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% Author: Simon-Pierre Boucher
% Contact: contact@spboucher.ai
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% Data and Variable Construction
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\section{Data and Variable Construction}\label{sec:data}

\subsection{Data Sources}

\paragraph{Options.}
I use a comprehensive end-of-day options database containing 3,831,907,488 records covering 11,077 underlyings from January~4, 2010 to December~31, 2025 (4,025 trading days). Each record includes trade date, strike, expiry, call/put flag, bid/ask prices, bid/ask implied volatilities, open interest, volume, and the full Greek sensitivities ($\delta$, $\Gamma$, $\mathcal{V}$, $\Theta$, $\rho$). Data quality is high: 92--95\% of records contain valid Greeks.

\paragraph{Intraday OHLCV.}
I use 5-minute OHLCV data across six asset classes: 7,789 US equities (1.23B bars), 4,302 ETFs (301M bars), 125 indices, 131 futures, 78 FX pairs, and 74 cryptocurrencies. Total: approximately 11.5~billion intraday observations.

\begin{table}[H]
\centering
\caption{Data coverage.}
\label{tab:data}
\begin{threeparttable}
\begin{tabular}{@{}lrrrl@{}}
\toprule
Asset class & Symbols & Rows (millions) & Freq.\ & Period \\
\midrule
\multicolumn{5}{@{}l}{\textit{Panel A: Options}} \\
Option contracts & 11,077 & 3,832 & Daily & 2010--2025 \\[6pt]
\multicolumn{5}{@{}l}{\textit{Panel B: Intraday OHLCV}} \\
US equities     & 7,789 & 1,225 & 5-min & 2000--2026 \\
ETFs            & 4,302 &   301 & 5-min & 2000--2026 \\
Equity indices  &   125 &    40 & 5-min & 2008--2026 \\
Futures         &   131 &    73 & 5-min & 2008--2026 \\
FX              &    78 &    83 & 5-min & 2010--2026 \\
Crypto          &    74 &    39 & 5-min & 2013--2026 \\
\midrule
\textbf{Total}  &       & \textbf{$\approx$11,500} & & \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}

\subsection{Sample Construction}

The analysis sample consists of 69 underlyings with overlapping options and OHLCV coverage: 49 major US stocks, 17 ETFs, and 3 broad market indices (SPX, NDX, RUT). After merging option-derived features with realized volatility, the sample comprises \textbf{264,383 ticker-day observations} (January 2010--December 2025).

\subsection{Variable Definitions}

\subsubsection{Option-Implied Features}

I construct ten daily features for each underlying from the raw option chain:

\begin{table}[H]
\centering
\caption{Option-implied variables.}
\label{tab:variables}
\begin{threeparttable}
\small
\begin{tabular}{@{}lp{10cm}@{}}
\toprule
Variable & Definition \\
\midrule
$IV_{ATM,30d}$ & Average mid-IV of calls with $|\delta - 0.5| < 0.10$, 20--40 DTE \\
$IV_{ATM,90d}$ & Same, 80--100 DTE \\
IV term slope & $IV_{ATM,90d} - IV_{ATM,30d}$ \\
Skew$_{25\delta}$ & $IV_{25\delta P,30d} - IV_{25\delta C,30d}$ \\
Implied skewness & $(IV_{10\delta P} - IV_{10\delta C})/IV_{ATM}$ \\
Implied kurtosis & $\overline{IV}_{wings} / IV_{ATM}$, with $0.03 < |\delta| < 0.15$ \\
PC vol.\ ratio & $\sum V^{put} / \sum V^{call}$ \\
PC OI ratio & $\sum OI^{put} / \sum OI^{call}$ \\
Net gamma exp.\ & $\sum \Gamma^{call} \cdot OI^{call} - \sum \Gamma^{put} \cdot OI^{put}$ \\
\bottomrule
\end{tabular}
\end{threeparttable}
\end{table}

\subsubsection{Realized Volatility Measures}

Following \citet{andersen2003modeling}, daily realized variance is:
\begin{equation}\label{eq:rv}
    RV_t = \sum_{i=1}^{N_t} r_{t,i}^{2},
    \qquad r_{t,i} = \ln\!\bigl(P_{t,i}\,/\,P_{t,i-1}\bigr).
\end{equation}
I also compute the weekly and monthly rolling averages $RV_t^{(w)}$ and $RV_t^{(m)}$, as well as forward targets $RV_{t+1}$ (1-day) and $\sum_{j=1}^{5}RV_{t+j}$ (5-day).

\subsection{Descriptive Statistics}

Table~\ref{tab:desc_groups} presents summary statistics by asset group.

\begin{table}[H]
\centering
\caption{Descriptive statistics by asset group.}
\label{tab:desc_groups}
\begin{threeparttable}
\begin{tabular}{@{}ld{3.4}d{3.4}d{3.4}d{3.4}@{}}
\toprule
& \multicolumn{1}{c}{Stocks} & \multicolumn{1}{c}{ETFs} & \multicolumn{1}{c}{Indices} & \multicolumn{1}{c}{All} \\
\midrule
$N$ (ticker-days)        & 188093  & 64256  & 12034  & 264383 \\
$N$ tickers              & 49      & 17     & 3      & 69 \\
Mean $IV_{ATM,30d}$      & 0.2664  & 0.1784 & 0.1823 & 0.2406 \\
Mean skew$_{25\delta}$   & 0.0361  & 0.0425 & 0.0562 & 0.0387 \\
Mean PC vol.\ ratio      & 0.890   & 2.345  & 1.754  & 1.283 \\
Mean RV (daily)          & 0.0005  & 0.0002 & 0.0001 & 0.0004 \\
Mean daily return (\%)   & 0.054   & 0.035  & 0.047  & 0.049 \\
Std daily return (\%)    & 1.90    & 1.25   & 1.29   & 1.74 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes.} Sample period: January 2010--December 2025. $IV_{ATM}$ is the 30-day at-the-money call implied volatility. Skew is the 25-delta put minus 25-delta call IV. PC vol.\ ratio is total put volume divided by total call volume. RV is realized variance from 5-minute returns.
\end{tablenotes}
\end{threeparttable}
\end{table}

Stocks exhibit higher ATM implied volatility (26.6\%) and return volatility (1.90\% daily) than ETFs (17.8\%, 1.25\%) and indices (18.2\%, 1.29\%). The put-call volume ratio is notably higher for ETFs (2.35) and indices (1.75) than for stocks (0.89), consistent with institutional portfolio hedging.

Table~\ref{tab:autocorr} reports the average autocorrelation structure. ATM implied volatility is highly persistent (lag-1: 0.959, lag-22: 0.538), while daily returns exhibit slight negative autocorrelation ($-0.062$).

\begin{table}[H]
\centering
\caption{Average autocorrelation structure.}
\label{tab:autocorr}
\begin{threeparttable}
\begin{tabular}{@{}ld{1.3}d{1.3}d{1.3}d{1.3}@{}}
\toprule
Variable & \multicolumn{1}{c}{Lag 1} & \multicolumn{1}{c}{Lag 5} & \multicolumn{1}{c}{Lag 10} & \multicolumn{1}{c}{Lag 22} \\
\midrule
ATM IV (30d)      & 0.959  & 0.861  & 0.753  & 0.538 \\
Skew (25$\delta$) & 0.832  & 0.711  & 0.593  & 0.416 \\
Realized variance & 0.320  & 0.211  & 0.139  & 0.057 \\
Daily return      & -0.062 & -0.010 & -0.006 & -0.030 \\
PC vol.\ ratio    & 0.299  & 0.218  & 0.172  & 0.118 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes.} Cross-sectional average of within-ticker autocorrelations.
\end{tablenotes}
\end{threeparttable}
\end{table}
