ABSTRACT
Given market prices for options on the same underlying asset with different maturities, one will typically obtain different implied volatilities for different maturities. Usually, this can still be reconciled with the Black/Scholes model if one allows for time–varying deterministic volatility. As the implied volatilities of options away from the money are often higher than those of at–the–money options, this is called the volatility smile. This chapter considers two approaches: implied distributions and stochastic volatility. Fitting a model to the implied risk neutral distribution is the most direct way of ensuring consistency with current market prices. In practice, market prices for options are available only for a small number of strikes, so it becomes necessary to interpolate. This interpolation could be implemented at any one of three levels: the option prices, the implied volatilities, or the implied distribution.
