 Solution

Consider a binomial market with dates $0,\dots ,n$, bank account $B_k=b^k$ for $b>0$, and stock price $S_{k+1}=S_{k}u$ or $S_{k}d$, where $0<d<u$. Both moves have positive physical probability, and the information at date $k$ is the history of the first $k$ moves. Absence of arbitrage requires and is implied by $d<b<u$. If $b\le d$, borrowing cash to buy stock has nonnegative payoff with a strict gain in one state; if $b\ge u$, the reverse trade does. In the interior range, the risk-neutral probability of an up move isIt makes ${\mathbb{E}}_Q[S_{k+1}\mid {\mathcal{F}}_k]=b{S}_k$, so $S_k/B_k$ is a martingale. This proves absence of arbitrage by pricing discounted gains under the equivalent measure.

At a node with stock price $s$, let the desired next-period portfolio values be $V_u,V_d$. Solve the two replication equations $\Delta su+\beta B_{k+1}=V_u$ and $\Delta sd+\beta B_{k+1}=V_d$. The replicating portfolio in a binomial market hasIts current value isRebalance at the next node using exactly the value supplied by the previous holdings, so the strategy is self-financing. Backward induction from any terminal contingent claim $H$ gives its hedge and the unique priceThus the finite binomial market is a complete market. The construction works for path-dependent claims on the full history tree, even when their values do not recombine merely as functions of the current stock price. For $H=f(S_n)$, independence under $Q$ gives the explicit time-zero priceThe stock hedge is a difference quotient of successor values, the discrete analogue of the delta hedge.