Kou Double-Exponential Jump-Diffusion Model

Jump-Diffusion

Extends Merton with asymmetric jump distribution

Uses asymmetric double-exponential distribution for jump sizes, allowing different characteristics for upward rallies vs downward crashes. Better captures the negative skewness observed in equity returns and implied volatility skew.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Drift
Diffusion volatility
Jump intensity
Probability of upward jump
Upward jump rate parameter
Downward jump rate parameter

Key Assumptions

  • •Asymmetric jumps (crashes vs rallies differ)
  • •Double-exponential (Laplace) jump distribution
  • •Better fits implied volatility skew than Merton
  • •Captures negative skewness in returns

Reference

Kou, S.G. (2002). "A Jump-Diffusion Model for Option Pricing." Management Science.

Model Hierarchy

Built on simpler models
SimplerStandard

Merton

Jump-Diffusion
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.

How Kou Compares to GBM

MetricKouGBMDifference
Log-Likelihood3463.43389.6+73.9
AIC-6915-6775−140
BIC-6885-6763−122

Kou Double-Exponential Jump-Diffusion Model achieves a log-likelihood of 3463.4, which is 73.9 higher than GBM's 3389.6. The AIC improves by 140 points (-6915 vs GBM's -6775), indicating a better fit even after penalizing for additional parameters. This represents a 2.2% 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 Kou Fits Better

Detected Jumps
43
Events per Year
10.8
Tail Fit
2.2% improvement in tail fit

Kou is preferred when discrete extreme events occur. In this dataset, the model detected 43 significant jumps, corresponding to approximately 10.8 extreme events per year. This provides 2.2% improvement in tail fit versus GBM. Captures asymmetric crash/rally dynamics.

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 Kou relates to observed return properties

SPY returns are negatively skewed—crashes are larger than rallies of equal probability. Kou models this asymmetry through double-exponential jumps with different decay rates for positive and negative moves. Like Merton, it cannot capture volatility clustering.

What Kou captures

  • Asymmetric tails
  • Negative skewness
  • Different crash vs rally magnitudes

What it cannot capture

  • Volatility persistence
  • Time-varying volatility
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

Kou Double-Exponential estimation on SPY ranks 2nd by AIC (ΔAIC = 15.7 from Merton Jump-Diffusion). Within the Jump-Diffusion category, this represents the strongest empirical fit. The model accurately captures extreme left-tail events (1% quantile ratio = 1.00). Compare with Merton Jump-Diffusion for the best overall fit.

2
Rank #2
Second best
AIC
-6914.8
ΔAIC
+15.7
Akaike Weight
0.0%
Best Model
Merton Jump-Diffusion
AIC: -6930.6
15.7 lower than this model
View full model comparison

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Run Simulation

Watch the Kou Double-Exponential 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 Merton with asymmetric double-exponential jumps. Captures volatility skew and different crash/rally dynamics. Based on Kou (2002).

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

Probability that a jump is upward (0 < p < 1)

Rate for upward jumps. Must be > 1 for finite mean. Higher = smaller jumps.

Rate for downward jumps. Higher = smaller 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.