Arbitrage Pricing of Russian Options and Perpetual Lookback Options

Abstract

Let $X = \{X_t,t \geq 0\}$ be the price process for a stock, with $X_0 = x > 0$. Given a constant $s \geq x$, let $S_t = \max\{s,\sup_{0\leq u \leq t} X_u\}$. Following the terminology of Shepp and Shiryaev, we consider a "Russian option," which pays $S_\tau$ dollars to its owner at whatever stopping time $\tau \in \lbrack 0,\infty)$ 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 $\delta X_t$, where $\delta \geq 0$. 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 $\delta$ is strictly positive, but is infinite when $\delta = 0$. 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.