The Russian Option: Reduced Regret

We propose a new put option and give its fair premium price based on the Black-Scholes model for fluctuations of the value of a stock or other asset. The option buyer receives the maximum price that the asset has ever traded at during the (open end) period, discounted ro the value of the time until exercised, and so the buyer feels less regret and need not keep close track of its price. A simple formula is given for the fair premium cost of this option and for the optimal exercise time the buyer, which is nto a fixed time, but depends on the fluctuations. The discount rate, lambda, must exceed the drift parameter, micron of the Black-Scholes model of the asset (or else the buyer can get an unbounded return), but is otherwise arbitrary. The premium depends on lambda, micron, and the volatility sigma in a simple way.