Pricing SOFR Futures and Understanding Convexity Adjustments

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


# 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?

## Answer by Daneel Olivaw (score 4, accepted)

https://quant.stackexchange.com/a/59522

(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.

## Answer by Frido (score 2)

https://quant.stackexchange.com/a/83673

A bit late to the party, but since I don't see formulas in the previous answers (which doesn't mean they're wrong!) I'll add those:

Let $r(t)$ be the SOFR rate.

Under a continuous time approximation you can write at time $t$ for the 1 month future, $$ \mathrm{1MFut} = \frac1\delta \mathbb E_t^\mathbb Q \left[ \int_T^{T+\delta} r(u) du \right], $$ and for the 3 month future, $$ \mathrm{3MFut} = \frac1\delta \mathbb E_t^\mathbb Q \left[ e^{\int_T^{T+\delta} r(u) du} \right], $$ where $\mathbb Q$ is the usual risk-neutral measure and $t \leq T+\delta$.

Lastly, one needs to choose a particular model to calculate the theoretical futures prices above.