  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.
