## DIMACS TR: 93-26

## A Dual Russian Option for Selling Short

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

**
ABSTRACT
**

We propose a new put asset where the option buyer receives the {\em maximum}
price (discounted) that the option has ever traded at during the time
period (which may be indefinitely long) between the purchase time and the
exercise time --- so that the buyer need look at the fluctuations 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 fluctuations follow the Black-Scholes
exponential Brownian motion model, 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 infinite.
We also solve the same problem under a different 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.

Paper available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-26.ps

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