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Mean-Reverting Stochastic Volatility

Author

Listed:
  • JEAN-PIERRE FOUQUE

    (Department of Mathematics, North Carolina State University, Raleigh NC 27695-8205, USA)

  • GEORGE PAPANICOLAOU

    (Department of Mathematics, Stanford University, Stanford CA 94305, USA)

  • K. RONNIE SIRCAR

    (Department of Mathematics, University of Michigan, Ann Arbor MI 48109-1109, USA)

Abstract
We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by-tick fluctuations of the index value, but it isfastmean-reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales.The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log-moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure, and the data produces estimates of the three important parameters which are found to be stable within periods where the underlying volatility is close to being stationary. These segments of stationarity are identified using a wavelet-based tool.The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk can be roughly estimated, but are not needed for the asymptotic pricing formulas for European derivatives. The extension to American and path-dependent contingent claims is the subject of future work.

Suggested Citation

  • Jean-Pierre Fouque & George Papanicolaou & K. Ronnie Sircar, 2000. "Mean-Reverting Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 101-142.
    • Handle: RePEc:wsi:ijtafx:v:03:y:2000:i:01:n:s0219024900000061
    DOI: 10.1142/S0219024900000061
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    Citations

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    Cited by:

    1. Kim, Hyun-Gyoon & Kim, Jeong-Hoon, 2023. "A stochastic-local volatility model with Le´vy jumps for pricing derivatives," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    2. Jean-Pierre Fouque & Sebastian Jaimungal & Yuri F. Saporito, 2021. "Optimal Trading with Signals and Stochastic Price Impact," Papers 2101.10053, arXiv.org, revised Aug 2023.
    3. Kim, Donghyun & Choi, Sun-Yong & Yoon, Ji-Hun, 2021. "Pricing of vulnerable options under hybrid stochastic and local volatility," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    4. Zhenyu Cui & J. Lars Kirkby & Guanghua Lian & Duy Nguyen, 2017. "Integral Representation Of Probability Density Of Stochastic Volatility Models And Timer Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(08), pages 1-32, December.
    5. Chuo Chang, 2020. "Dynamic correlations and distributions of stock returns on China's stock markets," Journal of Applied Finance & Banking, SCIENPRESS Ltd, vol. 10(1), pages 1-6.
    6. Nils Bertschinger & Oliver Pfante, 2015. "Inferring Volatility in the Heston Model and its Relatives -- an Information Theoretical Approach," Papers 1512.08381, arXiv.org.
    7. Alexander Lykov & Stepan Muzychka & Kirill Vaninsky, 2016. "Investor'S Sentiment In Multi-Agent Model Of The Continuous Double Auction," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(06), pages 1-29, September.
    8. Martin Tegnér & Rolf Poulsen, 2018. "Volatility Is Log-Normal—But Not for the Reason You Think," Risks, MDPI, vol. 6(2), pages 1-16, April.
    9. Frezza, Massimiliano & Bianchi, Sergio & Pianese, Augusto, 2021. "Fractal analysis of market (in)efficiency during the COVID-19," Finance Research Letters, Elsevier, vol. 38(C).
    10. Gordon R. Richards, 2004. "A fractal forecasting model for financial time series," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 23(8), pages 586-601.
    11. Alibeiki, Hedayat & Lotfaliei, Babak, 2022. "To expand and to abandon: Real options under asset variance risk premium," European Journal of Operational Research, Elsevier, vol. 300(2), pages 771-787.
    12. Leandro Maciel & Fernando Gomide & Rosangela Ballini, 2014. "An Evolving Fuzzy-Garch Approach Forfinancial Volatility Modeling And Forecasting," Anais do XL Encontro Nacional de Economia [Proceedings of the 40th Brazilian Economics Meeting] 138, ANPEC - Associação Nacional dos Centros de Pós-Graduação em Economia [Brazilian Association of Graduate Programs in Economics].
    13. Weixuan Xia, 2023. "Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences," Papers 2312.00266, arXiv.org.
    14. Maxim Bichuch & Jean-Pierre Fouque, 2019. "Optimal Investment with Correlated Stochastic Volatility Factors," Papers 1908.07626, arXiv.org, revised Nov 2022.
    15. Rebonato, Riccardo & Ronzani, Riccardo, 2021. "Is convexity efficiently priced? Evidence from international swap markets," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 392-413.
    16. Kim, Hyun-Gyoon & Kim, See-Woo & Kim, Jeong-Hoon, 2024. "Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model," The North American Journal of Economics and Finance, Elsevier, vol. 72(C).
    17. Perelló, Josep & Masoliver, Jaume & Anento, Napoleón, 2004. "A comparison between several correlated stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 134-137.
    18. Li, Hongshan & Huang, Zhongyi, 2020. "An iterative splitting method for pricing European options under the Heston model☆," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    19. Yipeng Yang & Allanus Tsoi, 2013. "Prospect Agents and the Feedback Effect on Price Fluctuations," Papers 1308.6759, arXiv.org, revised Jan 2014.
    20. Hongshan Li & Zhongyi Huang, 2020. "An iterative splitting method for pricing European options under the Heston model," Papers 2003.12934, arXiv.org.
    21. Ma, Chao & Ma, Qinghua & Yao, Haixiang & Hou, Tiancheng, 2018. "An accurate European option pricing model under Fractional Stable Process based on Feynman Path Integral," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 87-117.
    22. Lee, Min-Ku, 2016. "Asymptotic approach to the pricing of geometric asian options under the CEV model," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 544-548.
    23. Liu, Chang & Chang, Chuo, 2021. "Combination of transition probability distribution and stable Lorentz distribution in stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).

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