The Heston stochastic volatility model with piecewise constant parameters - efficient calibration and pricing of window barrier options

TL;DR

This paper introduces an efficient calibration method for the Heston stochastic volatility model with piecewise constant parameters, enabling better market fit and pricing of window barrier options through semi-analytical formulas and advanced numerical techniques.

Contribution

It develops a simple, numerically efficient calibration approach for the Heston model with time-dependent parameters using semi-analytical formulas and Gauss-Kronrod quadrature.

Findings

Improved calibration accuracy for FX options.

Enhanced pricing of window barrier options.

Demonstrated advantages over standard Heston model.

Abstract

The Heston stochastic volatility model is a standard model for valuing financial derivatives, since it can be calibrated using semi-analytical formulas and captures the most basic structure of the market for financial derivatives with simple structure in time-direction. However, extending the model to the case of time-dependent parameters, which would allow for a parametrization of the market at multiple timepoints, proves more challenging. We present a simple and numerically efficient approach to the calibration of the Heston stochastic volatility model with piecewise constant parameters. We show that semi-analytical formulas can also be derived in this more complex case and combine them with recent advances in computational techniques for the Heston model. Our numerical scheme is based on the calculation of the characteristic function using Gauss-Kronrod quadrature with an additional control variate that stabilizes the numerical integrals. We use our method to calibrate the Heston model with piecewise constant parameters to the foreign exchange (FX) options market. Finally, we demonstrate improvements of the Heston model with piecewise constant parameters upon the standard Heston model in selected cases.