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Register a new userA New Method to Estimate the Noise in Financial Correlation Matrices
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Financial correlation matrices measure the unsystematic correlations between stocks. Such information is important for risk management. The correlation matrices are known to be ``noise dressed. We develop a new and alternative method to estimate this noise. To this end, we simulate certain time series and random matrices which can model financial correlations. With our approach, different correlation structures buried under this noise can be detected. Moreover, we introduce a measure for the relation between noise and correlations. Our method is based on a power mapping which efficiently suppresses the noise. Neither further data processing nor additional input is needed.
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We investigate the herd behavior of returns for the yen-dollar exchange rate in the Japanese financial market. It is obtained that the probability distribution $P(R)$ of returns $R$ satisfies the power-law behavior $P(R) simeq R^{-beta}$ with the exponents $ beta=3.11$(the time interval $tau=$ one minute) and 3.36($tau=$ one day). The informational cascade regime appears in the herding parameter $Hge 2.33$ at $tau=$ one minute, while it occurs no herding at $tau=$ one day. Especially, we find that the distribution of normalized returns shows a crossover to a Gaussian distribution at one time step $Delta t=1$ day.
We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of price fluctuations. The central result of the present study is the remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). In particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management.
We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents $alpha$ of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Levy regime $alpha < 2$, the power law decay with large exponent ($alpha > 2$), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement.
The probability distribution of sequences with maximum entropy that satisfies a given amino acid composition at each site and a given pairwise amino acid frequency at each site pair is a Boltzmann distribution with $exp(-psi_N)$, where the total interaction $psi_N$ is represented as the sum of one body and pairwise interactions. A protein folding theory based on the random energy model (REM) indicates that the equilibrium ensemble of natural protein sequences is a canonical ensemble characterized by $exp(-Delta G_{ND}/k_B T_s)$ or by $exp(- G_{N}/k_B T_s)$ if an amino acid composition is kept constant, meaning $psi_N = Delta G_{ND}/k_B T_s +$ constant, where $Delta G_{ND} equiv G_N - G_D$, $G_N$ and $G_D$ are the native and denatured free energies, and $T_s$ is the effective temperature of natural selection. Here, we examine interaction changes ($Delta psi_N$) due to single nucleotide nonsynonymous mutations, and have found that the variance of their $Delta psi_N$ over all sites hardly depends on the $psi_N$ of each homologous sequence, indicating that the variance of $Delta G_N (= k_B T_s Delta psi_N)$ is nearly constant irrespective of protein families. As a result, $T_s$ is estimated from the ratio of the variance of $Delta psi_N$ to that of a reference protein, which is determined by a direct comparison between $DeltaDelta psi_{ND} (simeq Delta psi_N)$ and experimental $DeltaDelta G_{ND}$. Based on the REM, glass transition temperature $T_g$ and $Delta G_{ND}$ are estimated from $T_s$ and experimental melting temperatures ($T_m$) for 14 protein domains. The estimates of $Delta G_{ND}$ agree well with their experimental values for 5 proteins, and those of $T_s$ and $T_g$ are all within a reasonable range. This method is coarse-grained but much simpler in estimating $T_s$, $T_g$ and $DeltaDelta G_{ND}$ than previous methods.
In recent years there has been a closer interrelationship between several scientific areas trying to obtain a more realistic and rich explanation of the natural and social phenomena. Among these it should be emphasized the increasing interrelationship between physics and financial theory. In this field the analysis of uncertainty, which is crucial in financial analysis, can be made using measures of physics statistics and information theory, namely the Shannon entropy. One advantage of this approach is that the entropy is a more general measure than the variance, since it accounts for higher order moments of a probability distribution function. An empirical application was made using data collected from the Portuguese Stock Market.
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