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Why Spot and Futures Prices Converge Before Expiry

Article Quant Q&A · Author: Elizabeth
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Summary

The document asks why spot and futures prices tend to move toward one another before maturity, beyond the familiar no-arbitrage condition that requires equality at expiry. One response explains the process through arbitrage trades: when spot is above futures, traders can short the underlying and buy futures, putting downward pressure on spot and upward pressure on futures. This offers a market mechanism for reducing the price gap before delivery.

A second response uses a no-carry forward relation to show that the forward-to-spot ratio approaches one as time to maturity shrinks, because the financing component diminishes. It then cautions that futures have daily margining, so interim cash flows and changing interest rates can distinguish their pricing from forward pricing, particularly in some fixed-income and FX markets. These explanations are concise and rely on simplified assumptions; actual convergence and pricing can also depend on carry, trading frictions, and contract details, which the discussion does not examine.

Key ideas

  • Arbitrage trades can narrow a spot–futures price gap before the contract expires.
  • When spot exceeds futures, shorting the asset and buying futures can pressure the two prices toward each other.
  • For a no-carry forward, financing explains why the forward-to-spot ratio approaches one near maturity.
  • Daily futures margining can affect pricing relative to forwards when interest rates change.
  • The explanations use simplified assumptions and do not analyze trading frictions or contract specifics.

Tags

FuturesSpot marketsArbitrageDerivatives pricing
Full text
# Convergence of Spot and Futures prices


# Convergence of Spot and Futures prices












Any explanation I've found explaining why future and spot prices converge over time seem to only focus the explanation on why the spot and future price must be equal at maturity.

I understand that if the prices aren't the same at maturity, then an arbitrage opportunity will exist. But what's the explanation for the trend of convergence prior to maturity? I originally thought that this might be because the cost of carry decreases as time to maturity decreases, but doesn't this price behaviour hold true for non-commodity futures as well?

Thanks!

## Answer by MH.Q (score 5)

https://quant.stackexchange.com/a/33832

The trend of convergence is also due to arbitrageurs.

Prior to maturity, when the spot is higher than futures, arbitrageurs would short the underlying asset and long futures contracts. This in effect reduces the spot price due to increase in supply while raising the futures price due to increase in demand.

## Answer by fni (score 5)

https://quant.stackexchange.com/a/33834

Let’s start with a forward contract on some asset $S_t$ with no carry. There, it is obvious that at any point in time $F_t = \mathbb{E}^{Q_t}[S_T] = S_t e^{r(T-t)}$. You can see that $F_t/S_t = e^{r(T-t)} \to 1$ as $t \to T$. If you have a future contract, then you have to start thinking about margin requirements and the possibility that the daily risk-free rate can be different from the rate at which you entered in the trade. Nonetheless this should be relevant only for futures on Fixed Income and FX.

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.