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

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  • Daniel Guterding
  • Wolfram Boenkost

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.

Suggested Citation

  • Daniel Guterding & Wolfram Boenkost, 2018. "The Heston stochastic volatility model with piecewise constant parameters - efficient calibration and pricing of window barrier options," Papers 1805.04704, arXiv.org, revised Jan 2019.
    • Handle: RePEc:arx:papers:1805.04704
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    1. Anthonie W. Van Der Stoep & Lech A. Grzelak & Cornelis W. Oosterlee, 2014. "The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(07), pages 1-30.
    2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
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