Theoretical fair value of SOFR 1M and 3M Future contracts?

The fair value of Eurodollar future contracts is calculated using the no arbitrage pricing and the spot curve for LIBOR. How does one compute the theoretical fair value of 1M and 3M SOFR Future contracts?

(Edit 23.11.2020) [Note that my previous derivations were too hasty and had some issues, I will try to amend when time allows. In any case, note that those results were merely model-free: SOFR Futures have convexity adjustments and in practice you will need to specify a model for the forward rates to actually calculate them. Feel free to unmark as "answered".]

First off, let us recall how the CME group defines SOFR Future rates $$F$$:

• 1M SOFR future rate: "average daily SOFR interest during contract Delivery Month".
• 3M SOFR future rate: "compounded daily SOFR interest during contract Reference Quarter".

The settlement value of the Future is then equal to: $$100-F$$.

As @JanStuller explains, Futures are normally liquid instruments. They are used as building blocks for constructing interest rate curves, especially over the short-end that is maturities equal to or less than 1 year. Therefore, the Future rate is given by the market, rather than derived from a pricing model.

That being said, there might be circumstances under which you want to price a Future not observable in the market. For example, you might want to price a long-dated Future which is yet not being actively traded in the market. Another example is when computing valuation adjustments such as CVA: these require simulating Future rates at future times; you then normally simulate the interest rate curve and use a pricing model to obtain a simulated Future rate from your curve.

It is possible to derive a model-free expression for the SOFR Future rate. However bear in mind that, due to the neglect of discounting in Futures, there are convexity adjustments in the rate's calculation. To compute those adjustments you actually need a model for the rates.

• Tough to compete with a robot who looks every bit as human, but in fact possesses superior skills :). Superb answer! Nov 22 '20 at 16:10

Not sure if this will entirely answer your question, but the key concept here is that the Futures contracts are not priced via some theoretical model, but their price is entirely driven by supply and demand. In turn, the supply and demand reflect the market's expectation about future Libors (or SOFR, in case the underlying is SOFR).

Let's say that "today" you want to build a SOFR curve: the first point on this curve would be the spot SOFR rate (i.e. the SOFR rate that was published "today" in the morning by the New York Fed, reflecting "yesterday's average realized Secured Overnight Funding Rate transactions). The next point on the curve could be the 1-month point: this point would give today's market expectation of what the SOFR rate will be 1-month from today. This point could be taken from the 1-month SOFR futures contract. The next point could be the 3-month point: same thing (could be taken from the 3-month SOFR future).

So what I am trying to say is that the futures contract would be used as an input to build other curves, rather than the Futures themselves being an output of a no-arbitrage pricing model.

The value of the 1-month or the 3-month SOFR futures contract would change if suddenly many people want to short the contract (or buy the contract): that would push the contract price down (or up), rather than some pricing model.