Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model

TL;DR

This paper develops large deviation principles for stochastic volatility models incorporating reflecting diffusions, addressing misspecification issues in classical models and analyzing the asymptotic behavior of options in small-noise regimes.

Contribution

It introduces a novel class of stochastic volatility models with reflection and establishes large deviation principles for their log-price processes.

Findings

Large deviation principles for reflected stochastic volatility models.

Asymptotic analysis of barrier options and call prices.

Application to models with reflecting Ornstein-Uhlenbeck processes.

Abstract

We introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being because of a certain volatility misspecification in the classical Stein and Stein model. A version of this model that uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a special example of a stochastic volatility model with reflection. The main results obtained in the present paper are sample path and small-noise large deviation principles for the log-price process in a stochastic volatility model with reflection under rather mild restrictions. We use these results to study the asymptotic behavior of binary barrier options and call prices in the small-noise regime.