  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.
