Merton's 1976 jump-diffusion model adds a compound-Poisson jump term to geometric Brownian motion so the asset price has the SDE St−dSt=(μ−λκ)dt+σdWt+(eJ−1)dNt,
where Nt is a Poisson process with intensity λ, the log-jump J∼N(μJ,σJ2), and κ=E[eJ−1]=eμJ+σJ2/2−1 is the compensator that keeps the drift on S equal to μ. Integrating, St=S0exp((μ−21σ2−λκ)t+σWt+∑i=1NtJi) — between jumps the path is pure geometric Brownian motion; jumps add a discrete log-shift. The bright cyan path is the jump-diffusion realization with μ=0.08, σ=0.20, λ=1.2 /yr, μJ=−0.04, σJ=0.10; dashed vertical lines mark each jump (red = down, green = up). The faint grey companion is plain GBM with the same μ, σ and driven by the same Brownian increments, so any divergence between the two is purely jump-induced. The bottom-right inset bins daily log-returns and overlays the Gaussian density implied by the diffusion piece alone — even after a few hundred days you can see the empirical distribution sprouting the fat tails Merton's model was designed to reproduce, since equity returns in reality have far more ∣r∣>3σ events than Black-Scholes would predict. Click anywhere to reset and watch a new sample path build.