Incorporating Parameter Uncertainty into Copula Models: A Fuzzy Approach

This paper proposes a fuzzy copula-based optimization framework for modeling dependence structures and financial risk under parameter uncertainty. The parameters of selected copula families are represented as trapezoidal fuzzy numbers, and their α-cut intervals capture both the support and core ranges of plausible dependence values. This fuzzification transforms the estimation of copula parameters into a fuzzy optimization problem, enhancing robustness against sampling variability. The methodology is empirically applied to gold and oil futures (1 January 2015–1 January 2025), comparing symmetric copulas, i.e., Gaussian and Frank and asymmetric copulas, i.e., Clayton, Gumbel and Student-t. The results prove that the fuzzy copula framework provides richer insights than classical point estimation by explicitly expressing uncertainty in dependence measures (Kendall’s τ, Spearman’s ρ) and risk indicators (Value-at-Risk, Conditional Value-at-Risk). Rolling-window analyses reveal that fuzzy VaR and fuzzy CVaR effectively capture temporal dependence shifts and tail severity, with fuzzy CVaR consistently producing more conservative risk estimates. This study highlights the potential of fuzzy optimization and fuzzy dependence modeling as powerful tools for quantifying uncertainty and managing extreme co-movements in financial markets.