  We introduce time-inhomogeneous stochastic volatility models, in which the
volatility is described by a nonnegative function of a Volterra type continuous
Gaussian process that may have very rough sample paths. The main results
obtained in the paper are sample path and small-noise large deviation
principles for the log-price process in a time-inhomogeneous super rough
Gaussian model under very mild restrictions. We use these results to study the
asymptotic behavior of binary barrier options, exit time probability functions,
and call options.
