DIMACS TR: 93-25
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
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