# Chapter 6 Financial Networks

## 6.1 Introduction

Financial markets can be regarded as a complex network in which nodes represent different financial assets and edges represent one or many types of relationships among those assets. Filtered correlation-based networks have successfully been used in the literature to study financial markets structure particularly from observational data derived from empirical financial time series (Bardoscia et al. 2017; S. A. L. Tumminello Michele AND Miccichè 2011; R. N. Mantegna 1999; T. Aste, Shaw, and Di Matteo 2010; Michele Tumminello, Lillo, and Mantegna 2010, M. Tumminello et al. (2005)).

The underlying principle is to use correlations from empirical financial time series to construct a sparse network representing the most relevant connections. Analyses on filtered correlation-based networks for information extraction (Song, Aste, and Di Matteo 2008; T. Aste, Shaw, and Di Matteo 2010) have widely been used to explain market interconnectedness from high-dimensional data. Applications include asset allocation (Y. Li et al. 2018; Pozzi, Di Matteo, and Aste 2013), market stability assessments (Morales et al. 2012), hierarchical structure analyses (R. N. Mantegna 1999; T. Aste, Shaw, and Di Matteo 2010; Michele Tumminello, Lillo, and Mantegna 2010; Musmeci, Aste, and Matteo 2014; Song, Di Matteo, and Aste 2012) and the identification of lead-lag relationships (Curme, Stanley, and Vodenska 2015).

In this Chapter we will describe how to

- Construct and filter financial networks;
- Build price-based dynamic industry taxonomies;
- Implement a trading strategy based on financial network structure.

## 6.2 Network Construction

We selected \(N = 100\) of the most capitalized companies that were part of the S&P500 index from 09/05/2012 to 08/25/2017. The list of these companies’ ticker symbols is reported in the Appendix . For each stock \(i\) the financial variable was defined as the daily stock’s log-return \(R_i(\tau)\) at time \(\tau\).

Stock returns \(R_i\) and social media opinion scores \(O_i\) each amounted to a time series of length equals to 1251 trading days. These series were divided time-wise into \(M = 225\) windows \(t = 1, 2, \ldots, M\) of width \(T = 126\) trading days. A window step length parameter of \(\delta T = 5\) trading days defined the displacement of the window, i.e., the number of trading days between two consecutive windows. The choice of window width \(T\) and window step \(\delta T\) is arbitrary, and it is a trade-off between having analysis that is either too dynamic or too smooth. The smaller the window width and the larger the window steps, the more dynamic the data are.

To characterize the synchronous time evolution of assets, we used equal time Kendall’s rank coefficients between assets \(i\) and \(j\), defined as \[\begin{equation} \rho_{i, j}(t) = \sum\limits_{t' < \tau}sgn(V_i(t') - V_i(\tau))sgn(V_j(t') - V_j(\tau)), \end{equation}\]where \(t'\) and \(\tau\) are time indexes within the window \(t\) and \(V_i \in \{R_i, O_i\}\).

Kendall’s rank coefficients takes into account possible nonlinear (monotonic) relationships. It fulfill the condition \(-1 \leq \rho_{i, j} \leq 1\) and form the \(N \times N\) correlation matrix \(C(t)\) that served as the basis for the networks constructed in this work. To construct the asset-based financial and social networks, we defined a distance between a pair of stocks. This distance was associated with the edge connecting the stocks, and it reflected the level at which they were correlated. We used a simple non-linear transformation \(d_{i, j}(t) = \sqrt{2(1 - \rho_{i,j}(t))}\) to obtain distances with the property \(2 \geq d_{i,j} \geq 0\), forming a \(N \times N\) symmetric distance matrix \(D(t)\).

### 6.2.1 Network Filtering: Asset Graphs

We extract the \(N(N-1)/2\) distinct distance elements from the upper triangular part of the distance matrix \(D(t)\), which were then sorted in an ascending order to form an ordered sequence \(d_1(t), d_2(t), \ldots, d_{N(N-1)/2}(t)\). Since we require the graph to be representative of the market, it is natural to build the network by including only the strongest connections. This is a network filtering procedure that has been successfully applied in the construction of for the analyses of market structure . The number of edges to include is arbitrary, and we included those from the bottom quartile, which represented the 25% shortest edges in the graph (largest correlations), thus giving \(E(t) = \{d_1(t), d_2(t), \ldots, d_{\floor{N/4}}(t)\}\).

We denoted \(E^{F}(t)\) as the set of edges constructed from the distance matrix derived from stock returns \(R(t)\). The financial network considered is \(G^{F} = ( V, E^{F} )\), where \(V\) is the vertex set of stocks.

### 6.2.2 Network Filtering: MST

### 6.2.3 Network Filtering: PMFG

## 6.3 Applications

### 6.3.1 Industry Taxonomy

### 6.3.2 Portfolio Construction

### References

Aste, Tomaso, W. Shaw, and Tiziana Di Matteo. 2010. “Correlation Structure and Dynamics in Volatile Markets.” *New Journal of Physics* 12 (8). IOP Publishing: 085009.

Bardoscia, Marco, Stefano Battiston, Fabio Caccioli, and Guido Caldarelli. 2017. “Pathways Towards Instability in Financial Networks.” *Nature Communications* 8. Nature Publishing Group: 14416.

Curme, Chester, H Eugene Stanley, and Irena Vodenska. 2015. “Coupled Network Approach to Predictability of Financial Market Returns and News Sentiments.” *International Journal of Theoretical and Applied Finance* 18 (07). World Scientific Publishing Company: 1550043.

Li, Yan, Xiong-Fei Jiang, Yue Tian, Sai-Ping Li, and Bo Zheng. 2018. “Portfolio Optimization Based on Network Topology.” *Physica A: Statistical Mechanics and Its Applications*. doi:https://doi.org/10.1016/j.physa.2018.10.014.

Mantegna, R. N. 1999. “Hierarchical Structure in Financial Markets.” *The European Physical Journal B - Condensed Matter and Complex Systems* 11 (1): 193–97. doi:10.1007/s100510050929.

Morales, Raffaello, Tiziana Di Matteo, Ruggero Gramatica, and Tomaso Aste. 2012. “Dynamical Generalized Hurst Exponent as a Tool to Monitor Unstable Periods in Financial Time Series.” *Physica A: Statistical Mechanics and Its Applications* 391 (11). North-Holland: 3180–9.

Musmeci, Nicoló, Tomaso Aste, and Tiziana di Matteo. 2014. “Clustering and Hierarchy of Financial Markets Data: Advantages of the Dbht.” *CoRR*.

Pozzi, Francesco, Tiziana Di Matteo, and Tomaso Aste. 2013. “Spread of Risk Across Financial Markets: Better to Invest in the Peripheries.” *Scientific Reports* 3. Nature Publishing Group.

Song, W.-M., T. Aste, and T. Di Matteo. 2008. “Analysis on Filtered Correlation Graph for Information Extraction.” *Statistical Mechanics of Molecular Biophysics*, 88.

Song, W.-M., T. Di Matteo, and T. Aste. 2012. “Hierarchical Information Clustering by Means of Topologically Embedded Graphs.” *PLoS One* 7 (3). Public Library of Science: e31929.

Tumminello, M., T. Aste, T. Di Matteo, and R. N. Mantegna. 2005. “A Tool for Filtering Information in Complex Systems.” *Proceedings of the National Academy of Sciences of the United States of America* 102 (30): 10421–6. doi:10.1073/pnas.0500298102.

Tumminello, Michele, Fabrizio Lillo, and Rosario N. Mantegna. 2010. “Correlation, Hierarchies, and Networks in Financial Markets.” *Journal of Economic Behavior & Organization* 75 (1): 40–58. doi:http://dx.doi.org/10.1016/j.jebo.2010.01.004.

Tumminello, Salvatore AND Lillo, Michele AND Miccichè. 2011. “Statistically Validated Networks in Bipartite Complex Systems.” *PLoS ONE* 6 (3). Public Library of Science: 1–11. doi:10.1371/journal.pone.0017994.