Pricing Lookback Options and Dynamic Guarantees

Abstract

Pricing exotic options or guarantees in equity-indexed annuities can be problematic. The authors present closed-form formulas for pricing lookback options and dynamic guarantees that facilitate the hedging and reserving for such products. The principal tool used is a closed-form expression for B(u, T), the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process above a positive constant u in a finite time interval of length T. If the aggregate net income of a company is modeled with a Wiener process, then the excess of the running maximum above u can be interpreted as aggregate dividend payments, and the quantity B(u, T) is the expectation of the discounted value of the dividend payments up to time T. The formula for B(u, T) is used to price European lookback options (call and put, fixed and floating strike). It is also used to price dynamic fund protection, which is a guarantee on an investment fund: The number of units of the investment fund is increased whenever necessary, so that their total value does not fall below a guaranteed level. The guaranteed level can be stochastic, such as that given by a stock index. Some well-known results for the first passage time of the Wiener process are explained in the appendix.