Geometric Stopping of a Random Walk and Its Applications to Valuing Equity-linked Death Benefits

Abstract

We study discrete-time models in which death bene ts can depend on a stock price index, the logarithm of which is modeled as a random walk. Examples of such benefit payments include put and call options, barrier options, and lookback options. Because the distribution of the curtate-future-lifetime can be approximated by a linear combination of geometric distributions, it su ces to consider curtate-future-lifetimes with a geometric distribution. In binomial and trinomial tree models, closed-form expressions for the expectations of the discounted bene t payment are obtained for a series of options. They are based on results concerning geometric stopping of a random walk, in particular also on a version of the Wiener-Hopf factorization. This is a joint paper with Hans U. Gerber and Elias S.W. Shiu.