Stochastic Upper Bounds for Present Value Functions

In most practical cases, it is impossible to find an explicit expression for the distribution function of the present value of a sequence of cashflows that are discounted using a stochastic return process. In this article, the authors present an easily computable approximation for this distribution function. The approximation is a distribution function which is, in the sense of convex order, an upper bound for the original distribution function. Explicit examples are given for pricing stochastic annuities with a stochastic return process, for more general stochastic cash flows, as well as for pricing Asian options. Numerical results seem to indicate that the approximation will often be close to the original function. INTRODUCTION In several financial-actuarial problems, one is faced with the determination of the distribution function of random variables of the form