Bates Stochastic Volatility Jump-Diffusion Model

Jump-Diffusion

Extends Heston + Merton

Combines Heston stochastic volatility with Merton-style jumps, providing the most realistic single-asset dynamics. Captures both volatility clustering and sudden price movements for highest-fidelity equity and FX modeling.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Instantaneous variance
Variance mean reversion speed
Long-run variance
Vol-of-vol
Price-variance correlation
Jump intensity
Mean log-jump size
Jump size volatility

Key Assumptions

  • •Stochastic volatility with mean reversion
  • •Log-normal jumps for event risk
  • •Leverage effect and volatility clustering
  • •Suitable for short-dated option pricing

Reference

Bates, D.S. (1996). "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options." The Review of Financial Studies.

Model Hierarchy

Built on simpler models
SimplerStandard

Heston

Stochastic Vol
SimplerStandard

Merton

Jump-Diffusion
SimplerBaseline

GBM

Diffusion
Extended by more complex models
More ComplexMost Complex

DPS

Jump-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 Bates relates to observed return properties

SPY returns exhibit both volatility clustering and discrete jumps. Bates combines Heston stochastic volatility with Merton jumps, capturing both features. However, jump intensity is constant—it cannot reproduce the empirical pattern that crashes cluster in high-volatility regimes.

What Bates captures

  • Volatility clustering
  • Leverage effect
  • Fat tails from jumps
  • Combined SV and jump dynamics

What it cannot capture

  • State-dependent jump intensity
  • Crash clustering in high-vol regimes
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

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DPS

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The most comprehensive model: combines stochastic volatility with state-dependent jump intensity. Higher variance leads to more frequent jumps, captur...

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View all stochastic models

Run Simulation

Watch the Bates Stochastic Volatility Jump-Diffusion 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

Combines Heston stochastic volatility with Merton-style jumps. Highest-fidelity model capturing both volatility clustering and sudden price moves. Based on Bates (1996).

Category: stochastic_volatility_jump

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)

Expected number of jumps per year

Mean of log-jump size distribution

Std dev of log-jump size distribution

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.