By the conditional independence of X1, . . . , Xn with respect to the common factor V , which repre-sents the market-wide (or systematic) credit risk, we thus obtain, for every t1, . . . , tn∈ R+,
Q(τ1≤ t1, . . . , τn ≤ tn) = Z
R
Yn i=1
N Ã
N−1(Fi(ti)) − ρv p1 − ρ2
!
n(v) dv,
where n is the probability density function of V . It is worth noting that the components Vi are aimed to represent the firm-specific (or idiosyncratic) part of the credit risk for individual names in a credit portfolio. For numerical issues arising in implementations of the Li model, see Joshi and Kainth [102] and Chen and Glasserman [47].
5.6 Jarrow and Yu Model
Jarrow and Yu [95] (see also Yu [144]) approach can be considered as another attempt to develop a dynamic approach to dependence between default times by modeling directly the contagion effect.
For a given finite family of reference credit names, Jarrow and Yu [95] propose to make a distinction between primary and secondary firms. At the intuitive level, the rationale for their approach can be summarized as follows:
• the class of primary firms encompasses these entities whose probabilities of default are influ-enced by macroeconomic conditions, but not by the credit risk of counterparties; the pricing of bonds and other defaultable securities issued by primary firms is feasible through the standard intensity-based methodology,
• it is thus sufficient to focus on defaultable securities issued by a secondary firm, that is, a firm for which the intensity of default depends explicitly on the status of some other firms.
Let {1, 2, . . . , n} represent the set of all firms in our model and let F stand for some reference filtration. Jarrow and Yu [95] formally postulate that:
• for any firm from the set {1, 2, . . . , k} of primary firms, the ‘default intensity’ depends only on a reference filtration F,
• the ‘default intensity’ for any credit name that belongs to the class {k + 1, k + 2, . . . , n} of secondary firms may depend not only on the filtration F, but also on the status (default or no-default) of the primary firms.
5.6.1 Construction of Default Times
First step. We first construct default times for all primary firms. To this end, we assume that we are given a family of F-adapted ‘intensity processes’ λ1, . . . , λk and we produce a collection τ1, . . . , τk
of F-conditionally independent random times through the canonical method, that is, we set τi= infn
t ∈ R+: Z t
0
λiudu ≥ − ln ξi
o
5.6. JARROW AND YU MODEL 179 where ξi, i = 1, 2, . . . , k are mutually independent and identically distributed random variables with the uniform distribution on [0, 1] under the martingale measure Q.
Second step. In the second step, we are going to construct default times corresponding to secondary firms. To this end, we assume that:
• the probability space (Ω, G, Q) is large enough to support a family ξi, i = k + 1, k + 2, . . . , n of mutually independent random variables, with uniform distribution on [0, 1],
• these random variables are independent not only of the filtration F, but also of the already constructed in the first step default times τ1, . . . , τk of primary firms.
The default times τi, i = k + 1, k + 2, . . . , n are also defined by means of the standard formula τi= inf
n
t ∈ R+: Z t
0
λiudu ≥ − ln ξi
o .
However, the ‘intensity processes’ λi for i = k + 1, k + 2, . . . , n are now given by the following expression
λit= µit+ Xk
l=1
νti,l1{t≥τl},
where µi and νi,l are F-adapted stochastic processes. In case where the default of the jth primary firm does not affect the ‘default intensity’ of the ith secondary firm, we set νi,j= 0.
Let G = F ∨ H1∨ . . . ∨ Hn stand for the enlarged filtration and let bF = F ∨ Hk+1∨ . . . ∨ Hn be the filtration generated by the reference filtration F and the observations of defaults of secondary firms. Then:
• the default times τ1, . . . , τk of primary firms are conditionally independent with respect to F,
• the default times τ1, . . . , τk of primary firms are no longer conditionally independent when we replace the filtration F by bF,
• in general, the default intensity of a primary firm with respect to the filtration bF differs from the intensity λiwith respect to F.
5.6.2 Case of Two Credit Names
To illustrate the credit contagion effect, we will now consider the case of only two credit names, A and B say, and we postulate that A is a primary firm, whereas B is a secondary firm.
Let the constant process λ1t= λ1 represent the F-intensity of default time for firm A, so that τ1= inf
n
t ∈ R+: Z t
0
λ1udu = λ1t ≥ − ln ξ1
o ,
where ξ1 is a random variable independent of F with the uniform distribution on [0, 1]. For the second firm, the ‘default intensity’ is assumed to satisfy
λ2t = λ21{t<τ1}+ α21{t≥τ1}
for some positive constants λ2 and α2. We set τ2= inf
n
t ∈ R+: Z t
0
λ2udu ≥ − ln ξ2
o ,
where ξ2is a random variable with the uniform probability distribution, independent of F, and such that ξ1 and ξ2 are mutually independent. The following result summarizes properties of processes Λ1 and Λ2.
Lemma 5.6.1 The following properties hold:
(i) the process Λ1 is the hazard process of τ1 with respect to F, (ii) the process Λ2 is the hazard process of τ2 with respect to F ∨ H1,
(iii) the process Λ1 is not the hazard process of τ1 with respect to F ∨ H2 if the inequality λ26= α2
holds.
Assume for simplicity that r = 0. We wish to price a defaultable zero-coupon bond with the default time τi and with constant recovery payoff δi. We thus need to compute the following conditional expectation, for i = 1, 2,
Diδi(t, T ) = EQ(1{τi>T }+ δi1{τi≤T }| Gt), (5.3) where Gt = Ht1∨ H2t. To this end, we will first find the joint probability distribution of the pair (τ1, τ2). Let us denote G(s, t) = Q(τ1> s, τ2> t). We write ∆ = λ1+ λ2− α2 and we assume that
∆ 6= 0.
Lemma 5.6.2 The joint distribution of (τ1, τ2) under Q is given by, for every 0 ≤ t ≤ s, Q(τ1> s, τ2> t) = e−λ1s−λ2t
and, for every 0 ≤ s < t,
Q(τ1> s, τ2> t) = 1
∆λ1e−α2t¡
e−s∆− e−t∆¢
+ e−(λ1+λ2)t.
Proof. Let ψi= − ln ξi. For t < s, we have λ2t = λ2t on the set {s < τ1}. The equalities {τ1> s} ∩ {τ2> t} = {Λ1s< ψ1} ∩ {Λ2t < ψ2} = {λ1s < ψ1} ∩ {λ2t < ψ2} and the independence of ψ1 and ψ2lead to
Q(τ1> s, τ2> t) = e−λ1s−λ2t.
In particular, by setting t = 0, we obtain the equality Q(τ1> s) = e−λ1s for every s ∈ R+. For t > s, we have that
{τ1> s} ∩ {τ2> t} = {{t > τ1> s} ∩ {τ2> t}} ∪ {{τ1> t} ∩ {τ2> t}}
and
{t > τ1> s} ∩ {τ2> t} = {t > τ1> s} ∩ {Λ2t < ψ2}
= {t > τ1> s} ∩ {λ2τ1+ α2(t − τ1) < ψ2}.
The independence between ψ1and ψ2implies that the random variable τ1is independent of ψ2(note that τ1= (λ1)−1ψ1). Consequently,
Q(t > τ1> s, τ2> t) = EQ
³
1{t>τ1>s}e−(λ2τ1+α2(t−τ1))
´
= Z t
s
e−(λ2u+α2(t−u))λ1e−λ1udu
= 1
λ1+ λ2− α2λ1e−α2t
³
e−(λ1+λ2−α2)s− e−(λ1+λ2−α2)t
´ . Denoting ∆ = λ1+ λ2− α2, it follows that
Q(τ1> s, τ2> t) = 1
∆λ1e−α2t¡
e−∆s− e−∆t¢
+ e−(λ1+λ2)t. In particular, for s = 0, we obtain
Q(τ2> t) = 1
∆
³ λ1
³
e−α2t− e−(λ1+λ2)t
´
+ ∆e−(λ1+λ2)t
´ .
This completes the proof. ¤
5.6. JARROW AND YU MODEL 181
Bonds with Non-Zero Recovery
In view of (5.3), to find the price D1δ1(t, T ), it suffices to compute
Q(τ1> T | Gt) = Q(τ1> T | H1t∨ H2t) = 1{t<τ1}Q(τ1> T | H2t) Q(τ1> t | H2t). Observe that
Q(τ1> T | Gt) = 1{t<τ1}
µ
1{t≥τ2}∂2G(T, τ2)
∂2G(t, τ2) + 1{t<τ2}G(T, t) G(t, t)
¶ .
Similarly, valuation of Dδ22(t, T ) follows from the computation of
Q(τ2> T | Gt) = 1{t<τ2}Q(τ2> T | H1t) Q(τ2> t | H1t), where, by symmetry, we have that
Q(τ2> T | Gt) = 1{t<τ2} µ
1{t≥τ1}∂1G(τ1, T )
∂1G(τ1, t) + 1{t<τ1}G(t, T ) G(t, t)
¶ .
By straightforward computations, we obtain the following pricing result for defaultable bonds with non-zero recovery.
Corollary 5.6.1 The prices of defaultable bonds equal, for every t ∈ [0, T ] Dδ11(t, T ) = 1{t≥τ1}δ1+ 1{t<τ1}¡
e−λ1(T −t)+ δ1(1 − e−λ1(T −t))¢ and
D2δ2(t, T ) = δ2+ (1 − δ2)1{t<τ2}
½
1{t≥τ1}e−α2(T −t) + 1{t<τ1} 1
λ1+ λ2− α2
³
λ1e−α2(T −t)+ (λ2− α2)e−(λ1+λ2)(T −t)´¾ .
Bonds with Zero Recovery
Assume that λ1+λ2−α26= 0 and that the bond is subject to the zero recovery scheme. We maintain the assumption that r = 0 so that B(t, T ) = 1 for t ≤ T . Therefore, we have D02(t, T ) = Q(τ2 >
T | H1t∨ H2t) and thus the general formula yields
D20(t, T ) = 1{t<τ2}Q(τ2> T | H1t) Q(τ2> t | H1t).
The following pricing result is an immediate consequence of Corollary 5.6.1.
Corollary 5.6.2 Assume that the recovery δ2= 0. Then D2(t, T ) = 0 on the event {t ≥ τ2}. On the event {t < τ2} we have
D20(t, T ) = 1{t<τ1} 1 λ1+ λ2− α2
³
λ1e−α2(T −t)+ (λ2− α2)e−(λ1+λ2)(T −t)
´
+ 1{t≥τ1}e−α2(T −t).