Feynman-Kac Theorem

The Feynman-Kac theorem links partial differential equations to expectations of stochastic processes: the solution of a PDE equals the expected discounted payoff of a diffusion. This is the bridge that makes both pricing methods legitimate — it's why solving the Black-Scholes PDE and running a Monte Carlo simulation must give the same answer, and why a desk can choose whichever is cheaper to compute.