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

\section{Conclusion} \label{ch2:sec:conclusion}

Using high-frequency realized variance and a combination of HAR and Bayesian VAR models, this paper examines volatility transmission between four commodity ETFs and their underlying assets over the period 2010--2023. We find that volatility transmission is unidirectional from iNAV to ETF for physically-backed precious metals (GLD, SLV), while it is bidirectional (though still dominated by iNAV-to-ETF effects) for futures-based energy commodity funds (USO, UNG). Short-horizon transmission is far stronger in high-frequency data than in daily data, while longer-horizon effects are frequency-invariant; jump components transmit more strongly than continuous ones across all commodities; and these patterns are stable across sub-periods.

These results show how arbitrage operates in practice and how its effectiveness depends on market structure, extending the limits-to-arbitrage frameworks of \citet{ackert2000arbitrage}, \citet{pontiff1996costly}, and \citet{gromb2010limits}. The unidirectional transmission which we document for precious metals is consistent with physical-delivery frictions that limit ETF activity from influencing underlying prices. In contrast, the more balanced volatility transmission in energy commodity markets fits the concept of futures-based arbitrage with electronic settlement, as argued by \citet{basak2016model}. The frequency-specific results suggest that arbitrage operates primarily through high-frequency channels. The evidence of longer-horizon persistence reflects separate fundamental forces, building on the intraday microstructure analysis of \citet{hasbrouck2003intraday} and \citet{richie2008examination}. The dominance of jumps in the transmission of volatility, which we identify using the bipower variation decomposition due to \citet{barndorff2004power} and \citet{huang2005using}, indicates that models used for ETF pricing should pay separate attention to the continuous and discontinuous components of volatility.

In terms of methodology, our high-frequency iNAV series is the first that are built specifically for commodity ETFs over an extended sample period. These new series allow us to measure intraday arbitrage relationships that are unobservable when one uses end-of-day NAV \citep{petajisto2017inefficiencies}. Moreover, our comparison of HAR and BVAR models, together with a thorough analysis by sampling frequency, shows that temporal aggregation strongly affects the reliability of the empirical conclusions that can be drawn \citep{corsi2009simple, koop2011forecasting, andersen2007roughing}.

The findings in this paper carry practical implications for different market participants and extend the risk management discussion of \citet{madhavan2012exchange} and \citet{staer2017asset}. For investors and risk managers, we show that precious metals ETF volatility can be forecast from underlying volatility alone, while energy ETF models must account for bidirectional feedback. Both benefit from high-frequency information for short horizons. For market makers and authorized participants, arbitrage in precious metals flows mainly from underlying markets to ETFs, while energy markets contain more balanced bidirectional opportunities \citep{hendershott2013relationship}. For regulators, the asymmetries that we document suggest monitoring that is tailored to the commodity type: precious metals markets show little ETF feedback and thus have a lower destabilization risk, while energy markets exhibit stronger bidirectional links warranting closer monitoring during stress periods \citep{ohara2021etf, dannhauser2017effect}. The strength of the jump transmission further suggests that stress testing should account for discontinuous shock scenarios.


Finally, for future work, we note that the high-frequency iNAV methodology shown in this paper could be applied to equity, international, and fixed-income ETFs where similar arbitrage mechanisms operate. Future research could also investigate the relationship between jumps, which we find drive volatility transmission, to specific news events, order-flow imbalances, and market-maker inventory constraints.