How do jump-diffusion models improve on geometric Brownian motion for risk modeling?

Standard geometric Brownian motion (GBM) assumes prices follow a continuous, smooth path — like a drunk person wandering gradually. Real markets are more like a drunk person who occasionally trips and falls down a flight of stairs. Jump-diffusion models fix this.

Merton's Jump-Diffusion Model (1976):

The return process combines two components:

dS/S = (mu - lambdak)dt + sigmadW + JdN

Where:

• mu*dt = drift (normal expected return)
• sigma*dW = diffusion (continuous random movement, like GBM)
• J*dN = jump component (sudden, discrete price changes)
• lambda = expected number of jumps per year (Poisson intensity)
• k = expected jump size (usually log-normally distributed)
• dN = Poisson counting process (0 most of the time, 1 when a jump occurs)

Why GBM Fails for Risk:

GBM produces returns that are perfectly normal. But real financial returns exhibit:

1. Fat tails: Extreme events occur far more often than Normal predicts
2. Negative skewness: Large downside moves are more frequent than large upside moves
3. Excess kurtosis: The return distribution is more peaked with heavier tails

Example — Clearmont Asset Management:

Under GBM with sigma = 20%, a 4-sigma daily loss (about -5%) should occur once every 126 years. In reality, the S&P 500 has had roughly 20 such days since 1990 — about one every 1.7 years.

Merton's model with lambda = 2 jumps/year and average jump size of -3% produces much fatter tails:

FRM Implications:

• Jump risk is not diversifiable in the same way as diffusion risk — jumps tend to be systemic
• Option pricing under jump-diffusion produces higher prices for out-of-the-money puts (volatility smile)
• Standard delta hedging breaks down during jumps — you can't hedge continuously when prices gap
• Stress testing should incorporate jump scenarios, not just diffusion-based VaR

Explore advanced quantitative methods in our FRM Part I course.