Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T01:17:25.023Z Has data issue: false hasContentIssue false
  • English
  • Français

HYDRODYNAMIC LIMIT OF ORDER-BOOK DYNAMICS

Published online by Cambridge University Press:  28 November 2016

Xuefeng Gao  and
S.J. Deng
Xuefeng Gao
Affiliation:
Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong E-mail:[email protected]
S.J. Deng
Affiliation:
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA E-mail:[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we establish a fluid limit for a two-sided Markov order book model. The main result states that in a certain asymptotic regime, a pair of measure-valued processes representing the “sell-side shape” and “buy-side shape” of an order book converges to a pair of deterministic measure-valued processes in a certain sense. We also test the fluid approximation on data. The empirical results suggest that the approximation is reasonably good for liquidly traded stocks in certain time periods.

Keywords

applied probabilitymathematical financequeueing theorystochastic modeling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 1 , January 2018 , pp. 96 - 125
Copyright
Copyright © Cambridge University Press 2016 

References

1. Abergel, F. & Jedidi, A. (2013). On the stability and price scaling limit of a Hawkes process-based order book model. Working paper. Available at http://ssrn.com/abstract=2263162 Google Scholar
2. Aldous, D. (1981). Weak convergence and the general theory of processes. Incomplete draft of a monograph, http://www.stat.berkeley.edu/aldous/Papers/weak-gtp.pdf.Google Scholar
3. Bayer, C., Horst, U., & Qiu, J. (2014). A functional limit theorem for limit order books with state dependent price dynamics. arXiv preprint arXiv:1405.5230.Google Scholar
4. Billingsley, P. (1999). Convergence of probability measures. 2nd ed. New York: Wiley.Google Scholar
5. Blanchet, J. & Chen, X. (2013). Continuous-time modeling of bid-ask spread and price dynamics in limit order books. arXiv preprint arXiv:1310.1103.Google Scholar
6. Cont, R. (2015). High frequency dynamics of limit order markets: Multi-scale modeling and asymptotic analysis. Working paper. Slides available at https://workspace.imperial.ac.uk/quantitativefinance/Public/events/SLIDES%203RD%20IMPERIAL%20ETH/Rama%20Cont%202.pdf Google Scholar
7. Cont, R. & Larrard, De, A. (2011). Order book dynamics in liquid markets: Limit theorems and diffusion approximations. Working Paper. Available at http://ssrn.com/abstract=1757861 Google Scholar
8. Cont, R. & De Larrard, A. (2013). Price dynamics in a Markovian limit order market. SIAM Journal on Financial Mathematics, 4(1): 1–25.Google Scholar
9. Cont, R., Stoikov, S., & Talreja, R. (2010). A stochastic model for order book dynamics. Operations Research, 58(3): 549–563.Google Scholar
10. Dawson, D. (1993). Measure-valued Markov processes. Berlin, Heidelberg: Springer.Google Scholar
11. Driver, K.B. (2003). Analysis tools with applications. Lecture Notes. Available at http://www.math.ucsd.edu/~bdriver/240-01-02/Lecture_Notes/anal2p.pdf Google Scholar
12. Gould, M.D., Porter, M.A., Williams, S., McDonald, M., Fenn, D.J., & Howison, S.D. (2013). Limit order books. Quantitative Finance, 13(11): 1709–1742.CrossRefGoogle Scholar
13. Hautsch, N. & Huang, R. (2011). Limit order flow, market impact and optimal order sizes: Evidence from NASDAQ TotalView–ITCH data. Working Paper. Available at http://ssrn.com/abstract=1914293 Google Scholar
14. Horst, U. & Kreher, D. (2015). A weak law of large numbers for a limit order book model with fully state dependent order dynamics. arXiv preprint arXiv:1502.04359.Google Scholar
15. Horst, U. & Paulsen, M. (2015). A law of large numbers for limit order books. arXiv preprint arXiv:1501.00843.Google Scholar
16. Huang, H. & Kercheval, A.N. (2012). A generalized birth–death stochastic model for high-frequency order book dynamics. Quantitative Finance, 12(4): 547–557.Google Scholar
17. Jacod, J. & Shiryaev, A.N. (2003). Limit Theorems for stochastic processes. 2nd ed. Berlin: Springer.Google Scholar
18. Kipnis, C. & Landim, C. (1999). Scaling limits of interacting particle systems, Vol. 320. Berlin, Heidelberg: Springer.Google Scholar
19. Kirilenko, A., Sowers, R.B., & Meng, X. (2013). A multiscale model of high-frequency trading. Algorithmic Finance, 2(1): 59–98.Google Scholar
20. Kotelenez, P. (2010). Stochastic flows and signed measure valued stochastic partial differential equations. Theory of Stochastic Process, 16(2): 86–105.Google Scholar
21. Kruk, Ł. (2003). Functional limit theorems for a simple auction. Mathematics of Operations Research, 28(4): 716–751.Google Scholar
22. Kruk, Ł. (2012). Limiting distribution for a simple model of order book dynamics. Central European Journal of Mathematics, 10(6): 2283–2295.Google Scholar
23. Kurtz, T.G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8(2): 344–356.Google Scholar
24. Lakner, P., Reed, J., & Simatos, F. (2015). Scaling limits for a limit order book model using the regenerative characterization of Levy trees. http://people.stern.nyu.edu/jreed/papers/paper20.pdf Google Scholar
25. Lakner, P., Reed, J., & Stoikov, S. (2016). High frequency asymptotics for the limit order book. Market Microstructure and Liquidity. To appear. http://people.stern.nyu.edu/jreed/papers/paper19.pdf Google Scholar
26. Maglaras, C., Moallemi, C.C., & Zheng, H. (2015). Optimal execution in a limit order book and an associated microstructure market impact model. Available at https://ssrn.com/abstract=2610808 Google Scholar
27. Massey, W.A. (1987). Stochastic orderings for Markov processes on partially ordered spaces. Mathematics of Operations Research, 12(2): 350–367.Google Scholar
28. Parlour, C.A. & Seppi, D.J. (2008). Limit order markets: A survey. Handbook of financial intermediation and banking, 5: 63–95.Google Scholar
29. Protter, P.E. (2004). Stochastic integration and differential equations: Version 2.1, Vol. 21. Berlin, Heidelberg: Springer.Google Scholar
30. Seppäläinen, T. (2008). Translation invariant exclusion processes. Lecture notes. Available at http://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf.Google Scholar
31. Spiliopoulos, K., Sirignano, J.A., & Giesecke, K. (2014). Fluctuation analysis for the loss from default. Stochastic Processes and their Applications, 124(7): 2322–2362.Google Scholar
32. Tao, T. (2006). Nonlinear dispersive and wave equations: Local and global analysis. Number 106. American Mathematical Society.Google Scholar
33. Whitt, W. (2002). Stochastic-process limits. New York: Springer.Google Scholar