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Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance is defined by \begin{align*} \frac{252}{n}\sum_{i=1}^n \ln^2 \frac{S_{t_i}}{S_{t_{i-1}}}. \end{align*} Note that, for sufficiently small $x$, \begin{align*} ...


This is pretty much exactly the problem description for a standard over-the-counter FX option pricing tool from 10-20 years ago. (For more modern contexts, the data would almost surely contain also 10 delta RR and BF, and perhaps more points as well.) The best solution is, don't build this yourself, but instead use a prexisting tool. FX conventions are ...

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