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.

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

You’re quite welcome.