Applications of Statistical Mechanics to Finance (original) (raw)

Applications to Statistical Mechanics

2014

We discuss some apparently "universal" aspects observed in the empirical analysis of stock price dynamics in ÿnancial markets. Speciÿcally we consider (i) the empirical behavior of the return probability density function and (ii) the content of economic information in ÿnancial time series.

Econophysics: Financial Time Series from a Statistical Physics Point of View

In recent years, physicists have started applying concepts and methods of statistical physics to study economic problems. The word "Econophysics" is sometimes used to refer to this work. Much recent work is focused on understanding the statistical properties of ÿnancial time series. One reason for this interest is that ÿnancial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that ÿnancial time series viewed from a di erent perspective might yield new results. This article reviews the results of three recent phenomenological studies -(i) The probability distribution of stock price uctuations: Stock price uctuations occur in all magnitudes, in analogy to earthquakes -from tiny uctuations to drastic events, such as market crashes. The distribution of price uctuations decays with a power-law tail well outside the LÃ evy stable regime and describes uctuations that di er by as much as eight orders of magnitude. In addition, this distribution preserves its functional form for uctuations on time scales that di er by three orders of magnitude, from 1 min up to approximately 10 d. (ii) Correlations in ÿnancial time series: While price uctuations themselves have rapidly decaying correlations, the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power-law and persist for several months. (iii) Correlations among di erent companies: The third result bears on the application of random matrix theory to understand the correlations among price uctuations of any two di erent stocks. From a study of the eigenvalue statistics of the cross-correlation matrix constructed from price uctuations of the leading 1000 stocks, we ÿnd that the largest ≈ 1% of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix, whereas the rest of the eigenvalues conform to random matrix behavior -suggesting that these 1% of the eigenvalues contain system-speciÿc information about correlated time evolution of di erent companies.

Financial Time Series and Statistical Mechanics

2001

A few characteristic exponents describing power law behaviors of roughness, coherence and persistence in stochastic time series are compared to each other. Relevant techniques for analyzing such time series are recalled in order to distinguish how the various exponents are measured, and what basic differences exist between each one. Financial time series, like the JPY/DEM and USD/DEM exchange rates are used for illustration, but mathematical ones, like (fractional or not) Brownian walks can be used also as indicated.

A dynamical model describing stock market price distributions

Physica A: Statistical Mechanics and its Applications, 2000

High-frequency data in ÿnance have led to a deeper understanding on probability distributions of market prices. Several facts seem to be well established by empirical evidence. Speciÿcally, probability distributions have the following properties: (i) They are not Gaussian and their center is well adjusted by LÃ evy distributions. (ii) They are long-tailed but have ÿnite moments of any order. (iii) They are self-similar on many time scales. Finally, (iv) at small time scales, price volatility follows a non-di usive behavior. We extend Merton's ideas on speculative price formation and present a dynamical model resulting in a characteristic function that explains in a natural way all of the above features. The knowledge of such a distribution opens a new and useful way of quantifying ÿnancial risk. The results of the model agree -with high degree of accuracy -with empirical data taken from historical records of the Standard & Poor's 500 cash index.

Methods of Non-Extensive Statistical Physics in Analysis of Price Returns on Polish Stock Market

Acta Physica Polonica A, 2016

We use methods of non-extensive statistical physics to describe quantitatively the memory effect involved in returns of companies from WIG 30 index on the Warsaw Stock Exchange. The entropic approach based on the generalization of the Boltzmann-Gibbs entropy to non-additive Tsallis q-entropy is applied to fit fat tailed distribution of returns to q-normal (Tsallis) distribution. The existence of long term memory effects in price returns generated by two-point autocorrelations are checked via calculation of the Hurst exponent within detrended fluctuation analysis approach. The results are collected for diversified frequency of data sampling. We confirm the perfect inverse cubic power law for low time-lags (≈1 min) of returns for the main WIG 30 index as well as for the most of separate stocks, however this relationship does not hold for longer time-lags. The particular emphasis is given to a study of an independent fit of probability distribution of positive and negative returns to qnormal distribution. We discuss in this context the asymmetry between tails in terms of the Tsallis parameters q ±. A qualitative and quantitative relationship between the frequency of data sampling, the parameters q and q ± , and the corresponding main Hurst exponent H is provided to analyze the effect of memory in data caused by linear and nonlinear autocorrelations. A new quantifier based on asymmetry of the Tsallis index instead of skewness of distribution is proposed which we believe is able to describe the stage of market development and its robustness to speculation.

Statistical Mechanics of Financial Markets

The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multi-factor models including stochastic volatility. We use a previous development of a statistical mechanics of financial markets to model these issues. Daily Eurodollar futures prices and implied volatilities are fit to determine exponents of functional behavior of diffusions using methods of global optimization, Adaptive Simulated Annealing (ASA), to generate tight fits across moving time windows of Eurodollar contracts. These short-time fitted distributions are then developed into long-time distributions using a robust non-Monte Carlo path-integral algorithm, PATHINT, to generate prices and derivatives commonly used by option traders.

Econophysics Techniques and Their Applications on the Stock Market

Mathematics, 2022

Exact sciences have achieved many results, validated in practice. Although their application in economics is difficult due to the human factor involved, the lack of conservation laws, and experimental difficulties, it must be highlighted that the consistent bibliography gathered in recent years in this field encourages the econophysics approach. The objective of this article is to validate and/or define a few stock strategies, based on known results from mathematics, physics, and chemistry. The scope of this research demonstrates that statistics (in portfolio theory), geometry (in technical analysis), or financial mathematics can be used in the capital market. Many of the exact science results corresponded to strategies applicable to investors. Unlike the material world, financial markets have additional components that must be considered: human psychology, sociology at the firm level, and behavioral unpredictability. The findings obtained in this research enable the enormous vastne...

Econophysics: a new tool to investigate financial markets

2004

The relationship between physics and economics has a long and interesting history. Outstanding economists of the past explicitly inspired their work to the principles of Newtonian physics and statistical mechanics, attracted by the success of these theories. However, despite the existence of many problems of common interest, the interaction between statistical physicists and economists has never been strong.

Statistical Modeling of Stock Returns: A Historical Survey with Methodological Reà ‚ï¿ ½flections

This paper aims at identifying the motivating forces that gave birth to the statistical models of asset returns since the beginning of the twentieth century. The major question addressed is: Where do statistical models of asset returns come from?" This central question encompasses a number of secondary ones: What do these models do? Do they explain or simply describe the empirical regularities of asset returns, identi…ed at di¤erent historical periods? If explanation provides 'something', over and above description, then how can it be de…ned? Moreover, how is this re ‡ected on explanatory versus descriptive models of asset returns? In the context of the models identi…ed as explanatory, do these models o¤er an actual explanation for the regularities of interest or merely a potential explanation? Related to the last question, does the realism of the assumptions underlying the explanatory models matter? Has the literature adopted a realist or an instrumentalist attitude towards the explanatory models of asset returns? Our answers to these questions are being informed by our attempts to draw some analogies between the main issues concerning the statistical modelling of asset prices and those concerning the theoretical modelling of the Brownian motion in Physics.