% For submission to the Journal of Futures Markets (JFM)

\section{Data and Sample Construction} \label{sec:data}

Our data processing approach improves on what has been previously used in the ETF volatility literature, which typically relies on daily data and may not detect high-frequency transmission channels \citep{ben2018etfs, israeli2017etf}. By constructing minute-by-minute price and indicative NAV series for four commodity ETFs over thirteen years, we capture volatility dynamics at horizons not previously examined and compare volatility transmission across heterogeneous commodity market structures.

\subsection{Sample Selection and Data Sources}

We study four large single-commodity ETFs, each among the top funds in its category ranked by assets under management:

\begin{itemize}
\item \textbf{SPDR Gold Trust (GLD)} and \textbf{iShares Silver Trust (SLV)} are physically-backed precious-metals ETFs holding bullion in trust (over \$50 billion and \$10 billion in assets, respectively). Physical backing minimizes tracking error but creates arbitrage frictions tied to delivery, storage, and insurance.
\item \textbf{United States Oil Fund (USO)} and \textbf{United States Natural Gas Fund (UNG)} are futures-based energy ETFs tracking WTI crude oil and Henry Hub natural gas through NYMEX futures. Continuous rolling of expiring contracts generates tracking errors related to contango and backwardation \citep{todorov2021etf}, and natural gas adds extreme seasonality and storage constraints.
\end{itemize}

We select these ETFs for three reasons: high liquidity (top-five funds by assets in each category), continuous tradability over the full sample (January 2010 to January 2023) and-, most importantly, contrasting tracking mechanisms. Physically-backed ETFs (GLD, SLV) settle through physical delivery against London Bullion Market Association (LBMA) spot prices, with no rollover costs but slower arbitrage. Futures-based ETFs (USO, UNG) are arbitraged electronically against NYMEX futures, with faster settlement but involving roll costs, contango/backwardation patterns, and basis risk. Since these market structures create different frictions, transaction costs and lags, we expect them to generate different patterns of volatility transmission. We use single-commodity rather than index ETFs to avoid cross-commodity correlation effects.

Our primary data source is the Bloomberg terminal, which provides tick-by-tick trade prices, bid--ask quotes, and volume with millisecond timestamps for the ETFs and their underlying assets. We supplement this with futures contract price data from the Chicago Mercantile Exchange (CME) and spot price data from the LBMA. We begin the sample in 2010 for three reasons: By this date, the ETFs (launched 2004--2007) had plausibly reached the trading volume and market-making infrastructure needed for reliable high-frequency data and meaningful arbitrage. Moreover, starting in 2010 avoids the atypical microstructure effects during certain periods of the 2008--2009 crisis. Lastly, consistent tick-level data became available across all instruments in our sample. The resulting thirteen-year period spans the 2010--2012 European sovereign debt crisis, the 2014--2016 commodity collapse, the 2020 COVID-19 pandemic, and the 2021--2022 inflation surge, covering diverse volatility regimes.

\subsection{High-Frequency Data Construction and Cleaning}

High-frequency data require careful filtering to remove microstructure noise. Starting from raw tick data over regular U.S. market hours (9:30 AM--4:00 PM EST), we apply standard procedures adapted to the ETF market structure \citep{barndorff2009realized}. First, we remove outliers using the \citet{brownlees2014practical} method, deleting observations more than 10 standard deviations from a rolling 20-minute median (about 0.03\% of observations). Second, we apply the duration filter suggested by \citet{hansen2005realized}, excluding trades separated by more than 30 minutes, which typically indicates closures or technical failures (less than 0.1\% of observations). We then build synchronized price series at 1-, 5-, and 30-minute frequencies using previous-tick interpolation \citep{andersen2001distribution}. The final cleaned dataset contains roughly 45 million price observations. Data loss is minimal and concentrated in thinly traded periods.

\subsection{Indicative Net Asset Value (iNAV) Construction}

Official NAV is published only at the end of each trading day \citep{petajisto2017inefficiencies}. This variable cannot capture the intraday arbitrage activities that actually drive ETF pricing. We therefore construct an indicative NAV (iNAV) at a high frequency, which provides a real-time estimate of fundamental value from current underlying prices. This approach lets us measure arbitrage and transmission at the frequency where authorized participants make creation/redemption decisions, and let us separate fundamental ETF volatility (changes in underlying value) from non-fundamental volatility (liquidity shocks, inventory effects, or temporary arbitrage breakdowns).

For physically-backed ETFs (GLD, SLV), iNAV follows a composition-based approach:

\begin{equation}
\text{iNAV}_{t} = \frac{1}{N_t} \left[ \text{Cash}_t + \sum_{i} \left( P_{it} \cdot f_{it} \cdot q_{it} \cdot c_{it} \right) \right]
\end{equation}

where $N_t$ is the number of outstanding ETF shares, $\text{Cash}_t$ is the fund's cash holdings, $P_{it}$ is the price of underlying asset $i$ in local currency, $f_{it}$ the currency conversion factor, $q_{it}$ the quantity held, and $c_{it}$ an adjustment for accrued interest, dividends, or other cash flows. We use LBMA gold and silver prices converted to U.S. dollars at real-time exchange rates. Physical holdings are updated daily with creation/redemption activity, while intraday changes reflect only price movements.

For futures-based ETFs (USO, UNG), iNAV follows a futures-position model:

\begin{equation}
\text{iNAV}_{t} = \frac{1}{N_t} \left[ \text{Cash}_t + \sum_{j} \left( F_{jt} \cdot cc_{jt} \cdot q_{jt} \cdot m_{jt} \right) \right] \cdot FX_t
\end{equation}

where $F_{jt}$ is the price of futures contract $j$, $cc_{jt}$ the contract conversion factor, $q_{jt}$ the number of contracts held, $m_{jt}$ the contract multiplier, and $FX_t$ any currency conversion. We use real-time NYMEX WTI crude oil and Henry Hub natural gas futures, accounting for the funds' actual contract positions and monthly roll schedules.

A technical consideration is the mismatch between ETF trading hours and underlying market hours: gold and silver trade nearly around the clock in London and Asia, while energy futures have defined sessions. To handle this issue, we weight each market session by its share of price discovery and carry forward the most recent adjusted prices when an underlying market is closed. We validate the construction against published end-of-day NAV. We find that our iNAV has a correlation greater than 0.999 with the official NAV, and mean absolute deviations of less than 5 basis points. 

\subsection{Realized Variance Construction and Jump Detection}

We measure volatility using realized variance, i.e.,  the sum of squared intraday returns \citep{andersen2001distribution}. For asset $i$ on day $t$:

\begin{equation}
RV_{i,t} = \sum_{j=1}^{M} r_{i,t,j}^2
\end{equation}

where $r_{i,t,j} = \log(P_{i,t,j}) - \log(P_{i,t,j-1})$ and $M$ is the number of intraday returns, giving $M=390$ (1-minute), $M=78$ (5-minute), and $M=13$ (30-minute). Under standard conditions, realized variance converges to integrated variance as sampling intensifies \citep{barndorff2002econometric}. The choice of frequency trades statistical efficiency against microstructure bias from bid--ask bounce \citep{hansen2005realized, liu2015does}, so we use three frequencies to assess robustness.

To separate continuous movements from discrete jumps, we use the bipower variation \citep{barndorff2004power}:

\begin{equation}
BV_{i,t} = \mu_1^{-2} \sum_{j=2}^{M} |r_{i,t,j}| \cdot |r_{i,t,j-1}|
\end{equation}

where $\mu_1 = \sqrt{2/\pi} \approx 0.798$. Bipower variation consistently estimates integrated variance even in the presence of jumps. The jump component is

\begin{equation}
J_{i,t} = \max(RV_{i,t} - BV_{i,t}, 0)
\end{equation}

and the continuous component is $C_{i,t} = RV_{i,t} - J_{i,t}$ \citep{huang2005using}. This continuous component is reported as quadratic power variation (QPV) in the estimation tables. Separating the two matters because jumps---from supply disruptions or geopolitical events---may transmit across markets differently than smooth volatility.

\subsection{Descriptive Statistics and Stylized Facts}

Tables \ref{tab:desc_stats_5min}, \ref{tab:desc_stats_1min}, and \ref{tab:desc_stats_30min} report descriptive statistics for realized variance, bipower variation, and jumps across the three frequencies. Several stylized facts emerge. Realized variance is heterogeneous across commodities and between ETFs and their iNAVs. Natural gas and crude oil have the highest average volatility: the 5-minute mean realized variance is 0.104\%, and 0.081\% for the iNAV. For crude oil, the iNAV (0.081\%) is more volatile than the ETF value (0.066\%), as expected from the damping effect of arbitrage. The heaviest distributional tails occur in crude oil and gold: For crude oil, the iNAV realized variance reaches a maximum above 40\%. For gold, the ETF series shows occasional extreme outliers (maximum near 79\%), reflecting occasional disruptions in physical gold arbitrage adjustments. Precious metals are otherwise less volatile on average than energy, though for gold the ETF mean volatility (0.034\%) exceeds its iNAV mean (0.012\%). Jump activity is most pronounced for crude oil, consistent with this commodity's sensitivity to geopolitical and supply shocks, and it is weakest for precious metals, for which prices move more smoothly.

Figures \ref{fig:rv_uso}, \ref{fig:rv_gld}, \ref{fig:rv_slv}, and \ref{fig:rv_ung} show the evolution of realized variance over time. All series show strong volatility clustering, which is most pronounced during the 2014--2016 oil collapse, the 2016 Brexit referendum (precious metals), and the 2020 pandemic (for all commodities). ETF--iNAV synchronization is high for precious metals but more variable for energy commodities, where periods of close co-movement alternate with divergence. As these relationships are time-varying, a simple correlation analysis would fail to detect economically significant dynamics, which motivates the modeling framework in Section~\ref{sec:methodolog}.

Finally, all realized-variance series are stationary in both levels and logs (augmented Dickey--Fuller tests), and we validate the GLD iNAV against the NYSE indicative optimized portfolio value (IOPV), obtaining correlations above 0.995. Tests for structural breaks and the full set of robustness checks are reported with the econometric framework in Section~\ref{sec:methodolog}.