This paper investigates the dynamics of interest rate term structures in seven European markets (Austria, Belgium, Britain, France, Germany, Italy, and Spain) using daily zero-coupon bond data from 2017 to 2019. As a continuation of previous studies by Liu (JSIAM Letters, 2, 57–60 2010), Liu and Mancino (JSIAM Letters, 4, 17–20 2012), and Liu and Suzuki (2024), where the integrated volatility matrices are analyzed, in contrast to previous studies, we employ the Malliavin-Mancino method to analyze the dynamics of spot volatility matrices of both spot and forward rates. This method enables robust estimation in the presence of asynchronous observations and microstructure noise inherent in high-frequency data. Our empirical findings reveal three key insights into European interest rate markets. First, we show substantial differences in factor structures between spot and forward rates: while three factors explain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$80-95\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>80</mn> <mo>-</mo> <mn>95</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> of spot rate variability across all markets, forward rates exhibit significantly more complex dynamics requiring additional factors to achieve comparable explanatory power. Second, we find notable cross-country variations between British and continental European markets. British markets demonstrate the highest degree of factor concentration in spot rates (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$>95\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>></mo> <mn>95</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> explained by the first factor), while continental European markets show more diverse factor structures, with the first factor typically explaining 80-90% of variability. Third, we identify significant time variation in factor importance, especially pronounced in forward rates, where the first factor’s explanatory power fluctuates between <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$30-70\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>30</mn> <mo>-</mo> <mn>70</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>. These findings suggest that traditional three-factor models may be inadequate for forward rate modeling and highlight the importance of incorporating market-specific characteristics and time-varying dynamics in interest rate modeling frameworks.

Application of the Malliavin-Mancino Method to an Empirical Study of the Term Structure of Spot and Forward Interest Rates in Seven European Markets

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2025年7月19日

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Nien-Lin Liu,
Ryoichi Suzuki

摘要

This paper investigates the dynamics of interest rate term structures in seven European markets (Austria, Belgium, Britain, France, Germany, Italy, and Spain) using daily zero-coupon bond data from 2017 to 2019. As a continuation of previous studies by Liu (JSIAM Letters, 2, 57–60 2010), Liu and Mancino (JSIAM Letters, 4, 17–20 2012), and Liu and Suzuki (2024), where the integrated volatility matrices are analyzed, in contrast to previous studies, we employ the Malliavin-Mancino method to analyze the dynamics of spot volatility matrices of both spot and forward rates. This method enables robust estimation in the presence of asynchronous observations and microstructure noise inherent in high-frequency data. Our empirical findings reveal three key insights into European interest rate markets. First, we show substantial differences in factor structures between spot and forward rates: while three factors explain $80-95\%$ $80 - 95 %$ of spot rate variability across all markets, forward rates exhibit significantly more complex dynamics requiring additional factors to achieve comparable explanatory power. Second, we find notable cross-country variations between British and continental European markets. British markets demonstrate the highest degree of factor concentration in spot rates ( $>95\%$ $> 95 %$ explained by the first factor), while continental European markets show more diverse factor structures, with the first factor typically explaining 80-90% of variability. Third, we identify significant time variation in factor importance, especially pronounced in forward rates, where the first factor’s explanatory power fluctuates between $30-70\%$ $30 - 70 %$ . These findings suggest that traditional three-factor models may be inadequate for forward rate modeling and highlight the importance of incorporating market-specific characteristics and time-varying dynamics in interest rate modeling frameworks.