Topological Complexity and Phase Space Stability: A Persistent Homology Approach to Cryptocurrency Risk

TL;DR

This paper presents a novel risk measurement framework for cryptocurrencies using Topological Data Analysis, capturing market dynamics' geometric structure through persistent homology and phase space stability.

Contribution

It introduces a topological approach to quantify market risk by analyzing the stability of reconstructed phase spaces with persistent homology, offering a coordinate-free metric.

Findings

The method captures market regime changes via topological persistence.

The approach is robust to high-frequency noise in financial data.

A leverage heuristic based on topological cycles improves risk calibration.

Abstract

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.