Geometric stopping of a random walk and its applications to valuing equity-linked death benefits

We study discrete-time models in which death benefits 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 suffices to consider curtate-future-lifetimes with a geometric distribution. In binomial and trinomial tree models, closed-form expressions for the expectations of the discounted benefit 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. •Wiener–Hopf factorization for geometrically stopped random walks is derived.•Curtate-future-lifetime is approximated by combinations of geometric distributions.•The logarithm of the stock price process is modeled as a binomial or trinomial tree.•Closed-form formulas for various equity-linked death benefits are derived.