 Solution

For any finite contingent claim $C$ measurable at the final date in the natural binomial tree, backward induction givesAt a node with stock $S_j$, let $V_{j+1}^u,V_{j+1}^d$ be its two successor values. Hold, during the next period,The stock and bank holdings reproduce both successor values and cost ${\Delta }_j{S}_j+{\psi }_j{B}_j=R^{-1}[q{V}_{j+1}^u+(1-q)V_{j+1}^d]=V_j$. Rebalancing at every node is self-financing, and terminal value is $C$. Thus every such claim is replicated and the no-arbitrage price is unique.