Speed and biases of Fourier-based pricing choices: a numerical analysis

We compare the CPU effort and pricing biases of seven Fourier-based implementations. Our analyses show that truncation and discretization errors significantly increase as we move away from the Black-Scholes-Merton framework. We rank the speed and accuracy of the competing choices, showing which methods require smaller truncation ranges and which are the most efficient in terms of sampling densities. While all implementations converge well in the Bates jump-diffusion model, Attari's formula is the only Fourier-based method that does not blow up for any Variance Gamma parameter values. In terms of speed, the use of strike vector computations significantly improves the computational burden, rendering both fast Fourier transforms (FFT) and plain delta-probability decompositions inefficient. We conclude that the multi-strike version of the COS method is notably faster than any other implementation, whereas the strike-optimized Carr Madan's formula is simultaneously faster and more accurate than the FFT, thus questioning its use.