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Lessons

The Heston Stochastic Volatility Model8m
Girsanov & Radon–Nikodym: Pricing Under a New Measure8m

MatheLinux — quantitative finance, taught rigorously.

Course content credited to Ibrahim Lanre Adedimeji.

← Stochastic Volatility & Measure Change

The Heston Stochastic Volatility Model

Black–Scholes assumes volatility $\sigma$ 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.

The model

Let $S_t$ be the price and $v_t$ the instantaneous variance. Heston couples two SDEs:

d S_{t} = μ S_{t}

Knowledge check

Q1. In Heston, a negative correlation ρ between price and variance produces…

Q2. The Feller condition 2κθ ≥ ξ² guarantees what?

Mark this lesson complete to track progress

d

t

+

\sqrt{v_{t}}

S_{t}

d

W_{t}^{S}

,

dS_t = \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S,

d S_{t} = μ S_{t} d t + v_{t} S_{t} d W_{t}^{S},

dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^v,

with correlated Brownian drivers $\;d\langle W^S, W^v\rangle_t = \rho\,dt.$

Each parameter has a job:

Param	Name	Role
$\kappa$	mean-reversion speed	how fast variance pulls back
$\theta$	long-run variance	the level it reverts to
$\xi$	vol-of-vol	how erratic variance is
$\rho$	correlation	usually $<0$ : 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:

2\kappa\theta \ge \xi^2 .

Why it earns its keep

Negative $\rho$ produces the equity skew (puts richer than calls); $\xi$ 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 $(\kappa,\theta,\xi,\rho,v_0)$ 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.”