Merton Jump-Diffusion Model
Also known as: Merton jump-diffusion, jump-diffusion process, Atlama Difüzyon Modeli (Merton Jump-Diffusion)
The Merton Jump-Diffusion model, introduced by Robert C. Merton in 1976, extends Geometric Brownian Motion by adding sudden price jumps generated by a Poisson process. It captures the volatility smile and the fat-tailed return behaviour that standard Black-Scholes cannot explain, and is widely used in option pricing and risk management.
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When to use it
Use jump-diffusion when modelling a continuous asset price over time (a single time series with at least about 250 observations) whose returns show fat tails, abrupt moves, or a volatility smile that a pure Black-Scholes / Geometric Brownian Motion model misses. It is appropriate for option pricing and risk management where rare but large discontinuous moves matter. It assumes the price follows dS/S = μ dt + σ dW + J dN with Poisson-timed jumps and log-normal jump sizes, and that jump risk cannot be perfectly hedged away.
Strengths & limitations
- Captures fat tails and the volatility smile that standard Black-Scholes / Geometric Brownian Motion cannot reproduce.
- Models sudden, news-driven price jumps explicitly through a Poisson process rather than forcing them into a single diffusion term.
- Has a closed-form option-pricing solution (Merton, 1976) and supports both maximum-likelihood and option-implied calibration.
- Adds several extra parameters (jump intensity λ and the jump-size mean and volatility) that are hard to estimate stably and may be weakly identified.
- Requires a fairly long series (about 250 or more observations) because jumps are rare events.
- Jump risk cannot be perfectly hedged, so option prices depend on how the market prices that risk — an assumption, not an observable.
Frequently asked
How does jump-diffusion differ from Black-Scholes?
Black-Scholes assumes the price follows a purely continuous Geometric Brownian Motion. Jump-diffusion keeps that continuous diffusion but adds Poisson-timed jumps, so returns can move discontinuously and have fatter tails — which lets the model reproduce the volatility smile that Black-Scholes cannot.
What distribution do the jumps follow?
In Merton's (1976) formulation the jump times follow a Poisson process with intensity λ, and the jump sizes are log-normally distributed, described by a mean μ_J and a volatility σ_J. Each jump multiplies the price by a random log-normal factor.
How are the parameters estimated?
Either by maximum likelihood on the historical return series, or by calibrating the model so its option prices match quoted market prices (implied calibration). A long sample is needed because jumps are rare.
Why can't jump risk be hedged away?
Continuous delta-hedging can neutralise the diffusion part, but jumps are sudden and unpredictable in timing and size, so no continuous hedge can offset them. As a result the jump risk is borne — and priced — by the market.
Sources
- Merton, R. C. (1976). Option Pricing When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3(1–2), 125–144. DOI: 10.1016/0304-405X(76)90022-2 ↗
How to cite this page
ScholarGate. (2026, June 1). Merton Jump-Diffusion Model. ScholarGate. https://scholargate.app/en/finance/jump-diffusion-model
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