Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T19:59:50.121Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Strategy for Limit Order Book Submissions in High Frequency Trading

Part of: Game theory Computer system organization Special processes

Published online by Cambridge University Press:  12 May 2016

Na Song ,
Yue Xie ,
Wai-Ki Ching ,
Tak-Kuen Siu  and
Cedric Ka-Fai Yiu
Na Song*
Affiliation:
School of Management and Economics, University of Electronic Science and Technology, Chengdu, China
Yue Xie*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Wai-Ki Ching*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Tak-Kuen Siu*
Affiliation:
Department of Applied Finance and Actuarial Studies, Macquarie University, Macquarie University, Sydney, NSW 2109, Australia
Cedric Ka-Fai Yiu*
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
*
*Corresponding author. Email addresses:songna@uestc.edu.cn (N. Song), xieyue1988moon@163.com (Y. Xie), wching@hku.hk (W.-K. Ching), ktksiu2005@gmail.com (T.-K. Siu), macyiu@polyu.edu.hk (C. K.-F. Yiu)
*Corresponding author. Email addresses:songna@uestc.edu.cn (N. Song), xieyue1988moon@163.com (Y. Xie), wching@hku.hk (W.-K. Ching), ktksiu2005@gmail.com (T.-K. Siu), macyiu@polyu.edu.hk (C. K.-F. Yiu)
*Corresponding author. Email addresses:songna@uestc.edu.cn (N. Song), xieyue1988moon@163.com (Y. Xie), wching@hku.hk (W.-K. Ching), ktksiu2005@gmail.com (T.-K. Siu), macyiu@polyu.edu.hk (C. K.-F. Yiu)
*Corresponding author. Email addresses:songna@uestc.edu.cn (N. Song), xieyue1988moon@163.com (Y. Xie), wching@hku.hk (W.-K. Ching), ktksiu2005@gmail.com (T.-K. Siu), macyiu@polyu.edu.hk (C. K.-F. Yiu)
*Corresponding author. Email addresses:songna@uestc.edu.cn (N. Song), xieyue1988moon@163.com (Y. Xie), wching@hku.hk (W.-K. Ching), ktksiu2005@gmail.com (T.-K. Siu), macyiu@polyu.edu.hk (C. K.-F. Yiu)
Get access

Abstract

An optimal selection problem for bid and ask quotes subject to a stock inventory constraint is investigated, formulated as a constrained utility maximisation problem over a finite time horizon. The arrivals of buy and sell orders are governed by Poisson processes, and a diffusion approximation is employed on assuming the Poisson arrivals intensity is sufficiently large. Using the dynamic programming principle, we adopt an efficient numerical procedure to solve this constrained utility maximisation problem based on a successive approximation algorithm, and conduct numerical experiments to analyse the impacts of the inventory constraint on a dealer's terminal profit and stock inventory level. It is found that the stock inventory constraint significantly affects the terminal stock inventory level.

Keywords

High-frequency tradingLimit Order Book (LOB)Diffusion ApproximationHamilton-Jacobi-Bellman (HJB) Equation

MSC classification

Secondary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling 91A80: Applications of game theory
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 222 - 234
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avellaneda, M. and Stoikov, S., High-frequency trading in a limit order book, Quant. Finance 8, 217–242 (2008).CrossRefGoogle Scholar
[2]Chang, M. and Krishna, K., A successive approximation algorithm for stochastic control problems, Appl. Math. Comp. 18, 155–165 (1986).CrossRefGoogle Scholar
[3]Feller, W., An Introduction to Probability Theory and Its Applications, Second Ed. Wiley (1991).Google Scholar
[4]Grandell, J., Aspects of Risk Theory, Springer (1991).CrossRefGoogle Scholar
[5]Gabaix, X., Gopikrishnan, P., Plerou, V. and Stanley, H., A theory of large fluctuations in stock market activity, MIT Department of Economics Working Paper 8, No. 03-30 (2004).CrossRefGoogle Scholar
[6]Haldane, G.A., Patience and Finance, Bank of England (2010).Google Scholar
[7]Herzog, F., Peyrl, H. and Geering, H., Proof of the convergence of the Successive Approximation Algorithm for numerically solving the Hamilton-Jacobi-Bellman Equation, WSEAS Trans. Systems 12, 2238–2245 (2005).Google Scholar
[8]Ho, T. and Stoll, H., Optimal dealer pricing under transaction and return uncertainty, J. Financial Economics 9(1), 47–73 (1981).CrossRefGoogle Scholar
[9]Ho, T. and Stoll, H., On dealer markets under competition, J. Financial Economics 35(2), 259–267 (1980).Google Scholar
[10]Kobayashi, H., Application of the diffusion approximation to queueing networks I: Equilibrium queue distributions, J. ACM 21, 316–328 (1974).CrossRefGoogle Scholar
[11]Manoj, N., Public Commentary on SEC Market Structure Concept Release, Tradeworx, Inc (2010).Google Scholar
[12] Maslow, S. and Mills, M., Price fluctuations from the order book perspective: Empirical facts and a simple model, Physica A 299, 234–246 (2001).CrossRefGoogle Scholar
[13]Nagaev, A., Nagaev, S. and Kunst, R., A diffusion approximation to the Markov Chains Model of the financial market and the expected riskless profit underselling of call andput options, Economics (2005).Google Scholar
[14]Øksendal, B., Stochastic Differential Equations: An Introduction with Applications, Springer (2003).CrossRefGoogle Scholar
[15]Potters, M. and Bouchaud, J., More statistical properties of order books and price impact, Physica A: Statistical Mech. Appl. 324, 133–140 (2001).CrossRefGoogle Scholar
[16]Song, N., Ching, W., Siu, T. and Yiu, C., Optimal submission problem in a limit order book with VaR constraints, Proc. Fourth Int. Joint Conference on Computational Sciences and Optimization (CSO2012), 266–270, Harbin, Heilongjiang Province, China (2012).Google Scholar
[17]Strikwerda, J., Finite Differential Schemes and Partial Differential Equations, Wadsworth & Brooks, California (1989).Google Scholar