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\section{Introduction}

The exchange-traded fund (ETF) market has grown rapidly over the past two decades, with global assets under management (AUM) exceeding \$23 trillion as of May 2026. The growth in the ETF market has also altered the structure of underlying asset markets \citep{petajisto2017inefficiencies}. ETF ownership has been found to increase non-fundamental volatility in underlying equities \citep{ben2018etfs}, weaken the link between prices and fundamentals \citep{israeli2017etf}, and strengthen correlations during periods of market stress \citep{da2018exchange}. The number and economic size of commodity ETFs, such as the SPDR Gold Trust (GLD), has rapidly grown in recent years. These products offer exposure to commodity markets without the need for direct futures trading or physical storage \citep{gorton2006facts}, thus increasing the participation of financial investors, creating new economic links between commodity and equity prices, and establishing the ETF as an alternative venue for price discovery \citep{basak2016model, buyuksahin2014speculation}.

Commodities offer a particularly interesting setting to study ETF pricing and volatility transmission. Exchange-traded funds are usually studied as near-transparent wrappers around their holdings. For example, VOO and IVV replicate the U.S. S\&P~500 market index and hold the stocks that comprise this index according to their weights. For a typical equity ETF this framing is close to exact. The underlying securities trade on the same exchanges during the same hours, and creation-redemption arbitrage pins the fund's price to its net asset value quickly and in essentially one direction. Thus, for an equity ETF, asking whether the ETF leads the basket would be similar to asking whether a shadow leads the object that casts it. In contrast, commodity ETFs break this identity in a way that is not feasible in other major ETF classes. This is because their underlying may not be the physical good, but could be instead a derivative claim. For instance, if it is a futures position in crude oil or natural gas, there are economic implications for cost-of-carry, roll, and the shape of the term structure. In the case of a stored physical claim in gold and silver there is lease-and-storage economics to consider.

Thus, for commodity ETFs, the arbitrage relationship is not as simple as it is for equities or bonds, and volatility transmission becomes a genuine economic question rather than a purely mechanical one. To the point of this paper, it is important to investigate the fund's indicative net asset value (iNAV), which is the real-time fair value of that basket. Commodity ETFs also play a role in the financialization debate \citep{basak2016model}, as they are popular retail-accessible instruments.  It is important for investors, traders, hedgers and policymakers to understand how volatility travels between these vehicles and their underlyings. The direction of this volatility flow is not clear a priori. Indeed, commodity ETFs differ along another significant dimension, namely that unlike equities, in commodities the ETF may well be more liquid and more continuously accessible than its underlying. This feature provides a clear reason why the ETF might contribute to price discovery rather than simply inherit it. Whether it does, however, remains an empirical question.


The theoretical foundation of ETF pricing rests on arbitrage. Authorized participants (APs) maintain price alignment through creation and redemption, buying undervalued ETFs while selling their underlying constituents, or vice versa \citep{ackert2000arbitrage}. Classic arbitrage theory predicts that volatility transmission would be unidirectional, flowing from the underlying constituents (NAV) to the ETF: when underlying volatility rises, arbitrageurs trade more actively to maintain alignment, which transmits volatility to the ETF. Recent evidence challenges this prediction, documenting bidirectional transmission in which ETF trading influences underlying asset volatility \citep{ben2018etfs, da2018exchange}. This outcome arises because ETFs often trade more frequently and with smaller spreads than their constituents, making the ETF a primary venue for price formation \citep{glosten2021etf, pan2016etf}. The relative magnitude and direction of transmission therefore reveal which market dominates information discovery and whether arbitrage functions efficiently. These questions have direct implications for hedging (directional dependencies) and for regulation, e.g., as feedback effects may amplify volatility during crises \citep{madhavan2012exchange, petajisto2017inefficiencies}.

This paper studies how volatility is transmitted between commodity ETFs and their underlying assets using high-frequency realized variance over 2010--2023. We focus on four major single-commodity ETFs spanning two market structures: physically-backed precious metals (SPDR Gold Trust, GLD; iShares Silver Trust, SLV) and futures-based energy funds (United States Oil Fund, USO; United States Natural Gas Fund, UNG). Our central question is whether transmission is unidirectional or bidirectional, and how its direction, strength, and time horizon vary with commodity type, sampling frequency, and the continuous versus jump nature of volatility. We study volatility transmission rather than price discovery because it reveals how risk, not just information, propagates across linked markets and whether ETF arbitrage stabilizes or amplifies it---a question central to risk management and systemic-risk regulation.

The empirical literature on ETF volatility transmission is incomplete in three respects. First, it focuses on equity ETFs using daily data \citep{ben2018etfs, israeli2017etf}, leaving the intraday dynamics that govern arbitrage largely unexplored. Indeed, as arbitrage operates continuously through the day, daily aggregation may obscure rapid transmission. Second, the literature has not examined how spillovers differ across commodity types, even though precious metals trade in liquid global markets with physical arbitrage while energy commodities rely on futures with rollover costs and storage constraints---features that should generate systematically different transmission. Third, it has not distinguished the roles of underlying assets versus ETFs in driving volatility, leaving open the questionwhether ETFs passively follow their constituents or actively feed back into them.

We address these gaps by constructing minute-by-minute indicative Net Asset Value (iNAV) series for the four ETFs over thirteen years spanning the European sovereign debt crisis, the 2014--2016 commodity collapse, the COVID-19 pandemic, and the subsequent inflation surge. We combine Heterogeneous Autoregressive (HAR) models that capture the long memory of realized volatility across daily, weekly, and monthly horizons \citep{corsi2009simple} with Bayesian Vector Autoregression (BVAR) models to accommodate time-varying dependence while avoiding overfitting \citep{koop2011forecasting}. We measure realized variance from high-frequency returns \citep{andersen2001distribution} and decompose it into continuous and jump components \citep{barndorff2004power}. We separate jumps because commodity prices respond to discrete shocks---geopolitical events, supply disruptions---that may transmit through different channels than smooth price movements, with distinct consequences for tail-risk hedging. On this basis, we test four hypotheses, stated formally in Section~\ref{sec:methodolog}: (i) whether transmission is uni- or bidirectional and varies by commodity type; (ii) whether high-frequency sampling reveals dynamics hidden in daily data; (iii) whether jumps or the continuous component dominate transmission; and (iv) whether transmission is stable over time.

Our results reveal substantial heterogeneity across commodity types and sampling frequencies. For precious metals (GLD, SLV), transmission is strongly unidirectional from iNAV to ETF, with spillover coefficients ranging from 0.42 (silver) to 0.63 (gold) at one-minute frequency and negligible reverse effects. Energy ETFs (USO, UNG) show bidirectional transmission, with the iNAV-to-ETF direction dominant for crude oil (about four to one) and the two directions comparable for natural gas. Sampling frequency matters: the daily iNAV-to-ETF spillover is up to roughly twice as large in one-minute as in thirty-minute data. Jump components dominate continuous transmission, especially for precious metals. The BVAR analysis confirms these patterns through impulse responses and variance decompositions in which underlying volatility rivals or exceeds ETF self-persistence in explaining ETF volatility.

The paper makes two contributions. Methodologically, we construct the first comprehensive high-frequency iNAV series for commodity ETFs over an extended period, enabling precise measurement of intraday arbitrage relationships that are unobservable with daily NAV data. Empirically, we document systematic differences in transmission across commodity categories and sampling frequencies, showing that market structure shapes information transmission and that empirical inference is affected by temporal aggregation. Thus, we extend the literature on ETF volatility transmission from equity to commodity markets \citep{ben2018etfs, israeli2017etf}. We further connect it to the microstructure literature \citep{hasbrouck2003intraday, richie2008examination}, and we contribute to research on commodity-ETF financialization \citep{todorov2021etf, buyuksahin2014speculation} by showing that the effects vary in systematic ways between physically-backed and futures-based ETFs. The remainder of the paper is as follows. Section~\ref{sec:data} describes the data and the construction of high-frequency iNAV and realized-volatility series. Section~\ref{sec:methodolog} presents the econometric framework and states the hypotheses. Section~\ref{sec:results} reports the empirical results. Section~\ref{sec:conclusion} concludes.
