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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.

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Check Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance

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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.

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    $\begingroup$ Interesting premise: a market metric based on Jones polynomial! I wonder if its behavior differs much from standard correlation metrics. Based on your username, the paper is your own -- it would be polite to mention that in your answer. $\endgroup$
    – Brian B
    May 1, 2014 at 13:07
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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.

enter image description here

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There are several relatively recent papers that use topological methods (specifically persistent homology) to study financial time series.

  1. 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)
  2. 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)
  3. 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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