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

## TL;DR

This paper evaluates the speed and accuracy of seven Fourier-based option pricing methods, highlighting their biases, convergence properties, and computational efficiency across different models, with practical recommendations for implementation.

## Contribution

It provides a comprehensive comparison of Fourier-based pricing methods, identifying the most efficient and accurate approaches, especially the multi-strike COS method and strike-optimized Carr Madan's formula.

## Key findings

- Attari's formula remains stable across Variance Gamma parameters.
- Multi-strike COS method is the fastest among tested methods.
- Strike-optimized Carr Madan's formula offers a superior speed-accuracy trade-off.

## Abstract

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.