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Recent market events have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the medium-term log-return dynamics in a market with both fundamentaland technicaltraders. This is based on a trade arrival model with variable size orders and a general arrival-time distribution. With simplifications we are led in the jump-free case to a local volatility model defined by ahybridSDE mixingboth arithmetic and geometric or CIRBrownian motions, whose solution in the geometric case is given by a class of integrals of exponentials of one Brownian motion against another, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of ‘Pearson diffusion’, or, equivalently, a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under mean-reverting circumstances we arrive naturally at an equilibrium fat-tailed return distribution with a Student or Pearson Type~IV form. Under less-restrictive assumptions, richer dynamics are possible, including time-dependent Johnson-SU distributions and bimodal structures. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might havea prioribeen expected, so that ‘25σ’ events are significantly more probable. We exhibit simple example solutions of the Fokker–Planck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviors. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity. We touch briefly on the extent to which such a model is consistent with the dynamics of a ‘flash-crash’ event, and briefly explore the statistical evidence for our model. [ABSTRACT FROM AUTHOR]

In this paper we introduce an empirical approximation of the log-optimal investment strategy that guarantees an almost optimal growth rate of investments. The proposed strategy also considers the effects of portfolio rearrangement costs on growth optimality and recommends a suboptimal portfolio for discrete investment periods. We do not assume any parametric structure for the market process, only a first-order Markov property. The model introduced is based on kernel-based agents' (experts') approximation of the maximum theoretical growth rate with transaction costs. Although the optimal solution is theoretically a complex Bellman programming problem, our suboptimal empirical result appears to be attractive for Dow Jones 30 shares. The paper presents a performance analysis where the return of the empirical log-optimal portfolio is compared with passive portfolio counterparts compiled from similar components using the CAPM, the three-factor model and the four-factor model. The proposed methods, in the presence of transaction costs, provide a significant positive abnormal return compared with the preceding equilibrium models, and is even a survivorship bias-free setup. [ABSTRACT FROM PUBLISHER]

In this article, we consider a modification of the Karatzas–Pikovsky model of insider trading. Specifically, we suppose that the insider agent influences the long/medium-term evolution of Black–Scholes type model through the drift of the stochastic differential equation. We say that the insider agent is using a portfolio leading to a partial equilibrium if the following three properties are satisfied: (a) the portfolio used by the insider leads to a stock price which is a semimartingale under his/her own filtration and his/her own filtration enlarged with the final price; (b) the portfolio used by the insider is optimal in the sense that it maximises the logarithmic utility for the insider when his/her filtration is fixed; and (c) the optimal logarithmic utility in (b) is finite. We give sufficient conditions for the existence of a partial equilibrium and show in some explicit models how to apply these general results. [ABSTRACT FROM AUTHOR]

The Minority Game (MG), the Majority Game (MAJG) and the Dollar Game ($G) are important and closely related versions of market-entry games designed to model different features of real-world financial markets. In a variant of these games, agents measure the performance of their available strategies over a fixed-length rolling window of prior time-steps. These are the Time Horizon MG/MAJG/$Gs (THMG, THMAJG, TH$G). Their probabilistic dynamics may be completely characterized in Markov-chain formulation. Games of both the standard and TH variants generate time-series that may be understood as arising from a stochastically perturbed determinism because a coin toss is used to break ties. The average over the binomially distributed coin tosses yields the underlying determinism. In order to quantify the degree of this determinism and of higher-order perturbations, we decompose the sign of the time-series they generate (analogous to a market price time-series) into a superposition of weighted Hamiltonian cycles on graphs—exactly in the TH variants and approximately in the standard versions. The cycle decomposition also provides a ‘dissection’ of the internal dynamics of the games and a quantitative measure of the degree of determinism. We discuss how the outperformance of strategies relative to agents in the THMG—the ‘illusion of control’—and the reverse in the THMAJG and TH$G, i.e. genuine control, may be understood on a cycle-by-cycle basis. The decomposition offers a new metric for comparing different game dynamics with real-world financial time-series and a method for generating predictors. We apply the cycle predictor to a real-world market, with significantly positive returns for the latter. Part I provides an overview of the paper and its methodologies with an appendix for the mathematical details of the Markov analysis of the THMG, THMAJG and TH$G. Part I also describes the cycle predictor and applies it to real-world financial series. Part II performs further analyses of the cycle decomposition method as applied to the time-series generated by agent-based models to gain insight into the ‘illusion of control’ that certain of these games demonstrate, i.e. the fact that the strategies outperform the agents that deploy them. Part II also illustrates both numerical and analytic methods for extracting cycles from a given time-series and applies the method to a number of different real-world data sets, in conjunction with an analysis of persistence. [ABSTRACT FROM AUTHOR]

A representative investor generates realistic and complex security price paths by following this trading strategy: if, a few ticks ago, the market asset had two consecutive upticks or two consecutive downticks, then sell, and otherwise buy. This simple, unique, and robust model is the smallest possible deterministic model of financial complexity, and its generalization leads to complex variety. Compared to a random walk, the minimal model generates time series with fatter tails and more frequent crashes, thus more closely matching the real world. It does all this without any parameter fitting. [ABSTRACT FROM AUTHOR]

This article is the second part of a review of recent empirical and theoretical developments usually grouped under the heading Econophysics. In the first part, we reviewed the statistical properties of financial time series, the statistics exhibited in order books and discussed some studies of correlations of asset prices and returns. This second part deals with models in Econophysics from the point of view of agent-based modeling. Of the large number of multi-agent-based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioral finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that we discuss extensively here. Second, kinetic theory models designed to explain certain empirical facts concerning wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems. [ABSTRACT FROM AUTHOR]

We study market efficiency from a computational viewpoint. Borrowing from theoretical computer science, we define a market to be efficient with respect to resources S (e.g., time, memory) if no strategy using resources S can make a profit. As a first step, we consider memory-m strategies whose action at time t depends only on the m previous observations at times t - m, ... , t - 1. We introduce and study a simple model of market evolution, where strategies impact the market by their decision to buy or sell. We show that the effect of optimal strategies using memory m can lead to 'market conditions' that were not present initially, such as (1) market spikes and (2) the possibility for a strategy using memory m' > m to make a bigger profit than was initially possible. We suggest ours as a framework to rationalize the technological arms race of quantitative trading firms. [ABSTRACT FROM AUTHOR]

We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as real, asymmetric random matrices where the time-shift superimposes some structure. We demonstrate that, for large matrices, the associated eigenvalue spectrum is circular symmetric in the complex plane. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach numerically. Theoretical findings are then compared with eigenvalue densities obtained from actual high-frequency (5 min) data of the S&P 500 and the observed deviations are discussed. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks into causal sectors. We finally comment on the stability of the observed patterns. [ABSTRACT FROM AUTHOR]