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Small-time asymptotics for fast mean-reverting stochastic volatility models

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  • Jin Feng
  • Jean-Pierre Fouque
  • Rohini Kumar

Abstract

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment generating function computation in the particular case of the Heston model.

Suggested Citation

  • Jin Feng & Jean-Pierre Fouque & Rohini Kumar, 2010. "Small-time asymptotics for fast mean-reverting stochastic volatility models," Papers 1009.2782, arXiv.org, revised Aug 2012.
    • Handle: RePEc:arx:papers:1009.2782
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    Cited by:

    1. Konstantinos Spiliopoulos & Alexandra Chronopoulou, 2013. "Maximum likelihood estimation for small noise multiscale diffusions," Statistical Inference for Stochastic Processes, Springer, vol. 16(3), pages 237-266, October.
    2. Popovic, Lea, 2019. "Large deviations of Markov chains with multiple time-scales," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3319-3359.
    3. Jos'e E. Figueroa-L'opez & Ruoting Gong & Christian Houdr'e, 2013. "Third-Order Short-Time Expansions for Close-to-the-Money Option Prices under the CGMY Model," Papers 1305.4719, arXiv.org, revised Nov 2017.
    4. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    5. Antoine Jacquier & Konstantinos Spiliopoulos, 2018. "Pathwise moderate deviations for option pricing," Papers 1803.04483, arXiv.org, revised Dec 2018.
    6. Gailus, Siragan & Spiliopoulos, Konstantinos, 2017. "Statistical inference for perturbed multiscale dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 419-448.
    7. Bezemek, Z.W. & Spiliopoulos, K., 2023. "Large deviations for interacting multiscale particle systems," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 27-108.
    8. Dan Pirjol & Lingjiong Zhu, 2017. "Short Maturity Asian Options for the CEV Model," Papers 1702.03382, arXiv.org.
    9. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Small-time asymptotics for Gaussian self-similar stochastic volatility models," Papers 1505.05256, arXiv.org, revised Mar 2016.
    10. Jos'e E. Figueroa-L'opez & Ruoting Gong & Christian Houdr'e, 2012. "High-order short-time expansions for ATM option prices of exponential L\'evy models," Papers 1208.5520, arXiv.org, revised Apr 2014.
    11. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    12. Hossein Jafari & Ghazaleh Rahimi, 2019. "Small-Time Asymptotics In Geometric Asian Options For A Stochastic Volatility Jump-Diffusion Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(02), pages 1-19, March.
    13. Blessing, Jonas & Kupper, Michael & Nendel, Max, 2023. "Convergence of Infintesimal Generators and Stability of Convex Montone Semigroups," Center for Mathematical Economics Working Papers 680, Center for Mathematical Economics, Bielefeld University.
    14. Kumar, Rohini & Popovic, Lea, 2017. "Large deviations for multi-scale jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1297-1320.

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