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.
Broadie, Mark, and Y. Yamamoto. "A Double-Exponential Fast Gauss Transform for Pricing Discrete Path-Dependent Options." Operations Research 53, no. 5 (2005): 764-79.
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