## The Russian Option: Reduced Regret

### Authors: Larry Shepp and A. N. Shiryaev

ABSTRACT

We propose a new {\em call} option where the option {\em seller pays} the {\em minimum} price (in {\em inflated} dollars) that the asset has ever traded at during the time period (which may be indefinitely long) between the {\em selling} time and the {\em delivery} time (to be chosen by the {\em seller}). This option is the dual of the {\em put} option where the option {\em buyer receives} the {\em maximum} price (in {\em discounted} dollars) that the asset has ever traded at during the time period (which may be indefinitely long) between the {\em buying} time and the {\em exercise} time (to be chosen by the {\em buyer}). Because the settlement payoff is at the minimum (for the call) or the maximum (for the put) there is reduced regret in the sense that it is not necessary for the option holder to worry about missing a good price in the recent past (of course he may regret not holding on longer) since he gets the best price up to the settlement time. We give the 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 fluctuations follow the Black-Scholes exponential Brownian motion model, widely accepted. The Russian put option was studied earlier.

We show that in both cases, puts or calls, the simpler option where the settlement is based on the {\em current} price at the time of exercise rather than on the maximum or minimum price to date (i.e. without the Russian'' feature) does not give rise to an interesting stopping problem and there can be no trading on such an option.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-25.ps