STRATIFIED SAMPLING FOR RISK MANAGEMENT

Abstract. The authors discuss the approximation of Value at Risk (VaR) and other quantities relevant to risk management. One of the core problems in this context is the approximation of the distribution of quadratic forms of Gaussian vectors. It appears as an intermediate problem in the variance reduction techniques proposed by Glasserman et. al. as well as in so-called ∆-Γ-normal approaches. The purpose of this paper is to show that sampling methods are faster than Fourier inversion for a range of practical problems. In fact, the asymptotic cost to achieve a given accuracy is O(d) for Monte-Carlo and O(d3) for Fourier inversion in a specific setting. The theoretical results are supported by a case study based on real-life problems, showing that sampling methods are faster than Fourier inversion for all but the smallest problems. Thus the use of randomized methods is recommended and the issue of variance reduction becomes important. Stratification methods – especially randomized orthogonal arrays – turn out to lead to the most effec-tive methods in terms of accuracy per computational cost. 1. Problem formulation Given a portfolio, the typical task of a bank’s risk controller is to model the change in the portfolio’s value over a certain time horizon h in the future. If Sit denotes the value of asset i at time t and w i t the number of units of assets of type i in the portfolio, then V (wt, St) = M∑ i=1 witS i t is the portfolio’s value at time t. The object of interest is the hypo-thetical profit or loss (P&L) from holding the frozen portfolio (wt) over the time interval [t, t+ h]