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% Author: Simon-Pierre Boucher
% Contact: contact@spboucher.ai
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% Methodology
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\section{Methodology}\label{sec:method}

\subsection{Return Predictability (RQ1)}

I estimate pooled panel regressions:
\begin{equation}\label{eq:rq1}
    r_{i,t+h} = \alpha + \boldsymbol{\beta}'\,\mathbf{X}_{i,t}^{opt}
    + \gamma\, RV_{i,t} + \delta\, RV_{i,t}^{(w)} + \varepsilon_{i,t+h},
    \qquad h \in \{1,\,5\}\text{ days},
\end{equation}
with heteroskedasticity-robust (HC1) standard errors \citep{white1980heteroskedasticity}. All variables are winsorized at the 1st and 99th percentiles and standardized. Robustness uses Newey-West HAC(5) \citep{newey1987simple}, double-clustered (ticker $+$ date) standard errors \citep{cameron2011robust,petersen2009estimating}, Fama-MacBeth cross-sectional regressions, quantile regressions \citep{koenker1978regression}, rank-based information coefficients, decile sorts, leave-one-year-out estimation, and a within-ticker permutation placebo.

For economic significance, I construct daily-rebalanced equal-weighted quintile portfolios by cross-sectional sorting on each option-implied variable and evaluate annualized Sharpe ratios.

\subsection{RV Forecasting (RQ2)}

I compare five nested models for $RV_{t+1}$:
\begin{align}
    \text{GARCH proxy:}\quad & RV_{t+1} = \alpha + \beta_1\,RV_t + \beta_2\,r_t^2 + \varepsilon_{t+1} \label{eq:garch} \\
    \text{HAR-RV:}\quad & RV_{t+1} = \alpha + \beta_d\,RV_t + \beta_w\,RV_t^{(w)} + \beta_m\,RV_t^{(m)} + \varepsilon_{t+1} \label{eq:har}
\end{align}
with IV-only, IV-surface, and HAR-RV~$+$~IV-surface extensions. Out-of-sample evaluation uses rolling windows (500~training, 250~test) with MSE, MAE, $R^2_{OOS}$, and QLIKE loss. Model comparison uses the \citet{diebold1995comparing} test.

\subsection{Implied Correlation (RQ3)}

Following the CBOE methodology:
\begin{equation}\label{eq:implcorr}
    \hat{\rho}_{impl,t}
    = \frac{\sigma_{SPX,t}^{2} - n^{-1}\,\overline{\sigma_{i,t}^{2}}}
           {\bigl(1 - n^{-1}\bigr)\,\bar{\sigma}_{i,t}^{2}},
\end{equation}
where $\sigma_{SPX,t}$ is 30-day ATM IV of SPX, $\overline{\sigma_{i,t}^{2}}$ is the mean squared ATM IV of 30 constituents, and $\bar{\sigma}_{i,t}$ is the mean ATM IV. Realized correlation uses the 22-day rolling pairwise matrix. Stress is defined as VIX $> 25$ or 5-day VIX change $> 20\%$.

\subsection{Greeks Information Decay and Price Magnets (RQ4)}

I partition options into six DTE buckets (1w, 2w, 1m, 2m, 3m, 6m) and estimate predictive regressions within each. For the price magnet test:
\begin{equation}\label{eq:magnet}
    \mathrm{Magnet}_{i,t} = \mathbf{1}\!\Bigl[\,
    \frac{|P_{close,t} - K_t^{*}|}{P_{close,t}}
    <
    \frac{|P_{open,t}  - K_t^{*}|}{P_{open,t}}
    \Bigr],
\end{equation}
where $K_t^{*}$ is the max-OI strike. Under the null, $\Pr(\mathrm{Magnet}=1) = 0.5$.

\subsection{Machine Learning (RQ5)}

I construct 21 daily features from the SPX surface (listed in Appendix~\ref{app:features}) and train Random Forest (200~trees, max depth~10) and Gradient Boosting (200~trees, learning rate~0.05) on 2010--2019, testing on 2020--2025. The benchmark is $\widehat{RV}_t^{VIX} = (VIX_t/100)^2 / 252$.

\subsection{Granger Causality and VAR}

Bivariate Granger $F$-tests (5~lags) estimated per ticker. Bivariate VAR(5) models yield impulse response functions (IRFs) and forecast error variance decomposition (FEVD).