Credit market participants are often interested in the ability to compute prices of non-standard single tranches on index portfolios such as iTraxx and CDX. A widely accepted method used in pricing non-standard tranches consists of interpolating a series of implied correlations of base tranches using market attachments and detachments. In this Learning Curve, I propose an alternative approach to pricing non-standard tranches that is based on computing the term structure of loss distributions of the index portfolio in a self-consistent way.
Once the term structure of loss distribution is found, any tranche, whether standard or non-standard, can be straightforwardly priced. The appeal of this approach is that it produces a term structure of loss distributions that fit all market tranche prices at all maturities simultaneously. To work with a full term structure is important because it addresses the issue of distribution consistency from one maturity to another for the same index portfolio. While it is entirely feasible to work directly with a single loss variable, I prefer to work in a loss-equivalent bi-dimensional domain whose variables are the number of defaulted names and a discretized recovery rate.
Loss Distribution Information
Derivatives contracts based on realized recovery rates are becoming increasingly popular and the pricing information they contain may be used to better reconstruct the loss distribution. In many practical instances, there is some information available on the unknown risk-neutral loss distribution p in the form of distribution function q. In the credit-default swap index case, the function q can be derived either from historical data or from market data. Similarly, information on recovery rates is available from published historical studies or over-the-counter derivatives contracts that address recovery rate outcomes. If no information is available, one can use a uniform distribution function or a best-guess function.
To compute the portfolio loss distribution p implied by tranche prices that mostly resemble q I minimize the Kullback cross-entropy measure:
The cross-entropy measure is minimized subject to a set of linear constraints represented by the market tranche pricing equations
The optimization problem can be solved using the method of Lagrange multipliers.
This problem is often recast in its unconstrained dual form which can be solved numerically using standard Newton gradient methods.
The computation of the loss functions is done by bootstrapping each function one at a time starting from the shortest maturity and moving to the next maturity until the final maturity is reached. In order to bootstrap a term structure the pricing equations Ep(CF) = Ep(L) for all tranches and maturities are manipulated to derive expected tranche losses as functions of expected tranche losses at previous maturities. Because loss timing enters into CDS pricing, the pricing equations require knowledge of the loss distribution in continuous time to compute exact discount factors. In practice, loss distributions can only be computed at standardized maturities and intermediate expected losses must therefore be interpolated unless some further assumptions are made.
For a given index portfolio and maturity, standard CDS tranche price quotes are typically available on a daily basis. In addition, CDS premiums on single obligors that comprise the index portfolio are also available from dealers and pricing services. One possible choice for q is to assume that the loss distribution will not be too dissimilar from a loss distribution implied by a given copula. This allows us to incorporate marginal distribution information on individual obligors to narrow the set of candidate distributions.
The financial modeler has a wide variety of choices for q. It is important to select a function whose implied tranche prices do not differ significantly from the actual market prices, otherwise the optimization search may not converge.
Here I include a standard application to the three-year CDX Index for illustration purposes. In this example I selected the a priori distribution to be a multinomial distribution function obtained from single-obligor risk-neutral default probabilities. I also assumed a uniform distribution for recovery rates.
The optimized function as a function of the number of defaults exhibits a shape very similar to our initial guess, except for a small hump at higher defaults.
The fact that the two functions are similar and the relatively smooth behavior of the optimized function suggest that our initial choice for the a priori distribution is a good one. The inability of certain copulas to obtain a close fit to market prices stems from the fact that they are not flexible enough to generate these types of shapes. In conclusion, loss distributions using entropy maximization turn out to be quite powerful in reproducing market prices.
This week's Learning Curve was written by Luigi Vacca,head of quantitative analytics at Radian Asset Assurancein New York.
