Binomial Trees
Backward induction on a recombining tree makes replication concrete and converges to Black–Scholes as the steps shrink.
Binomial trees replace continuous time with discrete steps: at each step the price moves up or down by a chosen factor, and the option value is found by working backward from the terminal payoff. The method needs no calculus: delta hedging within a single step pins down a unique risk-neutral probability, and discounting expected payoffs step by step yields the price. Because replication is explicit at every node, trees make risk-neutral pricing visible rather than mystical. Let the step size shrink and the tree converges to Black–Scholes, which is also why trees remain the standard tool for American and exotic options with early-exercise features.