Uniform Importance Sampling and Quantile Estimation

Specialeforsvar: Theis Louis Bøegh Hansen

Titel: Uniform Importance Sampling and Quantile Estimation

Resume: The calculation of quantiles is widely used in the financial sector, though often disguised as the risk measure Value-at-Risk. This thesis looks at the problem of estimating quantiles using importance sampling methods. A typical importance sampling problem is to estimate the tail probability P(S_n/n > a) for a single point a, where S_n is the sum of n i.i.d. random variables. In the setting of this thesis, the probability P(S_n/n > a) is estimated for all a in a closed interval [a1, a2] and then used to estimate the quantiles in this interval. It will be shown that importance sampling with an exponentially shifted distribution is only efficient for estimating a single point. Therefore a method using mixtures of exponentially shifted distributions is developed and used to estimate the probabilities in the interval. The numerical examples in the thesis will indicate that the mixture methods introduced only work for choices of very high quantiles in practice. As
an extension, the theory developed in the thesis will be empirically tested in a multidimensional setting. The results indicate that the methods do seem to extend nicely to multiple dimensions

Vejleder: Jeffrey F. Collamore
Censor:  Anders Hedegaard Jessen