Abstract:
The stock market is a fantastic example of a “complex system” that exhibits vibrant correlation patterns and behaviours of the price (or return) time series. The direct impact of it
on the economic ecosystem of a country, as well as the abundant availability of structured
data, makes this system interesting for empirical studies. This thesis presents a general
and robust methodology to extract information about the “disorder” (or randomness) in the
market and its eigen modes, using the entropy measure “eigen-entropy” H, computed from
the eigen-centralities (ranks) of different stocks in the correlation-network. We have used
correlation matrix constructed using the log-return of adjusted closing price of two different data sets containing stocks from United States of American S&P-500 index (USA) and
Japanese Nikkei-225 index (JPN), spanning across a sufficiently long period of 32 years, to
demonstrate its robustness. Further, the eigenvalue decomposition of the correlation matrix into partial correlation - market, group and random modes, and the relative-entropy
measures computed from these eigen modes enabled us to construct a phase space, where
the different market events undergo phase-separation and display “order-disorder” transitions. Our proposed methodology may help us to understand the market events and their
dynamics, as well as find the time-ordering and appearances of the bubbles and crashes,
separated by normal periods. We have studied the evolution of events around major crashes
and bubbles (from historical records in USA and JPN). Furthermore, the relative entropy
with respect to the market mode H − H_M, displayed “universal scaling” behavior with respect to the mean market correlation µ; a data-collapse was observed when plotted in a
linear-logarithmic scale, which suggested that the fluctuations and co-movements in price
returns for different financial assets and varying across countries are governed by the same
statistical law. In addition, our study may lead to a deeper and broader understanding of
scaling and universality phenomena in complex systems, in general.
