Harmonic Analysis in Value at Risk Calculations

Value at Risk is a measure of risk exposure of a portfolio and is de ned as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain con dence, typically 95%. Full valuation of a portfolio under a large number of scenarios is a lengthy process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a Gaussian integral over a region bounded by a quadric. We use methods from harmonic analysis to nd approximate analytic formulas for the Value at Risk as a function of time and of the con dence level. In this framework, the calculation is reduced to the problem of evaluating linear algebra invariants such as traces of products of matrices, which arise from a Feynmann expansion. The use of Fourier transforms is crucial to resum the expansions and to obtain formulas that smoothly interpolate between low and large con dence levels, as well as between short and long time horizons. 1 e-mail: [email protected] 2 Sloan Fellow. e-mail: [email protected] 1 2 C. Albanese, L. Seco