Pricing Fixed-Strike Lookback Options

Abstract

A fixed-strike lookback option is an option whose payoff is determined by the maximum (or minimum) price of the underlying asset within the option's life. Under the Black-Scholes framework, the time-t price of an equity asset follows a geometric Brownian motion. Applying the method of Esscher transforms, this paper will derive explicit pricing formulas for fixed-strike lookback call and put options, respectively. In addition, this paper will show a relationship (duality property) betwen the pricing formulas of the call and put options. Finally, this paper will derive explicit pricing formulas for the fixed-strike lookback options when their underlying asset pays dividends continuously at a rate proportional to its price.