Random walk duality and the valuation of discrete lookback options FARID AITSAHLIA 1 and TZE LEUNG LAI 2 1 Hewlett-Packard Laboratories, 1501 Page Mill Road MS 4U-1, Palo Alto, CA 94304, USA 2 Department of Statistics, Stanford University, Stanford, CA 94305, USA Received June 1997. Accepted September 1998. Use is made of the duality property of random walks to develop a numerical method for the valuation of discrete-time lookback options. This method leads to a recursive numerical integration procedure which is fast, accurate and easy to implement. Keywords: exotic options, lookback options, recursive numerical integration, random walk duality 1. Introduction 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 discrete- time 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. 2. Literature review Heynen and Kat (1995) derived pricing formulas for the valuation of discretely monitored lookbacks. These formulas involve multivariate normal integrals. Speci®cally, if m is the number of price ®xings, then one has to evaluate ( m  1)-variate normal distribution functions in these formulas, and Monte Applied Mathematical Finance 5, 227±240 (1998) 1350±486X # 1998 Routledge