We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

A paradigm of statistical mechanics of financial markets (SMFM) using nonlinear nonequilibrium algorithms, first published in L. Ingber, Mathematical Modelling, 5, 343-361 (1984), is fit to multivariate financial markets using Adaptive Simulated Annealing (ASA), a global optimization algorithm, to perform maximum likelihood fits of Lagrangians defined by path integrals of multivariate conditional probabilities. Canonical momenta are thereby derived and used as technical indicators in a recursive ASA optimization process to tune trading rules. These trading rules are then used on out-ofsample data, to demonstrate that they can profit from the SMFM model, to illustrate that these markets are likely not efficient.

Abstract—A paradigm of statistical mechanics of financial markets (SMFM) is fit to multivariate financial markets using Adaptive Simulated Annealing (ASA), a global optimization algorithm, to perform maximum likelihood fits of Lagrangians defined by path integrals of multivariate conditional probabilities. Canonical momenta are thereby derived and used as technical indicators in a recursive ASA optimization process to tune trading rules. These trading rules are then used on out-of-sample data, to demonstrate that they can profit from the SMFM model, to illustrate that these markets are likely not efficient. This methodology can be extended to other systems, e.g., electroencephalography. This approach to complex systems emphasizes the utility of blending an intuitive and powerful mathematical-physics formalism to generate indicators which are used by AI-type rule-based models of management. 1.

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. 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.

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Few will argue that the epi-phenomena of biological systems and socio-economic systems are anything but complex. The purpose of this Feature is to examine critically and contribute to the burgeoning multi-disciplinary literature on markets as complex adaptive systems (CAS). The new sciences of complexity, the principles of self-organisation and emergence along with the methods of evolutionary computation and artificially intelligent agent models have been developed in a multi-disciplinary fashion. The cognoscenti here consider that complex systems whether natural or artificial, physical, biological or socio-economic can be characterised by a unifying set of principles. Further, it is held that these principles mark a paradigm shift from earlier ways of viewing such phenomenon. The articles in this Feature aim to provide detailed insights and examples of both the challenges and the prospects for economics that are offered by the new methods of the complexity sciences. The applicability or not of the optimisation framework of conventional economics depends on the domain of the problem and in particular the modern theories behind non-computability are outlined to explain why adaptive or emergent methods of computation and agent-based

A modern calculus of multivariate nonlinear multiplicative Gaussian-Markovian systems provides models of many complex systems faithful to their nature, e.g., by not prematurely applying quasi-linear approximations for the sole purpose of easing analysis. To handle these complex algebraic constructs, sophisticated numerical tools have been developed, e.g., methods of adaptive simulated annealing (ASA) global optimization and of path integration (PATHINT). In-depth application to three quite different complex systems have yielded some insights into the benefits to be obtained by application of these algorithms and tools, in statistical mechanical descriptions of neocortex (short-term memory and electroencephalography), financial markets (interest-rate and trading models), and combat analysis (baselining simulations to exercise data).

S&P 500 index data sampled at one-minute intervals over the course of 11.5 years (January 1989- May 2000) is analyzed, and in particular the Hurst parameter over segments of stationarity (the time period over which the Hurst parameter is almost constant) is estimated.

Summary. Time series of financial asset returns often exhibit the volatility clustering property: large changes in prices tend to cluster together, resulting in persistence of the amplitudes of price changes. After recalling various methods for quantifying and modeling this phenomenon, we discuss several economic mechanisms which have been proposed to explain the origin of this volatility clustering in terms of behavior of market participants and the news arrival process. A common feature of these models seems to be a switching between low and high activity regimes with heavytailed durations of regimes. Finally, we discuss a simple agent-based model which links such variations in market activity to threshold behavior of market participants and suggests a link between volatility clustering and investor inertia. 1

We describe an end-to-end real-time S&P futures trading system. Inner-shell stochastic nonlinear dynamic models are developed, and Canonical Momenta Indicators (CMI) are derived from a fitted Lagrangian used by outer-shell trading models dependent on these indicators. Recursive and adaptive optimization using Adaptive Simulated Annealing (ASA) is used for fitting parameters shared across these shells of dynamic and trading models.