Explain pathwise valuation within a binomial interest rate model, and use it to value a fixed-income instrument from its cash flows on every path

Pathwise valuation lists every possible path of rates through the tree, discounts the instrument's cash flows along each path at that path's rates, and averages the results. For a tree with n one-year periods there are 2 raised to (n − 1) paths for a cash flow at year n, so a three-year instrument has four paths: up-up, up-down, down-up and down-down. Because each path is equally likely, the simple average equals the backward-induction value. The method matters for path-dependent securities, where cash flows depend on history. The common error is counting paths by final nodes, which gives the middle node too little weight since two paths lead there.