Abstract
We present a mathematical study of the order book as a multidimensional continuous-time Markov chain where the order flow is modelled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent-based modelling), and the Stochastic Differential Equations approach used classically to describe price evolution in macroscopic time scales. To do this we rely on the theory of infinitesimal generators. We motivate our approach using an elementary example where the spread is kept constant (“perfect market making”). Then we compute the infinitesimal generator associated with the order book in a general setting, and link the price dynamics to the instantaneous state of the order book. In the last section, we prove the stationarity of the order book and give some hints about the behaviour of the price process in long time scales.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
W. Whitt. Stochastic — Process Limits. Springer, 2002
E. Smith, D. Farmer, L. Guillemot, and S. Krishnamurthy. Statistical theory of the continuous double auction. Quantitative Finance, 2003
R. Cont, S. Stoikov, and R. Talreja. A stochastic model for order book dynamics. Operations Research, 2008
P. Robert. Stochastic Networks and Queues. Springer, 2003
F. Abergel and A. Jedidi. In preparation
T. Preis, S. Golke, W. Paul, and J.J. Schneider. Multi-agent-based order book model of financial markets. Europhysics Letters, 2006
A. Ferraris. Equity market impact models (Presentation). 2008
S. Asmussen. Applied Probability and Queues. Springer, 2003
M. Bossy and N. Champagnat. Markov processes and parabolic partial differential equations. Encyclopedia of Quantitative Finance. R. Cont (Ed). Wiley, 2010
J.P. Bouchaud, M. Mézard, and M. Potters. Statistical properties of stock order books: empirical results and models. Quantitative Finance, 2002
S. Ethier and T. Kurtz. Markov Processes: Characterization and Convergence. Wiley, 1986
J. Gatheral and R. Oomen. Zero-intelligence realized variance estimation. Finance and Stochastics, 2007
F. Klebaner. Introduction to Stochastic Calculus with Applications. Imperial College Press, 2005
L.C.G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales. Cambridge University Press, 2000
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Italia
About this chapter
Cite this chapter
Abergel, F., Jedidi, A. (2011). A Mathematical Approach to Order Book Modelling. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds) Econophysics of Order-driven Markets. New Economic Windows. Springer, Milano. https://doi.org/10.1007/978-88-470-1766-5_7
Download citation
DOI: https://doi.org/10.1007/978-88-470-1766-5_7
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-1765-8
Online ISBN: 978-88-470-1766-5
eBook Packages: Business and EconomicsEconomics and Finance (R0)