ABSTRACT

In this chapter, we will discuss the use of transform techniques for pricing derivatives. As discussed in Section 1.1, one of the primary representations of the distribution of prices for a given asset is its characteristic function. The characteristic function of the distribution of asset prices is merely the Fourier transform of its probability distribution function (PDF). Thus its probability distribution function can be recovered from the characteristic function through Fourier inversion. This is particularly important for many classes of models which, as discussed in Section 1.2, have a closed form only in their characteristic function representation. We will outline techniques for pricing derivatives under a variety of different models using transform methods, focusing on fast Fourier transform (FFT) based techniques, fractional fast Fourier transforms, and the recently developed Fourier cosine (COS) method. Finally we will consider the saddlepoint method.