USV-Affine Models Without Derivatives: A Bayesian Time-Series Approach

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this study, the authors used USV-ATSMs method on emerging market yield curve without derivative data. From this exercise on four-factor USV, the result implies that the volatility could be extracted from bonds alone, which brings important values for markets lack of options. This study was conducted in a solid structure and is scientifically sound with consistency in robustness checks. This result is well-of-interested to fix income researchers, emerging market economists and risk managers. Overall, it provides a meaningful empirical contribution to the relevant research field.

Besides, the literature review covers classical pieces of research (Duffee 2002; Collin-Dufresne & Goldstein 2002; Collin-Dufresne et al. 2009; Bikbov & Chernov 2004) and also includes the latest development in 2024 and 2025. Only one suggestion- it is better to provide clearer summary of what value this paper adds, to set a reasonable expectation for readers.

Line 417, there is typo. “Septermber 2024” should read “September 2024”. And line 621, it writes” suprisingly”

Also, could you please make consistent notation across the article, like sometimes short rate variance while elsewhere volatility factor? I assume these are the same but please use one term to avoid confusion.

The third point in Section 5 discusses simulated ATM volatility. Could you clarify what was done with the “simulated at-the-money bond futures implied volatility”? If this is explored, provide details; if not, consider removing that bullet. Currently in the forecasting paragraph, you clarified the future state path but without stating how to translate those into an implied volatility. Even in the later sections, they focus on GARCH and macro proxies instead.

This paper mentioned MPR (market price of risk) differences between 3-factor and 4-factor models. It would be better with some context on economic significance of higher and more variable MPR under A1(4) USV. Also, it would be very exciting as well to interplay with other macroeconomic factors, for now Table 7 has already explored the correlation between filtered volatility and FX or oil shock, a deeper discussion is welcomed.

In conclusion, I would suggest including a statement on how this study is relevant to practitioner/modelers, which could articulate the broader significance of this study.

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Overall

Overall, this paper is very well researched, extensively considering technical nuances as the model is built out. The model is developed diligently.

Issue #1 - Readability

However, there is confusion in the flow of the presentation which affects readability and research insights. It almost feels like two papers combined into one. The first flow being a general literature review of the field of study without a particular conclusion being drawn. This is exacerbated by the non-committal nature of the title, as well the approach of packing a very large number of technical points into a ‘literature review'. Each of the important technical/structural nuances that matter to the build of the model should be explicitly addressed in properly titled subsections. The current flow comes across as meandering from technical point to technical point without a sense of why and where things are going. The second flow feels like there is an assertion that includes (1) "even without derivative instruments, it is possible to identify and interpret risk-neutral dynamics and volatility risk using observable time-series data”, and (2) “Our aim is to test their ability to generate accurate cross-sectional behavior and time series dynamics of bond yields. The paper does not convey clearly the extent to which the authors believe they have accomplished this and how.

Issue #2 - Clear Unique Contribution

The contribution of this paper relative to the closest similar research needs to be made more clear. There are many strong references backing the build out of the model, but it is not made clear what other research has been done that compares closest to the overall model/approach. Overall similar studies with relative strengths and weaknesses have been incorporated. This pertains mostly to the statements noted above: (1) "even without derivative instruments, it is possible to identify and interpret risk-neutral dynamics and volatility risk using observable time-series data”, and (2) “Our aim is to test their ability to generate accurate cross-sectional behavior and time series dynamics of bond yields.

Beyond this research being more forcefully placed in its context relative to the closest similar advances in the field, there is a also general need to be more clear in the conclusive insights of this paper. That should certainly be captured in a more impactful title that conveys the progress made.

Conclusions and Applications

The analysis conducted in this study is robust, so it feels like a lost opportunity that there is not more time spent in the conclusion to lay out clear practical applications of this research in real world scenarios. If researchers and policy makers are to leverage the insights of the research, it must be made more explicit in real-world application.

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Peer Review Report

Title: Affine Term Structure Models with Unspanned Stochastic Volatility in Illiquid Derivative Markets
Authors: Malefane Molibeli and Gary van Vuuren
Journal: Journal of Risk and Financial Management

Summary and Strengths

This paper explores the empirical performance of affine term structure models (ATSMs) that incorporate unspanned stochastic volatility (USV), with a focus on markets lacking derivative data—specifically, the South African government bond market. The authors implement both A1(3) and A1(4) USV models using a Bayesian MCMC-Kalman filtering framework and assess model quality through various statistical and economic metrics.

The work is methodologically rigorous and addresses an important niche in fixed-income modeling: understanding volatility risks in markets with sparse derivative instruments.

Key Strengths:

1. Extensive and Practical Evaluation of Model Performance
   The paper presents a comprehensive empirical evaluation using:

   • In-sample and out-of-sample RMSE and bias,

   • Posterior simulation of volatility,

   • Regression of the filtered latent volatility against macro factors.
     These components form a robust and practical toolkit for testing model effectiveness, especially under market incompleteness.

2. Clear Economic Intuition in Modeling and Evaluation Choices
   The modeling choices are grounded in sound intuition. The use of A1(3) versus A1(4), the rationale for Bayesian estimation, and the incorporation of oil and FX volatility are all explained with a clear understanding of the empirical context and modeling goals.

3. Mathematical Rigor and Transparency
   The paper is technically well-constructed. The stochastic dynamics of the A1(3) and A1(4) models are clearly laid out. Risk-neutral and physical measure dynamics, and parameter restrictions for USV, are all well-documented and logically structured.

4. Relevance to Emerging Markets
   By focusing on illiquid bond markets, the paper addresses real-world modeling limitations. The use of actual South African government bond data, rather than synthetic instruments, enhances practical relevance and policy applicability.

Areas for Improvement

While the paper makes a valuable contribution, several areas require clarification or expansion:

1. Language and Notation Consistency

• Some expressions are ambiguous or notationally inconsistent:

  • Line 223: Redefining XtX_tXt​ appears redundant unless a different meaning is intended.

  • Line 238: TTT is referenced but not defined in the corresponding equation.

  • Equations (5)–(6): Inconsistencies in notation (e.g., capital vs. lowercase κ\kappaκ, missing transposes) should be corrected to match earlier definitions (e.g., Equation (1)).

  • Line 251: Redundant reappearance of θQ\theta^QθQ requires clarification.

  • Equation (13): gVg_VgV​ is introduced without definition—every symbol in all model equations must be clearly defined.

2. Model Structure and Comparative Intuition

• The structural and empirical differences between A1(3) and A1(4) warrant deeper discussion:

  • Short-term yield prediction: A1(3) underperforms A1(4) for short maturities. The paper should provide intuition on which dynamic terms (e.g., curvature, volatility interaction) explain this divergence.

  • Lines 319–333: The restricted parameters for USV are stated without intuitive explanation or citations. Parameters such as ρrθ\rho_{r\theta}ρrθ​ and ρμθ\rho_{\mu\theta}ρμθ​ are referenced but not defined in equations.

  • Line 317: The paper should articulate what structural or identifiability conditions lead a model to exhibit USV.

  • Posterior intervals: Tables 2 and 3 show that A1(4)’s posteriors are significantly tighter than A1(3)’s. This may indicate underfitting in A1(3)—some discussion is warranted.

  • Model extensibility: If A1(4) is still underfitting, could higher-dimensional ATSMs (e.g., A1(5)) offer better fit? The authors could briefly comment on this possibility.

3. Principal Component Analysis (PCA) Transparency

• The PCA analysis needs clarification:

  • What input data is PCA applied to—raw yields, changes, or returns?

  • The factors do not obviously align with level, slope, or curvature, and some loadings seem erratic. This raises concerns of overfitting or lack of interpretability.

  • The paper should justify the choice of three components and discuss the economic interpretability and robustness of PCA across subperiods.

4. MCMC Implementation and Specification

• The MCMC procedure lacks detail for full reproducibility:

  • Algorithmic details (e.g., step-by-step outline, acceptance rate, convergence diagnostics) should be included.

  • Prior distributions are not specified. Are they conjugate to the model structure? What are their assumed forms (e.g., normal, inverse-gamma)?

  • The paper does not show the full conditional or joint posterior distributions. Without this, it’s difficult to evaluate the correctness of the Bayesian implementation.

  • The authors should also comment on whether more efficient samplers (e.g., Hamiltonian Monte Carlo) could improve computational performance over Metropolis-Hastings.

5. Data and Market Context Justification

• Several assumptions around the data need clarification:

  • Line 672 claims that weekly data is preferable to high-frequency data, but this is not substantiated. Are there references or empirical benchmarks to support this?

  • Is high-frequency data available in the SA bond market? If so, has it been tried in related models?

  • External applicability: Have the authors tested the models on other markets (e.g., US or EU)? If not, do the findings generalize, or are they specific to structural features of South Africa’s fixed income market?

  • The paper should comment on the availability and limitations of options data in SA. Why are implied volatility surfaces unreliable in practice? Could the methodology be compared to US markets with deep derivative data to validate the USV identification claim?

 

Author Response

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Author Response File: Author Response.docx