ABSTRACT
This chapter is devoted to the study of the volatility process. From one side, we see how to estimate the spot and the integrated volatilities from market asset prices, via realized variances and via Fourier-Malliavin estimators. From the other side, we introduce the main volatility models introduced in the literature to replicate real market data. In particular, we see how classical results as Gy.ongy's lemma and the Dupire formula allow us to construct a local volatility model (that is, a model where the volatility is a function of the asset price) that replicates market European option prices. We also study the so-called stochastic volatility models, where the volatility is a diffusion process, like in the SABR or the Heston models; and stochastic-local volatility models. We also discuss rough volatility models, driven by a fractional Brownian motion, and their role in volatility modeling. Finally, we introduce the main volatilty derivatives as variance or volatility swaps.
