The Local-Stochastic Volatility Model

The local-stochastic volatility (SLV) model is the production standard on equity exotic desks. Its construction follows directly from the SLV leverage constraint: once the leverage function $L(S,t)$ satisfying $L^2 = \sigma_L^2/\mathbb{E}[v|S]$ is in hand, the SLV model achieves what pure local vol and pure stochastic vol models each fail to achieve alone — an exact calibration to the vanilla surface and realistic forward smile dynamics.

This note presents the SLV SDE in detail, analyses how $L$ shapes the model's spot-vol dynamics, derives the forward smile improvement qualitatively and quantitatively, and identifies the three practical design choices — Heston parameter pre-calibration, correlation $\rho$, and the degree of vol-of-vol $\eta$ — that determine how much of the smile comes from stochastic vol versus leverage. The calibration algorithm itself (solving the Fokker–Planck PDE for $L$) is deferred to Lesson 3.

Learning objectives. By the end of this note you should be able to: