MST for Financial Market Analysis

• Building Minimal Spanning Tree (MST) using correlations between DOW30 stocks.
• Feb. 1, 2022 ~ Aug. 1, 2022

     

"Analysis of the correlation network in the US stock market during January 2020."

Paper

Abstract In January 2020, our study delved into the US stock market’s dynamics as COVID-19 began to affect the global economy. We scrutinized the Dow Jones Industrial Average (DJI) stocks, focusing on the correlations of their returns. We discerned patterns and anomalies through a structural and dynamic analysis of the correlation network facilitated by a distance function applied to the correlation coefficients. The study emphasized the significance of the minimum spanning tree (MST) in shaping the network’s structure and influencing the expansion of subnetworks. Central nodes with high connectivity in the MST emerged as crucial, particularly when the market exhibited abnormal behavior. These nodes’ daily variations and correlation structures provided insights into the market’s evolving nature. We observed that the MST’s radius was particularly reactive to market abnormalities, serving as a potential crisis indicator. Our analysis connected the alterations in the MST’s central nodes and the overall network structure with shifts in the four fundamental statistical moments of the correlation coefficients and distance weights. These elements proved to be instrumental in detecting and analyzing market irregularities.

• This paper is accepted by JKPS indexed in SCIE.

     

Overview

• We use the minimal spanning tree (MST) to identify the structure and dynamics of the US stock market. In particular, by using Dow Jones stock price dataset in January 2020, we analyze the financial market movements when the COVID-19 just outbreak.

• In this study, we generate several sub-networks not only by using the baseline MST structure, but also by adding important correlations to the MST in turn. By doing this, we analyze several measurements for MST and its sub-networks:

  • Average size of the MSTs and all sub-networks observed in the daily propagations.
  • Central node of the MSTs and all sub-networks observed in the daily propagations.
  • Degree of the central node of the MSTs for the daily propagations.
  • Radius of the MST for the daily propagations.
  • Mean, variance, skewness, kurtosis for the correlation coefficients and distances of MSTs for the daily propagation.

     

Dataset

• Dow Jones Industrial Average observed every minute during January 2020 (21 days)
• We calculate the returns per minute of the closing prices of the stocks. The return per minute of i stock at time t on date k follows:

$$ r^t _i = \frac{P^k _i (t+1) - P^k _i (t)}{P^k _i (t)}, \\ \text{where $t+1$ is the time one minute after time $t$.} $$

• And the average prices per day is calculated with the returns per minute:

$$ y^k _ i = \frac{1}{N} \sum^N _{t=1} r^k _i (t), \\ \text{where $N$ is the total number of returns in one day.} $$

➔ As a result, we get a matrix of size 21 by 30. (21 is the number of days in January 2020 and 30 is the number of stocks)

• We set the window width as 10 days, shifting 1 day. That is, we perform the daily propagation (DP) 12 times for our dataset.

     

Minimum Spanning Tree (MST)

• We analyze the correlation-based networks (Mantegna, 1999), which has been the basis of many works for the last two decades.

• First, we obtain the correlation coefficients from the returns of all the stocks:
  ,where <...>_m indicates a time average over the consecutive m days.

• And, we convert the correlation coefficients into distances:

$$ d_{ij}=\sqrt{2(1-\rho_{ij})}. $$

➔ This allows us to get a distance matrix of 30 by 30 size, which is symmetric. That is, considering the entries above(below) the main diagonal of this matrix, a total of $435 (30*30/2 - 15)$ distances can be obtained.

• For the distances, we construct the minimal spanning tree (MST) connecting the n stocks using the Prim's algorithm. This network is a connected graph with 29 edges.

• We mainly used the igraph package of R to implement MSTs.

• The MST structures observed for the dataset of the first 10 days:
  
  ➔ The red node is central vertex defined as the node with the largest degree in this study.

• The average size of MSTs and the radius of MSTs follow:

➔ The average size is a modified version of the normalized tree length (Mantegna, 1999) and defined as follows:

$$ A_s (N) = \frac{1}{n-1} \sum_{d_{ij}\in N} d_{ij}, \text{where $N$ is the network and $n$ is the number of the nodes in tree $N$.} $$

➔ The radius is given by:

$$ R (MST) = \sum_{d_{ij}\in \uptau_r} d_{ij}, \text{where $\uptau_{ms}$ and $d_{ij}$ are the trail path from the central node of MST.} $$

• Comparison of four moments between the correlation coefficients and distances of MSTs
  • Four moments of the correlation coefficients:
    
  • Four moments of the distances of MSTs:
    

     

Sub-networks derived from MST

• The sub-network of rank k is obtained by adding n-1 new links increasing distance weights to the subnetwork of rank $k-1$, $(k=0,...,14)$.

➔ n is the number of nodes in tree and the sub-network of rank 0 is the same as MST.

• All possible sub-networks of the first 10 days:

• The average size of all sub-networks and mean of average size of all sub-networks follow:

• Degree of central vertex for all sub-networks and degree distribution of all sub-networks of the first 10 days:
  

Info

• Authors : Doobae Jun, Seoyoung Oh and Gwangil Kim.
• Journal : Journal of the Korean Physical Society, JKPS
• Year : 2024