Large deviations for some fast stochastic volatility models by viscosity methods

TL;DR

This paper investigates the short-term behavior of stochastic systems with fast-evolving stochastic volatility using viscosity methods, deriving large deviation principles for different regimes and applying results to option price asymptotics.

Contribution

It introduces a novel analysis of stochastic volatility models at different oscillation speeds using homogenisation and viscosity techniques, establishing large deviation principles.

Findings

Large deviation principles for three volatility regimes

Asymptotic formulas for option prices near maturity

Application of viscosity methods to stochastic volatility models

Abstract

We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.