# Derivatives future and spot prices

Hello everyone, this is my first post and is a question about derivatives
I am having a hard time trying to understand the fact that “At expiration, the futures price and the spot price converges”
What spot price does it mean? The spot price that moves every day or the spot price at the date future contract was agreed

It cannot be the spot price that changes every day because the formula to compute the value of the future at expiration date: St - F(0,1) would always be zero or near zero.

Suppose that on 2/15/20 you enter into a futures contract that expires on 5/15/20. There will be a difference between the spot price on 2/15/20 and the contracted price (the futures price):

F_T=S_{2/15/20}\left(1+r_{f_{2/15/20}}\right)^{90/365}
F_T-S_{2/15/20}=S_{2/15/20}\left(1+r_{f_{2/15/20}}\right)^{90/365}-S_{2/15/20}=S_{2/15/20}\left[\left(1+r_{f_{2/15/20}}\right)^{90/365}-1\right]

Suppose that on 3/15/20 you enter into another futures contract that also expires on 5/15/20 with the same underlying. There will be a difference between the spot price on 3/15/20 and the contracted price (the futures price):

F_T=S_{3/15/20}\left(1+r_{f_{3/15/20}}\right)^{61/365}
F_T-S_{3/15/20}=S_{3/15/20}\left(1+r_{f_{3/15/20}}\right)^{61/365}-S_{3/15/20}=S_{3/15/20}\left[\left(1+r_{f_{3/15/20}}\right)^{61/365}-1\right]

Suppose that on 4/15/20 you enter into yet another futures contract that also expires on 5/15/20 with the same underlying, and you do the same on 4/30/20, 5/7/20, and 5/14/20. There will be a difference between the spot prices on those dates and the contracted price (the futures price):

F_T-S_{4/15/20}=S_{4/15/20}\left[\left(1+r_{f_{4/15/20}}\right)^{30/365}-1\right]
F_T-S_{4/30/20}=S_{4/30/20}\left[\left(1+r_{f_{4/30/20}}\right)^{15/365}-1\right]
F_T-S_{5/8/20}=S_{45/8/20}\left[\left(1+r_{f_{5/8/20}}\right)^{7/365}-1\right]
F_T-S_{5/14/20}=S_{5/14/20}\left[\left(1+r_{f_{5/14/20}}\right)^{1/365}-1\right]

Take a look at the factors on the right side of those equations:

\left(1+r_f\right)^{90/365}-1\\ \left(1+r_f\right)^{61/365}-1\\ \left(1+r_f\right)^{30/365}-1\\ \left(1+r_f\right)^{15/365}-1\\ \left(1+r_f\right)^{7/365}-1\\ \left(1+r_f\right)^{1/365}-1

As you approach the expiration date, those factors approach zero: the difference between the futures price and the spot price approaches zero; the futures price and the spot price converge.

That’s what they mean.

They’re not saying that the spot price will converge to your original contracted futures price, nor that the price of your original futures contract will converge to the spot price. It’s more complicated than that.

1 Like

thanks you so much S2000magician!!
now I have a better understanding of the why of that!

You’re quite welcome.

Hi …how r …I just want to ask a quet related to it…it’s ok there will be a conversion between spot and future prices but what about the contract itself …I still long or short the asset in the contract …so if the price of future is 40 and spot at expiration is 40 so what about if the fixed price in the contract to buy or sell at 50 for example …what will be the situation

Thanks

The long will have a loss of 10 and the short will have a profit of 10.

Ok …I have the contract … If I’m long on that contract I buy the contract now for 40 and obligated to buy x at 50 SO paying another 50 then sell x for 40 so I end by paying 50 so what I get from the contract and if I was short the contrary will happen …I know there is something I miss here …so If u plz can explain

I have no idea what this means.

Is the agreed price 40 or 50?

The agreed price is 50 but the price of the future contract which I think is different from the agreed price must converge to the spot price which is 40 …so what about executing the contract at 50

So . . . the forward price when the contract originated was 50, and Bob entered into it in the long position, say.

Now, two months later, say, are you saying that the value of the contract is 40? The price of the contract doesn’t change; it started at 50 and it will remain at 50. The price of a forward contract is, by definition, the agreed price that the long will pay and the short will accept when the contract expires.

So…how the convergen will happen the price is fixed at 50 …does the spot gave to converge to it …I don’t think so as the spot depends on supply and demand …so how the convergen will happen … does the value will convert to spot or what …sorry for that …but I have a problem all day because of this convergen

Did you read my first response (#2, above)? I explained there what that statement means.

I got it from the example u explained but what about the contract itself that carry an obligation for both sides at specific price …think with me we have a future contract that it’s price in the exchange must converge to the spot and also when I buy it from the exchange I carried this obligation …so there must be a difference between the price of the contract itself and the forward price which is fixed

What about it?

The price on that contract doesn’t change.

The price of that contract is the forward price (as of the inception of the contract).

Two things change, neither of which have anything to do with the price of your futures contract:

• The spot price of the asset underlying your futures contract
• The prices on new futures contracts that expire on the same day that yours expires.

Those are the prices that converge to each other.