We propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. The dynamics of the default intensity are governed by the equation dλt = κ(ρ(Lt, t) − λt)dt + σ √ λtdWt. The function ρ depends both on time t and accumulated loss Lt, providing sufficient freedom to calibrate the model to a generic distribution of loss. We develop a computationally efficient method for model calibration to the market of synthetic single tranche CDOs. The method is based on the Markovian projection technique which reduces the full model to a one-step Markov chain having the same marginal distributions of loss. We show that once the intensity function of the effective Markov chain consistent with the loss distribution implied by the tranches is found, the function ρ can be recovered with a very moderate computational effort. Because our model is Markovian and has low dimensionality, it offers a convenient framework for the pricing of dynamic credit instruments, such as options on indices and tranches, by backward induction. We calibrate the model to a set of recent market quotes on CDX index tranches and apply it to the pricing of tranche options.

Abstract. The pricing of collateralized debt obligations and other basket credit derivatives is contingent upon (i) a realistic modeling of the firms ’ default times and the correlation between them, and (ii) efficient computational methods for computing the portfolio loss distribution from the individual firms ’ default time distributions. Factor models, a widely-used class of pricing models, are computationally tractable despite the large dimension of the pricing problem, thus satisfying issue (ii), but to have any hope of calibrating CDO data, numerically intense versions of these models are required. We revisit the intensity-based modeling setup for basket credit derivatives and, with the aforementioned issues in mind, we propose improvements (a) via incorporating fast mean-reverting stochastic volatility in the default intensity processes, and (b) by considering homogeneous groups within the original set of firms. This can be thought of as a hybrid of top-down and bottom-up approaches. We present a calibration example, and discuss the relative performance of the framework. 1.

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In this paper we brie‡y describe dynamic and static factor models for credit correlation, and show how the static model can be calibrated to the market and used for the pricing of standard and bespoke tranches including tranchelets.

the Institute Laue-Langevin, where some of this work was carried out, for their hospitality; and Julien Houdain and Fortis Investments for providing the market prices used in the examples of this article. 4The support of the Natural Sciences and Engineering Research Council of Canada is acknowledged. This article describes a new approach to the risk-neutral valuation of CDO tranches, based on a general specification of the loss distribution, and the expected loss at zero default, for the reference portfolio. The new approach can describe tranche term-structures, and the generality with which the basic distributions are specified allows it to be perfectly calibrated to any set of market prices (for any number of tranches and maturities) that is arbitrage-free. Also, given a set of arbitrage-free market prices, arbitrage-free interpolated term structures (plots of tranche price versus maturity for a given tranche) and tranche structures (plots of tranche price versus tranche for a given maturity), as well as implied loss distributions, can be obtained, allowing bespoke tranches to be priced. The marking to market of tranche prices, and the establishment of forward-start premiums for the index are discussed. An efficient linear programming approach to valuation, essential to the implementation, is also described. The article also makes use of a new, simple yet general, approach to the problem of unequal notionals, and of random, time-dependent, risk-neutral, recovery rates.

Now that the market for cash and synthetic CDOs is well established, there is increased interest in trading forward contracts and options on CDO tranches. This article develops models for valuing these instruments. The model for valuing European options on CDO tranches has similarities to the standard market model for valuing European swap options and to the model for valuing options on credit default swaps. Once default probabilities, the expected recovery rates and the degree to which defaults tend to cluster have been estimated, it enables traders to calculate option prices from CDO tranche swap spread volatilities and vice versa. 2 FORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES A credit default swap (CDS) provides protection against default by some particular entity. During the life of the contract the buyer of protection makes periodic payments at some rate, s, times the notional until default or maturity whichever comes first. Typically these payments are made quarterly in arrears. In the event of a default the seller of protection

A forward starting CDO is a single tranche CDO with a specified premium starting at a specified future time. Pricing and hedging forward starting CDOs has become an active research topic. We present a method for pricing a forward starting CDO by converting it to an equivalent synthetic CDO. The value of the forward starting CDO can then be computed by the well developed methods for pricing the equivalent synthetic one. We illustrate our method using the industry-standard Gaussian-factor-copula model. Numerical results demonstrate the accuracy and efficiency of our method. 1

A basket default swap (BDS) is a credit derivative with contingent payments that are triggered by a combination of default events of the reference entities. A forward-starting basket default swap (FBDS) is a BDS starting at a specified future time. Existing analytic or semi-analytic methods for pricing FBDS are time consuming due to the large number of possible default combinations before the BDS starts. This paper develops a fast approximation method for FBDS based on the conditional independence framework. The method converts the pricing of a FBDS to an equivalent BDS pricing problem and combines Monte Carlo simulation with an analytic approach to achieve an effective method. This hybrid method is a novel technique which can be viewed either as a means to accelerate the convergence of Monte Carlo simulation or as a way to estimate parameters in an analytic method that are difficult to compute directly. Numerical results demonstrate the accuracy and efficiency of the proposed hybrid method.

We investigate different well-known multi-step credit portfolio models, all endowed with cross dependency, in their time-discretized versions. Refining then the time discretization we show that the correlation structure of only one model is invariant under this refinement whereas the others are not. In particular, the correlation structure of Markov-Chain migration model converges to the limit of no cross correlation, whereas the discrete barrier model converges to an unknown continuous limit. This has important implication on the risk assessment and pricing of portfolios of structured credit products when using these models, as it means that for consistent valuation the correlation structure of these models has to be calibrated to a given time horizon and time discretization of the respective implementation. Key words: multiperod models, credit portfolio risk, CDO

The Gaussian factor copula model is the market standard model for multi-name credit derivatives. Its main drawback is that factor copula models exhibit correlation smiles when calibrating against market tranche quotes. We introduce a multi-period factor copula model to overcome the calibration deficiency of factor copula models by allowing the factor loadings to be time-dependent. Usually, multi-period factor copula models require multi-dimensional integration, typically computed by Monte Carlo simulation, which makes calibration extremely time consuming. In our model, the portfolio loss of a completely homogeneous pool possesses the Markov property, thus we can compute the portfolio loss distribution analytically without multi-dimensional integration. Numerical results demonstrate the efficiency and flexibility of our model to match market quotes. 1