The Annals of Applied Probability

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

Jin Feng, Jean-Pierre Fouque, and Rohini Kumar

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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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Ann. Appl. Probab., Volume 22, Number 4 (2012), 1541-1575.

First available in Project Euclid: 10 August 2012

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Primary: 60F10: Large deviations 91B70: Stochastic models 49L25: Viscosity solutions

Stochastic volatility multi-scale asymptotic large deviation principle implied volatility smile/skew


Feng, Jin; Fouque, Jean-Pierre; Kumar, Rohini. Small-time asymptotics for fast mean-reverting stochastic volatility models. Ann. Appl. Probab. 22 (2012), no. 4, 1541--1575. doi:10.1214/11-AAP801.

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