A local-volatility model fits today’s vanilla surface exactly but makes a specific, empirically wrong prediction about the future smile: it flattens. This note traces that defect from the start—where the skew comes from, how diffusion averages it away over a forward-start horizon, and why a stochastic-volatility process does not suffer the same fate—and then builds the cure. Local stochastic volatility (LSV) marries the two: a stochastic variance for realistic dynamics, scaled by a deterministic leverage function that restores the exact vanilla fit. We motivate the leverage function through Gyöngy’s mimicking theorem, show why calibrating it is a fixed-point problem solved by bootstrapping forward in time, and develop the two work-horse solvers—the forward-PDE (Fokker–Planck) method and the particle (Monte-Carlo) method—in detail, with pseudocode for each. Appendices cover the supporting numerics: explicit versus implicit finite differences and the Craig–Sneyd ADI scheme, Euler–Maruyama time stepping, and kernel-weighted regression. Every smile and density is computed, not drawn by hand.
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Long-form technical notes — written in LaTeX, generated to the web from the same source.