More likely than not

The likelihood ratio method [BG96] provides a tool for the calculation of hedge ratios and risk parameters with Monte Carlo calculations. In this article, we present the extension of existing likelihood ratio methods to multiple underlyings and to higher order sensitivities such as the crossderivatives on the underlyings, and how the method can be used in a displaced diffusion [Rub83] framework. Next, an alternative likelihood ratio formula that can be used to compute sensitivities with respect to market adjustments of forward contracts is given. We also derive the respective first and second order hedge quantities with respect to the underlyings’ hazard rates for the correlatedtime-of-default model [Li00], as well as the corresponding correlation coefficient risk. In addition, we show how the sensitivities with respect to model parameters can be converted into quantities that are more meaningful to traders: exposures with respect to implied volatilities, or with respect to credit spreads. In a nutshell: this is a collection of likelihood ratio formulæ.