Option Pricing Formulae using Fourier Transform: Theory and Application

Fourier inversion methods are an important addition to the tool set for derivatives pricing applications. The characteristic function of a random variable is the Fourier transform of its probability density, and the density is the inverse Fourier transform of the characteristic function. This is especially helpful for distributions which do not possess a known density function but are tractable via their characteristic functions. Possible applications include Lévy processes, affine (jump) diffusions and regime switching models. The main focus of this paper is on the implementational issues and computational efficiency.


Fourier transform techniques are playing an increasingly important role in Mathematical Finance. For arbitrary stochastic price processes for which the characteristic functions are tractable either analytically or numerically, prices for a wide range of derivatives contracts are readily available by means of Fourier inversion methods. In this paper we first review the convenient mathematical properties of Fourier transforms and characteristic functions, survey the most popular pricing algorithms and finally compare numerical quadratures for the evaluation of density functions and option prices. At the end, we discuss practical implementation details and possible refinements with respect to computational efficiency.


option pricing, characteristic functions, Fourier inversion methods, Lévy processes, stochastic volatility, density approximation, numerical quadratures.

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Schmelzle [2010] Fourier Pricing