Summary of key results · M09 Volatility trading

Result	Formula
Close-to-close variance, s.e.	$\hat\sigma^2_{CC}$ (9.1); $\operatorname{Var}(r^2)=2\sigma^4$
Brownian range moments	$E R^p=4E\lvert Z\rvert^p\eta(p-1)(\sigma\sqrt T)^p$ ; $E R^2=4\ln2\,\sigma^2T$ , $E R^4=9\zeta(3)\sigma^4T^2$
Parkinson	$(\ln H/L)^2/(4\ln2)$ ; efficiency 4.91
Garman–Klass	$\tfrac12(\ln H/L)^2-(2\ln2-1)(\ln C/O)^2$ ; ≈ 7.4
Rogers–Satchell	$u(u-c)+d(d-c)$ ; drift-robust
Yang–Zhang	$V_O+kV_C+(1-k)V_{RS}$ , $k=0.34/(1.34+\frac{n+1}{n-1})$
Hedged P&L (long)	$d(e^{-rt}X)=\tfrac12e^{-rt}\Gamma^iS^2(\sigma_t^2-\sigma_i^2)dt$
Daily gamma–theta	$\tfrac12\Gamma S^2[(\Delta S/S)^2-\sigma_i^2\Delta t]$ ; breakeven move $\sigma_i\sqrt{\Delta t}$
Expected hedged P&L, GBM	(9.14); equals $BS(\sigma_r)-BS(\sigma_i)$ if $\mu=r$
Hedge at realised	P&L $=BS(\sigma_r)-BS(\sigma_i)$ , deterministic
Robustness	$\sigma_t\le\sigma_h$ , convex payoff ⇒ short hedged at $\sigma_h$ has P&L ≥ 0
VRP	$E^{\mathbb Q}[RV]-E^{\mathbb P}[RV]=\operatorname{Cov}(M,RV)/E[M]$
Breeden–Litzenberger	$q(K)=e^{rT}\partial^2_KC$
Smile quotes	$RR_{25}=\sigma_{25C}-\sigma_{25P}$ , $BF_{25}=\tfrac12(\sigma_{25C}+\sigma_{25P})-\sigma_{ATM}$
Sticky-delta delta	$\Delta=\Delta_{BS}+\mathcal V\,\partial\sigma/\partial S$ , $\partial\sigma/\partial S=-(K/S)\partial\sigma/\partial K$
Dupire	$\sigma^2_{loc}=\dfrac{\partial_TC+(r-q)K\partial_KC+qC}{\frac12K^2\partial^2_KC}$
Variance swap	$K_{var}=\dfrac{2e^{rT}}{T}\displaystyle\int_0^\infty\frac{Q(K)}{K^2}dK$ ; VIX discretisation (9.25)
ATM gamma, theta	$\Gamma\approx1/(\sigma S\sqrt{2\pi\tau})$ , $\Theta\approx-\sigma S/(2\sqrt{2\pi\tau})$
Kelly with skew	$\mu_3\le0\Rightarrow f^<\mu_1/\mu_2<\mu/\sigma^2$ ; always $f^<1/L$