# 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.

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).

• Very nice, thank you! I decided to accept your answer, since it seems to guide me to the relevant literature as well.
– Juho
Jan 24, 2019 at 11:46

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.

• 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.

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: 