Mathematical Finance (q-fin.MF)

Abstract: 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, explicit pricing formulas for European options have been recently obtained and applied to establish a power law behaviour of the implied volatility skew in the case when the threshold is taken at the money. These results have been obtained by techniques based on the Laplace transform. The purpose of the present paper is to demonstrate how to obtain the same results by another method. This alternative approach is based on the natural relationship of the model with Skew Brownian motion and consists in the systematic use of the joint distribution of this stochastic process and some of its functionals.