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An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

Author

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  • Lorella Fatone
  • Francesca Mariani
  • Maria Cristina Recchioni
  • Francesco Zirilli

Abstract

We introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univ pm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm. it/recchioni/finance . © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009

Suggested Citation

  • Lorella Fatone & Francesca Mariani & Maria Cristina Recchioni & Francesco Zirilli, 2009. "An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 29(9), pages 862-893, September.
  • Handle: RePEc:wly:jfutmk:v:29:y:2009:i:9:p:862-893
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    Cited by:

    1. Maria Cristina Recchioni & Yu Sun & Gabriele Tedeschi, 2017. "Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 17(8), pages 1257-1275, August.
    2. Gifty Malhotra & R. Srivastava & H. C. Taneja, 2018. "Quadratic approximation of the slow factor of volatility in a multifactor stochastic volatility model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(5), pages 607-624, May.
    3. Seungho Yang & Jaewook Lee, 2014. "Do affine jump-diffusion models require global calibration? Empirical studies from option markets," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 111-123, January.
    4. Recchioni, M.C. & Sun, Y., 2016. "An explicitly solvable Heston model with stochastic interest rate," European Journal of Operational Research, Elsevier, vol. 249(1), pages 359-377.
    5. Bianca Reichert & Adriano Mendon a Souza, 2022. "Can the Heston Model Forecast Energy Generation? A Systematic Literature Review," International Journal of Energy Economics and Policy, Econjournals, vol. 12(1), pages 289-295.
    6. Jang, H. & Lee, J., 2019. "Machine learning versus econometric jump models in predictability and domain adaptability of index options," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 74-86.
    7. Tao Pang & Katherine Varga, 2019. "Portfolio Optimization for Assets with Stochastic Yields and Stochastic Volatility," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 691-729, August.
    8. Issouf Soumaré & Ernest Tafolong, 2017. "Risk-based capital for credit insurers with business cycles and dynamic leverage," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 597-612, April.

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