A Generalized Fourier Transform Approach to Risk Measures

TL;DR

This paper introduces a generalized Fourier transform framework for efficiently computing popular risk measures like VaR and Expected Shortfall, applicable to complex non-Gaussian financial models using characteristic functions.

Contribution

The paper develops a novel Fourier-based method for risk measurement that extends to non-Gaussian models, improving computational efficiency and accuracy in risk analysis.

Findings

Outperforms standard Log-Normal risk predictions

Aligns closely with benchmark historical risk estimates

Effective for Levy noise and stochastic volatility models

Abstract

We introduce the formalism of generalized Fourier transforms in the context of risk management. We develop a general framework to efficiently compute the most popular risk measures, Value-at-Risk and Expected Shortfall (also known as Conditional Value-at-Risk). The only ingredient required by our approach is the knowledge of the characteristic function describing the financial data in use. This allows to extend risk analysis to those non-Gaussian models defined in the Fourier space, such as Levy noise driven processes and stochastic volatility models. We test our analytical results on data sets coming from various financial indexes, finding that our predictions outperform those provided by the standard Log-Normal dynamics and are in remarkable agreement with those of the benchmark historical approach.