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September  2022, 18(5): 3215-3231. doi: 10.3934/jimo.2021109

Generalized optimal liquidation problems across multiple trading venues

1. 

Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

2. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

3. 

Department of Actuarial Studies and Business Analytics, Macquarie Business School, Macquarie University, Sydney, Australia

Received  October 2019 Revised  April 2021 Published  September 2022 Early access  June 2021

In this paper, we generalize the Almgren-Chriss's market impact model to a more realistic and flexible framework and employ it to derive and analyze some aspects of optimal liquidation problem in a security market. We illustrate how a trader's liquidation strategy alters when multiple venues and extra information are brought into the security market and detected by the trader. This study gives some new insights into the relationship between liquidation strategy and market liquidity, and provides a multi-scale approach to the optimal liquidation problem with randomly varying volatility.

Citation: Qing-Qing Yang, Wai-Ki Ching, Jia-Wen Gu, Tak-Kuen Siu. Generalized optimal liquidation problems across multiple trading venues. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3215-3231. doi: 10.3934/jimo.2021109
References:
[1]

R. Almgren, Optimal execution of portfolio transactions, Risk, 3 (2001), 5-40. 

[2]

R. Almgren, Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Mathematical Finance, 10 (2003), 1-18. 

[3]

R. Almgren, Optimal trading with stochastic liquidity and volatility, SIAM J. Financial Math., 3 (2012), 163-181.  doi: 10.1137/090763470.

[4]

R. Almgren and N. Chriss, Value under liquidation, Risk, 12 (1999), 61-63. 

[5]

P. Brugiere, Optimal portfolio and optimal trading in a dynamic continuous time framework, 6th AFIR Colloquium. Nurenberg, Germany, 12 (1996), 89.

[6]

M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[7]

J. FouqueG. Papanicolaou and R. Sircar, Mean-reverting stochastic volatility, International Journal of Theoretical and Applied Finance, 30 (2000), 101-142. 

[8]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Sølna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge: Cambridge University Press, 2011. doi: 10.1017/CBO9781139020534.

[9]

H. M. SonerS. E. Shreve and J. Cvitanić, There is no nontrivial hedging portfolio for option pricing with transaction costs, Ann. Appl. Probab., 5 (1995), 327-355. 

[10]

S. T. TseP. A. ForsythJ. S. Kennedy and H. Windcliff, Comparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies, Appl. Math. Finance, 20 (2013), 415-449.  doi: 10.1080/1350486X.2012.755817.

[11]

Q.-Q. YangW.-K. ChingJ.-W. Gu and T.-K. Siu, Market-making strategy with asymmetric information and regime-switching, J. Econom. Dynam. Control, 90 (2018), 408-433.  doi: 10.1016/j.jedc.2018.04.003.

show all references

References:
[1]

R. Almgren, Optimal execution of portfolio transactions, Risk, 3 (2001), 5-40. 

[2]

R. Almgren, Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Mathematical Finance, 10 (2003), 1-18. 

[3]

R. Almgren, Optimal trading with stochastic liquidity and volatility, SIAM J. Financial Math., 3 (2012), 163-181.  doi: 10.1137/090763470.

[4]

R. Almgren and N. Chriss, Value under liquidation, Risk, 12 (1999), 61-63. 

[5]

P. Brugiere, Optimal portfolio and optimal trading in a dynamic continuous time framework, 6th AFIR Colloquium. Nurenberg, Germany, 12 (1996), 89.

[6]

M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[7]

J. FouqueG. Papanicolaou and R. Sircar, Mean-reverting stochastic volatility, International Journal of Theoretical and Applied Finance, 30 (2000), 101-142. 

[8]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Sølna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge: Cambridge University Press, 2011. doi: 10.1017/CBO9781139020534.

[9]

H. M. SonerS. E. Shreve and J. Cvitanić, There is no nontrivial hedging portfolio for option pricing with transaction costs, Ann. Appl. Probab., 5 (1995), 327-355. 

[10]

S. T. TseP. A. ForsythJ. S. Kennedy and H. Windcliff, Comparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies, Appl. Math. Finance, 20 (2013), 415-449.  doi: 10.1080/1350486X.2012.755817.

[11]

Q.-Q. YangW.-K. ChingJ.-W. Gu and T.-K. Siu, Market-making strategy with asymmetric information and regime-switching, J. Econom. Dynam. Control, 90 (2018), 408-433.  doi: 10.1016/j.jedc.2018.04.003.

Figure 1.  The performance of the "Constant-Vol" strategy
Table 1.  1, 000 simulations with $ Q = 100, s = 15, $ w.r.t. $ \mathcal R_T $
Statistics Constant-Vol Vol-adjust
Mean -300.70 -288.46
Std 54.76 27.45
Skewness 1.03 0.24
Kurtosis 5.26 3.05
Objective function -577.58 -560.36
Statistics Constant-Vol Vol-adjust
Mean -300.70 -288.46
Std 54.76 27.45
Skewness 1.03 0.24
Kurtosis 5.26 3.05
Objective function -577.58 -560.36
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