Approximating Arithmetic Forwards with Weighted Overnight Forwards

Summary

This note examines why arithmetic forwards can be approximated by weighted averages of overnight forwards. It presents an expression in which each overnight forward is adjusted by a model-dependent arithmetic factor. Because those factors can be numerically stable and close to one in some market conditions, simpler approximations may reduce computational cost while retaining useful accuracy.

The analysis explores theoretical bounds and closed-form factor expressions within Gaussian Heath-Jarrow-Morton models. It also compares one derived form with an approximation proposed in prior work on valuing arithmetic averages of Fed Funds rates. The excerpt offers theoretical and model-based justification, but does not report a broad empirical validation or specify how approximation accuracy varies across market conditions; practical use therefore depends on the assumptions and scenarios under which the factors remain well behaved.

Key ideas

• An arithmetic forward can be represented using weighted overnight forwards and adjustment factors.
• Simplified factors close to one can support cheaper computational approximations in certain scenarios.
• The note derives theoretical bounds and closed forms under Gaussian HJM models.
• One resulting approximation is compared with a previously suggested Fed Funds rate method.
• The excerpt describes theoretical justification but gives no broad empirical accuracy study.

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# Note on a Theoretical Justification for Approximations of Arithmetic Forwards


# Note on a Theoretical Justification for Approximations of Arithmetic Forwards









This note explores the theoretical justification for some approximations of arithmetic forwards ($F_a$) with weighted averages of overnight (ON) forwards ($F_k$). The central equation presented in this analysis is: \begin{equation*} F_a(0;T_s,T_e)=\frac{1}{τ(T_s,T_e)}\sum_{k=1}^K τ_k \mathcal{A}_k F_k\,, \end{equation*} with $\mathcal{A}_k$ being explicit model-dependent quantities, numerically stable and close to one under certain market scenarios. We will present computationally cheaper methods that approximate $F_a$, i.e., we will define some $\{\tilde{\mathcal{A}}_k\}_{k=1}^K$ such that \begin{equation*} F_a(0;T_s,T_e)\approx \frac{1}{τ(T_s,T_e)}\sum_{k=1}^K τ_k \tilde{\mathcal{A}}_k F_k\,, \end{equation*} thereby gaining some intuition about the arithmetic factors $\mathcal{A}_k$. Additionally, theoretical bounds and closed-form expressions for the arithmetic factors $\mathcal{A}_k$ in the context of Gaussian HJM models are explored. Finally, we demonstrate that one of these forms can be closely aligned with an approximation suggested by Katsumi Takada in his work on the valuation of arithmetic averages of Fed Funds rates.

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