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 payoffAt a support point $s_j$, it equals zero for $j\le i$ and one for $j\ge i+1$. Consequently any specified terminal payoff values $h_j$ satisfyThe 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.^[ref]

The pathwise identity ${\sum }_{t=1}^T(\Delta S_t{)}^2=S_T^2-S_0^2-2{\sum }_{t=1}^T{S}_{t-1}\Delta S_t$ turns a squared-increment claim into a terminal-square claim and predictable stock gains. On integer terminal support $\{0,\dots ,N\}$, $S_T^2=S_T+2{\sum }_{K=1}^N(S_T-K{)}^{+}$, so a static call strip plus a dynamic stock hedge replicates the claim.