Arbitrage Pricing of Russian Options and Perpetual Lookback Options

Let X={Xt,t.0} be the price process for a stock, with X0=x>0. Given a constant s.x, let St=max{s,sup0.u.tXu}. Following the terminology of Shepp and Shiryaev, we consider a "Russian option," which pays S. dollars to its owner at whatever stopping time ..[0,.) the owner may select. As in the option pricing theory of Black and Scholes, we assume a frictionless market model in which the stock price process X is a geometric Brownian motion and investors can either borrow or lend at a known riskless interest rate r>0. The stock pays dividends continuously at the rate .Xt, where ..0. Building on the optimal stopping analysis of Shepp and Shiryaev, we use arbitrage arguments to derive a rational economic value for the Russian option. That value is finite when the dividend payout rate . is strictly positive, but is infinite when .=0. Finally, the analysis is extended to perpetual lookback options. The problems discussed here are rather exotic, involving infinite horizons, discretionary times of exercise and path-dependent payouts. They are also perfectly concrete, which allows an explicit, constructive treatment. Thus, although no new theory is developed, the paper may serve as a useful tutorial on option pricing concepts.