Note on a Theoretical Justification for Approximations of Arithmetic Forwards

Paper: arXiv 2406.09488

Authors: Álvaro Romaniega

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}{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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of arithmetic forwards and provides closed-form approximations under the Gaussian HJM framework. However, it relies primarily on theoretical derivations and basic numerical illustrative simulations rather than extensive empirical backtesting on market data.

Research Flowchart

  flowchart TD
    A["Research Goal: Justify & approximate arithmetic forwards (F_a) using weighted ON forwards (F_k)"] --> B["Inputs: Market Data & Gaussian HJM Framework"]
    B --> C["Derive Exact Formula: F_a = (1/τ) ∑ τ_k A_k F_k with Model-Dependent Factors A_k"]
    C --> D["Develop Approximations: Define Computational Factors Ã_k ≈ A_k"]
    C --> E["Derive Bounds & Closed-Form Expressions under Gaussian HJM"]
    D --> F["Key Outcome: Cheaper, stable F_a approximations & theoretical alignment with Takada's Fed Funds model"]
    E --> F