Saddlepoint approximation of risk measures

Abstract

This thesis outlines the application of saddlepoint approximations to portfolio theory. Knowledge about the distribution of a profit and loss random variable is of practical importance. Unfortunately, its density and distribution function are often unknown. In principle, knowledge about the moment-generating or cumulant-generating function permits to obtain both functions using certain integral inversion formulas. In practice, though, the complexity of the integration involved may be unduly costly. Fortunately, the inversion integrals can be approximated, which is where saddlepoint approximations come into play. Among various approximations, they stand out by being accurate in the tail distribution and with small sample sizes or merely a single observation.