Skip to main content
SpringerLink
Log in

Explicit Solution for Constrained Optimal Execution Problem with General Correlated Market Depth

  • Published:
Journal of the Operations Research Society of China Aims and scope Submit manuscript

Abstract

This work studies the constrained optimal execution problem with a random market depth in the limit order market. Motivated from the real trading activities, our execution model considers the execution bounds and allows the random market depth to be statistically correlated in different periods. Usually, it is difficult to achieve the analytical solution for this class of constrained dynamic decision problem. Thanks to the special structure of this model, by applying the proposed state separation theorem and dynamic programming, we successfully obtain the analytical execution policy. The revealed policy is of feedback nature. Examples are provided to illustrate our solution methods. Simulation results demonstrate the advantages of our model comparing with the classical execution policy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Note that \(p_k\) is the price before the trading happens at stage k.

  2. By using this model, the filtration \(\mathcal {F}_k\) can be specified as the sigma-algebra generated by the realization of the random variables until time \(k-1\), i.e., the realizations of \(\theta _0,\varepsilon _0\), \(\theta _1, \varepsilon _1, \cdots , \theta _{k-1}, \varepsilon _{k-1}\).

References

  1. Almgren, R., Chriss, N.: Value under liquidation. Risk 12, 61–63 (1999)

    Google Scholar 

  2. Bertsimas, D., Lo, A.W.: Optimal control of execution costs. J. Financ. Mark. 1(1), 1–50 (1998)

    Article  Google Scholar 

  3. Almgren, R., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3(2), 5–39 (2001)

    Article  Google Scholar 

  4. Almgren, R.: Optimal execution with nonlinear impact functions and trading-enhanced risk. Appl. Math. Finance 10(1), 1–18 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Almgren, R., Thum, C., Hauptmann, E., Li, E.: Equity market impact. Risk 7, 57–62 (2005)

    Article  Google Scholar 

  6. Chen, J., Feng, L., Peng, J., Ye, Y.: Analytical results and efficient algorithm for optimal portfolio deleveraging with market impact. Oper. Res. 62(1), 195–206 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cont, R., Stoikov, S., Talreja, R.: A stochastic model for order book dynamics. Oper. Res. 58(3), 549–563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gould, M.D., Porter, M.A., Williams, S., McDonald, M.D., Fenn, J., Howison, S.D.: Limit order books. Quant. Finance 13(11), 1709–1742 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rosu, I.: A dynamic model of the limit order book. Rev. Financ. Stud. 22(11), 4601–4641 (2009)

    Article  Google Scholar 

  10. Obizhaeva, A.A., Wang, J.: Optimal trading strategy and supply/demand dynamics. J. Financ. Mark. 16(1), 1–32 (2013)

    Article  Google Scholar 

  11. Kavajecz, K.A.: A specialists quoted depth and the limit order book. J. Finance 54(2), 747–771 (1999)

    Article  Google Scholar 

  12. Kempf, A., Korn, O.: Market depth and order size. J. Financ. Mark. 2(1), 29–48 (1999)

    Article  Google Scholar 

  13. Ranaldo, A.: Order aggressiveness in limit order book markets. J. Financ. Mark. 7(1), 53–74 (2004)

    Article  Google Scholar 

  14. Alfonsi, A., Fruth, A., Schied, A.: Optimal execution strategies in limit order books with general shape functions. Quant. Finance 10(2), 143–157 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tsoukalas, G., Wang, J., Giesecke, K.: Dynamic portfolio execution. Manag. Sci. (2017)

  16. Chen, N.Y., Kou, S., Wang, C.: A partitioning algorithm for markov decision processes and its applications to market microstructure. Manag. Sci. (2017). https://doi.org/10.1287/mnsc.2016.2639

  17. Potters, M., Bouchaud, J.P.: More statistical properties of order books and price impact. Phys. A 324, 133–140 (2003)

    Article  MATH  Google Scholar 

  18. Bouchaud, J., Gefen, Y., Potters, M., Wyart, M.: Fluctuations and response in financial markets: the subtle nature of random price changes. Quant. Finance 4(2), 176–190 (2004)

    Article  Google Scholar 

  19. Weber, P., Rosenow, B.: Order book approach to price impact. Quant. Finance 5(4), 357–364 (2005)

    Article  MATH  Google Scholar 

  20. Biais, B., Hillion, P., Spatt, C.: An empirical analysis of the limit order book and the order flow in the paris bourse. J. Finance 50(5), 1655–1689 (1995)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Jian-Jun Gao

  2. Department of Automation, Shanghai Jiao Tong University, Shanghai, 200241, China

    Wei-Ping Wu

Authors
  1. Wei-Ping Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian-Jun Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-Jun Gao.

Additional information

This paper is dedicated to Professor Duan Li in celebration of his 65th birthday.

This research is partially supported by the National Natural Science Foundation of China (No.61573244).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, WP., Gao, JJ. Explicit Solution for Constrained Optimal Execution Problem with General Correlated Market Depth. J. Oper. Res. Soc. China 6, 159–174 (2018). https://doi.org/10.1007/s40305-018-0197-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40305-018-0197-3

Keywords

Mathematics Subject Classification

Search

Navigation