Accelerating the Calibration of Stochastic Volatility Models

Abstract

Option valuation models keep advancing toward greater versatility and realism, by introducing more-flexible stochastic processes and more-complex payoff structures. Allowing stochastically time-varying volatility and jumps in the returns process, the volatility process, or both greatly broadens the range of asset price behavior that can be incorporated in the pricing model. One of the key mathematical tools for solving more-complicated models is the use of characteristic functions and related transform methods. But these tools require computing integrals in the complex domain, which is complicated and has some potential pitfalls. Several different numerical solution techniques are used, including direct integration with an approach such as Gaussian quadrature, the Fast Fourier Transform (FFT), or fractional FFT. In this article, Kilin reviews these methods and proposes a computational enhancement to the basic Gaussian quadrature approach that speeds it up substantially by storing intermediate results and reusing the saved values rather than recomputing them each time. In the direct comparison reported here, the modified quadrature approach was around 30 times faster than FFT and 15 times faster than fractional FFT.