Option prices don’t always change smoothly—sometimes they jump suddenly. An earnings surprise, a flash crash, or a sudden policy shift can cause a jump that smooth‑pricing models miss. Merton’s jump‑diffusion model captures these events, but the pricing equation becomes a partial integro‑differential equation (PIDE)—a PDE with an integral term that reaches across all prices. Solving it directly looks intimidating, but in this chapter we’ll break it down, uncover its hidden convolution pattern, and build fast, practical numerical methods.