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We investigate the statistical properties of the correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the correlation matrix is positively skewed and changes over time. We find that the eigenvalue distribution of original correlation matrix deviates from the eigenvalues predicted by the RMT, and the largest eigenvalue is 52 times larger than the maximum value among the eigenvalues predicted by the RMT. The $beta_{473}$ coefficient, which reflect the largest eigenvalue property, is 0.8, while one of the eigenvalues in the RMT is approximately zero. Notably, we show that the entropy function $E(sigma)$ with the portfolio risk $sigma$ for the original and filtered correlation matrices are consistent with a power-law function, $E(sigma) sim sigma^{-gamma}$, with the exponent $gamma sim 2.92$ and those for Asian currency crisis decreases significantly.
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The stock market has been known to form homogeneous stock groups with a higher correlation among different stocks according to common economic factors that influence individual stocks. We investigate the role of common economic factors in the market in the formation of stock networks, using the arbitrage pricing model reflecting essential properties of common economic factors. We find that the degree of consistency between real and model stock networks increases as additional common economic factors are incorporated into our model. Furthermore, we find that individual stocks with a large number of links to other stocks in a network are more highly correlated with common economic factors than those with a small number of links. This suggests that common economic factors in the stock market can be understood in terms of deterministic factors.
In this study, we have investigated empirically the effects of market properties on the degree of diversification of investment weights among stocks in a portfolio. The weights of stocks within a portfolio were determined on the basis of Markowitzs portfolio theory. We identified that there was a negative relationship between the influence of market properties and the degree of diversification of the weights among stocks in a portfolio. Furthermore, we noted that the random matrix theory method could control the properties of correlation matrix between stocks; this may be useful in improving portfolio management for practical application.
During any unique crisis, panic sell-off leads to a massive stock market crash that may continue for more than a day, termed as mainshock. The effect of a mainshock in the form of aftershocks can be felt throughout the recovery phase of stock price. As the market remains in stress during recovery, any small perturbation leads to a relatively smaller aftershock. The duration of the recovery phase has been estimated using structural break analysis. We have carried out statistical analyses of the 1987 stock market crash, 2008 financial crisis and 2020 COVID-19 pandemic considering the actual crash-times of the mainshock and aftershocks. Earlier, such analyses were done considering an absolute one-day return, which cannot capture a crash properly. The results show that the mainshock and aftershock in the stock market follow the Gutenberg-Richter (GR) power law. Further, we obtained a higher $beta$ value for the COVID-19 crash compared to the financial-crisis-2008 from the GR law. This implies that the recovery of stock price during COVID-19 may be faster than the financial-crisis-2008. The result is consistent with the present recovery of the market from the COVID-19 pandemic. The analysis shows that the high magnitude aftershocks are rare, and low magnitude aftershocks are frequent during the recovery phase. The analysis also shows that the distribution $P(tau_i)$ follows the generalized Pareto distribution, i.e., $displaystyle~P(tau_i)proptofrac{1}{{1+lambda(q-1)tau_i}^{frac{1}{(q-1)}}}$, where $lambda$ and $q$ are constants and $tau_i$ is the inter-occurrence time. This analysis may help investors to restructure their portfolios during a market crash.
We propose an approach to generate realistic and high-fidelity stock market data based on generative adversarial networks (GANs). Our Stock-GAN model employs a conditional Wasserstein GAN to capture history dependence of orders. The generator design includes specially crafted aspects including components that approximate the markets auction mechanism, augmenting the order history with order-book constructions to improve the generation task. We perform an ablation study to verify the usefulness of aspects of our network structure. We provide a mathematical characterization of distribution learned by the generator. We also propose statistics to measure the quality of generated orders. We test our approach with synthetic and actual market data, compare to many baseline generative models, and find the generated data to be close to real data.
In this study, we attempted to determine how eigenvalues change, according to random matrix theory (RMT), in stock market data as the number of stocks comprising the correlation matrix changes. Specifically, we tested for changes in the eigenvalue properties as a function of the number and type of stocks in the correlation matrix. We determined that the value of the eigenvalue increases in proportion with the number of stocks. Furthermore, we noted that the largest eigenvalue maintains its identical properties, regardless of the number and type, whereas other eigenvalues evidence different features.
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