A Pseudo-Boolean Approach to the Market Graph Analysis by Means of the p-Median Model

In the course of recent 10 years algorithms and technologies for network structure analysis have been applied to financial markets among other approaches. The first step of such an analysis is to describe the considered financial market via the correlation matrix of stocks prices over a certain period of time. The second step is to build a graph in which vertices represent stocks and edge weights represent correlation coefficients between the corresponding stocks. In this paper we suggest a new method of analyzing stock markets based on dividing a market into several substructures (called stars) in which all stocks are strongly correlated with a leading (central, median) stock. Our method is based on the p-median model a feasible solution to which is represented by a collection of stars. Our reduction of the adjusted p-Median Problem to the Mixed Boolean pseudo-Boolean Linear Programming Problem is able to find an exact optimal solution for markets with at most 1,000 stocks by means of general purpose solvers like CPLEX. We have designed and implemented a high-quality greedy-type heuristic for large-sized (many thousands of stocks) markets. We observed an important “median nesting” property of returned solutions: the p leading stocks, or medians, of the stars are repeated in the solution for p + 1 stars. Moreover, many leading stocks (medians), for example, in the USA stock market are the well-known market indices and funds such as the Dow Jones, S&P which form the largest stars (clusters).