Why Futures Converge to Spot and When Arbitrage Can Fail
Summary
The document explains why futures and spot prices tend to converge at expiry through replication and arbitrage. If a futures contract is mispriced relative to the cost of holding the underlying, traders can buy the cheaper exposure and sell the more expensive one. For equity index futures with cash settlement, the replicating trade can use the underlying stock basket; cash settlement does not itself prevent convergence.
The argument depends on being able to trade and carry the spot asset through delivery, with transaction costs and other frictions set aside in the explanation. Convergence may fail when the underlying cannot be held or delivered in a way that replicates the futures payoff. The examples distinguish equity indexes from VIX, whose spot level cannot be bought and stored, and fresh eggs, which cannot remain deliverable in their original condition over a long contract period. The note is conceptual and does not quantify these frictions or provide a full pricing model.
Key ideas
- Arbitrage can align futures prices with the cost of holding the underlying toward expiry.
- Cash settlement can still support replication when the underlying basket can be traded.
- The convergence argument assumes traders can buy and carry the spot exposure.
- A mismatch between the deliverable and the spot asset can weaken convergence.
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Full text
# Why does a futures price converge to a spot price?
# 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?
## Answer by FinanceGuyThatCantCode (score 1)
https://quant.stackexchange.com/a/34383
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.