Modeling the Theoretical Price of SOFR Swap Futures

# SOFR swap futures


# SOFR swap futures












A few questions about the price of Eris SOFR swap futures (https://www.cmegroup.com/markets/interest-rates/files/eris-sofr-swap-futures-overview.pdf) as I'm not familiar yet with these instruments.

First the main features of these futures contracts:

- The futures are listed (start trading) 9 months before the contract effective date, where the contract effective dates are quarterly IMM dates.

- The tenors of the underlying swap are 1,2,3,4,5,7,10,12,15,20,30 years.

- The (futures) contracts do not expire, but continue trading until the swap expiry date.

- The daily settlement price is $P(t) = 100 + A(t) + B(t) - C(t)$, where $A(t)$ is the NPV of future cashflows discounted at SOFR, $B(t)$ is accumulated fixed and floating payments and $C(t)$ is accumulated daily SOFR interest on fixed and floating payments, also known as Price Alignment Interest.

The questions I have are:

a. Is the initial price of the swap futures contract when it is listed 100? And how does this price correspond to the MAC IMM coupon rate, i.e. the fixed rate of the IMM (forward start) swap (https://www.cmegroup.com/trading/interest-rates/swap-futures/mac-standard.html)?

b. In the period between the date the futures is listed and the contract effective date the futures price will obviously fluctuate. My understanding is that once the MAC is published (9 months and 1 week before the contract effective date) it remains fixed. So based on the futures p/l one can deduce the mid-market / fair (fixed) rate of a forward starting swap, correct?

## Answer by Frido (score 1)

https://quant.stackexchange.com/a/83538

As there are no answers yet, and I've had some time to mull over it, I'll try to partially answer this, in particular the $1 + A(t)$ part of the futures price breakdown. Other answers are more than welcome.

I think at the end of the day it boils down to having to choose a model.

First of all let $t_0$ be the listing date of the swap futures contract, $T_0$ the (forward) start date of the swap, $T_1,\ldots, T_N$ the coupon dates and $t$ with $t_0 < t < T_0$ the valuation/pricing date of the futures contract. So we have $t_0 < t < T_0 < T_1 < \ldots < T_N$ and let $\Delta := T_{i+1} - T_i$.

I'll also assume the existence of zero coupon bonds $p(\cdot,\cdot)$. The expectation at time $t$ under the money market measure is $E_t^\mathbb Q [\cdot]$ and under the $T$ forward measure is $E_t^T [\cdot]$.

Now of course in practice the market will price the futures contract, but I'd like to know its theoretical price.

So let $R(t_0)$ be the par (forward start) swap rate at $t_0$. The value of the swap at $T_0$ will be

$$ 1 + V(T_0) = p(T_0,T_N) + R(t_0) \Delta \sum_{i=1}^N p(T_0,T_i) $$

The theoretical swap futures price is then

$$ f(t) = 1 + E_t^\mathbb Q [ V(T_0)] $$

This can be calculated if one assumes a model for the short rate / instantaneous forward rate. As far as I can see the above expression cannot be simplified any further.