I wish to summarize the contents of a relatively old paper with the above title by authors Chi K. Tse, Jing Liu and Francis C.M. Lau.
It describes a nice method to construct stock market networks and also define new market indexes based on the same. The motivation for this article was derived from our recent lectures by Prof. Marsili in financial networks and the idea to understand some good methods as an extension to our course work.
Introduction
Fluctuations of stock prices are not independent, but are
highly inter-coupled with strong correlations with the business sectors and
industries to which the stocks belong. Network models have been proposed for
studying the correlations of stock prices. The usual approach involves a
procedure of finding correlation between each pair of time series of stock
prices, and a subsequent procedure of constructing a network that connects the
individual stocks based on the levels of correlation. The resulting networks
are usually very large and their analysis is rather complex.
The approach used by authors basically examines the time
series of the daily stock prices and establish connections between any pair of
stocks. If the cross correlation of the time series of the daily stock prices
of two stocks is greater than a threshold (e.g. 0.9), we consider that the two
stocks are “connected”.
Because power-law distributions have been found in the stock
prices, we know that a small number of stocks are having strong influence over
the entire market, and we therefore propose that stocks corresponding to nodes
of high degrees can be used to compose a new index that can naturally and
adequately reflect the market variation.
Network Construction
For each pair of stocks (nodes), we will evaluate the cross
correlation of the time series of their daily stock prices, daily price returns
and daily trading volumes.
Let pi(t) be the closing price of stock i on day t
and vi(t) be the trading volume of stock i on day t. Then, the price
return of stock i on day t, denoted by ri(t), is defined as
Suppose xi(t) and xj(t) are the daily
prices or price returns or trading volumes of stock i and stock j,
respectively, over the period t=0 to N−1. We now compare the two time series with
no relative delay. In other words, xi and xj are compared
from i=0 to N−1 with no relative time shift. The cross correlation between xi
and xj with no time shift is given by Cohen et al. (2003).
We begin with relatively large values of ρ as our objective is to construct stock networks that reflect
connections of highly correlated stock price time series. It is found that the
degree distributions display scale free characteristics when ρ is sufficiently large. Applying the least squares method with
data points in the straight line segment of the log–log degree distribution
plots, the power-law exponent is found to vary between 1 and 3. We also
calculate the mean fitting error to examine the fitness of the power-law
distribution over the data points.
Specifically, suppose the distribution of p(k) vs. k has been
approximated by a power-law function P(k)=αe−γk,
and the values of α and γ can be found from any fitting method. Here, we define fitting
error (ε), as follows:
For ρ below about a certain value, the
networks do not show clear scale free characteristics. This is because with
small ρ, the network tends to be randomly connected. In the case
where ρ is high as the network so formed would connect stocks of
closely resembling daily price fluctuations.
Conclusion
As the stock network is scale free and displays a
power-law degree distribution, we may conclude that stocks having close
resemblance with a large number of other stocks are relatively few. This implies
that any stock market is essentially influenced by a relatively small number of
stocks. Thus new indexes that reflect on the performance of the majority stock may be defined based on a relatively small number of highly connected stocks.
The financial paper is quite new though. Is it a coincidence that this article came after our finance microcredit?
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