## Methodology

Spot Volatility Index (SV): A measure of spot return volatility obtained from the cross-section of short-dated out-of-the-money option prices.

The index is constructed from near- and next-term options with tenor Tnear and Tnext,

$%\text{SV}_t = 100 \times \sqrt{\frac{\sum_{m = 1}^{M_t} V_{t,T_m}}{M_t}} \text{SV}_t ~=~ 100 \, \times \, \sqrt{\frac{~ V_{t,T_\text{near}} + V_{t,T_\text{next} } ~}{2}}$ where
\begin{align*} V_{t,T_\text{near}} \, &= \text{ Spot Variance Estimator from Near-Term Options}, \\ V_{t,T_\text{next}} \, &= \text{ Spot Variance Estimator based on Next-Term Options}. \end{align*}
Option-based Spot Variance Estimator at time t with Tenor T:

\begin{align*}\label{eq:ole} & \text{V}_{t,T} ~=~ - \, \frac{2}{\,T\, {\widehat{u}}_{t,T}^2 \,} \, \Re\left\{\,\log(\mathcal{L}_{t,T}({\widehat{u}}_{t,T})) \, \right\},\\ & \mathcal{L}_{t,T}(u) ~=~ 1 - (u^2+iu)\sum_{j=1}^{N_{t,T}}e^{iu\,[\log(K_{j-1})-\log(F_{t,T})] } ~ \frac{O_{t,T}(K_{j-1})}{K_{j-1}^2} \,\Delta_j,~~u\in\mathbb{R}, \end{align*} where \begin{align*} & \Re = \text{ real part of a complex number}, \\ & i = \sqrt{-1} \text{ is the imaginary unit},\\ & N_{t,T} = \text{ number of OTM option strikes with maturity $T$}, \\ & F_{t,T} = \text{ forward price with maturity $T$}, \\ & K_j = \text{ strike price}, \\ & O_{t,T}(K_j) = \text{ option price for strike $K_j$}, \\ & \Delta_j = \text{ strike increment, $K_j-K_{j-1}$}, \\ & T = \text{ time-to-maturity (annualized)}. \end{align*}
For an estimate of the characteristic function $$\widehat{\mathcal{L}}_{t,T}(u)$$, the characteristic exponent is set according to, $$\widehat{u}_{t,T} \,=\, \widehat{u}_{t,T}^{(1)} \wedge \widehat{u}_{t,T}^{(2)}$$ for,

\begin{align*} & \widehat{u}_{t,T}^{(1)} ~=~ \inf\left\{u\geq 0: \, |\widehat{\mathcal{L}}_{t,T}(u)|\leq 0.2\right\}, \\ & \widehat{u}_{t,T}^{(2)} ~=~ \textrm{argmin}_{u\in[0,\overline{u}]} \, |\widehat{\mathcal{L}}_{t,T}(u)|, ~ \text{ with }~~\overline{u}_{t,T} \,=\, \sqrt{\, \frac{2}{T}\, \frac{\log(1/0.05)}{\widehat{\text{BSIV}}^2_{t,T}} \, }, \end{align*}
where, $$\widehat{\text{BSIV}}_{t,T}$$ = Black-Scholes Implied Volatility for Strike closest to the Forward.