No-Arbitrage Bounds for Forward and Futures Prices

Summary

The document explains why a forward and a futures contract on the same asset need not have identical prices when interest rates are stochastic. With deterministic rates, their prices generally agree in the idealized model; random rates can affect the relationship, especially through how rates co-move with the underlying asset. The discussion focuses on whether opposing positions in the contracts guarantee arbitrage when their initial prices differ.

At expiration, both contract prices are tied to the underlying asset in the idealized framework, but that endpoint alone does not establish a riskless profit from a price gap earlier. Real market details can also weaken convergence: for example, physical settlement in one contract and cash settlement in the other can impose costs on traders trying to arbitrage the difference. The answer therefore frames no-arbitrage as imposing bounds on price processes, with exact convergence at expiry applying only under simplifying assumptions. It offers conceptual guidance rather than a worked model or empirical evidence, and does not quantify the bounds or transaction costs.

Key ideas

• With deterministic interest rates, forward and futures prices generally agree in standard models.
• Stochastic rates can change the relationship between forward and futures prices.
• Contract prices converge to the underlying at expiry in idealized settings.
• Settlement differences and trading costs can leave a basis near expiration.
• No-arbitrage typically constrains prices to a range rather than forcing equality in every market setting.

Tags

# Futures and Forwards in Relation to No-Arbitrage Axiom


# Futures and Forwards in Relation to No-Arbitrage Axiom












Is it possible to make an arbitrage profit by taking a long position in the futures contract and a short position in the forward contract when Forward Contract F(0,0) > Futures Contract G(0,0)? That is, the forward contract and futures contract at time = 0 and state = 0.

I believe it has to do with the fact that under the No-Arbitrage Axiom, we must have that F(N,j) = S(N,j) (foward price equal to asset price) and G(N,j) = S(N,j) (futures price equal to asset price) when the interest rates are deterministic, but what if it isn't?

## Answer by river_rat (score 1)

https://quant.stackexchange.com/a/58893

Forwards and futures only need to agree on price in a world with deterministic interest rates as a general rule (it is possible to cook up examples of random rate models where the correlation between the underlying and the rate process force the same relationship but they are rather contrived). That covers the general case but your questions revolves around the expiry process and that can be different even if rates are deterministic due to market microstructure issues. For example, the forward may be physically settled while the future is cash settled. This would allow a basis to continue into expiry as anyone trying to arbitrage it would be forced to open and close spot positions at a cost to themselves.

In general arbitrage conditions puts bounds on price processes, so there is some range that a forward and future need to coincide in at expiry but it is only a single point in idealized mathematical models.

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.