  This paper concerns a local volatility model in which volatility takes two
possible values, and the specific value depends on whether the underlying price
is above or below a given threshold value. The model is known, and a number of
results have been obtained for it. In particular, option pricing formulas and a
power law behaviour of the implied volatility skew have been established in the
case when the threshold is taken at the money. In this paper we derive an
alternative representation of option pricing formulas. In addition, we obtain
an approximation of option prices by the corresponding Black-Scholes prices.
Using this approximation streamlines obtaining the aforementioned behaviour of
the skew. Our approach is based on the natural relationship of the model with
Skew Brownian motion and consists of the systematic use of the joint
distribution of this stochastic process and some of its functionals.
