Heston Stochastic Volatility Model

Stochastic Vol

Models volatility as a mean-reverting stochastic process correlated with returns. Captures volatility clustering (high vol follows high vol), leverage effect (negative correlation between returns and volatility), and the implied volatility smile.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Instantaneous variance at time t
Drift (expected return)
Mean reversion speed
Long-run variance level
Volatility of variance (vol-of-vol)
Correlation between price and variance
Feller condition (ensures v_t > 0)

Key Assumptions

  • •Variance follows a CIR (Cox-Ingersoll-Ross) process
  • •Mean reversion in volatility
  • •Leverage effect via correlation ρ < 0
  • •Feller condition ensures strictly positive variance

Reference

Heston, S.L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility." The Review of Financial Studies.

Model Hierarchy

Built on simpler models
SimplerBaseline

GBM

Diffusion
Extended by more complex models
More ComplexComplex

Bates

Jump-Diffusion
More ComplexAdvanced

Rough Heston

Stochastic Vol
More ComplexAdvanced

RSSV

Stochastic Vol
More ComplexComplex

SLV

Local Vol

Complexity increases from left to right. More complex models capture additional market phenomena but require more parameters and may be harder to estimate.

Empirical Context (SPY Returns)

How Heston relates to observed return properties

SPY returns show persistent volatility regimes and negative correlation between returns and volatility changes. Heston captures this through mean-reverting stochastic volatility with leverage, but cannot reproduce the sudden discontinuous moves that drive extreme tail quantiles.

What Heston captures

  • Volatility clustering
  • Mean reversion in volatility
  • Leverage effect (negative correlation)

What it cannot capture

  • Sudden jumps
  • Extreme tail quantiles (1%/99%)
  • Discrete event risk
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

Parameter Estimation (P-Measure)

Estimate Heston model parameters from historical SPY data using Quasi-Maximum Likelihood Estimation (QMLE).

Related Models

Bates

Jump-Diffusion

Combines Heston stochastic volatility with Merton-style jumps, providing the most realistic single-asset dynamics. Captures both volatility clustering...

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Rough Heston

Stochastic Vol

Rough volatility model where variance follows a fractional Volterra integral equation with Hurst exponent H < 0.5. The key property is that H < 0.5 cr...

Compare with Rough Heston →

GBM

Diffusion

The foundational model for stock prices. GBM assumes log-normal price distributions with constant drift and volatility. It underlies the Black-Scholes...

Compare with GBM →
View all stochastic models

Run Simulation

Watch the Heston Stochastic Volatility Model in action. Adjust parameters and observe how the price path evolves in real-time.

Charts:

Price Path Simulation

Select a model and click Start to begin simulation

Log Returns

Log returns will appear here

Model

Stochastic volatility model with mean-reverting variance. Captures volatility smile/skew and leverage effect observed in equity markets.

Category: stochastic_volatility

Parameters

Starting asset price

Expected annual return (e.g., 0.05 = 5%)

Starting variance level (0.04 = 20% volatility)

Speed at which variance reverts to theta

Long-run average variance level

Volatility of variance process

Correlation between price and variance shocks (-1 to 1)

Simulation time in years

Total simulation steps (more = smoother path)

Simulation

0.25x

Adjust speed before or during simulation

Status:idle

Disclaimer: These simulations are for educational purposes only. They demonstrate the behavior of mathematical models and should not be used for trading decisions. Real market dynamics are significantly more complex than any single stochastic model can capture.