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% Author: Simon-Pierre Boucher
% Contact: contact@spboucher.ai
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% Appendix
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\section{Implied Correlation Around Crises}\label{app:crises}

Table~\ref{tab:app_crises} details the behavior of the implied-correlation index around the nine major market events of the sample period. Implied correlation rises after every event, with the largest jumps around Volmageddon ($+0.328$) and the August 2015 China devaluation ($+0.258$).

\begin{table}[H]
\centering
\caption{Implied correlation around major market events.}
\label{tab:app_crises}
\begin{threeparttable}
\small
\begin{tabular}{@{}ld{1.3}d{1.3}d{1.3}@{}}
\toprule
Crisis & \multicolumn{1}{c}{Pre IC} & \multicolumn{1}{c}{Post IC} & \multicolumn{1}{c}{$\Delta$IC} \\
\midrule
Flash Crash (May 2010)     & 0.223 & 0.404 & +0.181 \\
Euro Crisis (Aug 2011)     & 0.313 & 0.469 & +0.156 \\
China Deval.\ (Aug 2015)   & 0.231 & 0.489 & +0.258 \\
Volmageddon (Feb 2018)     & 0.108 & 0.436 & +0.328 \\
COVID-19 (Mar 2020)        & 0.473 & 0.648 & +0.175 \\
Meme Stocks (Jan 2021)     & 0.228 & 0.326 & +0.098 \\
Rate Shock (Jun 2022)      & 0.417 & 0.447 & +0.030 \\
SVB Crisis (Mar 2023)      & 0.328 & 0.373 & +0.045 \\
VIX Spike (Aug 2024)       & 0.138 & 0.316 & +0.178 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes.} IC = Implied Correlation. Pre = 30-day window before the event; Post = 10-day window after.
\end{tablenotes}
\end{threeparttable}
\end{table}


\section{SPX Options Surface Features (RQ5)}\label{app:features}

The 21 SPX surface features used in Section~\ref{sec:results} are: $IV_{ATM}$ at six tenors (1w, 2w, 1m, 2m, 3m, 6m); 25$\delta$ skew (1m, 3m); 10$\delta$ deep OTM skew (1m); butterfly ratio (1m); term-structure slopes (3m$-$1m, 6m$-$1m); total gamma\,$\times$\,OI; net gamma; total vega\,$\times$\,OI; average $\Theta$; put-call volume ratio; put-call OI ratio; total option volume; total OI; and the average bid-ask spread percentage.

\section{Random Forest Feature Importance (RQ5)}\label{app:importance}

Table~\ref{tab:app_importance} reports the full Random Forest importance ranking underlying the RQ5 discussion; the two shortest ATM tenors jointly account for two thirds of total importance.

\begin{table}[H]
\centering
\caption{Random Forest feature importance (1-day RV target, 2010--2019 training).}
\label{tab:app_importance}
\begin{threeparttable}
\begin{tabular}{@{}lr@{}}
\toprule
Feature & Importance \\
\midrule
$IV_{ATM,2w}$        & 0.508 \\
$IV_{ATM,1w}$        & 0.151 \\
Total option volume  & 0.083 \\
PC vol.\ ratio       & 0.067 \\
Net gamma exposure   & 0.037 \\
$IV_{ATM,1m}$        & 0.023 \\
$RV_{weekly}$        & 0.022 \\
Skew (25$\delta$, 3m)& 0.013 \\
Avg.\ $\Theta$       & 0.012 \\
$IV_{ATM,2m}$        & 0.011 \\
\midrule
\textit{All others (14 features)} & \textit{0.073} \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}


\section{Forecast Error Variance Decomposition}\label{app:fevd}

Table~\ref{tab:app_fevd} reports the full horizon profile of the forecast error variance decomposition summarized in Section~\ref{sec:results}: the share of realized-variance forecast error variance attributable to implied-volatility shocks rises monotonically from zero on impact to 73.8\% at 20 days.

\begin{table}[H]
\centering
\caption{FEVD: share of RV forecast error variance explained by IV shocks (\%).}
\label{tab:app_fevd}
\begin{threeparttable}
\begin{tabular}{@{}rcc@{}}
\toprule
Horizon (days) & IV shocks (\%) & RV shocks (\%) \\
\midrule
 1 &  0.0 & 100.0 \\
 2 & 28.0 &  72.0 \\
 5 & 56.1 &  43.9 \\
10 & 66.4 &  33.6 \\
15 & 71.1 &  28.9 \\
20 & 73.8 &  26.2 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes.} From the bivariate VAR(5) in standardized ATM IV and RV, averaged across 20 tickers.
\end{tablenotes}
\end{threeparttable}
\end{table}
