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On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

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  • Roman V. Ivanov

    (Laboratory of Control under Incomplete Information, V.A. Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya 65, 117997 Moscow, Russia)

Abstract

This paper discusses the generalized Black-Scholes-Merton model, where the volatility coefficient, the drift coefficient of stocks, and the interest rate are time-dependent deterministic functions. Together with it, we make the assumption that the volatility, the drift, and the interest rate depend on a gamma or inverse-gamma random variable. This model includes the models of skew Student’s t- and variance-gamma-distributed stock log-returns. The price of the European forward-start call option is derived from the considered models in closed form. The obtained formulas are compared with the Black-Scholes formula through examples.

Suggested Citation

  • Roman V. Ivanov, 2023. "On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model," Risks, MDPI, vol. 11(6), pages 1-23, June.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:6:p:111-:d:1167116
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    References listed on IDEAS

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    1. Sharif Mozumder & Ghulam Sorwar & Kevin Dowd, 2015. "Revisiting variance gamma pricing: An application to S&P500 index options," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-24.
    2. Albert N. Shiryaev & Jan Kallsen, 2002. "The cumulant process and Esscher's change of measure," Finance and Stochastics, Springer, vol. 6(4), pages 397-428.
    3. Zhongxian Men & Tony S. Wirjanto & Adam W. Kolkiewicz, 2021. "Multiscale Stochastic Volatility Model with Heavy Tails and Leverage Effects," JRFM, MDPI, vol. 14(5), pages 1-28, May.
    4. Ernst Eberlein & Antonis Papapantoleon & Albert Shiryaev, 2008. "On the duality principle in option pricing: semimartingale setting," Finance and Stochastics, Springer, vol. 12(2), pages 265-292, April.
    5. Roman V. Ivanov, 2018. "A Credit-Risk Valuation under the Variance-Gamma Asset Return," Risks, MDPI, vol. 6(2), pages 1-25, May.
    6. Damiano Brigo & Fabio Mercurio, 2002. "Lognormal-Mixture Dynamics And Calibration To Market Volatility Smiles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(04), pages 427-446.
    7. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models," Risks, MDPI, vol. 8(4), pages 1-31, September.
    8. Shackleton, Mark B. & Taylor, Stephen J. & Yu, Peng, 2010. "A multi-horizon comparison of density forecasts for the S&P 500 using index returns and option prices," Journal of Banking & Finance, Elsevier, vol. 34(11), pages 2678-2693, November.
    9. Ernst Eberlein & Antonis Papapantoleon & Albert N. Shiryaev, 2008. "Esscher transform and the duality principle for multidimensional semimartingales," Papers 0809.0301, arXiv.org, revised Nov 2009.
    10. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A generative adversarial network approach to calibration of local stochastic volatility models," Papers 2005.02505, arXiv.org, revised Sep 2020.
    11. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 361-393.
    12. Thomas Fung & Eugene Seneta, 2010. "Modelling and Estimation for Bivariate Financial Returns," International Statistical Review, International Statistical Institute, vol. 78(1), pages 117-133, April.
    13. Seok Young Hong & Ingmar Nolte & Stephen J Taylor & Xiaolu Zhao, 2023. "Volatility Estimation and Forecasts Based on Price Durations," Journal of Financial Econometrics, Oxford University Press, vol. 21(1), pages 106-144.
    14. Ronald J. Mahieu & Peter C. Schotman, 1998. "An empirical application of stochastic volatility models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 13(4), pages 333-360.
    15. Shuaiqiang Liu & Cornelis W. Oosterlee & Sander M. Bohte, 2019. "Pricing Options and Computing Implied Volatilities using Neural Networks," Risks, MDPI, vol. 7(1), pages 1-22, February.
    16. Damiano Brigo & Fabio Mercurio & Giulio Sartorelli, 2003. "Alternative asset-price dynamics and volatility smile," Quantitative Finance, Taylor & Francis Journals, vol. 3(3), pages 173-183.
    17. Roman V. Ivanov, 2022. "The Risk Measurement under the Variance-Gamma Process with Drift Switching," JRFM, MDPI, vol. 15(1), pages 1-27, January.
    18. Kjersti Aas & Ingrid Hobaek Haff, 2006. "The Generalized Hyperbolic Skew Student's t-Distribution," Journal of Financial Econometrics, Oxford University Press, vol. 4(2), pages 275-309.
    19. Takahashi, Makoto & Watanabe, Toshiaki & Omori, Yasuhiro, 2021. "Forecasting Daily Volatility of Stock Price Index Using Daily Returns and Realized Volatility," Discussion paper series HIAS-E-104, Hitotsubashi Institute for Advanced Study, Hitotsubashi University.
    20. 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.
    21. Andersen, Torben G. & Bollerslev, Tim & Dobrev, Dobrislav, 2007. "No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and i.i.d. noise: Theory and testable distributional implications," Journal of Econometrics, Elsevier, vol. 138(1), pages 125-180, May.
    22. Jouchi Nakajima, 2020. "Skew selection for factor stochastic volatility models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 47(4), pages 582-601, March.
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    Cited by:

    1. Rajeev Rajaram & Nathan Ritchey, 2023. "Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy," Mathematics, MDPI, vol. 11(15), pages 1-16, July.

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