Rough Heston Stochastic Volatility Model

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 creates "rough" volatility paths with Hölder regularity H, matching empirical observations that realized volatility has irregular, jagged paths rather than smooth ones.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Hurst exponent (roughness parameter)
Instantaneous variance at time t
Mean reversion speed
Long-run variance level
Volatility of variance
Correlation
Fractional kernel function

Key Assumptions

  • •Fractional Brownian motion for volatility
  • •H < 0.5 creates rough (irregular) volatility paths
  • •Empirical estimates suggest H ≈ 0.1
  • •Matches observed volatility surface dynamics better than classical Heston
  • •Memory effects through Volterra integral

Reference

Gatheral, J., Jaisson, T., Rosenbaum, M. (2018). "Volatility is rough." Quantitative Finance.

Model Hierarchy

Built on simpler models
SimplerStandard

Heston

Stochastic Vol
SimplerBaseline

GBM

Diffusion

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 Rough Heston relates to observed return properties

Empirical realized volatility exhibits rough, irregular paths rather than smooth mean reversion. Rough Heston matches this behavior with fractional dynamics (H ≈ 0.1), but like Heston, it lacks a jump component to capture discrete market shocks.

What Rough Heston captures

  • Rough volatility paths
  • Volatility clustering
  • Short-term volatility dynamics

What it cannot capture

  • Sudden jumps
  • Extreme tail events from discrete shocks
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

Related Models

Heston

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Run Simulation

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

Charts:

Price Path Simulation

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Log Returns

Log returns will appear here

Model

Rough volatility model with fractional Volterra variance process. Hurst exponent H < 0.5 creates rough paths matching empirical vol observations. Based on Gatheral, Jaisson, Rosenbaum (2018).

Category: stochastic_volatility

Parameters

Roughness parameter (0-0.5). Lower values = rougher paths. Empirical estimates ~0.1

Starting asset price

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

Starting variance level (0.04 = 20% volatility)

Speed at which variance reverts to θ

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, better roughness estimation)

Optional seed for reproducibility (leave empty for random)

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.