Black–Scholes assumes volatility is constant. It isn't — implied vol changes with strike and maturity (the smile), and realized vol clusters. Heston's fix: make variance itself a random process.
Let be the price and the instantaneous variance. Heston couples two SDEs:
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Lesson close · Stochastic Volatility & Measure Change
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Continue to Girsanov & Radon–Nikodym: Pricing Under a New Measure, the next prerequisite step in this course.
with correlated Brownian drivers
Each parameter has a job:
| Param | Name | Role |
|---|---|---|
| mean-reversion speed | how fast variance pulls back | |
| long-run variance | the level it reverts to | |
| vol-of-vol | how erratic variance is | |
| correlation | usually : prices down ⇒ vol up (the leverage effect) |
The variance SDE is a Cox–Ingersoll–Ross process: mean-reverting and non-negative. It stays positive as long as the Feller condition holds:
Negative produces the equity skew (puts richer than calls); controls the smile's curvature. Crucially, Heston has a semi-closed-form characteristic function, so European options price by Fourier inversion in milliseconds — fast enough to calibrate to a live surface. The next lesson supplies the measure-change machinery that makes such pricing legitimate.
Free preview of the Stochastic Volatility course.
Based on Ibrahim Lanre Adedimeji’s “Heston Stochastic Volatility Model.”
Interactive · Heston path sketch
Knowledge check
Q1. In Heston, a negative correlation ρ between price and variance produces…
Q2. The Feller condition 2κθ ≥ ξ² guarantees what?
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