= Solution
Consider a <binomial market> with dates $0,\ldots,n$, bank account $B_k=b^k$ for $b>0$, and <stock> price $S_{k+1}=S_ku$ or $S_kd$, 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 $b\leq d$, borrowing cash to buy <stock> has nonnegative payoff with a strict gain in one state; if $b\geq u$, the reverse trade does. In the interior range, the <risk-neutral probability> of an up move is
$$
\boxed{q=\frac{b-d}{u-d}\in(0,1).}
$$
It makes $\mathbb E_Q[S_{k+1}\mid\mathcal F_k]=bS_k$, 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> has
$$
\boxed{\Delta_k=\frac{V_u-V_d}{s(u-d)},\qquad\beta_kB_k=\frac{uV_d-dV_u}{b(u-d)}.}
$$
Its current value is
$$
\boxed{V_k=\Delta_ks+\beta_kB_k=\frac{qV_u+(1-q)V_d}{b}.}
$$
Rebalance 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 price
$$
V_k=B_k\mathbb E_Q[H/B_n\mid\mathcal F_k].
$$
Thus 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 price
$$
\boxed{V_0=b^{-n}\sum_{j=0}^n\binom njq^j(1-q)^{n-j}f(S_0u^jd^{n-j}).}
$$
The <stock> hedge is a difference quotient of successor values, the discrete analogue of the <delta hedge>.
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