ABSTRACT
This chapter introduces two representative models of stochastic volatility: the stochastic alpha, beta and rho model (SABR) model and the London Interbank Offer Rates version of the S. Heston's model by Wu and Zhang. In modern literature of option pricing, the notion of smiles means non-flat curves of implied Black's volatilities. On top of the constant-elasticity-variance model, Andersen and Brotherton-Ratcliffe superimposed an independent square-root volatility process that serves to generate additional curvature to the otherwise monotonic volatility skews. The SABR model can capture various shapes of implied volatility curves, and its approximate closed-form formula for vanilla options in terms of the Black implied volatilities enables efficient calculation of Greeks, the sensitivity parameters. After rigorous justifications were published in 1997, the market model, which is based on lognormal assumption for forward rates, established itself as the benchmark model for interest rate derivatives.
