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I am aware why the price of the underlying security/commodity and its futures contract price would converge at expiration, i.e. if the underlying price was lower than the futures price, an arbitrageur could short the contract, buy the underlying at a lower price in the physical marketplace and deliver at the (higher) contract price. Vice versa for when the underlying price is higher than the contract price. This ties up well when the contract is physically settled.

However, what causes the price of the underlying and its futures to converge at expiry in case of a cash settled contract? What stops the two from behaving like two separate price items altogether? I was unable to find a satisfactory theory for this.

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I think you can make the exact same argument for the cash-settled futures. The only difference is that you have to sell the underlying at expiry and delivery the cash instead of delivering the actual physical underlying.

To keep things simple let us disregard financial frictions (transaction costs, cost-of-carry, dividends and interest rates). Let us assume that the position is not marked-to-market daily, so we only have to considered when we enter and exit the position.

If we buy the underlying and sell the cash-settled futures, because the underlying is trading at a discount, then it will cost us $S_t$ at time $t$. The futures is costless to enter and has fixed price $F_t$.

At expiry we have to deliver a cash amount of $S_T-F_t$, i.e. we receive $F_t-S_T$. As we own the physical underlying worth $S_T$, we can sell it to receive $F_t-S_T+S_T=F_t$ at time $T$. The payoff of this strategy is thus $F_t-S_t$, which is positive due to our assumption, which makes it an arbitrage opportunity. So for the market to be arbitrage-free we must have $S_t=F_t$. This equation will of course only hold when there is no financial frictions, but the same method can be applied in case there were.

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It depends how the settlement level of the contract is determined.

If the excahnge determines the settlement level based on something linked to the physical market (for example ICE Brent futures final settlement), then the settlement level before the expiry of the future can trade wherever it wants, based on the vwap in the window. But if you hold the future to expiry, then it will settle according to the ICE Brent Index, which is (in a somewhat convoluted way) linked to the physical oil price in the north sea. In this case, it doesn't matter if where the futures trade*, since the value is based on the value of oil and not on the value the futures trade at.

ETFs can also trade at a premium or discount to their NAV, depending on suply and demand essentially (for example, see here to read about the premium/discount on the GBTC etf).

Here is another interesting story about the DGAZF etf going from \$400 to \$24,000 in a number of days, on basically no volume, and no related move in natgas.

Another example from last year happened when the physical gold price because detached from the futures price - this is because the contracts are not the same, as the gold futures are physically settled in new york, while the physical gold price is set in london - there are infrastrcutural contraints to moving gold between the countries, as well as changing the format of the bars acceptable for delivery. Last year during the height of covid, these infrastructure issues because even harder to solve and allowed the gold futures price in new york to deviate from the physical value of gold set in london.

The important differences in these examples is in how you're able to turn your holdings back into cash, and additionally what liabilities they can provide you with.

*However, it's worth bearing in mind that if the futures trade significantly above the actual value of oil, then someone able to trade the physical market may step in and sell the futures while buying physical, where they'd be happy to buy physical oil at any price up to where they can sell the futures, so the spot may be lifted in this scenario.

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