European Option Pricing Using Fourier Transform

Fourier transform techniques are playing an increasingly important role in Mathematical Finance. Some sophisticated price dynamics like affine jump diffusions and Levy processes often do not possess density functions in closed form or have quite complicated analytical expressions involving special functions and infinite summations. However, for many of the more advanced asset price models characteristic functions are available in closed form. By analogy, if the characteristic function of the underlying is tractable, option prices can also be obtained by a single integration.The method of Fourier transform can obviously simplify the pricing process.In this paper we first review the convenient mathematical properties of Fourier transforms and characteristic functions, then develop two different form of pricing formulae by Fourier transform.The first approach is reducing the probability density function of underlying by Fourier transform,and then reducing the value of options.The second approach is reducing the value of options directly by Fourier transform.We discretise the formulae developed by the second approach and find out the expression can be easily applied by computer with Fast Fourier transform(FFT) method.At last we describe the characteristic functions of some different modles for option pricing like the BS modle and the VG modle.And use history data to calculate the value of option.The result shows that applying Fourier transform to value options is accurate and efficiency.