Efficient Options Pricing Using the Fast Fourier Transform

Abstract

We review the commonly used numerical algorithms for option pricing under Levy process via Fast Fourier transform (FFT) calculations. By treating option price analogous to a probability density function, option prices across the whole spectrum of strikes can be obtained via FFT calculations. We also show how the property of the Fourier transform of a convolution product can be used to value various types of option pricing models. In particular, we show how one can price the Bermudan style options under Levy processes using FFT techniques in an efficient manner by reformulating the risk neutral valuation formulation as a convolution. By extending the finite state Markov chain approach in option pricing, we illustrate an innovative FFT-based network tree approach for option pricing under Levy process. Similar to the forward shooting grid technique in the usual lattice tree algorithms, the approach can be adapted to valuation of options with exotic path dependence. We also show how to apply the Fourier space time stepping techniques that solve the partial differential-integral equation for option pricing under Levy process. This versatile approach can handle various forms of path dependence of the asset price process and embedded features in the option models. Sampling errors and truncation errors in numerical implementation of the FFT calculations in option pricing are also discussed.