From pricing to rating structured credit products and vice-versa

Chapter 2

Modelling and pricing CSO

tranches

After choosing the pool of single-name CDS and defining the characteristics of the CSO tranche (attachment and detachment points), we want to determine a fair spread to be paid to the tranche buyer (i.e. the protection seller) as a fair reward for bearing this credit risk so that the present value of his investment is zero at inception (assuming no transaction costs nor fees to be paid to the arranging bank). Such a fair spread will eventually depend on the portfolio's joint loss distribution function accross time horizon L(t) until the CSO's maturity.

2.1 Modelling a CSO tranche payoff

Given an underlying portoflio of single name CDS, we assume that we have access to the joint loss distribution function L(t) of the portfolio at any time t, 0 t T, where T denotes the maturity of the transaction. We call respectively K and K the attachment and detachment points of our tranche. Its initial nominal amount is equal to K - K and the cumulative losses M(t) that affect that tranche at any time t is given by the following formula:

M(t) = (L(t) - K) - (L(t) - K)

We now assume that the underlying CDS portfolio is made of N reference obligors, each with a nominal amount A_n and a recovery rate R_n for n = 1,2, .., N. Let L_n = (1 - Rn)An be the loss given default of obligor n. Let r_n be the default time of obligor n. Let N_n(t) = ¹{ô_n=t} define the counting process which jumps from 0 to 1 when the n^th obligor defaults. The portfolio loss function L(t) is then given by:

L(t) = _XN L_nN_n(t) (2.1)

n=1

We note that the functions L(t) and therefore M(t) are pure jump processes.

2.2 Default and premium legs of a CSO tranche

Similarly to the approach presented for valuing the fair spread of a single-name CDS,
we determine the fair premium W * of the CSO tranche by equalizing the present

value of the default leg DL and the premium leg PL(W) of the tranche: by definition, W* solves the following equation:

PL(W^*) - DL = 0 (2.2)

The existence of a liquid market for standard CSO tranches based on ITRAXX and CDX indices provides us with a satisfactory framework for pricing credit default correlation among obligors, hence CSO tranches, under the risk-neutral probability. From now on, we assume that all expectations are taken under the risk-neutral probability measure.

2.2.1 Default leg

Given that M(t) is an increasing function, we can define Stieltjes-Lebesgue integrals with respect to M(t). The discounted payoff corresponding to potential default payments can therefore be written as:

I0

T n

_X ( )

B(0, t)dM(t) := B(0, ôj)Nj(T ) M(ôj) - M(ô- _j )

j=1

Using Stieltjes integration by parts formula and Fubini's theorem, the price of the default leg under the risk neutral probability measure can be expressed as:

I T J T

DL = E[ B(0, t)dM(t)] = B(0, T ) E[M(T )] + E[M(t)]dB(0, t)

0 0

2.2.2 Premium leg

Similarly, the price of the premium leg of the CSO tranche under the risk neutral probability measure is given by the folowing expression, where discrete premium payments are assumed to take place on _(Tj)j=1..m with T0 is the start date of the tranche and T_m = T is its legal maturity date.

? ?

m J Tj

P L(W ) = E ? B(0, Tj) W (K - K - M(t))dt_?

j=1 Tj-1

_Xm
j=1

J Tj

B(0, Tj)W (K - K)(Tj - _Tj-1) - E[M(t)]dt

Tj-1