# Convergence of Futures Prices and Spot Prices

Hey everyone,

I am confused with the concept of Futures Prices converging with Spot prices near maturity. Here’s my confusion:

1. Are they talking about the Price of the Futures contract itself, or the price stated within the contract that is fixed and will be acted upon in maturity?

My confusion derives from my book’s statement: “If the futures price is greater than the spot price during the delivery period, an arbitrageur buys the asset, shorts a futures contract, and makes delivery for an immediate profit”

• This makes little sense to me because the price stated within the contract is the delivery price, not the price of the future contract itself.

Here’s an example I am trying to use to help me understand: Imagine today you enter a futures contract for \$80 on AAPL share in two months time. A day before the contract expires, the price of the futures contract is now \$89 and the Spot price of AAPL share is \$85.

• So how would I make profit here if these two prices did not converge by maturity?
• According to my book (my interpretation not theirs) I would buy the AAPL share for \$89, go short on the contract, sell the share for \$80? Do I sell at the price stated in the contract \$80 or the \$89? Is the new comitted price \$89 or \$80?
• Can someone be kind enough to set up a short example so I can clear the confusion in my head? I’d really appreciate it!

Thanks guys!

The price of a futures contract is the agreed price for the underlying to be paid at contract expiration. If you enter into a 6-month futures contract with price of \$80, then tomorrow the price of your contract will be \$80, one week from now the price of your contract will be \$80, three months from now the price of your contract will be \$80, five months from now the price of your contract will be \$80, and fifteen minutes before expiration the price of your contract will be \$80. It doesn’t change.

Suppose that a futures contract has a delivery period of any time within the next week, the price of the futures contract (recall from above: that’s the agreed price of the underlying) is \$80, and the spot price of the underlying is \$75, then an arbitrageur will buy the underling for \$75, enter into the short position of a futures contract with a price of \$80, deliver the underlying against the futures contract immediately, and receive \$80, for risk-free profit of \$5.

Your example is too complicated. You have three numbers to consider: the price of your futures contract, the price of a current futures contract expiring on the same day as yours, and the spot price of the underlying.

Furthermore, you did not specify on your futures contract whether you’re in the long position or the short position.

I need to run to an appointment, but I’ll return later and revisit this. In the interim, think about examples that involve only two of the three numbers you mentioned. There are three such examples.

But there’s something that is still bugging me

Prices of future contracts change every day, there’s an example in my book where an investor exits a long future contract after the price of the contract drops, so the price of the contract is still fixed?

Also, if you are saying the price is fixed, how is the spot price converging to a fixed price? Its like saying the price you and I agreed on is going to be the spot price (as if we predicted it beforehand)?

Also, if in your first paragraph you’re saying price will at \$80, what price is making the mark to market daily if you said that after days and months the price is staying the same?

Thank you

I encourage you to spend time understanding forward contracts first, then move to futures contracts. They’re fundamentally the same thing, but futures have the (at least) daily marking to market which complicates the discussion needlessly.

The price of your forward contract will not converge to the spot price, because the price of your forward contract doesn’t change; it’s the agreed price to be paid for the underlying when your forward contract expires.

The spot price will not converge to the price of your forward contract. First, the market doesn’t know what the price of your forward contract is. Second, even if it did, it wouldn’t care. The spot price is going to move around because of supply and demand in the spot market, not because of some stupid contract into which you entered three months ago.

When the curriculum says that the spot and forward prices converge, what they mean is that if you look at the prices of forward contracts versus the spot prices of the underlying, the shorter the amount of time until expiration, the smaller the difference between the spot price and the forward price.

Here’s what you might see for forward contracts expiring on 12/10/18:

• On 9/10/18 (91 days to expiry), the spot price is \$85.00 and the forward price is \$86.04; the difference is \$1.04
• On 10/10/18 (61 days to expiry), the spot price is \$87.00 and the forward price is \$87.71; the difference is \$0.71
• On 11/10/18 (30 days to expiry), the spot price is \$88.00 and the forward price is \$88.35; the difference is \$0.35
• On 11/20/18 (20 days to expiry), the spot price is \$87.50 and the forward price is \$87.73; the difference is \$0.23
• On 11/30/18 (10 days to expiry), the spot price is \$88.25 and the forward price is \$88.37; the difference is \$0.12
• On 12/05/18 (5 days to expiry), the spot price is \$88.35 and the forward price is \$88.41; the difference is \$0.06
• On 12/09/18 (1 day to expiry), the spot price is \$88.47 and the forward price is \$88.48; the difference is \$0.01

Note that the spot price one day before expiry is nowhere close to the forward price from 9/10, or the forward price from 10/10, or the forward price from 11/10, and so on.

Resuming where I left off, let’s take the numbers two at a time.

First, let’s assume that in the original contract you were long; i.e., you agreed to buy AAPL for \$80/share.

(By the way, AAPL’s selling for \$218.33 today, so your numbers are off a bit.)

If the forward price for AAPL (expiring on the same date as your original contract) is \$89/share, then you can enter into a new forward contract in the short position, agreeing to sell AAPL for \$89/share. When the two contracts expire, you buy AAPL for \$80/share from the counterparty of your first contract and sell AAPL for \$89/share to the counterparty of your second contract; you make \$9/share for the number of shares in the contracts.

If the spot price for AAPL is \$85, then you pay \$80/share to the counterparty of your original contract and sell the shares in the spot market at \$85/share; you make \$5/share for the number of shares in your original contract. (Note that you really have to wait one more day before you get to buy AAPL for \$80/share. That subtlety doesn’t change the main point.)

If you didn’t enter into the original forward contract, but you’re looking at a spot price of \$85/share and a forward price of \$89/share, then you buy shares in the spot market and enter into the short position in the forward contract to sell them (tomorrow) for \$89/share. You make \$4/share.

Hello Magician,

First off, I want to thank you sincerly for your time and effort to write out such a great response and great example, I appreciate it. I think i’ve caught up now to the idea, and I am ready to prepare a summary below:

Futures:

Magician and Koko enter into a futures contract on AAPL for \$200 today, which matures next month. Magician assumes a short position and Koko assumes a long position.

• Price of Future: \$200
• Price of Underlying: \$202

The next day: The spot price of AAPL moves up, and the price of the futures contract follows it merely:

• Price of Future: \$201
• Price of Underlying: \$202
• So far, Koko has made a profit of \$1 and Magician has made an equivalent loss of \$1. (Zero sum Game)

One week later. The spot price of AAPL moves down, and the price of the futures contract follows it

• Price of Future: \$190
• Price of Underlying: \$195
• So far, Koko has made a loss of \$10, and Magician has made a gain of \$10

Two weeks later: KoKo has decided he wants out of the contract after the price of the contract hit \$180, and leaves his position when Price of Future is \$180. David has decided he wants to take KoKo’s position:

• Price of Future: \$180
• Price of Underlying: \$178
• Koko left with cumulative loss of (200-180) =\$20 and Magician has now gained \$20.

It is now a day before expiry, and David and Magician are the only parties that are left in this futures contract

• Price of Future: \$175
• Price of Underlying: \$174
• Magician decides to exit the contract and has made a cumulative gain of (200-174) and David has made a loss of (180-174) and closes his position.

It is now the day of expiry, Junior (long) and Pablo (short) enter the contract in the beginning of its expiry date (assuming they can): at end of day:

• price of future: \$174
• price of underlying: \$174
• Pablo buys a share for \$174 and sells the share to Junior for \$200 OR Pablo buys a share for \$200 and sells the share to Junior for \$200

Forward:

Magician and Koko enter into a Forward contract on AAPL for \$200 today, which matures next month. Magician assumes a short position and Koko assumes a long position.

• Price of Forward: \$200
• Price of Underlying: \$202

On Expiry day:

• Price of Future: \$180
• Price of Underlying: \$180
• Koko left with cumulative loss of (200-180) =\$20 and Magician has now gained \$20

Correct??

_____________________________________________________________________________________________________

Both Forward and Futures lead to the same payoffs, except Magicians gain was spread over the time he held the contract in a futures contract, whereas in the forward contract he realized the whole gain on the final day!

Sorry Magician if I may ask, you said the price does not change, but in this example you set up there is a clear change you stated?

Stop right there!

First, futures contracts are not between two individuals; they’re between an individual and the clearinghouse of an exchange. (Remember, I said to focus on forwards, not futures.)

Second, unless the risk-free rate is negative, this pair of prices is impossible. The forward price has to be the spot price _ increased _ by the risk-free rate for the period of the forward contract. You have a futures (forward) price that is less than the spot price.

Ok sir,

Assuming the prices spot prices > futures price and assuming the two parties are individual and clearing house, is the rest of my example correct?

The price of a forward contract is, by definition, the agreed price for the underlying at the expiration of the contract. It doesn’t change.

Here, if you enter into a forward contract on 9/10/18, the price is \$86.04 of that forward contract, and it will remain \$86.04 for the next 91 days, whereupon the short will sell the underlying to the long for \$86.04.

If you enter into a forward contract on 10/10/18, the price of _ that _ forward contract is \$87.71. But that’s a different contract from the one into which you entered on 9/10/18.

Please rewrite the example with appropriate prices.

I’m not doing this to be mean; I want it indelibly carved into your consciousness that forward prices have to be higher than spot prices. The best way to ensure that is for you to compose an example in which the prices are correct.

Once you do that, I’ll happily critique it.

Oooh! That makes so much sense.

So essentially the payoff a forward will always be (St-K) for a long position or (K-St) for a short position, and the (K) depends on the date you entered.

Wish the world only entitled Forwards contract, they are far less confusing then futures.

Some would wish that the world allowed only automatic transmissions, as they’re a lot less confusing than manual transmissions.

But they’re also a lot less fun.

Hi mate. What book are you using to study these concepts?