Topological Data Analysis for Financial Markets
Summary
The document surveys ways topological methods have been applied to finance and points readers toward tools and research. It introduces Mapper as a way to turn data into simplicial complexes for visualization, and persistent homology as a method used to study the shape of financial time series. The examples include research on crash landscapes, early warning signals in Bitcoin, and forecasting adverse market regimes. It also mentions a book on noncommutative geometry and stochastic calculus, a paper representing stocks as braids and knots, and software for exploring topology and visualizing networks.
The evidence offered is a list of references and reported applications, not a comparative evaluation or a trading test. The author notes that topological methods appear especially promising for detecting crashes, but the document gives no performance figures or details about validation, data choices, or implementation. It is therefore a starting point for further study rather than evidence that these methods produce reliable signals or profitable strategies.
Key ideas
- Topological data analysis can represent financial data through structures such as simplicial complexes.
- Persistent homology has been used to study shapes and changes in financial time series.
- Several cited studies apply topological methods to crash detection and adverse market regimes.
- Mapper and network visualization software offer ways to explore topological data structures.
- The document provides research leads but no comparative evidence of trading performance.
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Full text
# Topological methods in finance # Topological methods in finance Recently a promising start-up (Ayasdi) has made headlines. They are a spin-off of the Applied and Computational Algebraic Topology group of Stanford University (ComTop). What they basically do is visualizing the topological structure of Big Data. Now I know that this idea of applied (algebraic) topology crops up from time to time also in finance, see e.g. this blog post from Quantivity: Manifold learning. There is even an R-package with which you can play around (the concept behind it is persistent homology which is a robust variant of the topological invariant): Phom. My question I am interested in applications of topological methods in finance, i.e. references (books, papers, articles) and software applications, that you can use to do some tests on your own. ## Answer by Alexey Kalmykov (score 6, accepted) https://quant.stackexchange.com/a/7115 Check Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance ## Answer by Ovidiu (score 2) https://quant.stackexchange.com/a/11126 Stocks in the market can be twisted in braids and knots according to this paper http://arxiv.org/abs/1404.6637 Is a direct way to apply topology in finance. ## Answer by chad39 (score 1) https://quant.stackexchange.com/a/22553 You might want to check into Python Mapper, a Python module written by some of the founders of Ayasdi. It can be used to generate simplicial complexes which can be used to construct visualizations like those at Ayasdi. I ended using Gephi, a network visualization software, to visualize the 1-simplexes generated from Mapper. As an example, the image below is the result of running mapper on a point cloud of data sampled from a torus. ## Answer by William Wu (score 1) https://quant.stackexchange.com/a/71616 There are several relatively recent papers that use topological methods (specifically persistent homology) to study financial time series. - Gidea, Marian, and Yuri Katz. “Topological data analysis of financial time series: Landscapes of crashes.” Physica A: Statistical Mechanics and its Applications 491 (2018): 820-834. (arXiv) - Ismail, Mohd Sabri, Saiful Izzuan Hussain, and Mohd Salmi Md Noorani. "Detecting early warning signals of major financial crashes in bitcoin using persistent homology." IEEE Access 8 (2020): 202042-202057. (IEEE) - Baitinger, Eduard, and Samuel Flegel. "The better turbulence index? Forecasting adverse financial markets regimes with persistent homology." Financial Markets and Portfolio Management 35.3 (2021): 277-308. (ResearchGate) An observation is that topology seems to be particularly effective at detecting financial crashes. Being very interested in this topic, I am also actively compiling a list on my personal website: https://blog.nus.edu.sg/wuchengyuan/topology-in-finance/
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