Graph Theory Applications in Finance and Portfolio Construction
Summary
The document surveys several ways discrete mathematics and graph theory appear in quantitative finance. Network analysis can model links among banks and trace how credit stress spreads, examine industry concentration and stock returns, and study how customer-supplier relationships affect asset pricing. These examples show that graph methods can describe economic connections that ordinary security-level models may miss.
Other applications include detecting triangular arbitrage as a negative-cycle search, allocating regulatory capital with combinatorial methods such as the Shapley value, and representing capital or collateral flows as networks. It also notes game theory in market making and algorithmic trading. For portfolio construction, hierarchical risk parity uses graph structure and machine learning on an asset correlation matrix; unlike some traditional risk-based allocations, it does not require that matrix to be invertible. The document is an overview assembled from brief examples and citations, not a tutorial or comparative empirical study, so it provides little detail about implementation or performance.
Key ideas
- Financial networks can represent connections among banks and help analyze credit contagion.
- Industry and customer-supplier networks can have implications for stock returns and asset pricing.
- Triangular arbitrage can be formulated as a negative-cycle detection problem in a graph.
- Shapley-value methods provide a combinatorial approach to allocating regulatory capital.
- Hierarchical risk parity uses graph structure and correlation estimates without requiring an invertible covariance matrix.
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Full text
# Examples of discrete math and graph theory within quantitative finance # Examples of discrete math and graph theory within quantitative finance The Wikipedia article on quants mentions discrete mathematics as a possible piece of their mathematical background. Are there good examples of problems within quantitative finance that are heavily combinatorial or discrete in nature? Bonus points for problems involving graph theory, since that is a crucial subarea of discrete math, but I've never encountered it in this context. ## Answer by skoestlmeier (score 5, accepted) https://quant.stackexchange.com/a/43667 There is a huge strand of literature on graph theory in finance which analyzes networks summarized here: > Allen, F., and A. Babus (2009): “Networks in Finance,” in Network-based Strategies and Competencies, ed. by P. Kleindorfer, and J. Wind, pp. 367–382. First applications of graph theory in networks focused on credit-risk in interconnected banks and how the spill-over spreads through the network (e.g. Acemoglu et al. (2013)). Furthermore, first papers also used networks to analyze concentrated industries and their implications for stock returns (e.g. Hou/Robinson (2006)). Besides this micro-foundation and financial contagion, asset pricing implications from customer-supplier networks are described in Herskovic (2018). ## Answer by Bob Jansen (score 4) https://quant.stackexchange.com/a/43681 A famous example of using graph theory in finance is the detection of triangle arbitrage by finding a negative cycle in a graph. More on this problem and the solution on Math.SE. ## Answer by Attack68 (score 2) https://quant.stackexchange.com/a/43666 - The allocation of regulatory capital to a set of trades that makes up the portfolios and sub portfolios of a bank. One theory is Shapley value which is combinatorial index, but complexity runs far deeper. - Encryption one might consider inherent to a lot of finance. - I suspect one might consider graph theory in terms of money flows / capital flows from different geographic regions to another or different funds to another. Whilst visibility of this is limited the potential value of assigning structural importance to graphs of this nature, particularly of collateral flows is probably significant and valuable if interpreted. E.g. "OFR Working Paper: A map of Collateral Uses and Flows" - What about game theory? Its application to market making and of algorithmic trading I suspect is prevalent. I have certainly used it. My answers vague and speculative but I do not perceive it unusual to ask for a requirement to some other forms of math other than financial calculus, linear algebra and statistics. ## Answer by develarist (score 1) https://quant.stackexchange.com/a/59112 The Hierarchical risk parity (HRP) portfolio, introduced by Lopez de Prado (2016), applies graph theory and machine learning to build a diversified portfolio. Like the traditional risk based allocation methods, HRP is also a function of the estimate of the covariance matrix, but it doesn't require its invertibility. Complete graph and tree graph figure, regarding asset return correlation matrices, taken from his textbook Advances in Financial Machine Learning of the same article:
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