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I found that for the exchange traded Futures, we can deliver the Futures before the last trading day, namely you can sell a Future then deliver it immediately, which definitely has a arbitrage opportunity if spot price and Future price are different.

So do I misunderstand some rules of exchanged Futures?

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    $\begingroup$ I think we should clarify some things first, to ensure that we do not mess up things. Futures as standardized forward contracts could be traded over an exchange at any time as long as you find a buyer. The future itself could be delivered (transferred) immediately or usually within 3 bank days. However this does not mean that the underlying is (physically) delivered. This is possible on expiry only (in contrast to options). Considering this, I do not see the arbitrage opportunity you mentioned. But maybe I missunderstood your question? $\endgroup$ – Fokko Mar 19 '19 at 14:05
  • $\begingroup$ Usually there is a Delivery Period (more than one day long) when the contract approaches expiration, where you can deliver. The Exchange allows this so that convergence of future price to spot can take place ahead of expiration, while the exchange monitors the situation. They don't want any "surprises" on the last day of trading. $\endgroup$ – Alex C Mar 19 '19 at 23:06
  • $\begingroup$ @Fokko Here i mean the underlying delivery but not the offset of position. For the arbitrage, if Future price > spot price, you can sell a Future and buy a spot, then delivery it immediately, otherwise you can do the reverse way. $\endgroup$ – user6703592 Mar 20 '19 at 8:36
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    $\begingroup$ @user6703592 This is not the way it works as a future is a contract of two parties: the long position holder and the short position holder. The seller has the obligation to deliver (in your example you are the seller) and the buyer to buy. So if you sell your future as a short position holder the new owner of the short position steps into your legal position and is obliged to deliver. Your example implies that you are long and short position holder at once - so leaving your long position and then deliver from your short position. This is in fact not possible. $\endgroup$ – Fokko Mar 21 '19 at 10:23
  • $\begingroup$ @Fokko I may confuse with short position and seller. Ok how about if Future price > spot, you borrow money to buy a spot and enter a short position of Future. Then deliver Future immediately, you deliver the spot and receive the money(=Future price), then you repay the money( =spot )you borrow. Finally you make the profit. $\endgroup$ – user6703592 Mar 21 '19 at 13:43
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If a Futures contract has two separate dates $T_d$ (when deliveries may begin) and $T_f$ (which is the last date by which delivery must take place) then you cannot use the formula found in introductory books such as Hull:

$$F=S e^{rT}$$

For example for Bond Futures $T_d$ is about one month before $T_f$. In that case which value of T would you use? The formula has been derived under the assumption of a single delivery date (which is true for example for Stock Index Futures).

For such cases the choice of the delivery date itself becomes a variable in the problem, and there are various (complicated) theories and methods available (generally described as Delivery Option analysis). Markets participants are certainly aware that the delivery date is uncertain for these kinds of futures; some (such as myself) simply avoid trading the future between $T_d$ and $T_f$ when the delivery may take place, and leave the "delivery timing game" to the specialists. Between time 0 and $T_d$ they assume delivery will take place at whichever of $T_d$ or $T_f$ is least favorable for their position. Others use much more complex optimization models.

But yes, real world futures can be more challenging that those presented in introductory courses.

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  • $\begingroup$ As far as "convergence of the futures price to spot", the convergence occurs at time $T_d$ and is maintained thereafter until $T_f$. $\endgroup$ – noob2 May 23 at 8:53
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You can't "deliver" a futures contract. Its not a thing - it is a legal contract with the exchange.

A physically settled futures contract cannot be physically settled early unless the rules permit it. Think about it - you can't turn up early and dump loads of stuff at the delivery point early and say you are done. Storing it until the delivery date will cost money. The warehouse owner will charge you, etc. And you can't say that the exchange owns it, because until the delivery date, legally they don't. You could disown it, but then the warehouse will sell it on the spot market and keep the proceeeds.

So in short, there is no arbitrage opportunity in this respect.

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    $\begingroup$ I think what was meant by “deliver” in this instance was delivery of the underlying asset. This DOES occur with futures contracts, it is not the literal delivery of the contract. And yes when delivery of the asset is not possible then settlement occurs... $\endgroup$ – JazKaz May 21 at 22:54
  • $\begingroup$ Being accurate with the correct and well understood words / phrases / names is critical to being able to understand and operate in financial markets. Not all futures contracts have an "underlying asset" - this is a better concept for options rather than futures contracts. This phrase makes no sense: "And yes when delivery of the asset is not possible then settlement occurs..". Are you referring to final setlement? Cash settled or physically settled? etc $\endgroup$ – ThatDataGuy May 23 at 12:44
  • $\begingroup$ All derivative values or more precisely prices are “derived from an underlying asset” hence why they are called derivatives... if you are referring to the different types of futures, commodity, interest or currency futures, the commodity, the interest rate and the currency would be the “underlying asset” the item of which the futures value is derived. As for settlement, yes final settlement in being cash settled when the underlying asset cannot be “delivered”, there is settlement of the contract paid upon expiry date. $\endgroup$ – JazKaz May 23 at 14:44
  • $\begingroup$ As for what I meant by settlement, yes final settlement in being cash settled when the underlying asset cannot be “delivered”, there is settlement of the contract paid upon expiry date. I am aware of the options to “settle futures contracts”: closing out, cash settlement and physical delivery. When physical delivery is not possible, and one hasn’t sold the contract before expiry, then the contract is settled via cash, which I believe would be most appropriate given the case mentioned above $\endgroup$ – JazKaz May 23 at 14:53
  • $\begingroup$ I've not heard of a phsically delivered futures contract allowing cash final settlement. How would the cash value be determined? Can you link to a document from an exchange that allows this? Not all futures contracts have underlying assets. For example weather futures on the CME. They reference an index of weather observations, which is not an asset. $\endgroup$ – ThatDataGuy May 23 at 18:33

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