ).
It is worth noting that the value of the DZC is discontinuous at default time τ , since we have, on the event {τ ≤ T },
Dδ(τ, T ) − Dδ(τ −, T ) = δB(τ, T ) − eDδ(τ, T ) = (δ − 1)Bγ(t, T ) < 0,
where the last inequality holds for any δ < 1. Recall that for δ = 1 the DZC is simply a default-free zero-coupon bond.
For practical purposes, equality (2.2) can be rewritten as follows Deδ(t, T ) = B(t, T )(1 − LGD × DP),
where the loss given default (LGD) is defined as 1 − δ and the conditional default probability (DP) is given by the formula
DP = Q(t < τ ≤ T )
Q(t < τ ) = Q(τ ≤ T | t < τ ).
2.1. ELEMENTARY MARKET MODEL 39
If the hazard rate γ ≥ 0 is constant then the pre-default credit spread equals S(t, T ) =e 1
T − tln B(t, T )
Deδ(t, T ) = γ − 1 T − tln³
1 + δ(eγ(T −t)− 1)´ .
It is thus easily seen that the pre-default credit spread converges to the constant γ(1 − δ) when time to maturity T − t tends to zero. It is thus strictly positive when γ > 0 and 0 ≤ δ < 1.
Recall that for δ = 0, the equality eD0(t, T ) = Bγ(t, T ) is valid. Hence the short-term interest rate has simply to be adjusted by adding the credit spread (equal here to γ) in order to price DZCs with zero recovery using the formula for default-free bonds. The default-risk-adjusted interest rate equals b
r = r + γ and thus it is higher than the risk-free interest rate r if γ is positive. This corresponds to the real-life feature that the value of a DZC with zero recovery is strictly smaller than the value of a default-free zero-coupon with the same par value and maturity provided, of course, that the real-life probability of default event during the bond’s lifetime is positive.
General Recovery at Maturity
Let us now assume that the payment is a deterministic function of the default time, denoted as δ : R+→ R. Then the value at time 0 of this defaultable zero-coupon is
Dδ(0, T ) = B(0, T ) EQ
¡1{T <τ }+ δ(τ )1{τ ≤T }¢ or, more explicitly,
Dδ(0, T ) = B(0, T )
³ G(T ) +
Z T
0
δ(s)f (s) ds
´ ,
where, as before, G(t) = 1−F (t) stands for the survival probability. More generally, the ex-dividend price is given by the formula, for every t ∈ [0, T [,
Dδ(t, T ) = B(t, T ) EQ
¡1{T <τ }+ δ(τ )1{τ ≤T }¯
¯ Ht
¢.
The following result furnishes an explicit representation for the bond’s price in the present setup.
Lemma 2.1.2 The price of the bond satisfies, for every t ∈ [0, T [,
Dδ(t, T ) = 1{t<τ }Deδ(t, T ) + 1{t≥τ }δ(τ )B(t, T ), (2.3) where the pre-default value eDδ(t, T ) equals
Deδ(t, T ) = B(t, T ) EQ
¡1{T <τ }+ δ(τ )1{τ ≤T }¯
¯ t < τ¢
= B(t, T )G(T )
G(t) +B(t, T ) G(t)
Z T
t
δ(u)f (u) du
= Bγ(t, T ) +Bγ(t, T ) G(T )
Z T
t
δ(u)f (u) du
= Bγ(t, T ) + Bγ(t, T ) Z T
t
δ(u)γ(u)eRuTγ(v) dvdu.
The dynamics of the process ( eDδ(t, T ), t ∈ [0, T ]) are
d eDδ(t, T ) = (r(t) + γ(t)) eDδ(t, T ) dt − B(t, T )γ(t)δ(t) dt. (2.4) The proof of the lemma is based on straightforward computations. To derive the dynamics of Deδ(t, T ), it is useful to observe, in particular, that
dBγ(t, T ) = (r(t) + γ(t))Bγ(t, T ) dt.
The risk-neutral dynamics of the discontinuous process Dδ(t, T ) involve also the H-martingale M introduced in Section 2.2 below (see Example 2.2.2).