First-Order Convexity Adjustment for SOFR Futures Under Affine Models

Summary

The document asks how to approximate the difference between a SOFR futures rate and the corresponding forward term rate in a continuous-time affine term-structure model. It writes the bond price using deterministic functions of maturity and the short rate, then expresses the futures and forward rates as expectations under the money-market and forward measures, respectively.

A first-order Taylor expansion leads to an approximate adjustment proportional to the bond-price sensitivity to the short rate and the difference between the two measures’ expected short rates at the futures fixing date. The document poses this derivation as a question rather than providing a proof, numerical example, or empirical validation. The approximation is therefore presented as a proposed quick estimate; its accuracy depends on the model and the quality of the linearization, and the document does not examine higher-order terms or specific SOFR contract conventions.

Key ideas

• The bond price in the affine model depends exponentially on the short rate through a deterministic sensitivity function.
• The futures and forward term rates use expectations under different probability measures.
• A first-order expansion approximates their difference using the gap between expected short rates under those measures.
• The proposed adjustment is a quick approximation whose accuracy is not established in the document.

Tags

# Approx. for futures / forward convexity adjustment in ATS models


# Approx. for futures / forward convexity adjustment in ATS models












I'm not an IRD expert, so bear with me. I'd like to have a quick and dirty approximation for the (SOFR) futures/forward convexity adjustment under affine term structure models in continuous time.

Under an ATS model we can write for the $T$-price of a discount bond with maturity $T+\delta$: $$ p(T,T+ \delta) = e^{A(T,T+\delta) - B(T,T+\delta) r(T)} $$ where $r(T)$ is the instantaneous short rate at time $T$ and $A,B$ are deterministic functions.

Now let $t<T$. The continuous time approximation for a SOFR future is $$ f(t,T,T+\delta) = \mathbb E_t^\mathbb Q \left[ \frac1\delta \left( \frac{1}{p(T,T+ \delta)} - 1\right) \right] $$ and the corresponding SOFR term rate is $$ F(t,T,T+\delta) = \mathbb E_t^{T+\delta} \left[ \frac1\delta \left( \frac{1}{p(T,T+ \delta)} - 1\right) \right] $$ Here $\mathbb Q$ is the money market measure and $T+\delta$ as a superscript for the expectation is the $T+\delta$ forward measure.

A first order Taylor expansion then gives for the convexity adjustment $$ F(t,T,T+\delta) - f(t,T,T+\delta) \approx \frac{B(T,T+\delta)}{\delta} \left( \mathbb E_t^{T+\delta} [r(T)] - \mathbb E_t^\mathbb Q [r(T)] \right). $$

Is this correct?

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