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In a one-factor market whose filtration is the usual augmentation of the natural Brownian filtration, with nonzero spot volatility and local martingale deflator $Y$, the Brownian martingale representation theorem constructs a nonnegative replicating strategy for a bounded nonnegative contingent claim. The minimal initial cost among nonnegative self-financing portfolios is $\mathbb{E}[Y_T{\xi }_T]$.

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.

The model has a unique equivalent martingale measure $\mathbb{Q}$ under whichand the discounted stock is a martingale. The time-zero value of an integrable time-$T$ contingent claim $C$ is therefore

A replicating strategy is a self-financing portfolio whose terminal wealth equals the prescribed payoff of a contingent claim. Admissibility specifies the permitted wealth bounds and integrability of its holdings.