Small-Time Option Pricing in Fast Mean-Reverting Stochastic Volatility Models

Summary

This paper studies option pricing when the option maturity is short but still long relative to the mean-reversion time of a stochastic volatility factor. It treats the problem as an averaging or homogenization problem for nonlinear Hamilton–Jacobi–Bellman equations, with a fast factor evolving over a noncompact state space.

The authors develop a general argument using viscosity solutions and apply it to two regimes. They derive a large deviation principle and use it to obtain asymptotic prices for out-of-the-money calls and puts, together with their implied volatilities. The abstract does not specify the two regimes in detail or report numerical tests. It presents the results as a generalization of earlier work that used moment-generating functions for the Heston model, so applicability depends on the assumptions of the regimes and model rather than establishing universal pricing accuracy.

Key ideas

• The analysis considers maturities that are short but exceed the volatility factor’s mean-reversion timescale.
• The fast stochastic volatility factor is handled through an averaging problem for a nonlinear HJB equation.
• Viscosity-solution methods support the argument when the fast variable lies in a noncompact space.
• A large deviation principle yields asymptotic prices and implied volatilities for out-of-the-money calls and puts.
• The framework extends a result previously obtained for the Heston model using moment-generating functions.

Tags

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


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









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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