Full and fast calibration of the Heston stochastic volatility model

TL;DR

This paper introduces a rapid and comprehensive calibration algorithm for the Heston stochastic volatility model, utilizing an analytical gradient approach to significantly improve efficiency and robustness in practical trading scenarios.

Contribution

The paper develops the fastest calibration method for the Heston model by deriving an analytical gradient and modifying the characteristic function to avoid discontinuities.

Findings

Calibration is around ten times faster than previous methods.

The method avoids multiple local minima, ensuring reliable convergence.

The objective function has a narrow valley with a flat bottom, facilitating optimization.

Abstract

This paper presents an algorithm for a complete and efficient calibration of the Heston stochastic volatility model. We express the calibration as a nonlinear least squares problem. We exploit a suitable representation of the Heston characteristic function and modify it to avoid discontinuities caused by branch switchings of complex functions. Using this representation, we obtain the analytical gradient of the price of a vanilla option with respect to the model parameters, which is the key element of all variants of the objective function. The interdependency between the components of the gradient enables an efficient implementation which is around ten times faster than a numerical gradient. We choose the Levenberg-Marquardt method to calibrate the model and do not observe multiple local minima reported in previous research. Two-dimensional sections show that the objective function is shaped as a narrow valley with a flat bottom. Our method is the fastest calibration of the Heston model developed so far and meets the speed requirement of practical trading.