Random walk duality and the valuation of discrete lookback options

Lookback options are popular in OTC markets for currency hedging. The payoff of a lookback option depends on the minimum or maximum price of the underlying asset over the life of the contract. When the extreme values are continuously monitored, these options can be valued analytically (Conze and Viswanathan, 1991; Goldman et al., 1979a,b). On the other hand, when the maximum or the minimum is only monitored at speci®c (discrete) dates, mispricing occurs if one uses continuous-time formulas, as illustrated by Broadie et al. (1998) and Heynen and Kat (1995), but pricing these options via discretetime methods presents computational challenges in both speed and accuracy. In this paper we introduce a new method for the valuation of lookback options where the monitoring dates can be as frequent as daily ®xings. This method, based on the duality property of random walks, results in a fast and accurate recursive scheme which requires only univariate numerical integration. The paper is organized as follows. We begin with a brief review of the literature on numerical methods for pricing these options in Section 2. Section 3 provides the details of the procedure. Section 4 illustrates the numerical integration algorithm with a few examples and Section 5 gives some concluding remarks.