East Asian Journal on Applied Mathematics, 1 (2011), pp. 20-34. Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes Raymond H. Chan 1*, Tao Wu 11 Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR, PR China. Received 2 March 2010; Accepted (in revised version) 12 April 2010 Published Online: 26 October, 2010. doi:10.4208/eajam.020310.120410a Abstract This paper concerns the Monte Carlo method in pricing American-style options under the general class of exponential L\'evy models. Traditionally, one must store all the intermediate asset prices so that they can be used for the backward pricing in the least squares algorithm. Therefore the storage requirement grows like $\mathcal{O}(mn)$, where $m$ is the number of time steps and $n$ is the number of simulated paths. In this paper, we propose a simulation method where the storage requirement is only $\mathcal{O}(m+n)$. The total computational cost is less than twice that of the traditional method. For machines with limited memory, one can now enlarge $m$ and $n$ to improve the accuracy in pricing the options. In numerical experiments, we illustrate the efficiency and accuracy of our method by pricing American options where the log-prices of the underlying assets follow typical L\'evy processes such as Brownian motion, lognormal jump-diffusion process, and variance gamma process. Key words: American options, Monte Carlo simulation, memory reduction, exponential L\'evy processes *Corresponding author. Email: rchan@math.cuhk.edu.hk (R. H. Chan), twu@math.cuhk.edu.hk (T. Wu)