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A hybrid approach for the implementation of the Heston model

Author

Listed:
  • Maya Briani

    (IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone" - CNR - National Research Council of Italy | Consiglio Nazionale delle Ricerche)

  • Lucia Caramellino

    (Dipartimento di Matematica [Rome] - Università degli Studi di Roma Tor Vergata [Roma] = University of Rome Tor Vergata)

  • Antonino Zanette

    (MATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech - Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique)

Abstract

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm.

Suggested Citation

  • Maya Briani & Lucia Caramellino & Antonino Zanette, 2017. "A hybrid approach for the implementation of the Heston model," Post-Print hal-00916440, HAL.
    • Handle: RePEc:hal:journl:hal-00916440
    DOI: 10.1093/imaman/dpv032
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    References listed on IDEAS

    as
    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    3. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
    4. 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.
    5. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    7. Lim Kian Guan & Guo Xiaoqiang, 2000. "Pricing American options with stochastic volatility: Evidence from S&P 500 futures options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 20(7), pages 625-659, August.
    8. 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.
    9. Ionuţ Florescu & Frederi Viens, 2008. "Stochastic Volatility: Option Pricing using a Multinomial Recombining Tree," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 151-181.
    10. Carl Chiarella & Boda Kang & Gunter H. Meyer, 2010. "The Evaluation Of Barrier Option Prices Under Stochastic Volatility," Research Paper Series 266, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:

    1. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2019. "Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate," Papers 1903.00369, arXiv.org, revised Jul 2019.
    2. 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.
    3. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2023. "Backward Hedging for American Options with Transaction Costs," Papers 2305.06805, arXiv.org, revised Jun 2023.
    4. Ludovic Goudenège & Andrea Molent & Antonino Zanette, 2020. "Computing credit valuation adjustment solving coupled PIDEs in the Bates model," Computational Management Science, Springer, vol. 17(2), pages 163-178, June.
    5. Ludovic Goudenege & Andrea Molent & Antonino Zanette, 2019. "Pricing and hedging GMWB in the Heston and in the Black–Scholes with stochastic interest rate models," Computational Management Science, Springer, vol. 16(1), pages 217-248, February.
    6. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2018. "Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model," Papers 1809.05328, arXiv.org.

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