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**1 - 1**of**1**### FAST SIMULATION OF LÉVY PROCESSES

"... Abstract. We present a robust method for simulating an increment of a Lévy process, based on decomposing the jump part of the process into the sum of its positive and negative jump components. The characteristic exponent of a spectrally one-sided Lévy process has excellent analytic properties, whi ..."

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Abstract. We present a robust method for simulating an increment of a Lévy process, based on decomposing the jump part of the process into the sum of its positive and negative jump components. The characteristic exponent of a spectrally one-sided Lévy process has excellent analytic properties, which we exploit to design a fast and accurate algorithm for calculating the cumulative distribution function of an increment of such a process. This algorithm is based on the parabolic inverse Fourier transform method introduced by S. Boyarchenko and S. Levendorskĭi, while the method of simulating a random variable using the values of its cumulative distribution function goes back to the work of P. Glasserman and Z. Liu. For Lévy processes of class KoBoL (a.k.a. the CGMY model), our simulation method typically performs faster than the method introduced by Madan and Yor by a factor of 10–100, and sometimes even higher, depending on the type of the option. Acknowledgments. We thank Sergei Levendorskĭi for the suggestion to apply the parabolic iFT method to the calculation of cumulative distribution functions. We are also grateful to the participants of the Mathematical Finance Seminar at Columbia University, where this work was presented.