On a Transform Method for the Efficient Computation of Conditional VaR (and VaR) with Application to Loss Models with Jumps and Stochastic Volatility

TL;DR

This paper introduces a Fourier transform-based method for efficiently computing Value-at-Risk and Conditional Value-at-Risk for loss models with jumps and stochastic volatility, leveraging their convex optimization structure.

Contribution

It develops a novel Fourier transform approach utilizing Fast and Fractional Fourier transforms for calculating risk measures in complex loss models.

Findings

Efficient computation of VaR and CVaR using Fourier techniques.

Applicable to models driven by Lévy processes and stochastic volatility.

Demonstrates practical application to univariate loss models with jumps.

Abstract

In this paper we consider Fourier transform techniques to efficiently compute the Value-at-Risk and the Conditional Value-at-Risk of an arbitrary loss random variable, characterized by having a computable generalized characteristic function. We exploit the property of these risk measures of being the solution of an elementary optimization problem of convex type in one dimension for which Fast and Fractional Fourier transform can be implemented. An application to univariate loss models driven by L\'{e}vy or stochastic volatility risk factors dynamic is finally reported.