Stochastic Volatility (Heston)
Heston makes variance itself random and mean-reverting, giving realistic smile dynamics at the price of a heavy five-parameter calibration.
Stochastic volatility makes variance itself a random process. In the Heston model, variance follows a mean-reverting square-root diffusion whose own volatility, the vol of vol, generates smile curvature, while correlation between variance and spot shocks produces the skew. A closed-form characteristic function makes vanilla prices quick through Fourier inversion, and the model delivers realistic smile dynamics — skews that steepen in selloffs and flatten in rallies — that local volatility cannot. The price is calibration: five parameters create a non-convex objective full of local minima, equity fits routinely violate the Feller condition, and sensible initial guesses plus regularization become essential.