= Static replication on a finite terminal support
Suppose the terminal <stock> price has support $s_1<\cdots<s_m$. For adjacent support points define the normalized <call spread> payoff
$$
R_i(s)=\frac{(s-s_i)^+-(s-s_{i+1})^+}{s_{i+1}-s_i}.
$$
At a support point $s_j$, it equals zero for $j\leq i$ and one for $j\geq i+1$. Consequently any specified terminal payoff values $h_j$ satisfy
$$
h_j=h_1+\sum_{i=1}^{m-1}(h_{i+1}-h_i)R_i(s_j).
$$
The sum telescopes. Cash paying $h_1$ at maturity plus the displayed <call spreads> gives a static <replicating strategy>; the cash cost is discounted to today. With $m=3$ and payoffs $(2,5,1)$, use cash paying two, plus three units of $R_1$ minus four units of $R_2$.
The identity is required only on the stated support. It replicates every function of $S_T$, while a <path-dependent contingent claim> can distinguish histories with the same terminal price. Thus this spanning statement alone does not establish completeness for all history-dependent claims. The call-spread construction is the finite-support counterpart of strike-curvature recovery.<Prices of State-Contingent Claims Implicit in Option Prices by Douglas T. Breeden and Robert H. Litzenberger>[sectionII, printed pp625–626, discrete terminal values and call portfolios]{ref}
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