Inferring SOFR Futures Convexity from Futures and Swap Curves
Summary
The document explains a simplified way to estimate the adjustment between short-term interest rate futures and cash rates. The quoted approach compares a futures-implied rate with a reference rate derived from a swap curve; the observed difference is treated as a combined convexity adjustment and futures-swap basis. The example provides SOFR futures settlement prices, but does not work through a numerical curve calculation.
The distinction matters because the market-observed spread is not a pure measure of convexity. It can also reflect liquidity, supply and demand, and differences in clearing arrangements. Some basis components may be observable through cross-clearing prices, while others remain latent. Recovering true convexity requires modeling the relevant rate volatility, and the task is more complicated for multi-curve markets such as Euribor. A convexity trade also carries basis exposure unless its payoff can be replicated with swaptions.
Key ideas
- Comparing futures-implied rates with swap-curve rates gives an observable combined convexity and basis spread.
- The observed spread does not isolate true convexity from futures-swap basis.
- Liquidity, supply and demand, and clearing differences can contribute to the basis.
- Estimating true convexity requires a volatility model, with added complexity in multi-curve markets.
- A convexity position generally retains basis exposure unless replicated with swaptions.
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Full text
# Model-free convexity adjustment
# Model-free convexity adjustment
I have the settlement prices of 3-month SOFR IMM futures and I'm trying to compute the forward curve to replicate FactSet's results, but I have trouble understanding how they do the convexity adjustment. According to their white paper,
> At FactSet, we are using a simplified model-free approach to infer the convexity adjustment from the Future market price where $ConvexAdj(t) = 100 − f(t,S,T) − P_{market}(t)$. Notice that the convexity adjustment here contains both the true convexity adjustment and future/cash basis spread, which is why this is a simplified approach.
```
expiration, settlement
01/24, 94.6875
02/24, 94.7800
03/24, 94.9250
04/24, 95.0600
05/24, 95.2150
06/24, 95.3550
09/24, 95.7500
12/24, 96.1000
03/25, 96.3900
06/25, 96.5900
09/25, 96.7000
12/25, 96.7350
```
I have $P_{market}(t)$ but this still doesn't explain to me how I can get $ConvexAdj(t)$ term in a model-free manner. How are they getting the convexity adjustment?
## Answer by Attack68 (score 6, accepted)
https://quant.stackexchange.com/a/77952
The convexity adjustment that is referred to here is the difference between the rate implied on a period by the STIR futures market and the Interest rate swap market. This is observable (in a model free sense) becuase you can derive a curve based on futures and a curve based on swaps and compare the two rates (this is the formula presented).
It is mentioned that this contains the true convexity adjustment and the futures/swaps basis becuase in real markets there may be other reasons why these products deviate in price. This is typically caused by supply/demand imbalances driven by low liquidity or by different jurisdictions, e.g. swaps predominantly clearing in one clearing house and the futures settling on another exchange without margin netting.
It may be possible to observe a part of the futures/swaps basis relevant to cross clearing margin by analysing the swaps cross-clearing house basis prices, but some elements of it, such as that caused by low liquidity or general supply demand will not be observable and thus must be considered a latent variable.
The true convexity adjustment can only be obtained by modelling the appropriate volatility, which is difficult to do. A bit easier for SOFR which has only one curve, and more difficult for Euriobor which has an IBOR forecasting and RFR discounting curve.
Note that you are not able to trade true convexity since any convexity trade will always also include some element of the future/swap basis as part of the transaction, unless you can exactly replicate it with swaptions, in which case the price of the true convexity is equal to the value of that set of swaptions.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.