Stochastic Volatility (Heston) and Characteristic Functions
Heston couples price and variance dynamics; its characteristic function gives semi-closed-form prices.
Heston models stochastic variance with mean reversion, vol-of-vol, and correlation with the price process.
Its characteristic function allows Fourier-based option pricing and fast calibration to smiles.
Parameter combinations control smile shape: skew, curvature, and term structure of implied volatility.
Heston couples the price and variance dynamics: volatility itself moves over time and can be correlated with price shocks. Here you see a single simulated path and the implied-volatility smile it induces.
Increasing vol-of-vol makes the green volatility path move more and deepens the curvature of the smile. Changing ρ tilts the smile: negative correlation (price down when vol up) produces the familiar equity-style left skew, while positive ρ pushes mass to the right.
The top panel shows how price and volatility co-move over time (scaled to the same frame). The bottom panel summarises this in one number per strike: the implied volatility that would be consistent with these stochastic-volatility dynamics at a fixed maturity.