2.4 Back to CSO tranche pricing: computing the expected
portfolio loss
We shall now present three numerical methods in order to
evaluate the expected tranche loss function E[M(t)], Vt E [0, T]. We stress the
fact that the following methods apply within the framework of the one factor
gaussian copula model to a finite heterogeneous portfolio (i.e. in terms of
individual nominal weights and recovery rates) of obligors with deterministic
recovery rates. We shall not detail the well known analytic results that can be
derived from Large Homogeneous Portfolios (LHP).
2.4.1 Monte-Carlo simulations
The Monte-Carlo approach is probably the most straightforward
method to price a CSO tranche fair premium:
1. Simulate N +1 independent standard gaussian variables (Z,
Z1, ..ZN) by using, for instance, the Box-Müller transform, the polar
method or even by drawing in the random variable Ö-1(U), where
U is a uniform random variable; within the one factor gaussian copula model,
the random vector (X1, .., XN) = ( /ñZ + /1 - ñZ1, .., /ñZ
+ /1 - ñZN) is therefore a gaussian vector with correlation matrix R.
2. Simulate (ô1, .., ôN) by drawing in the random
vector
(Q-1
1 (Ö(X1)), .., Q-1
N (Ö(XN))
3. Evaluate the tranche loss function M(t), Vt E [0, T] along
this loss scenario;
4. Repeat steps 1 & 2 and evaluate the tranche loss function
along this new loss scenario;
5. Loop on step 4 until you feel comfortable (confidence
interval or variance criteria) with the convergence of the empirical estimate
of E(M(t)), Vt E [0, T];
6. Evaluate the CSO tranche fair premium detailed in equation
(2.3). Such a simple anf flexible method comes at a high computation cost
though, because one has to draw millions of random variables in the case of a
reasonably large portfolio (between 100 and 200 obligors).
2.4.2 Evaluating the loss characteristic function
We first determine the expression of pj(t|Z), the cumulative
distribution function of default time ôj, j = 1..N conditional on the
common factor Z.
?j ? {1, .., N}, ?t ? [0, T], pj(t|Z) : = P(ôj = t|Z)
= P (Q6 1(Ö(Xj)) t|Z)
~ = PQ-1(Ö(vñZ + ñZj)) = t|Z)
= P (Zj G 4Ö-1 (Qj(t)) v
ñZ |Z)
(Ö-1(Qj(t)) ? ?ñZ
v1 ? ñ )
- ñ
= Ö
We then derive the total loss characteristic function conditional
on the common factor Z:
?u ? R, ÖL(t)(u|V ) : = E [exp(iuL(t))|Z]
= E [exp(iu ELnNn(t))|Z1
n=1
=
E [fl exp(iuLnNn(t)) | Z1
n=1
YN E [exp(iuLnNn(t))|Z]
n=1
YN [1 + pn(t|Z)(exp(iuLn) -
1)]
n=1
where we have used that (N1(t),..,NN(t)) are mutually
independent conditionally
on the common factor Z.
We now integrate the conditional characteristic function over the
common gaussian factor Z to retrieve the unconditional characteristic function
ÖL(t):
+8
= IÖL(t)(u|z)dÖ(z)
?u ? R, ÖL(t)(u) : = E [ÖL(t)(u|Z)]
Once we have found the loss characteristic function, we can
use the Fast Fourier Transform (FFT) to recover the loss distribution function
itself, which we then plug into equation (2.3) to derive the CSO tranche fair
premium.
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