Lecture 2: Fitting the Volatility Skew

∗I am grateful to Peter Friz for carefully reading these notes, providing corrections and suggesting useful improvements. 5 Getting Implied Volatility from Local Volatil-ities 5.1 Motivation For a model to be useful in practice, it needs to return (at least approxi-mately) the current market prices of European options. That implies that we need to fit the parameters of our model (whether stochastic or local volatility model) to market implied volatilities. A number of ways have been suggested to do this for local volatility models. For example, we could work with the European option prices directly in a trinomial tree framework as in Derman, Kani, and Chriss (1996) or we could maximize relative entropy (of missing information) as in Avellaneda, Friedman, Holmes, and Samperi (1997). These methods are non-parametric (assuming actual option prices are used, not interpolated or extrapolated values). On the other hand, w