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TL;DR

A model combining continuous diffusion with discrete jumps: \( dS/S = \mu dt + \sigma dW + J dN \) where N is a Poisson process and J is the random jump size. Merton's model assumes log-normal jumps.

By Valenke Exam Prep Team·Last updated 2026-06-03

Jump-Diffusion Model

A model combining continuous diffusion with discrete jumps: \( dS/S = \mu dt + \sigma dW + J dN \) where N is a Poisson process and J is the random jump size. Merton's model assumes log-normal jumps.

Why it matters for interviews

Jump-diffusion models capture sudden market movements (crashes, earnings jumps) that pure diffusion models miss. They produce volatility smiles and are used for pricing options on assets with jump risk.

Definition and Mathematical Foundation

A model combining continuous diffusion with discrete jumps: \( dS/S = \mu dt + \sigma dW + J dN \) where N is a Poisson process and J is the random jump size. Merton's model assumes log-normal jumps.

Application in Quantitative Finance

Jump-diffusion models capture sudden market movements (crashes, earnings jumps) that pure diffusion models miss. They produce volatility smiles and are used for pricing options on assets with jump risk.

Related Concepts

Poisson ProcessBrownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

What is Merton's jump-diffusion model?▼
Stock follows GBM plus Poisson jumps with log-normal sizes: \( \ln(1+J) \sim N(\mu_J, \sigma_J^2) \). The option price is a weighted sum of Black-Scholes prices across different jump scenarios, making computation relatively simple.
How do jumps affect option prices?▼
Jumps produce excess kurtosis in returns, creating a volatility smile. Short-dated OTM puts become more valuable (crash risk), which pure diffusion models underprice. The smile flattens with maturity as diffusion dominates.
Can jump risk be hedged?▼
Jumps make the market incomplete -- perfect replication is impossible because you cannot hedge the discontinuous component. Pricing requires choosing a risk premium for jump risk, typically calibrated to market option prices.