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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
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    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.

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    File URL: http://arxiv.org/pdf/1009.2782
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1009.2782.

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    Date of creation: Sep 2010
    Date of revision: Aug 2012
    Publication status: Published in Annals of Applied Probability 2012, Vol. 22, No. 4, 1541-1575
    Handle: RePEc:arx:papers:1009.2782

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    Web page: http://arxiv.org/

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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, Springer, vol. 16(3), pages 237-266, October.
    2. Jos\'e E. Figueroa-L\'opez & Ruoting Gong & Christian Houdr\'e, 2013. "A note on high-order short-time expansions for close-to-the-money option prices under the CGMY model," Papers 1305.4719, arXiv.org, revised Aug 2014.

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