Summary of key results · M09A Options desk

Result	Formula
Parity-implied forward	$F=K+e^{r\tau}(C-P)$ (9A.1)
Option price change	$\Delta V=\Delta\,\Delta S+\tfrac12\Gamma\Delta S^2+\Theta\Delta t+\mathcal V\Delta\sigma+R$ (9A.2)
Put–call ratios	$\sum p_K/\sum c_K$ (OI), $\sum v^p_K/\sum v^c_K$ (volume) (9A.3)
Holders' payout; max pain	$W(x)=\sum_K[c_K(x-K)^++p_K(K-x)^+]$ , $x^\ast=\arg\min W$ (9A.4)
Max pain condition (Prop. 9A.3)	$\sum_{K<x^\ast}c_K\le\sum_{K\ge x^\ast}p_K$ and $\sum_{K\le x^\ast}c_K\ge\sum_{K>x^\ast}p_K$ (9A.5)
Hedging feedback (Prop. 9A.4)	$\sigma_{\text{eff}}=\sigma/(1+\lambda G)$ (9A.6)
ATM straddle (Prop. 9A.5)	$2e^{-r\tau}F[2N(\sigma\sqrt\tau/2)-1]=e^{-r\tau}F\sigma\sqrt\tau\sqrt{2/\pi}\,(1-\sigma^2\tau/24+\dots)$ (9A.7)–(9A.8)
Expected absolute move (Prop. 9A.6)	$S\sigma\sqrt\tau\sqrt{2/\pi}$ ; $P(\lvert\text{move}\rvert<\text{straddle})=0.575$ under normality (9A.9)
IV rank / percentile	(9A.10), (9A.11); percentile uniform under exchangeability (Prop. 9A.7)
Forward variance; calendar monotonicity	$\sigma_{1,2}^2=(w_2-w_1)/(\tau_2-\tau_1)\ge0$ (9A.12), Prop. 9A.8
Event variance, two expiries after	$e=\tau_1\tau_2(\sigma_1^2-\sigma_2^2)/(\tau_2-\tau_1)$ , $\sigma_d^2=(\sigma_2^2\tau_2-\sigma_1^2\tau_1)/(\tau_2-\tau_1)$ (9A.13)
Event variance, before/after	$e=\sigma_a^2\tau_a-\sigma_b^2\tau_b-\sigma_d^2(\tau_a-\tau_b)$ (9A.14)
Participant OI identity	$\sum_cL_c=\sum_cS_c$ per instrument (9A.15)
Position Greeks	$\sum_jq_j\cdot$ Greek$_j$ (9A.16)
Scenario margin (Prop. 9A.12)	$\max(0,\max_kw_k\ell_k)$ ; subadditive; $\le$ max loss for defined risk (9A.17)
Adjustments (Prop. 9A.13)	$\mathbb E[P+A]=\mathbb E[P]+\mathbb E[A]-c(A)$ (9A.18)
31 Nifty expiries, Apr–Nov 2026	max pain no better than previous close; no strike clustering; realised $\lvert$ move$\rvert$/straddle 0.84