Modeling High-Frequency Order Flow Imbalance by Functional Limit Theorems for Two-Sided Risk Processes

A micro-scale model is proposed for the evolution of the limit order book. Within this model, the flows of orders (claims) are described by doubly stochastic Poisson processes taking account of the stochastic character of intensities of bid and ask orders that determine the price discovery mechanism in financial markets. The process of order flow imbalance (OFI) is studied. This process is a sensitive indicator of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. The OFI process tracks best bid and ask queues and change much faster than prices. It incorporates information about build-ups and depletions of order queues so that it can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The two-sided risk processes are suggested as mathematical models of the OFI process. The multiplicative model is proposed for the stochastic intensities making it possible to analyze the characteristics of order flows as well as the instantaneous proportion of the forces of buyers and sellers, that is, the intensity imbalance (II) process, without modeling the external information background. The proposed model gives the opportunity to link the micro-scale (high-frequency) dynamics of the limit order book with the macro-scale models of stock price processes of the form of subordinated Wiener processes by means of functional limit theorems of probability theory and hence, to give a deeper insight in the nature of popular subordinated Wiener processes such as generalized hyperbolic Levy processes as models of the evolution of characteristics of financial markets. In the proposed models, the subordinator is determined by the evolution of the stochastic intensity of the external information flow.

[1]  Frédéric Abergel,et al.  Econophysics: Empirical facts and agent-based models , 2009, 0909.1974.

[2]  O. Barndorff-Nielsen,et al.  Models for non-Gaussian variation, with applications to turbulence , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[3]  Vladimir E. Bening,et al.  Generalized Poisson Models and their Applications in Insurance and Finance , 2002 .

[4]  Rama Cont,et al.  The Price Impact of Order Book Events , 2010, 1011.6402.

[5]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[6]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[7]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[8]  V. Korolev,et al.  Convergence of Random Sequences with Independent Random Indices II , 1995 .

[9]  V. Korolev,et al.  On Convergence of Distributions of Compound Cox Processes to Stable Laws , 1999 .

[10]  Ioane Muni Toke,et al.  "Market making" behaviour in an order book model and its impact on the bid-ask spread , 2010, 1003.3796.

[11]  O. Barndorff-Nielsen.  Exponentially decreasing distributions for the logarithm of particle size , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  Rama Cont,et al.  Price Dynamics in a Markovian Limit Order Market , 2011, SIAM J. Financial Math..

[13]  Rama Cont,et al.  Order Book Dynamics in Liquid Markets: Limit Theorems and Diffusion Approximations , 2012, 1202.6412.

[14]  P. Clark.  A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices , 1973 .

[15]  Alexander I. Zeifman,et al.  The decay function of nonhomogeneous birth-death processes, with application to mean-field models , 1997 .

[16]  Michio Suzuki.  A transfer theorem , 1978 .

[17]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[18]  Alexander I. Zeifman,et al.  Modelling stock order flows with non-homogeneous intensities from high-frequency data , 2013 .

[19]  E. Eberlein,et al.  New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model , 1998 .

[20]  E. Eberlein,et al.  The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures , 2002 .

[21]  Christine A. Parlour,et al.  Equilibrium in a Dynamic Limit Order Market , 2003 .

[22]  Karsten Prause,et al.  Modelling Financial Data Using Generalized Hyperbolic Distributions , 1997 .

[23]  E. Eberlein.  Application of Generalized Hyperbolic Lévy Motions to Finance , 2001 .

[24]  Christine A. Parlour.  Price Dynamics in Limit Order Markets , 1998 .

[25]  Thierry Foucault,et al.  Order flow composition and trading costs in a dynamic limit order market 1 I am grateful to Bruno Bi , 1999 .

[26]  O. Barndorff-Nielsen,et al.  A parsimonious and universal description of turbulent velocity increments , 2004 .

[27]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[28]  B. Gnedenko,et al.  Random Summation: Limit Theorems and Applications , 1996 .

[29]  O. Barndorff-Nielsen,et al.  Normal Variance-Mean Mixtures and z Distributions , 1982 .

[30]  Rama Cont,et al.  A Stochastic Model for Order Book Dynamics , 2008, Oper. Res..

[31]  J. Grandell.  Doubly stochastic Poisson processes , 1976 .

[32]  V. Seshadri,et al.  Halphen's Laws , 2006 .

[33]  S. Kotz,et al.  Encyclopedia of Statistical Sciences: Update Volume 1 , 1998 .

[34]  Andrey Gorshenin,et al.  Statistical properties of the dynamics of order books: empirical results , 2012 .

[35]  A. Shiryaev.  Essentials of stochastic finance , 1999 .

[36]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[37]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[38]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .

[39]  B. Mandelbrot.  Fractals and Scaling In Finance: Discontinuity, Concentration, Risk , 2010 .

[40]  V. Korolev,et al.  Generalized Hyperbolic Laws as Limit Distributions for Random Sums , 2014 .

[41]  Alexander I. Zeifman.  Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes , 1995 .

[42]  V. Yu. Korolev.  Asymptotic Properties of Extrema of Compound Cox Processes and Their Applications to Some Problems of Financial Mathematics , 2001 .

[43]  He Huang,et al.  A generalized birth–death stochastic model for high-frequency order book dynamics , 2012 .

[44]  Ioanid Roşu.  A Dynamic Model of the Limit Order Book , 2008 .

[45]  Victor Korolev,et al.  Modeling High-Frequency Non-Homogeneous Order Flows by Compound Cox Processes* , 2014, Journal of Mathematical Sciences.

[46]  V. Korolev,et al.  ON THE CONVERGENCE OF DISTRIBUTIONS OF RANDOM SUMS OF INDEPENDENT RANDOM VARIABLES TO STABLE LAWS , 1998 .

[47]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[48]  M. Avellaneda,et al.  High-frequency trading in a limit order book , 2008 .

[49]  M. Mézard,et al.  Statistical Properties of Stock Order Books: Empirical Results and Models , 2002 .

[50]  Aymen Jedidi,et al.  A Mathematical Approach to Order Book Modeling , 2010, 1010.5136.

[51]  Hélyette Geman,et al.  Pure Jump Levy Processes for Asset Price Modelling , 2002 .

[52]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[53]  J. Farmer,et al.  The power of patience: a behavioural regularity in limit-order placement , 2002, cond-mat/0206280.

[54]  O. Barndorff-Nielsen,et al.  Lévy processes : theory and applications , 2001 .

[55]  Alexander Zeifman,et al.  On convergence of random walks generated by compound Cox processes to Lévy processes , 2013 .