# The effective pay off of mispriced futures

Hi all, I have a doubt with regards to futures… considering a mispriced future contract (i mean a contract where arbitrage is possible), is the net payoff of the contract the arbitrage on the contract+the payoff of the contract at maturity?? I am really sorry if I am asking a trivial doubt but I am a bit confused about this…

I would say no.

The arbitrage is the net payoff amount and it is guranteed (assuming no credit risk from any of the parties involved).

In a cash-and-carry arbitrage, you would have to pay back the the loan that was used to purchase the underlying.

In a reverse-cash-and-carry arbitrage, you would have to purchase the underlying to cover the short sale.

Oh, is it?? Let us suppose that the spot cost of an asset is \$1000, continuously compunded risk free rate is 4% (for all maturities) and no cash flows expected from the asset… Hence, the no arbitrage 6 month forward price of the asset would be 1000*e^0.04(0.05) = \$1020.2… If for some reason, a 6 month future contract on the asset is priced at \$1025, one can go for a cash and carry arbitrage on the asset for a riskless profit of \$4.8… but, if at maturity of the future contract, the spot rate of the asset is \$1040, should the net payoff to the short not be 4.8 - (1040-1025) = a loss of \$10.2??

In this case, you will not earn a profit on a cash-and-carry with respect to the spot price and the futures contract.

In the cash and carry, you buy the underlying asset with borrowed money, you then must deliver the asset at maturity for the \$1025 agreed upon price. So there will be no profit from the spot trade because you have to give it up and a predetrmined price anyways.

In this case, you will not earn a profit on a cash-and-carry with respect to the spot price and the futures contract.

In the cash and carry, you buy the underlying asset with borrowed money, you then must deliver the asset at maturity for the \$1025 agreed upon price. So there will be no profit from the spot trade because you have to give it up and a predetrmined price anyways.

I did not quite get you… one can borrow the money, buy the asset today and sell it at 1025 on expiry of the contract ie at a rate lower than the market rate of 1040… and then repay the amount of 1000 + risk free interest on the amount… thus one makes arbitrage profit but sells at lower than market rate… pls correct me if I am wrong…

At the initiation of the contract: Step 1: Borrow \$1000 at 4% annually continuous compound. Step 2: Buy the underlying asset for \$1000. Step 3: Enter into a short futures contract to deliver the underlying for \$1025 at maturity.

At expiration of the contract: Step 4: Settle the forward contract and receive \$1025 when you deliver the underlying to the long. Step 5: Pay back the borrowed money: \$1000e^(0.04)(0.5) = \$1,020.20

Net profit: \$1025.00 - 1,020.20 = \$4.80.

As you can see, there is no gain on the spot price of the asset because you have locked in the price of delivery.

ya, very same, except that the long on the futures contract gets to purchase the underlying at lower than market rate (1025 instead of 1040)…

also, how different would the payoff been to the short had it been a cash settled contract??

I think you are missing the point. This arbitrage example is in the eyes of the short so we do not care about the long position.

As far as the long is concerned, he “got lucky”. You picked only 1 example where the spot price is above the futures price at expiration. It is entirely possible that the spot price is 1000 at expiration and the investor has to purchase the underlying above the spot price.

But as for as the short is concerned, it does not matter where the spot price is at expiration, they will earn the arbitrage regardless of where the spot price is at maturity.

oh great, got it finally!! thank you so much

You’re welcome.