Shepp and Shiryayev defined the Russian option in pure mathematical terms. Consider a share whose price

follows a geometric Brownian motion with drift

and volatility

(they also consider the case where the share price follows a Brownian motion with drift—i.e., the Bachelier model). Let

and let

be the payoff of the option at time

. The task is to maximize the expected value of the discounted payoff

over all stopping times

. In financial terms the problem can be expressed as follows. The owner of a Russian option chooses an exercise date, represented by a stopping time

, and then receives either

or the maximum stock price achieved up to this exercise time, whichever is larger, discounted by

. Shepp and Shiryaev showed that there is a number

that depends only on

,

, and

, such that the optimal strategy is to exercise the option at the first time

such that

(and the payoff is

). This they define as the fair value of the option. It is crucial that

is larger than

, otherwise it is never optimal to exercise the option. In terms of arbitrage theory one can assume that the stock pays a continuous dividend

and take

. In this case, the Shepp and Shirayev "fair price" is also the "arbitrage price" of the option.

The original proof of Shepp and Shirayev was based basically on "guessing" the correct formula for the option price by being guided by Kolmogorov's principle of smooth fit (which they say was a part of the reason for the name of the option) and then proving that the conjectured answer was the right one. Subsequently several other proofs have been given, for example in [2], in which a formula for the expected waiting time for optimal exercise is also given. This value is represented by the blue dot on the (horizontal) time axis.