ABSTRACT
In this chapter we highlight the most common stochastic models used to model the volatility surface in practice. We begin with simple models that fail to accurately incorporate the dynamics we observe in different volatility surfaces, such as the Bachelier, Black-Scholes and CEV model. We then consider three different approaches for modeling the volatility surface, stochastic volatility models, jump-diffusion models, and local volatility models. When discussing stochastic volatility models, we consider the Heston and SABR models. Similarly, for jump-diffusion models, we consider Variance Gamma, Merton's Jump Diffusion model and others. Additionally, many characteristic functions for the stochastic volatility and jump-diffusion models are presented so that these models can be leveraged using the FFT techniques introduced in the next chapter.
