We propose an efficient pricing method for arithmetic and geometric Asian options under exponential Lévy processes based on Fourier cosine expansions and Clenshaw-Curtis quadrature. The pricing method is developed for both European-style and American-style Asian options and for discretely and continuously monitored versions. In the present paper we focus on the European-style Asian options. The exponential convergence rates of Fourier cosine expansions and Clenshaw-Curtis quadrature reduces the

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... CPU time of the method to milliseconds for geometric Asian options and a few seconds for arithmetic Asian options. The method's accuracy is illustrated by a detailed error analysis and by various numerical examples. EFFICIENT PRICING OF ASIAN OPTIONS 401 in section 4, and numerical results are presented in section 5. We compare our results to those presented in [15] . The ASCOS method is extended to pricing American-style Asian options in another paper [23] . What is key here is that instead of recovering the density function, like in [7, 15, 3, 19, 14] , the characteristic function is recovered, which enables us to also price American-style Asian options. Here we focus on fixed-strike Asian options. The extension to floating-strike Asian options follows directly from the symmetry between floating-strike and fixed-strike Asian options, as explained in [16, 11] . ASCOS method for European-style geometric Asian options. The ASCOS pricing technique for geometric and arithmetic Asian options is described in sections 2 and 3, respectively. The characteristic function of the geometric or arithmetic mean value of the underlying is recovered, which is then used to calculate the Asian option value by Fourier cosine expansions. For geometric Asian options, the characteristic function of the logarithm of the geometric average of the underlying asset at the monitoring dates is known analytically for exponential Lévy processes, as we will see below. . Denote the (identical) characteristic functions of these increments by ϕ(u, τ ), i.e., and ϕ(u, τ ) is known analytically for different Lévy processes, for which we refer the reader to [13] . The characteristic function of y given x 0 is given by Downloaded 05/27/13 to 131.180.131.253. Redistribution subject to SIAM license or copyright; see EFFICIENT PRICING OF ASIAN OPTIONS 403 + for a call, K − S 0 (1 + exp (y)) M + 1 + for a put.