# Why does a futures price converge to a spot price?

I've sort of get the arbitrage logic of it, i.e if the futures price is more expensive than spot price, then investors would short the contract and buy the asset for delivery. Correct me if i'm wrong. The investor enters into a short position on the contract and buys the asset for delivery. So does this mean it works only in the case of physical delivery of asset and not cash settlements?

• This question was also asked here not long ago. quant.stackexchange.com/questions/33831/… May 24, 2017 at 11:38
• Futures could be higher or lower than the spot. You must factor in the carrying cost to determine if an arb exists though. If you are only looking at Future and spot and nothing more you will the world is full of Arb's! :) This post may shed some light too: quant.stackexchange.com/questions/33493/… May 24, 2017 at 11:47

on edit: oops - didn't read this question carefully! Futures price can be above or below spot price. In equities, for example, if dividends are more than the risk free rate, then the futures will be below the spot if the risk free rate is greater than the dividends, then the futures will be above the spot. Assuming continuous rates and div yields, $F=Se^{(r-q)T}$ where we can assume that $q$ can absorb all funding costs and borrow costs for the stock for borrowing/lending the stock for short selling. If the equation above is an inequality, buy the cheaper one and sell the more expensive one to capture the convergence. Follow the usual replication argument.

end of edit.

It works for cash deliveries too. For SPX futures, you have to buy the basket of stocks and sell the futures or vice versa. Cash delivery happens in the end, but that cash - assuming no transaction costs - can be retrieved by selling the basket.

The cases where the arbitrage fails is when you can't actually buy the spot item today and deliver it at expiration.

Probably the most liquid version of this is VIX - you cannot buy VIX spot and hold it to delivery for a VIX futures contract.

Agricultural commodities sometimes have a convergence problem:

http://farmdocdaily.illinois.edu/2013/08/solving-markets-non-convergence-puzzle.html

If there were 6M futures on fresh eggs, you couldn't buy physical eggs today for delivery in 6M and call them fresh!

• +1 for the link to the paper showing there has been some non-convergence in agricultural futures. Interesting. Life is more complicated than the textbook says. May 24, 2017 at 23:15