Graph theory trading

Graph Theory and Finance in Mathematica—Wolfram Blog

The asset values within a well-diversified portfolio do not move up and down in perfect synchrony. A simple way to explore diversification within the stock market is to invest in stocks from different sectors or different geographic regions. Beyond stocks, investors can consider diversification in different asset classes such as bonds, commodities, or real estate.

Cumulative return shows how much an investment changes over time. The numerical correlation is nearly 1, an indication that their returns track each other extremely well.

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Below, I have selected three asset classes—stock, commodity, and fixed income. The correlation matrix shows how pairs of assets are related: a value of 1 indicates that the corresponding pair of assets go up and down in perfect synchronization, tic- tac- toe for binary options value of 0 indicates there is no relationship between their fluctuations, and a value of -1 indicates that when one goes up, the other graph theory trading down by the same amount.

Absent the visual clues that are often highlighted by graphical images, it is easy for the analyst to overlook important changes in relationships. DOW 30 Index Member Stocks Correlation Graph In this example I have selected a universe of the Dow 30 stocks, together with a sample of commodities and bonds and compiled a database of daily returns over the period from Jan to Dec

A matrix is helpful in that it provides precise information. However, once we expand our search to a wider investment universe, a large matrix will be cumbersome to interpret.

Can we visually represent this correlation information?

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Graph theory provides a good solution. To illustrate, I first define matrix m: In this matrix, 1 means a pair of vertices is connected, while 0 means that it is not.

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An arrow connects the two nodes of the corresponding graph exactly when there is a 1 in the corresponding location of the adjacency matrix. In matrix m above, vertex 1 reading from the column of matrix and vertex 2 reading from the row of matrix are connected. Correspondingly, an arrow is drawn from vertex 1 to vertex 2 in the adjacency graph.

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Stepping through the rows and columns of matrix m, the adjacency graph of matrix m is completed. To visualize relationships between returns in the portfolio above as a graph, I start with the correlation matrix.

Since the correlation matrix is already in a matrix form, all I need to do is to turn it into something that can be represented by AdjacencyGraph.

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To do so, we can first define a threshold below which the entries in the correlation matrix will be 0 and above which the entries will be 1. Here I define the threshold to be 0. From those thresholded correlations, AdjacencyGraph yields a graph. I have first defined a bigger portfolio. It consists of all the members of the Dow Jones, a few commodities, and a few bonds. I am only interested in the correlation of returns since the beginning of this year.

In the graph below, I have chosen a correlation coefficient threshold of 0. If the return correlation between two assets is below 0. Thus, a pair of investments is connected if they are highly correlated. As we can see, there are some distinct features of this graph.

Graph Theory and Finance in Mathematica—Wolfram Blog

First of all, there are many assets whose returns are not strongly correlated with any other asset. They are the individual components within the graph. Secondly, there are five connected graphs. We can display those connected subgraphs for a graph theory trading inspection.

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The first subgraph consists of a few members of the Dow Jones Industrials. The second subgraph consists of members of a few bond funds.

Marjan Orang Abstract Efficient Market Hypothesis EMH is the cornerstone of the modern financial theory and it states that it is impossible to predict the price of any stock using any trend, fundamental or technical analysis.

The third and forth subgraphs are connected graphs with two vertices each. These findings make sense. Traditionally, asset allocation between equity and bond provides a good diversification graph theory trading. In the subgraphs, equities are indeed separated from bonds.

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Gold and silver are separated from the rest of the commodities. One of the immediate analyses we can perform on the subgraph is to find out which groups of stocks tend to move in sync. To find subgroups of stocks in which every pair of stocks is connected, I can ask for the maximum clique within g1 using FindClique. The implication is that since the beginning of the year, those stocks have all tended to move in sync with each other. We can verify this claim from the cumulative return plots below.

However, for diversification purposes, we are looking for the assets whose returns are not highly correlated and therefore have correlation coefficients that are below graph theory trading preset threshold value.

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In a graph representation, there will be no connection between these diversified assets. Without a connection to other vertices, the diversified assets turn out to be the vertices that are never incident to the same edge. There are many such independent sets. To include as many as assets as possible, we can use FindIndependentVertexSet, which finds a maximum number of independent vertices. Graph theory has helped to determine which asset classes are highly correlated with one another and which are not.

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From the graph representation, the relationships between asset classes can be easily seen, the assets that are highly correlated with others can be quickly identified, and diversification between assets can be understood more intuitively.

We can certainly expand this type of analysis to include more asset classes and a longer or different time period. You can find all the necessary code in this post so you may explore more of the finance web.

A big thanks goes to our graph theory developer Charles Pooh for all the very helpful discussions on this blog post.

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