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High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids

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  • Bertram During
  • Michel Fourni'e
  • Christof Heuer

Abstract

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiments a comparative standard second-order discretisation is significantly outperformed. We conduct a numerical stability study which indicates unconditional stability of the scheme.

Suggested Citation

  • Bertram During & Michel Fourni'e & Christof Heuer, 2014. "High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids," Papers 1404.5138, arXiv.org.
    • Handle: RePEc:arx:papers:1404.5138
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    References listed on IDEAS

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    1. 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.
    2. Nigel Clarke & Kevin Parrott, 1999. "Multigrid for American option pricing with stochastic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 177-195.
    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. Jin‐Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32, January.
    5. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv.
    6. Bertram Düring, 2009. "Asset pricing under information with stochastic volatility," Review of Derivatives Research, Springer, vol. 12(2), pages 141-167, July.
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    Citations

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    Cited by:

    1. Sinem Kozp{i}nar & Murat Uzunca & Bulent Karasozen, 2016. "Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements," Papers 1606.08381, arXiv.org, revised Mar 2020.
    2. Bertram During & James Miles, 2015. "High-order ADI scheme for option pricing in stochastic volatility models," Papers 1512.02529, arXiv.org.
    3. Bertram During & Christof Heuer, 2016. "Essentially high-order compact schemes with application to stochastic volatility models on non-uniform grids," Papers 1611.00316, arXiv.org.
    4. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach to compute the first-passage probability density function in a time-changed Brownian model: Applications in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 463(C), pages 330-344.
    5. Bertram During & Alexander Pitkin, 2017. "High-order compact finite difference scheme for option pricing in stochastic volatility jump models," Papers 1704.05308, arXiv.org, revised Feb 2019.
    6. Kemper, Annika & Schmeck, Maren Diane & Kh.Balci, Anna, 2022. "The market price of risk for delivery periods: Pricing swaps and options in electricity markets," Energy Economics, Elsevier, vol. 113(C).
    7. Bertram During & Christof Heuer, 2015. "High-order compact schemes for Black-Scholes basket options," Papers 1505.07613, arXiv.org.
    8. Kemper, Annika & Schmeck, Maren Diane & Khripunova Balci, Anna, 2020. "The Market Price of Risk for Delivery Periods: Pricing Swaps and Options in Electricity Markets," Center for Mathematical Economics Working Papers 635, Center for Mathematical Economics, Bielefeld University.
    9. Kozpınar, Sinem & Uzunca, Murat & Karasözen, Bülent, 2020. "Pricing European and American options under Heston model using discontinuous Galerkin finite elements," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 568-587.
    10. Süleyman Cengizci & Ömür Uğur, 2025. "A Computational Study for Pricing European- and American-Type Options Under Heston’s Stochastic Volatility Model: Application of the SUPG-YZ $$\beta$$ β Formulation," Computational Economics, Springer;Society for Computational Economics, vol. 66(1), pages 179-206, July.
    11. Bertram During & Christian Hendricks & James Miles, 2016. "Sparse grid high-order ADI scheme for option pricing in stochastic volatility models," Papers 1611.01379, arXiv.org.

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