Replicating portfolio in a binomial market

For one-period successor claim values $V_u,V_d$ and stock prices $S_u,S_d$, the replicating stock holding isThe remaining value is placed in the risk-free asset. In a complete binomial market, backward replication determines the unique no-arbitrage claim price.

At each node of a binomial tree, a claim with next-period values $V_u,V_d$ has valuewhere $q$ is the local risk-neutral probability in a binomial market. For an American claim, replace this continuation value by the maximum of continuation and immediate exercise value.