Overview

The local volatility and stochastic volatility branches of the unit answer the same question — "what generates the smile?" — from opposite directions. Local vol encodes the surface in a deterministic function $\sigma_L(S,t)$ and fits vanilla prices exactly but predicts unrealistic forward smiles. Stochastic vol (Heston, SABR) generates realistic dynamics and calibrates approximately. This topic is about the synthesis: the local-stochastic volatility (SLV) model, which achieves exact surface calibration and realistic dynamics by multiplying a Heston-type process by a deterministic leverage function chosen via Gyöngy's theorem. After SLV, the topic surveys rough volatility, the research-frontier extension that replaces the Brownian driver in the variance process with a fractional Brownian motion ($H < \tfrac{1}{2}$) and achieves dramatically better fits to short-dated smiles.

The mathematical glue is Gyöngy's theorem: any Itô process has the same one-dimensional marginal distributions as a Markov diffusion whose squared diffusion coefficient is the conditional expectation of instantaneous variance. In finance this says any stochastic vol model calibrated to a surface is vanilla-equivalent to the local vol model extracted from that surface — a fact that explains both why local vol works for vanillas and why it fails for path-dependent products.