A Generalized Fourier Transform Approach to Risk Measures

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.

Feller's condition removed, some typos in Appendix A amended

References in corpus (3)

• Probability distribution of returns in the exponential Ornstein-Uhlenbeck model
• Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model
• Bayesian Analysis of Value-at-Risk with Product Partition Models

Cited by in corpus (5)

• Minimal model of financial stylized facts
• Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model
• Accounting for risk of non linear portfolios: a novel Fourier approach
• Maximum likelihood approach for several stochastic volatility models
• On a Transform Method for the Efficient Computation of Conditional VaR (and VaR) with Application to Loss Models with Jumps and Stochastic Volatility