A Fast Method for Pricing Early-Exercise Options with the FFT R. Lord 1 , F. Fang 2 , F. Bervoets 1 , and C.W. Oosterlee 2 1 Modeling and Research, Rabobank International, Utrecht, The Netherlands roger.lord@rabobank.com; frank.bervoets@rabobank.com 2 Delft University of Technology, Delft Institute of Applied Mathematics, Delft, The Netherlands f.fang@ewi.tudelft.nl; c.w.oosterlee@tudelft.nl Abstract. A fast and accurate method for pricing early exercise options in computational finance is presented in this paper. The main idea is to reformulate the well-known risk-neutral valuation formula by recognizing that it is a convolution. This novel pricing method, which we name the ‘CONV’ method for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially L´ evy models, including the exponentially affine jump-diffusion models. For an M-times exercisable Bermudan option, the overall complexity is O(MN log(N )) with N grid points used to discretize the price of the underlying asset. It is also shown that American options can be very efficiently computed by combining Richardson extrapolation to the CONV method. Keywords: Option pricing, L´ evy Process, Convolution, FFT, Transform. 1 Introduction When valuing and risk-managing exotic derivatives, practitioners demand fast and accurate prices and sensitivities. As the financial models and option con- tracts used in practice are becoming increasingly complex, efficient methods have to be developed to cope with such models. Aside from non-standard exotic deriv- atives, plain vanilla options in many stock markets are actually of the American type. As any pricing and risk management system has to be able to calibrate to these plain vanilla options, it is of the utmost importance to be able to value these American options quickly and accurately. In the past couple of years a vast body of literature has considered the model- ing of asset returns as infinite activity L´ evy processes, due to the ability of such processes to adequately describe the empirical features of asset returns and at the same time provide a reasonable fit to the implied volatility surfaces observed in option markets. Valuing American options in such models is however far from trivial, due to the weakly singular kernels of the integral terms appearing in the PIDE, as reported in, e.g., [2,6,10,11]. Y. Shi et al. (Eds.): ICCS 2007, Part II, LNCS 4488, pp. 415–422, 2007. c  Springer-Verlag Berlin Heidelberg 2007