A Geometric Witness Framework for Signed Multivariate Tail-Dependence Compatibility: Asymptotic Structure and Finite-Threshold Synthesis

Battle Count: 60

The paper is highly relevant to quantitative risk management, particularly for modeling signed tail dependence (both lower and upper tail co-movements) in multivariate portfolios. Applications include stress testing, scenario design, and tail-dependence compatibility for risk aggregation. The framework provides tools for calibrating extremal dependence from expert judgement or partial structural information. However, it is primarily a mathematical/constructive framework rather than a direct trading strategy tool. The signed multivariate tail dependence modeling is directly applicable to portfolio risk assessment where harmful deviations occur in both directions.

The core algorithm (triangular inversion) is straightforward linear algebra with O(3^d) complexity. However, the full framework involves: (1) combinatorial indexing of active sets and sign patterns, (2) construction of the signed incidence matrix, (3) triangular back-substitution, (4) LP formulation for partial/noisy cases, (5) canonical ray geometry simulation, and (6) finite-threshold ternary mass computation. The dimensionality curse (3^d-1 generators) limits practical application to moderate dimensions. The provided Python code handles all these components but requires understanding of the underlying combinatorial structure.

Full Python implementation provided in Appendix E, including the canonical witness sampler, LP solver, triangular inversion, and variable-threshold diagnostic helpers. Numerical experiments run in Python 3.13.5 with scipy 1.17.1 using HiGHS backend. Benchmark specifications are fully explicit with closed-form solutions. Monte Carlo validation protocol is clearly described (R=20 runs, M=5×10^5 samples).