Note on a Theoretical Justification for Approximations of Arithmetic Forwards

TL;DR

This paper provides a theoretical foundation for approximating arithmetic forwards using weighted averages of overnight forwards, introduces computationally efficient methods, and explores bounds and formulas within Gaussian HJM models.

Contribution

It offers a new theoretical justification for arithmetic forward approximations, introduces stable model-dependent factors, and connects these to existing valuation methods.

Findings

Explicit model-dependent factors are stable and close to one under certain scenarios.

Proposed approximations are computationally cheaper and theoretically justified.

Connections to Takada's approximation for Fed Funds rate valuations.

Abstract

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.