A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options

This paper develops algorithms for the pricing of discretely sampled barrier, lookback and hindsight options and discretely exercisable American options. Under the Black-Scholes framework, the pricing of these options can be reduced to evaluation of a series of convolutions of the Gaussian distribution and a known function. We compute these convolutions efficiently using the double-exponential integration formula and the fast Gauss transform. The resulting algorithms have computational complexity of O(nN), where the number of monitoring/exercise dates is n and the number of sample points at each date is N , and our results show the error decreases exponentially with N . We also extend the approach and provide results for Merton’s lognormal jump-diffusion model. Received January 2003; accepted September 2004. Subject classifications: Finance: asset pricing. Area of review: Financial engineering. ∗This work was partially supported by NSF grant DMS-0074637 and by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists, 16760053, 2004 and Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science.”