Intensity (and Variants) Gamma Alan Stacey (joint work with Mark Joshi and carried out in large part at the Quantitative Research Centre (QuaRC), Royal Bank of Scotland) Credit Risk Under Lévy Models Edinburgh, September 22nd 2006

The one-factor Gaussian copula The joint distribution of default times is determined from marginal distributions via a Gaussian copula. In the one-factor model, conditional on a single Gaussian random variable, Z, the default times are independent A single correlation number, , determines how much the default times are determined by the value of Z. If we restrict our attention to equity tranches, the map from  to price is strictly decreasing. So given the price of the 0 to x% equity tranche, there is a unique correlation (x%) which gives rise to this price. This is known as base correlation. The map, x→ (x), is the base correlation smile As usual, subject to recovery rate assumptions etc.

A standard but not a model This has become the market standard for quoting correlation. The price of a tranche can be quoted as a spread, or as the value of  which would imply that spread. However, it is very hard to infer new prices within the Gaussian copula framework. Even arbitrage-free interpolation of the base correlation curve is very difficult. In practice fairly sophisticated interpolation and mapping methodologies have been developed to obtain prices. The model is not based on any financial explanation of why defaults are correlated – it just correlates default times in a naïve way. No dynamics. Just as with Black-Scholes, when there is a smile, a model is needed to give those prices, so that other things can be priced. But this situation is somewhat worse. The Black-Scholes model may be a good first approximation to market behaviour, but the Gaussian copula model is not a dynamic model at all. The mapping and interpolation methodologies may work directly with base correlation. But there seems to be general agreement that they are unsatisfactory.

Desiderata of a correlation model Calibrates (more or less exactly) to relevant liquid market instruments: single name products (e.g. credit default swaps) quoted tranches of standardized baskets (iTraxx, CDX etc.) Deduce arbitrage-free prices of non-liquid instruments reasonably painlessly including tranches of standardized baskets with non-standard attachment points. bespoke CDOs with similar characteristics to an index hybrid CDOs, e.g. mix of regions or credit quality more general portfolio credit derivatives, e.g., CDO2

Desiderata (2) Realistic internal dynamics. Stable Greeks with good P&L explanatory power leading to good hedges. Important market changes (e.g. spread widening) taking place with the model (and hence within-the-model Greek calculation)

The basic Intensity Gamma model Based on stochastic or business time – the flow of information. If a lot happens in a given year then each firm has an increased chance of defaulting. One has an increasing business time process, t. Name i defaults with a constant hazard rate, ci, but defaults are driven by the business time process (common to all names) t, not calendar time. So conditional on the process (t), then for S ≤ T, the probability that a name survives to time T, given that it has survived to S, is exp(−ci(T− S)).

The gamma process We will take t to be stationary with independent increments. An increasing process with this property is known as a subordinator. The most well-known subordinator is the gamma process. Then t has a gamma distribution, with parameters t and  (for some ≥0, >0). This has density function It is helpful to think of this as the sum of t independent copies of an exponential random variable with parameter  (mean 1/). Of course, this is only strictly true when t is an integer. Stationarity is clearly desirable. One can argue about independent increments, depending on whether one thinks of this as a default process or genuinely an information process – we will return to this. The existence of a subordinator with this distribution for the increments needs to be proved. We do not go into this: look at a book on Levy processes. The gamma process is a pure jump process.

Calibrating to individual default probabilities The unconditional survival probability to T of a name which defaults with business time hazard rate c is just a Laplace transform: E(−cT). If we take business time to be a gamma process, this is just So calibrating each ci to a survival probability for name i is immediate.

Refining the very basic model For each name, we wish to match specified survival probabilities at a few different times. We take ci to be a piecewise constant function of calendar time. Note that if  in the specification of the gamma process changes, then each ci changes by the same factor. Effectively we have only one free parameter for the gamma process. We need more flexibility in our model for business time. For i=0,1, take (it) to be a gamma process with parameters i and i, the two processes being independent. Then set t = t0 + t1 + at for some constant drift a. We call this a multigamma process. And we call the model the intensity gamma model.

Pricing with intensity gamma Given a choice of multigamma parameters we then rapidly calibrate each ci to the survival probabilities for name i at a small number of times. These are inferred from CDS prices (and a recovery rate assumption). A product whose payoff is determined by the default times of a basket of names can then be priced by Monte Carlo. Draw a random path for business time, (t). Conditional on (t), the default times for each name are independent. Draw each of these. Compute the payoff and discount, assuming deterministic interest rates. Average over many paths. Some deterministic recovery rate assumption, as is fairly standard practice, is implicit throughout. There is a a much better way to draw the time path than just inverting the distribution function for a gamma random variables. It involves finding the large jumps precisely and then approximating the remaining jumps by a constant drift.

Matching the correlation market We aim to match the quoted prices of a single index. Prices are typically quoted for four or five tranches e.g. with detachment points 3%, 6%, 9%, 12%, 22%. Given multigamma parameters, we can price each tranche. We then find multigamma parameters which best match the market prices using an optimizer. Having chosen the multigamma parameters to match quoted tranche prices, we then use the same multigamma process for non-standard attachment/detachment points bespoke baskets with similar properties to the index to which we have calibrated: same region and similar levels of credit quality and diversity Significantly different maturities turn out to be more difficult. We think in terms of base correlation and equity tranches. We actually use the downhill simplex method. The calibration to the base correlation curve is the trickiest part of the process.

Matching an investment grade curve The quality of fit we see here is typical of investment grade indices, whether European or North American, both before and after May.

North American High-Yield index After May 2005, the high-yield curve became convex and non-monotonic. Unlike other market curves, it does not seem possible to fit this one perfectly with our multigamma model. However, the fit is still comfortably within bid-offer spread.

Extensions to the model (1) Some products depend on a basket of names corresponding to different indices, e.g., High Yield/Investment Grade hybrids different regions. Divide the names of interest into sub-baskets corresponding to different indices. Calibrate a different multigamma processes to each index. The defaults of each sub-basket are driven by the corresponding multigamma process. One needs a way to make the different multigamma processes strongly correlated.

Extensions to the model (2) Can introduce a random time lag between information arrival and default. This is more realistic. One way to do this is to have information arrive as a multigamma process, (t), as before, with the impact of the information spread out in a way that decays exponentially with parameter α. We then have a positive residual information process (Rt) satisfying and then an impact process, (It) driving defaults as before

Extensions to the model (3) Retains tractability and rapid calibration to individual names with benefits including no longer have simultaneous defaults, although if there is a big jump in business time one will get a lot of defaults in a short space of time better matching of the market across different time horizons credit spread widening (one-factor only) within the model Can use a different class of subordinators, e.g. tempered stable processes.

Summary of strengths Provided one can match the index prices, one can obtain arbitrage-free prices for products whose payoffs depend upon the default times of a basket of names. These are consistent with Single-name survival probabilities (typically derived from CDSs) Tranche prices for the corresponding CDO index (and, to some extent, multiple indices where appropriate). Once calibrated to the appropriate index, pricing is rapid and straightforward. No ad hoc interpolation or curve-mapping techniques are required.

Some limitations Intensity Gamma is only a default model. It does not model the dynamics of credit spread movements. Within the model, credit spreads are deterministic. (In the time-lag extension, however, systemic movements of spreads do occur.) Hedging of spread movements must be outside-the-model. Similarly, the multigamma parameters are fixed, but if the index tranche prices move then they must be re-calibrated. Not capable of matching the market prices of correlation products with different maturities.

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