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Russian options are essentially a generalised form of the American perpetual put option initially proposed by Shepp & Shiryaev (1993). This type of option is known as a "reduced regret option" in that a minimum payout m to the buyer is guaranteed. The payout is given as the discounted maximum price that the option has ever traded at during the life of the option and can be extremely beneficial for the option holder. In a sense, this option is a perpetual American style lookback option.
Introduction || Pricing || Canadized Form || Jump Diffusion || Other Known Names & Variants || References
Russian options are, although complicated sounding, not that complicated at all. They are, in fact, a variant of the American option. American options are relatively difficult to price under closed form frameworks, although there are notable cases where this pricing becomes somewhat simplified - that of the American perpetual put option.
Russian options are essentially a generalised form of the American perpetual put option initially proposed by Shepp & Shiryaev (1993). This type of option is known as a "reduced regret option" in that a minimum payout m to the buyer is guaranteed. The payout is given as the discounted maximum price that the option has ever traded at during the life of the option and can be extremely beneficial for the option holder. In a sense, this option is a perpetual American style lookback option.
These options give the buyer of the option "reduced regret" if he or she fails to exercise the option at an earlier time when the option was greater in the money and are more expensive than many of its relatives. Because of this reason however, this option is not readily traded in the existing market.
The Canadized version of Russian options are Russian options with an independent exponential variable which acts as an expiry variable. This can also be thought of as Russian options with finite expiry. The pricing of such options is similar to solving the optimal stopping problem as in the standard Russian option case (see below).
Shepp & Shiryaev (1993, 1995) consider the pricing of this option under a Black Scholes Merton framework by considering the underlying asset under Geometric Brownian motion and determined the optimal stopping time for the solution of the pricing problem in the case of the perpetual option; an explicit solution was found for this class of option.
A number of authors have examined the pricing of Russian options under a Levy jump process.
Gerber, Michaud & Shiu (1995) consider the case of Russian options under a compound Levy framework where the jumps are governed by a Poisson process. Analytical closed form solutions were given in this case where the perpetual options are governed by exponential negative jumps and a deterministic drift function.
Modecki & Moreira (2001), Avram, Kyprianou & Pistouris (1, 2) (2004) and Avram, Asmussen & Pistouris (2004) also consider the pricing of this class of options under an exponential jump diffusion framework. They find analytical solutions for the price of the options with both positive and negative jumps.
Avram, Kyprianou & Pistouris (2004) consider the case of spectrally negative jump diffusion whereby the underlying asset exhibits no positive jumps. The solution in this case can be found by solving the optimal stopping problem by determining the stopping time which optimizes the expected discounted claim under a particular (risk neutral) measure.
Avram, Asmussen & Pistouris (2004) also finds an explicit solution for phaze-type jumps by again solving the optimal stopping problem via Wiener-Hopf factorisation.
Canadized Russian Options
Additional/Useful List of resources Papers: Asmussen, S., Avram, F., & Pistorius, M., "Russian & American Put Options Under Exponential Phase-Type Levy Models", Stochastic Process and their Applications, 109, pp. 79-111 (2004) |