Abstract:

The focus of this dissertation is the development and implementation of Monte Carlo algorithms to calibrate, via relative entropy, point process models for pricing multi-name credit derivatives.

We first implement a Weighted Monte Carlo algorithm to calibrate a self-exciting point process model for defaults to CDX data. We find that it is an efficient algorithm for calibration to a term structure of CDO tranche spreads. Furthermore, we use the calibrated model to price bespoke CDO tranches in an arbitrage-free manner, thus providing an appealing alternative to the market standard base correlation.

Instead of calibrating to tranche spreads observed in the market, it is sometimes desirable to calibrate to expected losses of tranches. These expected losses are not observed in the market, however, and recovering them from observed spreads is non-trivial. We propose and implement a simulation-based minimum relative entropy algorithm to solve this inverse problem. Numerical tests reveal some interesting similarities between expected tranche losses computed with our algorithm and those produced by other proposals in the literature.

We then present the main theoretical contribution of this dissertation--a new Monte Carlo algorithm for solving the minimum relative entropy calibration problem, which, unlike Weighted Monte Carlo, enables simulation of sample paths directly under the calibrated measure. In developing this algorithm, we prove that the solution to the infinite-dimensional model calibration problem can be disintegrated into (i) a distribution that solves a finite-dimensional relative entropy minimization problem and (ii) a regular conditional probability. In the context of CDO pricing model calibration, this regular conditional probability represents the law of a pinned counting process, and we show how to characterize such a process using h -transform techniques. We conclude by implementing our algorithm and discussing its application to pricing exotic multi-name credit derivatives.