Pricing Options with Complex Fourier Series

Abstract

This paper develops a complex Fourier series (CFS) expansion pricing formula for evaluating European-type (call/put) options -- asset-or-nothing options, cash-or-nothing options, forward-start options, ratchet options and vanilla options -- when these risky assets are driven by Levy processes, time-changed Levy processes or stochastic volatility with or without jumps. This novel pricing formula is an extension of the original CFS expansion method proposed in Chan (2016). The major contribution of this paper is to derive a CFS pricing formula for both forward-start and ratchet options. In this paper, we also conduct an error-bound analysis of the CFS expansion pricing formula. In most cases, we can obtain an exponential convergence rate when we have an appropriate definite integration range for pricing European-type options. Moreover, based on the pricing formula, we can also derive different option Greeks. Finally, in follow-up papers, we will present the application of this pricing formula to options with early-exercise features (such as American options and barrier options) and path-dependent features (such as Asian options).