The Russian Option: Reduced Regret

We propose a new put asset where the option buyer receives the maximum price (discounted) that the option has ever traded at during the time period (which may be inde nitely long) between the purchase time and the exercise time | so that the buyer need look at the uctuations only occasionally and enjoys having little or no regret that he didn't exercise the option at an earlier time (except for the discounting). We give an exact simple formula for the optimal expected present value (fair price) that can be derived from the option and the (unique) optimal exercise strategy which achieves the optimum value under the assumption that the asset uctuations follow the Black-Scholes exponential Brownian motionmodel, widely accepted. It is important to note that the discounting is necessary; if it is omitted or even if it is less than the Black-Scholes drift then the value to the buyer under optimum performance is in nite. We also solve the same problem under a di erent model | that of the original Bachelier linear Brownian market with linear discounting; this model is no longer accepted, but of course the mathematics is consistent. To our knowledge no such regretless option is currently traded in any existing market despite its evident appeal. We call it the Russian option partly to distinguish it from the American and European options, where the term of the option is prescribed in advance and where no exact formula for the value has been given. The second author is a member of the Steklov Math Institute, Moscow; the work reported here was done while he was visiting Bell Labs. The Russian Option: Reduced Regret Larry Shepp A. N. Shiryaev* AT&T Bell Laboratories Murray Hill, New Jersey 07974