Discrete realized-variance replication identity
= Discrete realized-variance replication identity
{title2=$\sum(\Delta S_t)^2=S_T^2-S_0^2-2\sum S_{t-1}\Delta S_t$}
The pathwise identity $\sum_{t=1}^T(\Delta S_t)^2=S_T^2-S_0^2-2\sum_{t=1}^TS_{t-1}\Delta S_t$ turns a squared-increment claim into a terminal-square claim and predictable stock gains. On integer terminal support $\{0,\ldots,N\}$, $S_T^2=S_T+2\sum_{K=1}^N(S_T-K)^+$, so a static call strip plus a dynamic stock hedge replicates the claim.