Exchange option ($(S_1(T)-S_2(T){)}^{+}$)

An exchange option permits delivery of one asset in exchange for another at maturity, producing the positive-part payoff $(S_1(T)-S_2(T){)}^{+}$. In a complete finite market its price is the discounted expected value under the unique risk-neutral measure. For a two-period joint binomial model, evaluating this payoff at all four terminal paths and weighting by their risk-neutral path probabilities gives its price directly.