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Derivations Using the Fourier Transform

Abstract

The Heston model laid the foundation for the popularity of Fourier transforms in mathematical finance. Fourier transforms have now earned their place as crucial tools in pricing models for equity derivatives. This owes to the fact that, in most cases, the terminal price density has no analytic structure. The characteristic function, on the other hand, is often readily available for many models, including the Heston model. The Heston model is a perfect illustration of why Fourier transforms are useful. Indeed, if the characteristic function is available for the price process, then the Fourier transform can be used to extract the probabilities from the characteristic function and obtain the call price using the Black–Scholes–style representation described in Chapter 2.

The application of Fourier transforms to option pricing is not limited to obtaining probabilities, as is done in Heston's (1993) original derivation. As explained by Wu (2008), the literature approaches Fourier transforms in option pricing in two broad ways. The first approach considers option prices to be analogous to cumulative distribution functions. This is the approach adopted by Heston (1993), Carr and Madan (1999), Bakshi and Madan (2000), and others. The second approach considers option prices to be analogous to probability density functions. This is the approach ...

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