We are now interested in the evaluation of thedefaultable zero coupon. A defaultable zero coupon is the financial product where an investor receives 1 monetary unit at maturity T if no default occurs before T and 0 otherwise. Here, we don’t take into account the discounting impact and the risk-neutral evaluation. Hence the conditional probabilityP(τ > T|Gt) is the key term to evaluate. By previous discussions, we know that thisG-martingale coincides on the set {τ > t} with someF-adapted process and it can be calculated by
P(τ > T|Gt) = 11{τ >t}P(τ > T|Ft)
P(τ > t|Ft) = 11{τ >t}E[GT|Ft] Gt
= 11{τ >t}GeTt ,
where GeTt = E[GT|Ft]/Gt. Here the intensity is not the suitable tool to study the problem.
1.4.1 General Framework and abstract HJM
The multiplicative decomposition of G=L D in terms of the F-local martingale L= E(MfeΓ) and the F-predictable decreasing process D = E(−ΛF) is the natural tool to studyGeTt under the following assumption:
Hypothesis 1.4.1 Assume that ∆ΛF 6= 1, and that the exponential martingale L= E(MfeΓ) is strictly positive on [0, T]. Then, the change of probability measure
dQL=LT dP on FT (1.12)
is well-defined.
Remark 1.4.2 When the (H)-hypothesis holds between the filtrations F and G, the F-survival processG is the exponential of an adapted decreasing process −Φ. So we can work directly under the initial probability measureP, that is LT = 1.
Forward hazard process
We now study properties of GeTt in both directions, as a function of T or as a semi-martingale ont. Given the multiplicative decomposition of G, we have
GeTt = E
GT|Ft Gt = E
LT DT|Ft
LtDt =EQL
DT,t|Ft
where DT,t = DDT
t is FT-measurable. Since D is the Dol´eans-Dade exponential of
−ΛF, under our hypothesis,D is decreasing, and we can introduce another predictable increasing processLF such that
exp(−LFt) =E(−ΛFt) =Dt.
By analogy with zero-coupon modelling, we introduced the process ΓTt defined as the parallel of the logarithm of “defaultable zero-coupon bond”. This process is known as theforward hazard process
ΓTt =−lnGeTt =−lnEQL
exp −(LFT − LFt)
|Ft
. (1.13)
Then we have the abstract version of HJM framework for the G-survival probability before the default time.
Proposition 1.4.3 (Abstract HJM) We take previous notations.
1) For any t≤T, the process P(τ > T|Gt) = 11{τ >t}e−ΓTt, t≥0
is aG-martingale.
2) If Λ is continuous, then the process 11{τ >t}eΛt = 11{τ >t}eΛFt∧τ, t ≥ 0
is a G -martingale.
3) The process GbT defined byGbTt = exp−(LFt + ΓTt ) =EQL[DT|Ft]is a QL-martingale on [0, T] with respect to the filtration F.
4) Assume the processΛF to be absolutely continuous, withF-intensity processλregular enough to make valid differentiation under QL expectation.
Then the semimartingale process ΓT is also absolutely continuous with respect to T. Its “derivative” process is the F-forward intensity process γ(t, T) such that
exp
−RT
0 λu11[0,t](u) +γ(t, u)11[t,T](u) du
, t≥0
is an F-martingale on [0, T] with respect to the QL probability measure.
Proof. 1) is obvious by definition.
2) By the multiplicative decomposition, St = 11{τ >t} =E(−Λt)E(−Nt) where Λ is the G-compensator process ofτ andN is theG-martingale in the Doob-Meyer decomposi-tion ofS. If Λ is continuous, thenE(−Λt) = exp(−Λt). So (11{τ >t}eΛt =E(−Nt), t≥0) is a G-martingale.
3) is direct by (1.13).
4) Since ΛF is absolutely continuous and is of density λ, E(−ΛFt) = exp −Rt
0λudu . Then by definition, ΓTt =−lnEQL
e−RtTλuduFt
is absolutely continuous with respect to T. If we denote by γ(t, T) the F-forward intensity process — the derivative of ΓTt with respect toT, then the process defined in 3) is nothing but GbT, which implies the
desired result. 2
1.4.2 HJM model in the Brownian framework
As been shown above, the zero coupon prices gives us the necessary information on the G-conditional probability P(τ > T|Gt) for all T > t. We now explicate this point in the classical context of the HJM model. In the following of this subsection, we assume that the filtration F is generated by a QL Brownian motion Wc and that the F-predictable process ΛF is absolutely continuous. In this case, the processGand its martingale part L are continuous. In addition, we suppose that the Hypothesis 1.3.1 holds for the filtrations (F,G).
HJM approach was first developed by Heath, Jarrow and Morton [47] to describe the dynamics of the term structure of interest rate. Modelling the whole family of interest rate curves for all maturities appears to be a difficult problem with infinite dimension. While in fact under the condition of absence of arbitrage opportunity, the dynamics of the forward rate is totally determined by the short-term rate today and the volatility coefficient. Application of this approach on the credit study is introduced by Jarrow and Turnbull [50], Duffie [26] and Duffie and Singleton [29]. In Sch¨onbucher [72] and Bielecki and Rutkowski [9], one can find related descriptions of this approach applied to the defaultable term structure and defaultable bond of a single credit.
Sch¨onbucher [72] used the HJM framework to represent the term structure of the defaultable bond and give the arbitrage-free conditions. In the following, we proceed from a different point of view. First, we work under the modified probability QL. Thus we proceed similarly as in the interest rate modelling. On the other hand, as mentioned above, the only difference when the (H)-hypothesis holds is that there is no need to change the probability. Therefore, we can deal with the general case without extra effort than in the special case with (H)-hypothesis. Second, instead of supposing the dynamics of the forward rate, we here suppose to know the dynamics of theF-martingale GbT under the probabilityQL. This is exactly as in the interest rate modelling where the discounted zero coupon price has a martingale representation.
We deduce the HJM model under the probability QL. Then it’s easy to obtain GTt by multiplying the discounted factor Dt. We then deduce the dynamics of the F-survival processGand thus the G-conditional probabilityP(τ > T|Gt).
Proposition 1.4.4 Assume that for any T > 0, the process (GbTt ,0≤t≤T) satisfies the following equation:
dGbTt
GbTt = Ψ(t, T)dcWt (1.14)
where (Ψ(t, T), t ∈ [0, T]) is an F-adapted process which is differentiable with respect toT and cW is a Brownian motion under the probabilityQL. If, in addition, ψ(t, T) =
∂
∂TΨ(t, T) is bounded uniformly on (t, T), then we have 1)
GbTt =GbT0 exp Z t
0
Ψ(s, T)dcWs−1 2
Z t
0 |Ψ(s, T)|2ds
(1.15) 2)
b
γ(t, T) =bγ(0, T)− Z t
0
ψ(s, T)dWcs+ Z t
0
ψ(s, T)Ψ(s, T)∗ds. (1.16) 3) We haveΨ(u, u) = 0 and
Gbt = exp
− Z t
0 bγ(s, s)ds
. (1.17)
4)
b
γ(t, T) =bγ(T, T) + Z T
t
ψ(s, T)dcWs− Z T
t
ψ(s, T)Ψ(s, T)∗ds. (1.18) Proof. 1) The first equation is the explicit form of the solution of equation (1.14).
2) By definition,bγ(t, T) is obtained by taking the derivative of−lnGbTt with respect to T. Then combining (1.15), we get
b
3) Equality (1.15) implies that lnGbt = lnGbt0+
Moreover, we have from equation (1.16) that Z t Combining (1.19) and (1.20) we get
Gbt = exp
On the other side, Gb = D is a decreasing process, so its martingale part vanishes, which implies that Ψ(t, t) = 0 for anyt≥0.
4) is a direct result from 2). 2
Remark 1.4.5 The above result can be applied directly to the first default time in the multi-credits case since the condition we need for the filtrations (F,G) is the general Hypothesis 1.3.1 which is fulfilled in this case.