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

TL;DR

This paper analyzes small-time asymptotics in fast mean-reverting stochastic volatility models, deriving large deviation principles and asymptotic option prices, extending previous results to more general regimes.

Contribution

It introduces a viscosity solutions approach to derive large deviation principles for regimes with fast mean reversion, generalizing prior specific model results.

Findings

Derived large deviation principles for the models.

Obtained asymptotic prices for out-of-the-money options.

Extended previous Heston model results to broader regimes.

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.