The Heston model (1993) is the canonical stochastic volatility model. Rather than fixing (Black-Scholes) or making it a deterministic function of spot and time (local vol), Heston lets instantaneous variance follow a mean-reverting square-root process (a Cox-Ingersoll-Ross process) correlated with the stock. This one change — randomising the vol — makes the model produce a downward-sloping smile, a hump-shaped term structure of skew, and realistic forward dynamics. It is the starting point for almost every subsequent extension in the advanced derivatives literature.
This note presents the SDE system, its five parameters, the Feller condition that determines whether the variance process can touch zero, and the intuitive mapping from parameters to surface features. The heavy algebra — characteristic function derivation — is in the derivation sub-note.
Learning objectives. By the end of this note you should be able to:
- write the Heston SDE system for and identify all five parameters;
- state the Feller condition and explain what happens at the boundary when it fails;
- map each parameter to its primary effect on the vol surface (level, skew, curvature, term structure);
- compute the short-maturity and long-maturity ATM vol from .
In this lesson
- 1.1 The model
- 1.2 The Feller condition
- 1.3 Parameter-to-surface mapping
- 1.4 ATM vol formula
Before you start
- The Empirical Smile — Fat Tails, Skew, and the Limits of Black-Scholes
- The Forward Smile Problem and Why Local Vol Is Not Enough
The full lesson, with worked examples and exercise solutions, is for members.
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