How does Merton's jump-diffusion model price options?

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AcadiFi · Accepted Answer

Merton's jump-diffusion adds a compound Poisson jump process to geometric Brownian motion: dS/S = (mu - lambda*k)*dt + sigma*dW + (Y-1)*dN, where N is a Poisson process with intensity lambda, Y is the jump size (typically lognormal with mean 1+k and vol delta), and (mu - lambda*k) is the risk-adjusted drift.

Merton's trick: if jump risk is diversifiable (not priced), the European call price is a Poisson-weighted sum of Black-Scholes prices: C = sum over n of e^(-lambda*T) * (lambda*T)^n / n! * BS(S, K, T, sigma_n, r_n), where sigma_n and r_n are adjusted for having observed n jumps.

Impact on the vol surface:

1. Short-dated skew. For short T, the probability of a big jump dominates — so OTM puts command large premiums and ATM is close to the diffusive vol. This produces the steep short-dated skew Black-Scholes can't reproduce with constant sigma.

2. Fat tails. Unconditional returns have kurtosis > 3 because of the Poisson mixture. Sample VaR estimated from Gaussian assumptions underestimates true tail risk.

3. Long-dated convergence. For long T, the central limit theorem averages many jumps and the distribution approaches lognormal — skew flattens, consistent with market reality.

A worked example: Pinehurst Capital models a 1-month option on AnvilSoft. They estimate lambda=3/year (one jump per 4 months), mean jump -8%, jump vol 15%, and diffusive sigma=25%. The model produces a 1-month 90%-moneyness put implied vol of 34% vs 25% diffusive — matching the market skew. For FRM exam: jump-diffusion naturally produces negative skew when jumps are downward-biased (k < 0), fat tails raise VaR and ES, and Merton's risk-neutral pricing requires the jump risk to be either zero-premium (Merton) or priced separately.

How does Merton's jump-diffusion model price options?

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