Showing papers by "Parameswaran Gopikrishnan published in 2002"
TL;DR: A analysis of cross correlations between price fluctuations of different stocks using methods of random matrix theory finds that the largest eigenvalue corresponds to an influence common to all stocks, and discusses applications to the construction of portfolios of stocks that have a stable ratio of risk to return.
Abstract: We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis" - a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return. (Less)
914 citations
TL;DR: It is found that large price fluctuations occur when demand is very small-a fact that is reminiscent of large fluctuations that occur at critical points in spin systems, where the divergent nature of the response function leads to large fluctuations.
Abstract: We empirically address the question of how stock prices respond to changes in demand. We quantify the relations between price change G over a time interval Deltat and two different measures of demand fluctuations: (a) Phi, defined as the difference between the number of buyer-initiated and seller-initiated trades, and (b) Omega, defined as the difference in number of shares traded in buyer- and seller-initiated trades. We find that the conditional expectation functions of price change for a given Phi or Omega, (Phi) and (Omega) ("market impact function"), display concave functional forms that seem universal for all stocks. For small Omega, we find a power-law behavior (Omega) approximately Omega(1/8) with delta depending on Deltat (delta approximately 3 for Deltat=5 min, delta approximately 3/2 for Deltat=15 min and delta approximately 1 for large Deltat). We find that large price fluctuations occur when demand is very small-a fact that is reminiscent of large fluctuations that occur at critical points in spin systems, where the divergent nature of the response function leads to large fluctuations.
266 citations
TL;DR: Two newly discovered scaling results that appear to be universal, in the sense that they hold for widely different economies as well as for different time periods are reviewed.
Abstract: This article discusses some of the similarities between work being done by economists and by physicists seeking to contribute to economics. We also mention some of the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge. In particular, we review two newly discovered scaling results that appear to be universal, in the sense that they hold for widely different economies as well as for different time periods: (i) the fluctuation of price changes of any stock market is characterized by a probability density function, which is a simple power law with exponent −4 extending over 102 SDs (a factor of 108 on the y axis); this result is analogous to the Gutenberg–Richter power law describing the histogram of earthquakes of a given strength; and (ii) for a wide range of economic organizations, the histogram shows how size of organization is inversely correlated to fluctuations in size with an exponent ≈0.2. Neither of these two new empirical laws has a firm theoretical foundation. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behavior of the response function at the critical point (zero magnetic field) leads to large fluctuations.
78 citations
TL;DR: In this article, the problem of finding an investment in stocks which exposes invested funds to a minimum level of risk is analogous to finding the magnetization of a random magnet, and the interactions for this random magnet problem are given by the cross-correlation matrix C of stock returns.
Abstract: Diversification of an investment into independently fluctuating assets reduces its risk. In reality, movements of assets are mutually correlated and therefore knowledge of cross-correlations among asset price movements are of great importance. Our results support the possibility that the problem of finding an investment in stocks which exposes invested funds to a minimum level of risk is analogous to the problem of finding the magnetization of a random magnet. The interactions for this random magnet problem are given by the cross-correlation matrix C of stock returns. We find that random matrix theory allows us to make an estimate for C which outperforms the standard estimate in terms of constructing an investment which carries a minimum level of risk.
67 citations
01 Jan 2002
TL;DR: This article showed empirically a series of sharp patterns in stock market fluctuations, trading activity and their contemporaneous relationships, including the cubic law of returns and Zipf's law for mutual funds.
Abstract: We show empirically a series of sharp patterns in stock market fluctuations, trading activity and their contemporaneous relationships. We link together and explain the following facts: (i) the cubic law of returns: returns follow a power law distribution with exponent 3. This “cubic” law seems to hold both across time and internationally. Stock market “crashes” (e.g. the 1929 and 1987 crashes) are not outliers to this law; (ii) the half cubic law of volumes: volumes follow a power law distribution with exponent 3/2; (iii) the square root law of price impact (the price impact of a volume V is proportional to V 1/2 ); (iv) Zipf’s law for mutual funds: mutual funds size follow a power law distribution with exponent 1. The model also makes predictions about the crossconditional relationships between various trading variables. They all appear to be verified empirically. The model makes a series of other, out of sample, testable predictions. Finally, it shows that a Tobin tax, or circuit breakers, do not affect that size of extreme fluctuations. However, a tax that increases with the size of the transactions does reduce the magnitude of those very large fluctuations.
12 citations
TL;DR: In this article, the authors discuss the similarities between work being done by economists and by computational physicists seeking to contribute to economics and discuss the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge.
Abstract: This paper discusses some of the similarities between work being done by economists and by computational physicists seeking to contribute to economics. We also mention some of the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge. In particular, we review two such new results. Specifically, we discuss the two newly discovered scaling results that appear to be `universal', in the sense that they hold for widely different economies as well as for different time periods: (i) the fluctuation of price changes of any stock market is characterized by a probability density function, which is a simple power law with exponent -4 extending over 102 standard deviations (a factor of 108 on the y-axis); this result is analogous to the Gutenberg-Richter power law describing the histogram of earthquakes of a given strength; (ii) for a wide range of economic organizations, the histogram that shows how size of organization is inversely correlated to fluctuations in size with an exponent ≈0.2. Neither of these two new empirical laws has a firm theoretical foundation. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behaviour of the response function at the critical point (zero magnetic field) leads to large fluctuations. We discuss a curious `symmetry breaking' for values of Σ above a certain threshold value Σc; here Σ is defined to be the local first moment of the probability distribution of demand Ω - the difference between the number of shares traded in buyer-initiated and seller-initiated trades. This feature is qualitatively identical to the behaviour of the probability density of the magnetization for fixed values of the inverse temperature.
5 citations
TL;DR: In this article, the authors discuss the similarities between work being done by economists and by computational physicists seeking to contribute to economics and discuss the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge.
5 citations
01 Jan 2002
TL;DR: A heuristic argument is developed that serves to make more plausible the scaling and universality hypotheses and will help guide research on interpreting empirical data on economic fluctuations.
Abstract: This manuscript is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena—scale invariance and universality—can be useful in guiding research on interpreting empirical data on economic fluctuations. In particular, we shall develop a heuristic argument that serves to make more plausible the scaling and universality hypotheses.
4 citations
01 Jan 2002
TL;DR: It is found that RMT methods can distinguish random and non- random parts of C, and the non-random part of C which deviates from RMT results, provides information regarding genuine collective behavior among stocks.
Abstract: We review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory (RMT), developed in physics to describe the energy levels of complex systems. RMT makes predictions for “universal” properties that do not depend on the interactions between the elements comprising the system; deviations from RMT provide clues regarding system-specific properties. WE compare the statistics of the cross-correlation matrix C—whose elements C i,j are the correlation coefficients of price fluctuations of stock i and j — against a random matrix having the same symmetry properties. It is found that RMT methods can distinguish random and non-random parts of C. The non-random part of C which deviates from RMT results, provides information regarding genuine collective behavior among stocks.
2 citations
01 Jan 2002
TL;DR: It is demonstrated that the sectors identified are useful for the practical goal of finding an investment which earns a given return without exposure to unnecessary risk.
Abstract: We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994–95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. The eigenvectors whose eigenvalues deviate from the random matrix bound contain information about business sectors and are stable in time. Finally, we demonstrate that the sectors we identify are useful for the practical goal of finding an investment which earns a given return without exposure to unnecessary risk.
2 citations
01 Jan 2002
TL;DR: In this paper, the authors apply methods and concepts of statistical physics to the study of economic organizations and find the existence of scaling laws describing the growth of the size of these organizations.
Abstract: We apply methods and concepts of statistical physics to the study of economic organizations. We identify robust, universal, characteristics of the time evolution of economic organizations. Specifically, we find the existence of scaling laws describing the growth of the size of these organizations. We study a model assuming a complex evolving internal structure of an organization that is able to reproduce many of the empirical findings.