Multivariate Stochastic Volatility Models and Large Deviation Principles

TL;DR

This paper develops a comprehensive large deviation principle for multivariate stochastic volatility models, enabling asymptotic analysis of rare events and option pricing in complex financial models.

Contribution

It establishes a new sample path large deviation principle applicable to a wide class of multivariate stochastic volatility models, including Gaussian, fractional, and Volterra type models.

Findings

Large deviation formulas for first exit times of log-processes.

Asymptotic estimates for multidimensional binary barrier option prices.

Sample path LDP for solutions to Volterra stochastic integral equations.

Abstract

We establish a comprehensive sample path large deviation principle (LDP) for log-processes associated with multivariate time-inhomogeneous stochastic volatility models. Examples of models for which the new LDP holds include Gaussian models, non-Gaussian fractional models, mixed models, models with reflection, and models in which the volatility process is a solution to a Volterra type stochastic integral equation. The LDP for log-processes is used to obtain large deviation style asymptotic formulas for the distribution function of the first exit time of a log-process from an open set and for the price of a multidimensional binary barrier option. We also prove a sample path LDP for solutions to Volterra type stochastic integral equations with predictable coefficients depending on auxiliary stochastic processes.