Event variance from two expiries

(a) Prove Proposition 9A.9. (b) On 5 Oct 2026 the ATM IVs were 14.46% for the 13 Oct expiry (8 days) and 14.04% for 27 Oct (22 days) (bhavcopy). Compute $e$, the event standard deviation in percent and in points (index 22,555.75), $\sigma_d$, and the event-day total standard deviation. (c) On 6 Oct the same expiries (7 and 21 days) showed 12.15% and 12.55%. Compute $e$ and interpret. (d) The index moved from 22,776.10 to 22,603.05 on 7 Oct. Express the log move in units of the event-day standard deviation from (b). List three reasons the 5 Oct estimate may overstate the RBI event premium.