# Is the forward price equal to the future price?

If $$f^{T_1}(t)$$ is the price of a forward and $$F^{T_1}(t)$$ is the price of a future on some stock, both maturing at date $$T_1$$ and with the assumptions:

• no dividend
• constant interest rates
• no arbitrage

My notes say that the "constant interest rate" assumption implies the following:

$$f^{T_1}(t) = F^{T_1}(t)$$

I have 2 questions:

1. Isn't the above equality wrong due to counterparty risk? That is, we need a "no credit risk" assumption for the above equality to hold.

2. Even with "no credit risk" assumption, would the above equality hold?

• Yes agree, in practice you would expect a basis between the two, but textbooks can show the two are equal under certain assumptions Jun 3 '20 at 17:15
• Yes we need, and implicitly we are making, a "no credit risk" assumption in the above. We could justify it by saying that precautions (collateralization, screening of counterparties, existence of a futures clearing house) are in place to reduce or eliminate credit risk. Jun 3 '20 at 17:16
• And if we wanted to go down the rabbit hole, we could get into the specifics of the two parties to the forward and the correlation of the forward price wrt the creditworthiness of each... Jun 3 '20 at 18:50

$$1.)$$ Yes, you are right. The equation you mention in your question would only hold when you make certain assumptions. These come in the form of Illiquidity Risk and Counterparty risk. Because a forward contract is so customised, it can be hard to get out of at a fair price as they are traded OTC. Additionally, it could be cheaper for the counterparty to 'see you in court' rather than actually fulfill the obligation of that contract.

$$2.)$$ This where the No-arbitrage requirement comes in to play, I'll outline why this is relevant.

There are two ways to procure an asset for date $$T$$ delivery.

1. Buy a future or forward contract with $$T$$ years to delivery.
2. Buy the underlying assset and store for $$T$$ years.

In order for the no arbitrage requirement to hold, these two methods must equal each other. I'm sure the equation is described in your textbook (if it needs explaining let me know and I'll edit my answer). But the equation has the following symbols

• $$S_{0}$$: Spot price at time $$T_{0}$$
• $$F_{0}$$: Forward price agreed on at time $$T_{0}$$
• $$H_{0}$$: Futures price agreed on at time $$T_{0}$$
• $$r$$: Risk free rate
• $$FV$$: Net storage costs

As we could use either a futures or forward contract to fulfilll the requirement, this derives the equation $$H_{0} \simeq F_{0} = S_{0}e^{rT} + FV$$

Now that I've outlined the equation above and the importance of the no-arbitrage requirement, the emphasis is particularly on the $$\simeq$$ symbol. The $$\simeq$$ symbol implies the assumptions made regarding Counterparty risk and Illiquidity risk and hence, they are not properly encapsulated and can only be approximated. For the sake of textbook purposes, it makes these assumptions and does not take this into account. I hope this helps.