Dynamic Tail Copulas
Jun 27, 2025
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition explores how time-varying copulas are being integrated into portfolio frameworks to better capture asymmetric co-movement and stress dynamics. The research combines structured dependence models with regime filters, macro-volatility layers and robust distributional assumptions, leading to more adaptive allocation and tail-aware risk control.
Multivariate models of commodity futures markets: a dynamic copula approach
Sihong Chen, Qi Li, Qiaoyu Wang & Yu Yvette Zhang
We apply flexible multivariate dynamic models to capture the dependence structure of various US commodity futures across different sectors between 2004 and 2022; particular attention is paid to the 2008 financial crisis and the COVID- 19 pandemic. Our copula-based models allow for time- varying nonlinear and asymmetric dependence by integrating elliptical and skewed copulas with dynamic conditional correlation (DCC) and block dynamic equicorrelation (Block DECO). Flexible copula models that allow for multivariate asymmetry and tail dependence are found to provide the best performance in characterizing co- movements of commodity returns. We also find that the connectedness between commodities has dramatically increased during the financial distress and the COVID-19 pandemic. The impacts of the financial crisis appear to be more persistent than those of the pandemic. We apply our models to some risk management tasks in the commodity markets. Our results suggest that optimal portfolio weights based on dynamic copulas have persistently outperformed the equal-weighted portfolio, demonstrating the practicality and usefulness of our proposed models.
Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity
Zhengyang Fan, Ran Ji & Miguel A. Lejeune
We investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.
Dynamic asymmetric tail dependence structure among multi-asset classes for portfolio management: Dynamic skew- t copula approach
Kakeru Ito & Toshinao Yoshiba
This study proposes AC dynamic skew-𝑡 copula with cDCC model to capture the dynamic asymmetric tail dependence structure among multi-asset classes (government bonds, corporate bonds, equities, and REITs). We provide new evidence that lower tail dependence coefficients increased compared to upper ones for all pairs in the COVID-19 crash and the recent high inflation period, indicating that the diversification effect through multi-asset investment decreased. Our empirical analysis also shows that in terms of AIC and BIC, dynamic AC skew-𝑡 copula fits data of multi- asset classes better than other dynamic elliptical copulas because it can consider the above dependence structure characteristics. Furthermore, out-of-sample analysis reveals that considering an asymmetry of tail dependence structure at each point with an AC dynamic skew-𝑡 copula enhances expected shortfall (ES) estimation accuracy and the performance of a minimum ES portfolio. These results indicate that capturing dynamic asymmetric tail dependence is crucial for multi-asset portfolio management.
GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets
Can-Zhong Yao & Min-Jian Li
This study proposes a generalized autoregressive conditional heteroskedasticity (GARCH)-mixed data sampling (MIDAS)-generalized autoregressive score (GAS)- copula model to calculate conditional value at risk (CoVaR). Our approach leverages the GARCH-MIDAS model to enhance stock market volatility modeling and incorporates the GAS mechanism to create a copula with dynamic parameters. This approach allows for the precise calculation of both CoVaR and its changes over time (delta CoVaR). The results of our study demonstrate a significant improvement in CoVaR calculation accuracy compared to other models, showcasing the effectiveness of the GARCH-MIDAS-GAS- copula model. In addition, the CoVaR indicator provides a more comprehensive view of risk spillover relationships compared to value at risk (VaR), offering deeper insights into the asymmetrical risk transmission dynamics between the Chinese and US stock markets, providing valuable information for risk management and investment decisions.
Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation
Hitomi Mori & Koichi Miyamoto
Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.
Dynamic allocation: extremes, tail dependence, and regime Shifts
Yin Luo, Sheng Wang & Javed Jussa
By capturing outliers, volatility clustering, and tail dependence in the asset return distribution, we build a sophisticated model to predict the downside risk of the global financial market. We further develop a dynamic regime switching model that can forecast real-time risk regime of the market. Our GARCH-DCC-Copula risk model can significantly improve both risk- and alpha-based global tactical asset allocation strategies. Our risk regime has strong predictive power of quantitative equity factor performance, which can help equity investors to build better factor models and asset allocation managers to construct more efficient risk premia portfolios.
References
Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity. October 2024. Fan, Z., Ji, R. and Lejeune, M.A. J Optim Theory Appl 203, 2870–2907 (2024). Available at Springer Nature Link: https://doi.org/10.1007/s10957-024-02550-y
Dynamic allocation: extremes, tail dependence, and regime Shifts. June 2025. Luo, Y., Wang,S. and Jussa, J. Available at arXiv: https://doi.org/10.48550/arXiv.2506.12587
Dynamic asymmetric tail dependence structure among multi-asset classes for portfolio management: Dynamic skew-t copula approach. January 2025. Ito, K. and Yoshiba, T. International Review of Economics & Finance, 97 (2025). Available at ScienceDirect:
GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets. March 2023. Yao, C. and Li, M. The North American Journal of Economics and Finance, 66 (2023). Available at ScienceDirect: https://doi.org/10.1016/j.najef.2023.101910
Multivariate models of commodity futures markets: a dynamic copula approach. February 2023. Chen, S., Li, Q., Wang, Q. and Zhang Y.Y. Empir Econ 64, 3037–3057 (2023). Available at Springer Nature Link: https://doi.org/10.1007/s00181-023-02373-2
Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation. January 2025. Mori, H. and Koichi, M. Available at arXiv: https://doi.org/10.48550/arXiv.2404.10624
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition explores how time-varying copulas are being integrated into portfolio frameworks to better capture asymmetric co-movement and stress dynamics. The research combines structured dependence models with regime filters, macro-volatility layers and robust distributional assumptions, leading to more adaptive allocation and tail-aware risk control.
Multivariate models of commodity futures markets: a dynamic copula approach
Sihong Chen, Qi Li, Qiaoyu Wang & Yu Yvette Zhang
We apply flexible multivariate dynamic models to capture the dependence structure of various US commodity futures across different sectors between 2004 and 2022; particular attention is paid to the 2008 financial crisis and the COVID- 19 pandemic. Our copula-based models allow for time- varying nonlinear and asymmetric dependence by integrating elliptical and skewed copulas with dynamic conditional correlation (DCC) and block dynamic equicorrelation (Block DECO). Flexible copula models that allow for multivariate asymmetry and tail dependence are found to provide the best performance in characterizing co- movements of commodity returns. We also find that the connectedness between commodities has dramatically increased during the financial distress and the COVID-19 pandemic. The impacts of the financial crisis appear to be more persistent than those of the pandemic. We apply our models to some risk management tasks in the commodity markets. Our results suggest that optimal portfolio weights based on dynamic copulas have persistently outperformed the equal-weighted portfolio, demonstrating the practicality and usefulness of our proposed models.
Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity
Zhengyang Fan, Ran Ji & Miguel A. Lejeune
We investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.
Dynamic asymmetric tail dependence structure among multi-asset classes for portfolio management: Dynamic skew- t copula approach
Kakeru Ito & Toshinao Yoshiba
This study proposes AC dynamic skew-𝑡 copula with cDCC model to capture the dynamic asymmetric tail dependence structure among multi-asset classes (government bonds, corporate bonds, equities, and REITs). We provide new evidence that lower tail dependence coefficients increased compared to upper ones for all pairs in the COVID-19 crash and the recent high inflation period, indicating that the diversification effect through multi-asset investment decreased. Our empirical analysis also shows that in terms of AIC and BIC, dynamic AC skew-𝑡 copula fits data of multi- asset classes better than other dynamic elliptical copulas because it can consider the above dependence structure characteristics. Furthermore, out-of-sample analysis reveals that considering an asymmetry of tail dependence structure at each point with an AC dynamic skew-𝑡 copula enhances expected shortfall (ES) estimation accuracy and the performance of a minimum ES portfolio. These results indicate that capturing dynamic asymmetric tail dependence is crucial for multi-asset portfolio management.
GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets
Can-Zhong Yao & Min-Jian Li
This study proposes a generalized autoregressive conditional heteroskedasticity (GARCH)-mixed data sampling (MIDAS)-generalized autoregressive score (GAS)- copula model to calculate conditional value at risk (CoVaR). Our approach leverages the GARCH-MIDAS model to enhance stock market volatility modeling and incorporates the GAS mechanism to create a copula with dynamic parameters. This approach allows for the precise calculation of both CoVaR and its changes over time (delta CoVaR). The results of our study demonstrate a significant improvement in CoVaR calculation accuracy compared to other models, showcasing the effectiveness of the GARCH-MIDAS-GAS- copula model. In addition, the CoVaR indicator provides a more comprehensive view of risk spillover relationships compared to value at risk (VaR), offering deeper insights into the asymmetrical risk transmission dynamics between the Chinese and US stock markets, providing valuable information for risk management and investment decisions.
Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation
Hitomi Mori & Koichi Miyamoto
Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.
Dynamic allocation: extremes, tail dependence, and regime Shifts
Yin Luo, Sheng Wang & Javed Jussa
By capturing outliers, volatility clustering, and tail dependence in the asset return distribution, we build a sophisticated model to predict the downside risk of the global financial market. We further develop a dynamic regime switching model that can forecast real-time risk regime of the market. Our GARCH-DCC-Copula risk model can significantly improve both risk- and alpha-based global tactical asset allocation strategies. Our risk regime has strong predictive power of quantitative equity factor performance, which can help equity investors to build better factor models and asset allocation managers to construct more efficient risk premia portfolios.
References
Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity. October 2024. Fan, Z., Ji, R. and Lejeune, M.A. J Optim Theory Appl 203, 2870–2907 (2024). Available at Springer Nature Link: https://doi.org/10.1007/s10957-024-02550-y
Dynamic allocation: extremes, tail dependence, and regime Shifts. June 2025. Luo, Y., Wang,S. and Jussa, J. Available at arXiv: https://doi.org/10.48550/arXiv.2506.12587
Dynamic asymmetric tail dependence structure among multi-asset classes for portfolio management: Dynamic skew-t copula approach. January 2025. Ito, K. and Yoshiba, T. International Review of Economics & Finance, 97 (2025). Available at ScienceDirect:
GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets. March 2023. Yao, C. and Li, M. The North American Journal of Economics and Finance, 66 (2023). Available at ScienceDirect: https://doi.org/10.1016/j.najef.2023.101910
Multivariate models of commodity futures markets: a dynamic copula approach. February 2023. Chen, S., Li, Q., Wang, Q. and Zhang Y.Y. Empir Econ 64, 3037–3057 (2023). Available at Springer Nature Link: https://doi.org/10.1007/s00181-023-02373-2
Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation. January 2025. Mori, H. and Koichi, M. Available at arXiv: https://doi.org/10.48550/arXiv.2404.10624
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition explores how time-varying copulas are being integrated into portfolio frameworks to better capture asymmetric co-movement and stress dynamics. The research combines structured dependence models with regime filters, macro-volatility layers and robust distributional assumptions, leading to more adaptive allocation and tail-aware risk control.
Multivariate models of commodity futures markets: a dynamic copula approach
Sihong Chen, Qi Li, Qiaoyu Wang & Yu Yvette Zhang
We apply flexible multivariate dynamic models to capture the dependence structure of various US commodity futures across different sectors between 2004 and 2022; particular attention is paid to the 2008 financial crisis and the COVID- 19 pandemic. Our copula-based models allow for time- varying nonlinear and asymmetric dependence by integrating elliptical and skewed copulas with dynamic conditional correlation (DCC) and block dynamic equicorrelation (Block DECO). Flexible copula models that allow for multivariate asymmetry and tail dependence are found to provide the best performance in characterizing co- movements of commodity returns. We also find that the connectedness between commodities has dramatically increased during the financial distress and the COVID-19 pandemic. The impacts of the financial crisis appear to be more persistent than those of the pandemic. We apply our models to some risk management tasks in the commodity markets. Our results suggest that optimal portfolio weights based on dynamic copulas have persistently outperformed the equal-weighted portfolio, demonstrating the practicality and usefulness of our proposed models.
Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity
Zhengyang Fan, Ran Ji & Miguel A. Lejeune
We investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.
Dynamic asymmetric tail dependence structure among multi-asset classes for portfolio management: Dynamic skew- t copula approach
Kakeru Ito & Toshinao Yoshiba
This study proposes AC dynamic skew-𝑡 copula with cDCC model to capture the dynamic asymmetric tail dependence structure among multi-asset classes (government bonds, corporate bonds, equities, and REITs). We provide new evidence that lower tail dependence coefficients increased compared to upper ones for all pairs in the COVID-19 crash and the recent high inflation period, indicating that the diversification effect through multi-asset investment decreased. Our empirical analysis also shows that in terms of AIC and BIC, dynamic AC skew-𝑡 copula fits data of multi- asset classes better than other dynamic elliptical copulas because it can consider the above dependence structure characteristics. Furthermore, out-of-sample analysis reveals that considering an asymmetry of tail dependence structure at each point with an AC dynamic skew-𝑡 copula enhances expected shortfall (ES) estimation accuracy and the performance of a minimum ES portfolio. These results indicate that capturing dynamic asymmetric tail dependence is crucial for multi-asset portfolio management.
GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets
Can-Zhong Yao & Min-Jian Li
This study proposes a generalized autoregressive conditional heteroskedasticity (GARCH)-mixed data sampling (MIDAS)-generalized autoregressive score (GAS)- copula model to calculate conditional value at risk (CoVaR). Our approach leverages the GARCH-MIDAS model to enhance stock market volatility modeling and incorporates the GAS mechanism to create a copula with dynamic parameters. This approach allows for the precise calculation of both CoVaR and its changes over time (delta CoVaR). The results of our study demonstrate a significant improvement in CoVaR calculation accuracy compared to other models, showcasing the effectiveness of the GARCH-MIDAS-GAS- copula model. In addition, the CoVaR indicator provides a more comprehensive view of risk spillover relationships compared to value at risk (VaR), offering deeper insights into the asymmetrical risk transmission dynamics between the Chinese and US stock markets, providing valuable information for risk management and investment decisions.
Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation
Hitomi Mori & Koichi Miyamoto
Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.
Dynamic allocation: extremes, tail dependence, and regime Shifts
Yin Luo, Sheng Wang & Javed Jussa
By capturing outliers, volatility clustering, and tail dependence in the asset return distribution, we build a sophisticated model to predict the downside risk of the global financial market. We further develop a dynamic regime switching model that can forecast real-time risk regime of the market. Our GARCH-DCC-Copula risk model can significantly improve both risk- and alpha-based global tactical asset allocation strategies. Our risk regime has strong predictive power of quantitative equity factor performance, which can help equity investors to build better factor models and asset allocation managers to construct more efficient risk premia portfolios.
References
Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity. October 2024. Fan, Z., Ji, R. and Lejeune, M.A. J Optim Theory Appl 203, 2870–2907 (2024). Available at Springer Nature Link: https://doi.org/10.1007/s10957-024-02550-y
Dynamic allocation: extremes, tail dependence, and regime Shifts. June 2025. Luo, Y., Wang,S. and Jussa, J. Available at arXiv: https://doi.org/10.48550/arXiv.2506.12587
Dynamic asymmetric tail dependence structure among multi-asset classes for portfolio management: Dynamic skew-t copula approach. January 2025. Ito, K. and Yoshiba, T. International Review of Economics & Finance, 97 (2025). Available at ScienceDirect:
GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets. March 2023. Yao, C. and Li, M. The North American Journal of Economics and Finance, 66 (2023). Available at ScienceDirect: https://doi.org/10.1016/j.najef.2023.101910
Multivariate models of commodity futures markets: a dynamic copula approach. February 2023. Chen, S., Li, Q., Wang, Q. and Zhang Y.Y. Empir Econ 64, 3037–3057 (2023). Available at Springer Nature Link: https://doi.org/10.1007/s00181-023-02373-2
Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation. January 2025. Mori, H. and Koichi, M. Available at arXiv: https://doi.org/10.48550/arXiv.2404.10624
