  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.
