Stochastic Volatility and Local Volatility

Stochastic volatility (SV) models are useful because they explain in a self-consistent way why options with different strikes and expirations have different Black-Scholes implied volatilities—that is, the ‘‘volatility smile.’’ Moreover, unlike alternative models that can fit the smile (such as local volatility models, for example), SV models assume realistic dynamics for the underlying whereas local volatility models do not really represent a separate class of models; the idea is more to make a simplifying assumption that allows practitioners to price exotic options consistently with the known prices of vanilla options. This chapter explores the volatility surface by introducing stochastic volatility—the notion that volatility varies in a random fashion.