Merton jump-diffusion
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Extension of GBM with a compound-Poisson jump term: dS_t / S_t- = (μ - λ k)dt + σ,dW_t + (Y_t - 1),dN_t, where N_t is Poisson with intensity λ, jump sizes ln Y_t sim mathcalN(μ_J, σ_J^2), and k = mathbbE[Y - 1]. Closed-form option price as a Poisson-weighted sum of BSM prices: C = sum_n=0^∞ dfrace^-λ' T(λ' T)^nn!,C_BS(σ_n, r_n). Generates short-dated implied-vol skew that diffusion alone cannot.
Merton jump-diffusion in plain English
The classic stock model assumes prices drift and wiggle smoothly, but real markets sometimes LEAP — a surprise earnings miss or a shock headline makes a stock gap up or down instantly. Merton's jump-diffusion adds those sudden jumps on top of the normal jiggle: most of the time the price wanders as usual, but occasionally a random 'jolt' hits, arriving unpredictably like the ring of a phone you can't schedule. Cleverly, the option price becomes a weighted blend of ordinary formulas — one for 'zero jumps happened,' one for 'exactly one jump,' and so on, weighted by how likely each count is. This matters because those possible jumps explain why short-dated options are priced for more extreme moves than the smooth-only model could ever justify.
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