Merton Jump-Diffusion Model

Jump-Diffusion

Extends GBM with compound Poisson jumps

Extends GBM with compound Poisson jumps to capture sudden price movements from earnings announcements, market shocks, or other discontinuous events. The drift is adjusted to maintain arbitrage-free pricing.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Drift (expected return)
Diffusion volatility
Jump intensity (expected jumps per year)
Mean of log-jump size
Standard deviation of log-jump size
Poisson process with intensity λ
Drift compensation for arbitrage-free pricing

Key Assumptions

  • •Combines continuous diffusion with discrete jumps
  • •Log-normal jump sizes (symmetric up/down)
  • •Jump arrivals follow a Poisson process
  • •Captures fat tails and excess kurtosis

Reference

Merton, R. (1976). "Option Pricing When Underlying Stock Returns Are Discontinuous." Journal of Financial Economics.

Model Hierarchy

Built on simpler models
SimplerBaseline

GBM

Diffusion
Extended by more complex models
More ComplexAdvanced

Kou

Jump-Diffusion
More ComplexComplex

Bates

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.

How Merton Compares to GBM

MetricMertonGBMDifference
Log-Likelihood3470.33389.6+80.7
AIC-6931-6775−155
BIC-6906-6763−143

Merton Jump-Diffusion Model achieves a log-likelihood of 3470.3, which is 80.7 higher than GBM's 3389.6. The AIC improves by 155 points (-6931 vs GBM's -6775), indicating a better fit even after penalizing for additional parameters. This represents a 2.4% likelihood improvement over the baseline GBM model.

Lower AIC and BIC values indicate a better model fit after penalizing for complexity. A difference greater than 10 is generally considered strong evidence in favor of the better model.

When Merton Fits Better

Detected Jumps
45
Events per Year
11.4
Tail Fit
2.4% improvement in tail fit

Merton is preferred when discrete extreme events occur. In this dataset, the model detected 45 significant jumps, corresponding to approximately 11.4 extreme events per year. This provides 2.4% improvement in tail fit versus GBM. Captures symmetric extreme events.

Recommended Use Cases

  • •Short-dated options near earnings announcements
  • •Tail risk hedging and crash protection
  • •Credit derivatives with default events
  • •Markets prone to sudden discontinuous moves

Empirical Context (SPY Returns)

How Merton relates to observed return properties

SPY returns show fat tails with occasional extreme moves from earnings, macro shocks, and market events. Merton captures these through Poisson jumps, improving tail fit significantly. However, it assumes constant diffusion volatility, missing the clustering behavior visible in squared returns.

What Merton captures

  • Fat tails and excess kurtosis
  • Sudden discrete price movements
  • Extreme daily returns

What it cannot capture

  • Volatility clustering
  • Persistent high/low volatility regimes
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

Parameter Estimation

Real-world (P-measure) parameter estimates from historical SPY returns

Expand

Model Comparison

vs 4 other models on SPY

Merton Jump-Diffusion estimation on SPY achieves the best fit among all 5 tested models (AIC = -6930.6, Akaike weight = 100%). Within the Jump-Diffusion category, this represents the strongest empirical fit. The model accurately captures extreme left-tail events (1% quantile ratio = 1.00).

1
Rank #1
Best fit
AIC
-6930.6
Akaike Weight
100.0%
Best Model
🏆
This is the best model!
View full model comparison

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Bates

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Run Simulation

Watch the Merton Jump-Diffusion 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

Extends GBM with Poisson-distributed jumps. Captures sudden price movements from news events, earnings, or market shocks. Based on Merton (1976).

Category: jump-diffusion

Parameters

Starting asset price

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

Annual diffusion volatility (e.g., 0.2 = 20%)

Expected number of jumps per year

Mean of log-jump size distribution (negative = downward jumps)

Std dev of log-jump size (higher = more variable jumps)

Simulation time in years

Total simulation steps (more = smoother path, more candles)

Overlay equivalent GBM path using same random diffusion

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.