Valuation of European Call Options via the Fast Fourier Transform and the Improved Mellin Transform

Abstract

This paper considers the valuation of European call options via the fast Fourier transform and the improved Mellin transform. The Fourier valuation techniques and Fourier inversion methods for density calculations add a versatile tool to the set of advanced techniques for pricing and management of financial derivatives. The Fast Fourier transform is a numerical approach for pricing options which utilizes the characteristic function of the underlying instrument’s price process. The Mellin transform has the ability to reduce complicated functions by realization of its many properties. Mellin’s transformation is closely related to an extended form of other popular transforms, particularly the Laplace transform and the Fourier transform. We consider the fast Fourier transform for the valuation of European call options. We also extend a framework based on the Mellin transforms and show how to modify the method to value European call options. We obtain a new integral equation to determine the price of European call by means of the improved Mellin transform. We show that our integral equation for the price of the European call option reduces to the Black-Scholes-Merton formula. The numerical results show that the tremendous speed of the fast Fourier transform allows option prices for a huge number of strikes to be evaluated very rapidly but the damping factor or the integrability parameter must be carefully chosen since it controls the intensity of the fluctuations and the magnitude of the functional values. The improved Mellin transform is more accurate than the fast Fourier transform, converges faster to the Black- Scholes-Merton model, provides accurate comparable prices and the approach can be regarded as a good alternative to existing methods for the valuation of European call option on a dividend paying stock.