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Stochastic Intensity Modelling for Baskets. Pedro A. C. Tavares Alexander Chapovsky London, Sep 2006. Overview. Motivation Stochastic intensity basket modelling Model specifications & calibration Credit Options Conclusion. Motivation. To have an intuitive parameterisation
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Stochastic Intensity Modelling for Baskets Pedro A. C. TavaresAlexander ChapovskyLondon, Sep 2006
Overview • Motivation • Stochastic intensity basket modelling • Model specifications & calibration • Credit Options • Conclusion
Motivation • To have an intuitive parameterisation • Setup a modelling framework that can value loss products, path-dependent structures and options consistently: • CDO, CDO2 • Step-ups, forward starting tranches • Tranche options, leveraged super-senior • Fast and exact calibration to single-name levels • Efficient and accurate calibration to index tranche levels
Stochastic Intensity Setup To overcome the shortcomings of default copula models practitioners are turning to the modelling of default intensities. Duffie and Gârleanu proposed a model of defaults with the intensity specified as a Cox process: Where d is the default time. The intensity, h(t), is specified as a mean-reverting jump diffusion where the jump size parameter, j, is exponential.
Stochastic Intensity Setup • In general, if one proposes to price basket correlation using default intensities as a starting point, the following problems need to be overcome: • “Slow” processes (diffusions) are not capable of generating sufficient correlation between default events as to be compatible with observed prices. • The jump distribution needs to be rich enough to capture prices in the market. Recent work (Mortensen 2006) suggests an exponential jump characterised by a single parameter is not sufficiently flexible. • Efficient CDS calibration and basket loss distributions require careful implementation.
Basic Setting To ensure the best chance of an efficient calibration to the CDS curve it is useful to write the intensity such that we explicitly separate the forward intensity curve: where h(0)(t) is the (deterministic) forward intensity curve and h’(t) is a random process. Non-arbitrage requires that default probability must match the CDS implied value at all times: Since by construction,
Basic Setting And then, Given this last expression it is useful if h’(t) has a deterministic compensator. In that case: We have established only the very basic characteristics that our model should inherit. In order to carry things further we need to take into consideration the pricing of tranches on CDS baskets…
Basket Loss It’s been observed that CDO tranche cash flows are a function only of the pool loss distribution in time. In a Gaussian-copula setting we often condition on the central factors driving this distribution: Identically, in our proposed framework (we introduce the asset indexi =1…n): This is obviously very high dimensionality (infinite in principle). To reduce dimensionality one needs to reduce the number of factors driving the h processes. One should also realise that the loss depends on the intensity paths only through their integrals.
Model Specification The previous considerations motivate the following model specification where and y(t) is a suitable stochastic process common to all assets and H is the integral of the compensator introduced earlier. The ai(t) are asset specific weights. The conditional survival probability is then,
Jump Size Distribution The next step is to find the distribution of X(t). This can be done if one observes that eH is the Laplace transform of X. The probability distribution, P(X), can then be computed by taking the inverse Laplace transform of the left-hand side. Inverse Laplace transforms are notorious for being difficult to compute. See Isenger (2006) for a detailed description of efficient methods useful in this instance.
Intensity Process Specification I Finally we need to specify the common random process y(t). As mentioned earlier this factor must include jumps in order to generate sufficient default event correlation. A reasonable starting choice would be: Where jis a random jump size and N is the Poisson counting process. Under this choice the compensator term has the following form, given the intensity of jump, A, (Björk 1996):
Intensity Process Specification II Calibration results which we showed earlier suggest that we can add a volatility term to account for the finer structure of p(j). A specification which is still tractable can be written in the following form: Where k and mare respectively the mean reversion speed and level of the diffusion component with square-root volatilityv. The compensator now has a more complex form but is still analytic. See again Björk (1996) for further details of this calculation.
Intensity Process Specification III We observed that the jump structure exhibits a very dramatic term structure. In particular the jump contribution drops quickly between 5 and 10 years. In order to “soften” this structure we made the jump distribution itself mean-reverting. At this stage we lose much of the tractability and a numerical solution is required. The advantage is that such a solution allows great freedom in selecting the “skew” parameters. In particular we choose: The solution takes the opposite route to what we described before - we determine the probability distribution, P(X), using finite-difference methods, then compute H(t,u) through:
Credit Options • The framework described earlier allows one to price derivatives that depend on realised losses (CDO, CDO2) and those that depend on realised intensity (CPPI). Either semi-analytic or Monte-Carlo approaches can be used. • Derivatives where the payoff depends in a non-linear way on these quantities cannot be priced in this way (e.g. option dependent on tranche mark-to-market). A backward induction scheme is best suited to deal with these cases. • Obviously the fundamental problem is of very high dimensionality and is not suited to these methods. The question one then needs to answer is how to reduce the dimensionality of the pricing problem.
Credit Options - Dimensionality • Naturally, realised loss, l, and current intensities, encoded in y, must be included. • It is also highly desirable that loss distributions and therefore tranche prices are recovered as well. These have the representation: and so we need also to include X because loss distributions depend on realised survival probabilities and therefore on X. • A mixture of fine structure and large jumps requires large grids and slows down calculation. All index tranches are still priced in seconds.
Credit Options – Basic Setup Given our 3-dimensional setup, the value of a derivative with payoff equal to Vt(y, l, X), is given by: The function F is required where early exercise is included in the contract definition and Bt is a cash account (assumed independent). Given the model dynamics introduced earlier the backward induction equation becomes: where L is the differential operator as defined by the choice of model and f describes the trade cash flows.
Credit Options – Contract Example Many contracts can be encoded with this specification. For the purpose of illustration the default leg of a vanilla tranche with attachment a and exhaustion e is described in the following manner:
Conclusion • We introduced a class of models that is capable of pricing, within a consistent framework, products whose payoffs depend on realised losses and/or realised spreads. • This class of models can be specialised in such a way to allow efficient calibration to CDS (exact) and CDO (approximate) levels. • Model parameterisation is very rich and intelligent choices need to be made in calibration. • We showed several examples of calibration to index CDO prices.
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