Cen-tral Limit Theorem, instead (in fact the DeMoivre-Laplace Limit Theorem for binomial random variables), which implies that the conditional distribution of L = I−1P
i1Yi
7.5 Default correlation in reduced-form models
Recall that in the reduced form models of Section 5.1, the default of obligor i occurs at the first jump of a point process Nti with intensity λit. If we simply consider the combined process
is the compensator for Nt. However, it is not true in general that Nt is itself a counting process, since its jumps can have a magnitude greater than one in the presence of simul-taneous defaults. Moreover, even if simulsimul-taneous defaults are not allowed in the model, starting with Cox processes Nti does not guarantee that Nt is itself a Cox process with intensity λ1t + . . . λIt. In the next sections, we consider more specific assumptions on the intensity processes in order to appropriately model the possibility of joint defaults.
7.5.1 Doubly stochastic models
As we have seen, the doubly stochastic assumption of Section 5.1 needs to be modified in the presence of several default events. Now, a doubly stochastic model of correlated defaults means a model whereby, conditioned on the background filtration (G)t , which in particular contains the realization of the multidimensional intensity process (λ1t, . . . , λIt), the processes Ni are independent inhomogeneous Poisson processes, each with intensityλit. That is, the only interdependence between default events occurs through the correlations prevailing between the intensity processes. Once the intensities are revealed, the remaining stochastic processes triggering default are independent. Although this assumption seems a natural generalization of the single-name doubly stochastic setup, it leads to difficulties in obtaining realistic correlations for default events.
As an example, consider the joint default probability for obligors i, j for a fixed time
horizon T . That is
pij := P [τi ≤ T, τj ≤ T ]
= E[1{τi≤T }1{τj≤T }] = E[E[1{τi≤T }1{τj≤T }|GT]]
= E[(1 − e−R0Tλisds)(1 − e−R0Tλjsds)] (7.26)
= pi+ pj + E[e−R0T(λis+λjs)ds] − 1, (7.27) where pi = 1 − E[e−R0Tλisds]. This achieves its maximum value for perfectly correlated intensities, that is λi = λj in which case the correlation between default events is
ρ = pij − p2i
pi(1 − pi) = 2pi+ E[e−2R0Tλisds] − 1 − p2i pi(1 − pi)
= Var[e−R0Tλisds]
pi(1 − pi) . (7.28)
To estimate the order of magnitude of this correlation for intensities given by diffusion processes, let us assume that the integral RT
0 λisds is normally distributed with mean µ and variance σ2, 0 ≤ σ µ 1. We then obtain that
ρ = 1 − p
p (eσ2 − 1) ∼ σ2
µ. (7.29)
Usually this is too low to account for empirical data for correlated defaults. The only way to remedy this situation within the reduced form setting is to allow for joint large jumps in the intensity processes.
Consider the model of Section 5.5.3 with mean-reverting intensities with exponentially distributed jumps. Let us suppose for simplicity that all firms have identical characteris-tics, that is, the parameters k, θ, J, c are firm independent. One can create the possibility of strong default correlations by letting some jumps affect all firms, while other jumps are firm-specific. Thus we write
dλit = k(θ − λit)dt + dZi+ dZ (7.30) where the collection {Zi, Z} are independent compound Poisson processes with identi-cally distributed jump sizes, but arriving at rates c(1 − ρ) and ρc respectively. In this picture, jumps that negatively affect all firms simultaneously arrive at rate ρc, while the idiosyncratic jumps arrive at the rate consistent with the total arrival rate of jumps to be c.
More generally, one can imagine that the common adverse changes in credit quality that arise from Z affect only a random collection of firms each time. One simple way to realize this idea is by supposing that a common jump affects a given firm with probability p. To be consistent with the previous idea, one changes the rate of Z to ρc/p and at the time of each jump of Z one introduces a set of independent Bern(p) random variables Xi.Then the SDE for intensities becomes
dλit= k(θ − λit)dt + dZi+ XidZ (7.31)
7.5. DEFAULT CORRELATION IN REDUCED-FORM MODELS 107 As p tends to 0, the model becomes one of independent defaults, but for p = 1, ρ = 1 defaults are not perfectly correlated. While in this case, the firms have identical intensities, conditional on the intensities defaults are independent.
7.5.2 Joint default events
As an alternative to the doubly stochastic framework above, Duffie and Singleton (1998) suggest a model in which there are J credit events, each triggered by the first jump of a Cox process bNtj with intensity bλjt. Each of the events consist of multiple defaults, for example, in the case of two obligors A and B the credit events are default for A, default for B and simultaneous default for A and B, with the respective intensities bλA, bλB and bλAB. For consistency with single-names intensities, we must have
λA = bλA+ bλAB
λB = bλB+ bλAB. (7.32)
In principle, for I firms, one needs to consider all the 2I possible subsets of I, leading to unmanageable complexity. One could try to consider only the number of defaults in a certain credit event, making no distinction between the actual firms involved. Besides being too restrictive on the possible dependency structure, this assumption fails to capture the fact that, over a non-infinitesimal time horizon, the intensity of a k-default event must also depend on the occurrence of events with a smaller number of defaults.
More fundamentally, this approach leads to joint defaults happening most likely at the same time, possibly involving a large number of firms, but without affecting the surviving firms. Amongst other things, this precludes the emergence of crises, that is, periods during which default intensities are higher for all firms.
7.5.3 Infectious defaults
The idea of a large default event spreading its influence to surviving firms is expressed in the models proposed by Davis and Lo (2001) [7] and Jarrow and Yu (2001) [24]. Here we consider obligors with initially identical intensities λ(i)0 = λ, which get uniformly increased by a risk enhancement factor a ≥ 1 at each default and thereafter decay to λ after an exponentially distributed time with parameter µ. The main drawback of this intuitively appealing model is that its joint distribution of default over a time horizon T is hard to calculate. Moreover, the resulting joint process for default indicators is not a Cox process, since we can obtain information about the default times by observing the joint intensities alone.
7.5.4 Hawkes Processes
These processes extend the concept of Cox process to allow an additional feedback from the counting process Nt to the intensity λt, allowing them to jump simultaneously. The
simplest Hawkes process is a pair λt, Nt satisfying the SDE
dλt= κ(c − λt)dt + δdNt, t > 0, (7.33) and N is a counting process with compensator Λt=Rt
0 λtdt. The solution to (7.33) involves a Stieltjes integral:
λt= c + (λ0− c)e−κt+ δ Z t
0
e−κ(t−s)dNs
The following proposition gives us the analytical tool we need to handle such processes.
Proposition 7.5.1. λt, Λt, Nt are jointly affine, and the joint characteristic function g(T, λ, u, v, w) := EλeuλT+vΛT+wNT
(7.34) is exponential-affine, of the form eA(T ,u,v,w)+B(T ,u,v,w)λ where A, B solve Riccati equations:
∂TA = κcB, A(0) = 0
∂TB = v − κcB + ew+δB− 1, B(0) = u
(7.35)
Proof: Consider the martingale
Ft := EeuλT+vΛT+wNT|Ft
= evΛt+wNtEeuλT+v(ΛT−Λt)+w(NT−Nt)|Ft := evΛt+wNtf (t, T, λt)
Note that f has the given form because of the Markov property of the model: the random variables λT, NT − Nt, ΛT − Λt depend only on λt, and not the path of λt up to time t.
Applying the Itˆo formula to the implied Stieltjes integrals yields the differential
dFt = evΛt+wNt
"
(∂tf + vλtf + κ(c − λt)∂λf + (ewf (t, λt−+ δ) − f (t, λt−)) dNt
#
By Doob-Meyer, dNt− λtdt is a martingale differential, and thus
dFt= evΛt+wNt(∂tf + vλtf + L[f ]) dt + Martingale (7.36) where the Markov generator operates on functions h(λ) as
L[h](λ) = κ(c − λ)∂λh(λ) + λewh(t, λ + δ) − h(t, λ) (7.37) Since F is a martingale, we must have vanishing drift:
∂tf + vλf + L[f ] = 0
7.6. DEFINITION OF COPULA FUNCTIONS 109