The Computation of Risk Budgets Under the Lévy Process Assumption

Abstract

The Basle 2 Capital Accord issued by the Basle Committee on banking supervision has proposed a multiplier superior to 3 on banks' internal 99% 10-day Value-at-Risk calculated for market risk exposure. This ad hoc factor has not been fully explained and is poorly justified by arguing that the standard classical models of stock price dynamics do not adequately capture actual market risks. More generally, the current crisis has questioned a lot risk management practices about the determination of appropriate VaRs. In this paper, we revisit the computation of Value-at-Risk by introducing a new method based on Lévy processes. After a brief preliminary study where jump tests are performed in order to confirm the need for jump processes in financial modeling, we provide a new presentation of Variance Gamma Processes with Drift, that are reconstructed in an original way starting from the exponential distribution. Then, we display a fit of these processes on the American and French markets, before providing a new general Fourier formula that allows to compute VaR quickly and efficiently. Based on this formula, we conduct a study of the term structure of VaR, and provide a discussion of the Basle 2 and forthcoming Solvency 2 agreements.

Keywords: Value-at-Risk, Lévy Processes, Variance Gamma Processes, Fourier Transform, Basle 2, Solvency 2