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The Russian Option: Reduced Regret

A Russian option is a perpetual American-type option that pays the owner upon exercise the historical maximum price of the stock (this is supposed to "reduce the regret" of not exercising the option at the right time). The option was invented by L. Shepp and A. N. Shirayev and named "the Russian option". Although it is not traded in practice, several remarkable formulas for the option value, optimal exercise time, and the expected exercise time (under the assumption that the stock follows the Black–Scholes model) have been found and have had an important impact on probability theory.
In this Demonstration the orange line shows the movement of a single path of stock prices. You can choose whether the actual or discounted stock price should be shown. The initial stock price is fixed at 100. The blue dot on the vertical axis is the strike price, which you can vary. The blue line shows the discounted option payoff: it splits into two branches when the option is exercised—one, the horizontal line shows the discounted payoff after exercise; the other shows the value of the discounted payoff should you choose not to exercise the option. The horizontal black line shows the option value computed with the Shepp–Shirayev formula and the blue dot on the horizontal axis shows the expected exercise time found by Graversen and Peskir.
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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.
In [3] the problem of valuing a finite horizon Russian option was solved. As in the case of standard American options there are no explicit formulas and the option value is given as the solution of a nonlinear integral equation that has to be solved by numerical methods.
[1] L. Shepp and A. N. Shiryaev, "The Russian Option:Reduced Regret," The Annals of Applied Probability, 3, 1993 pp. 631–640.
[2] S. E. Graverseb and G. Peskir, "On the Russian Option: The Expected Waiting Time," Theory Probab. Appl., 42(3), 1997 pp. 564–575.
[3] G. Peskir, "The Russian Option: Finite Horizon," Finance Stochast., 9, 2005 pp. 251–267.
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