In this article we will explore how the Fourier Transform can be used to price options across a lot of strike prices in an efficient way following the method outlined in Carr and Madan (1999). It is important to understand that this technique doesn’t represent a model in itself, but rather, an elegant way to approximate solutions to the models that follow a Lévy process, but don’t have a closed-form solution or a “nice” Probability Density Function.

Of course, there is always the option to apply a Monte Carlo Simulation or some Numerical Method, however, this becomes very slow when you have to compute hundreds or thousands of prices at the same time, which is exactly what the Fourier Transform solves.

We will start with an understanding of the Fourier Transform (FT) and the Discrete Fourier Transform (DFT) starting from the Fourier Series. Afterwards, we will explore what characteristic functions are, and what they have to do with Probability Density Functions.

After exploring the background necessary to price options, we will outline and implement the framework on a simple model, the Black-Scholes Model, and we will measure its accuracy and computational speed.