A. Cerny, Introduction to Fast Fourier Transform in Finance, here.
Abstract. The Fourier transform is an important tool in Financial Economics. It delivers real time pricing while allowing for a realistic structure of asset returns, taking into account excess kurtosis and stochastic volatility. Fourier transform is also rather abstract and therefore off-putting to many practitioners. The purpose of this paper is to explain the working of the fast Fourier transform in the familiar binomial option pricing model. We argue that a good understanding of FFT requires no more than some high school mathematics and familiarity with roulette, bicycle wheel, or a similar circular object divided into equally sized segments. The returns to such a small intellectual investment are overwhelming.
Carr and Madan, Option valuation using the fast Fourier transform, here.
The Black±Scholes model and its extensions comprise one of the major developments in modern ®nance. Much of the recent literature on option valuation has successfully applied Fourier analysis to determine option prices (see e.g. Bakshi and Chen 1997, Scott 1997, Bates 1996, Heston 1993, Chen and Scott 1992). These authors numerically solve for the delta and for the risk-neutral prob- ability of ®nishing in-the-money, which can be easily combined with the stock price and the strike price to generate the option value. Unfortunately, this approach is unable to harness the considerable computational power of the fast Fourier transform (FFT) (Walker 1996), which represents one of the most fundamental advances in scientific computing. Furthermore, though the decomposition of an option price into probability elements is theoretically attractive, as explained by Bakshi and Madan (1999), it is numerically undesirable owing to discontinuity of the payoffs.