Three-state herding model of the financial markets

We propose a Markov jump process with the three-state herding interaction. We see our approach as an agent-based model for the financial markets. Under certain assumptions this agent-based model can be related to the stochastic description exhibiting sophisticated statistical features. Along with power-law probability density function of the absolute returns we are able to reproduce the fractured power spectral density, which is observed in the high-frequency financial market data. Given example of consistent agent-based and stochastic modeling will provide background for the further developments in the research of complex social systems.Comment: 11 pages, 3 figure

Modelling Financial High Frequency Data Using Point Processes

In this paper, we give an overview of the state-of-the-art in the econometric literature on the modeling of so-called financial point processes. The latter are associated with the random arrival of specific financial trading events, such as transactions, quote updates, limit orders or price changes observable based on financial high-frequency data. After discussing fundamental statistical concepts of point process theory, we review durationbased and intensity-based models of financial point processes. Whereas duration-based approaches are mostly preferable for univariate time series, intensity-based models provide powerful frameworks to model multivariate point processes in continuous time. We illustrate the most important properties of the individual models and discuss major empirical applications.Financial point processes, dynamic duration models, dynamic intensity models.

Modeling high frequency data using hawkes processes with power-law kernels

Those empirical properties exhibited by high frequency financial data, such as time-varying intensities and self-exciting features, make it a challenge to model appropriately the dynamics associated with, for instance, order arrival. To capture the microscopic structures pertaining to limit order books, this paper focuses on modeling high frequency financial data using Hawkes processes. Specifically, the model with power-law kernels is compared with the counterpart with exponential kernels, on the goodness of fit to the empirical data, based on a number of proposed quantities for statistical tests. Based on one-trading-day data of one representative stock, it is shown that Hawkes processes with power-law kernels are able to reproduce the intensity of jumps in the price processes more accurately, which suggests that they could serve as a realistic model for high frequency data on the level of microstructure

Financial markets as a complex system: A short time scale perspective

In this paper we want to discuss macroscopic and microscopic properties of financial markets. By analyzing quantitatively a database consisting of 13 minute per minute recorded financial time series, we identify some macroscopic statistical properties of the corresponding markets, with a special emphasize on temporal correlations. These analysis are performed by using both linear and nonlinear tools. Multivariate correlations are also tested for, which leads to the identification of a global coupling mechanism between the considered stock markets. The application of a new formalism, called transfer entropy, allows to measure the information flow between some financial time series. We then discuss some key aspects of recent attemps to model financial markets from a microscopic point of view. One model, that is based on the simulation of the order book, is described more in detail, and the results of its practical implementation are presented. We finally address some general aspects of forecasting and modeling, in particular the role of stochastic and nonlinear deterministic processes. --time series analysis,econophysics,simulated markets,temporal correlations,high-frequency data

Computational Finance

With the availability of new and more comprehensive financial market data, making headlines of massive public interest due to recent periods of extreme volatility and crashes, the field of computational finance is evolving ever faster thanks to significant advances made theoretically, and to the massive increase in accessible computational resources. This volume includes a wide variety of theoretical and empirical contributions that address a range of issues and topics related to computational finance. It collects contributions on the use of new and innovative techniques for modeling financial asset returns and volatility, on the use of novel computational methods for pricing, hedging, the risk management of financial instruments, and on the use of new high-dimensional or high-frequency data in multivariate applications in today’s complex world. The papers develop new multivariate models for financial returns and novel techniques for pricing derivatives in such flexible models, examine how pricing and hedging techniques can be used to assess the challenges faced by insurance companies, pension plan participants, and market participants in general, by changing the regulatory requirements. Additionally, they consider the issues related to high-frequency trading and statistical arbitrage in particular, and explore the use of such data to asses risk and volatility in financial markets

Statistical properties of a heterogeneous asset pricing model with time-varying second moment

Stability and bifurcation analysis of deterministic systems has been widely used in modeling financial markets. However, the impact of such dynamic phenomena on various statistical properties of the corresponding stochastic model, including skewness and excess kurtosis, various autocorrelation (AC) patterns of under and over reactions, and volatility clustering characterised by the long-range dependence of ACs, is not clear and has been very little studied. This paper aims to contribute to this issue. Through a simple behavioural asset pricing model with fundamentalists and chartists, we examine the statistical properties of the model and their connection to the dynamics of the underlying deterministic model. In particular, our analysis leads to some insights into various mechanisms that may generate some of the stylised facts, such as fat tails, skewness, high kurtosis and long memory, observed in high frequency financial data. © 2006 Springer-Verlag Berlin Heidelberg

Limit theorems for nearly unstable Hawkes processes

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the $L^1$ norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65-77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1005 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limit theorems for nearly unstable Hawkes processes: Version with technical appendix

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only "nearly unstable Hawkes processes" are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox Ingersoll Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes based price model introduced by Bacry et al. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well known stylized facts of prices, both at the microstructure level and at the macroscopic scale

Volatility modeling and limit-order book analytics with high-frequency data

The vast amount of information characterizing nowadays’s high-frequency financial datasets poses both opportunities and challenges. Among the opportunities, existing methods can be employed to provide new insights and better understanding of market’s complexity under different perspectives, while new methods, capable of fully-exploit all the information embedded in high-frequency datasets and addressing new issues, can be devised. Challenges are driven by data complexity: limit-order book datasets constitute of hundreds of thousands of events, interacting with each other, and affecting the event-flow dynamics. This dissertation aims at improving our understanding over the effective applicability of machine learning methods for mid-price movement prediction, over the nature of long-range autocorrelations in financial time-series, and over the econometric modeling and forecasting of volatility dynamics in high-frequency settings. Our results show that simple machine learning methods can be successfully employed for mid-price forecasting, moreover adopting methods that rely on the natural tensorrepresentation of financial time series, inter-temporal connections captured by this convenient representation are shown to be of relevance for the prediction of future mid-price movements. Furthermore, by using ultra-high-frequency order book data over a considerably long period, a quantitative characterization of the long-range autocorrelation is achieved by extracting the so-called scaling exponent. By jointly considering duration series of both inter- and cross- events, for different stocks, and separately for the bid and ask side, long-range autocorrelations are found to be ubiquitous and qualitatively homogeneous. With respect to the scaling exponent, evidence of three cross-overs is found, and complex heterogeneous associations with a number of relevant economic variables discussed. Lastly, the use of copulas as the main ingredient for modeling and forecasting realized measures of volatility is explored. The modeling background resembles but generalizes, the well-known Heterogeneous Autoregressive (HAR) model. In-sample and out-of-sample analyses, based on several performance measures, statistical tests, and robustness checks, show forecasting improvements of copula-based modeling over the HAR benchmark