Saddlepoint methods for conditional expectations with applications to risk management

TL;DR

This paper develops saddlepoint expansion techniques to efficiently approximate conditional expectations involving sample means, with applications to risk measures and financial sensitivities in risk management.

Contribution

It introduces novel saddlepoint expansions for conditional expectations of sample means, enabling fast and accurate computation of risk sensitivities in finance.

Findings

High accuracy of saddlepoint approximations demonstrated

Efficient computation of risk measure sensitivities achieved

Applicable to various risk management scenarios

Abstract

The paper derives saddlepoint expansions for conditional expectations in the form of $\mathsf{E}[\overline{X} | \overline{\mathbf Y} = {\mathbf a}]$ and $\mathsf{E}[\overline{X} | \overline{\mathbf Y} \geq {\mathbf a}]$ for the sample mean of a continuous random vector $(X, {\mathbf Y}^\top)$ whose joint moment generating function is available. Theses conditional expectations frequently appear in various applications, particularly in quantitative finance and risk management. Using the newly developed saddlepoint expansions, we propose fast and accurate methods to compute the sensitivities of risk measures such as value-at-risk and conditional value-at-risk, and the sensitivities of financial options with respect to a market parameter. Numerical studies are provided for the accuracy verification of the new approximations.