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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.13311 |
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| _version_ | 1866913032867676160 |
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| author | Santana, Gabriel Ramirez, Jemirson |
| author_facet | Santana, Gabriel Ramirez, Jemirson |
| contents | Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a rigorous mathematical framework utilizing Topological Data Analysis (TDA) to quantify risk as the structural instability of the reconstructed phase space. By applying Takens' Delay Embedding Theorem to cryptocurrency log-returns, we generate a point cloud representation of the underlying attractor. We analyze the evolution of the filtration of Vietoris-Rips complexes to compute persistent homology groups $H_k$. We define a "Topological Persistence Norm" to characterize market regimes and propose a leverage calibration heuristic based on the persistence of 1-dimensional cycles. This approach provides a coordinate-free, stability-invariant metric for risk assessment that is robust to high-frequency noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13311 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological Complexity and Phase Space Stability: A Persistent Homology Approach to Cryptocurrency Risk Santana, Gabriel Ramirez, Jemirson General Topology Computational Finance Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a rigorous mathematical framework utilizing Topological Data Analysis (TDA) to quantify risk as the structural instability of the reconstructed phase space. By applying Takens' Delay Embedding Theorem to cryptocurrency log-returns, we generate a point cloud representation of the underlying attractor. We analyze the evolution of the filtration of Vietoris-Rips complexes to compute persistent homology groups $H_k$. We define a "Topological Persistence Norm" to characterize market regimes and propose a leverage calibration heuristic based on the persistence of 1-dimensional cycles. This approach provides a coordinate-free, stability-invariant metric for risk assessment that is robust to high-frequency noise. |
| title | Topological Complexity and Phase Space Stability: A Persistent Homology Approach to Cryptocurrency Risk |
| topic | General Topology Computational Finance |
| url | https://arxiv.org/abs/2604.13311 |