Application of rotational invariant importance sampling

Abstract

The aim of this diploma thesis is to calculate risk measures like value at risk and expected shortfall using rotational invariant importance sampling. Rotational invariant importance sampling is a technique that increases the efficiency of rare event simulation.<br />We aim to simulate the loss distribution of a credit portfolio and calculate the corresponding risk measures. Through importance sampling, we change from the original density of underlying macroeconomic factors to an auxiliary density particularly causing losses in the relevant area of interest, i.e. the area around value at risk and expected shortfall. Therefore, a change of measure needs to be applied that is incorporated by the Radon-Nikodym derivative. In particular, rotational invariant importance sampling uses a rotational invariant measure. The advantage is that the rotational invariant measure accounts for several regions that are responsible for high losses in the portfolio. This is in contrast to simple mean shifting, where some areas are neglected, that potentially induce high losses. Under the new measure, the density depends on two parameters (c, s). In particular, an optimization procedure for the parameters involved is given