When their underlyings are driven by stochastic processes with known characteristic functions, options can be priced by applying Fourier inversion methods. Two main trends are found in the literature, applying Fourier inversion either to the cumulative distribution function, which leads to Black-Scholes-style formulas, or to the probability density function. Apart from using quadrature rules to directly evaluate the inversion of the Fourier integral, the integration can be performed by Fast Fourier Transform algorithms or series expansions. An advantage both methods, thee FFT method and the Fourier Cosine Expansion method, offer is their good paralleliseability on CPUs as well as on graphics processing units.
Basically one considers a truncated stock price domain and discretizes this domain to obtain equidistantly spaced grid points with corresponding instrument values V(y,T). To propagate the value of the option from t(m) to t(m-1) we apply a FFT to V(:,t(m)) multiply with the characteristic function and finally apply the inverse FFT to obtain V(:,t(m-1)).
For Fast Fourier Transformation (FFT), one transformation can be performed with a computational complexity of O(N log2 N). A major drawback of the the FFT algorithm is the restriction to an equidistantly spaced grid in the price domain. In addition, the pricing of American/Bermudan and other exotic, in particular path dependent, options can require long computation times.
In our friday blog post we will focus on the before mentioned Fourier Cosine Expansion technique.