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}{\tau(T_s,T_e)}\sum_{k=1}^K \tau_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}{\tau(T_s,T_e)}\sum_{k=1}^K \tau_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.