Arbitrage Pricing of Russian Options and Perpetual Lookback Options

Abstract

Let X = {{Xt},t ≥ 0} be the price process for a stock, with {$X_0} = x {\textgreater} 0$. Given a constant s ≥ x, let {$S_t} = {\textbackslash}max{\textbackslash}{s,{\textbackslash}sup_{0{\textbackslash}leq u {\textbackslash}leq t} X_u{\textbackslash}}$ . Following the terminology of Shepp and Shiryaev, we consider a {"Russian} option," which pays Sτ dollars to its owner at whatever stopping time τ ∈ [ 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 $r {\textgreater} 0$. The stock pays dividends continuously at the rate δ Xt, where δ ≥ 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 δ is strictly positive, but is infinite when δ = 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.