QuantMax

Volatility

4 lessons · 6 equations. Each lesson below gives its formulas and key rules; open the lesson for the full explanation.

Implied against realised, and the surface

Measuring realised volatility

σann=252n∑i=1nri2,ri=ln⁡SiSi−1\sigma_{\text{ann}} = \sqrt{\frac{252}{n}\sum_{i=1}^{n} r_i^2}, \qquad r_i = \ln\frac{S_i}{S_{i-1}}

Close-to-close volatility, annualised. Variance scales with time, so volatility scales with its square root.

Estimators that use the whole day

σ^P2=(ln⁡H/L)24ln⁡2,σ^GK2=12(ln⁡H/L)2−(2ln⁡2−1)(ln⁡C/O)2\hat\sigma^2_{\text{P}} = \frac{(\ln H/L)^2}{4\ln 2}, \qquad \hat\sigma^2_{\text{GK}} = \tfrac{1}{2}(\ln H/L)^2 - (2\ln 2 - 1)(\ln C/O)^2

Parkinson uses the high–low range; Garman–Klass adds the open and close. Both use more of the day’s information than close-to-close returns, so they estimate volatility several times more efficiently — at the cost of assuming continuous trading and no drift.

Remember

  • Realised looks backwards at returns; implied looks forwards out of a price.

Trading volatility: gamma scalping, events and the weekend

Isolating event variance

σevent2=(σwith2−σwithout2)T\sigma_{\mathrm{event}}^2=(\sigma_{\mathrm{with}}^2-\sigma_{\mathrm{without}}^2)T

Subtract total variances over the same horizon to isolate the variance attributed to an event.

Remember

  • Gamma scalping forces you to buy low and sell high; theta is the rent.

Variance swaps, the log contract and what the VIX is

The variance swap

payoff=Nvar(σrealised2−Kvar2)\text{payoff} = N_{\text{var}}\left(\sigma_{\text{realised}}^2 - K_{\text{var}}^2\right)

Linear in *variance*, not in volatility — which is what makes it replicable.

Why a volatility swap strikes below the variance swap

Kvol≈Kvar−Var⁡(σR2)8 Kvar3/2K_{\text{vol}} \approx \sqrt{K_{\text{var}}} - \frac{\operatorname{Var}(\sigma^2_R)}{8\,K_{\text{var}}^{3/2}}

Volatility is the square root of variance, and the square root is concave, so the expected volatility is below the square root of the expected variance (Jensen). The gap grows with the volatility of volatility, and it is what a volatility-swap seller keeps in exchange for not being able to hedge statically.

Remember

  • A variance swap is linear in variance and replicable by a 1/K21/K^2 strip of options.

Relative value: dispersion, skew trades and correlation

Index variance from its constituents

σI2=∑iwi2σi2+∑i≠jwiwjρijσiσj\sigma_I^2 = \sum_i w_i^2\sigma_i^2 + \sum_{i \ne j} w_iw_j\rho_{ij}\sigma_i\sigma_j

The index is a portfolio, so its variance carries a cross term — and that cross term is where correlation enters.

Remember

  • Index variance carries a cross term, and that term is the correlation.

Detailed formula cards