As shown in Schoutens et al. (2004), different dynamics for the underlying asset price are able to achieve a similarly good calibration to the market prices of European plain vanilla options. Consequently, these models are able to generate very similar marginal distributions for all future points in time. However, their path-behaviour differs thus often yielding significantly different prices for exotic options. In this post, we provide some intuition for the sign of the price differences between local and stochastic volatility models for barrier and American binary options. These arguments are not novel and we refer to e.g. Baker et al. (2004) for a more in-depth and rigorous discussion.
Jim Gatheral’s book “The Volatility Surface – A Pratitioner’s Guide” (Wiley, 2006) provides an excellent treatment on volatility modelling. Most of the proofs and derivations are only outlined in the book and it is left to the reader to do the intermediate steps. The attached document provides the missing steps in the derivation of the local volatility as a function of the implied volatility on pages 11 – 13 in the book.