Double Integrals over Rectangles and Fubini's Theorem

When you price a European option whose payoff depends on two correlated underlyings, the price is an integral of the payoff against a bivariate probability density over $\mathbb{R}^2$. To evaluate that integral you need to decompose it into two successive single-variable integrals — one asset at a time. Fubini's theorem is the theorem that licenses this decomposition: when $f$ is continuous on a rectangle, the double integral $\iint_R f\,dA$ equals either iterated integral, and they are equal to each other. This note builds the double integral from Riemann sums, proves (in essence) Fubini's theorem, and works through the iterated-integration technique.

Learning objectives. By the end of this note you should be able to: