July  2016, 21(5): 1401-1420. doi: 10.3934/dcdsb.2016002

Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact

1. 

Department of mathematics, Tongji University, Shanghai 200092

2. 

Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom

3. 

Department of Mathematics, Imperial College, London SW7 2AZ

Received  October 2013 Revised  March 2014 Published  April 2016

In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.
Citation: Baojun Bian, Nan Wu, Harry Zheng. Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1401-1420. doi: 10.3934/dcdsb.2016002
References:
[1]

P. Bank and D. Baum, Hedging and portfolio optimization in financial markets with a large trader, Mathematical Finance, 14 (2004), 1-18. doi: 10.1111/j.0960-1627.2004.00179.x.  Google Scholar

[2]

F. Black, Towards a fully automated exchange: Part 1, Financial Analyst Journal, 27 (1971), 29-34. Google Scholar

[3]

U. Çetin, R. A. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 8 (2004), 311-341. doi: 10.1007/s00780-004-0123-x.  Google Scholar

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W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2006.  Google Scholar

[5]

J. Cvitanic and I. Karatzas, Hedging and portfolio optimization under transaction costs: A martingale approach, Mathematical Finance, 6 (1996), 370-398. doi: 10.1214/aoap/1034968136.  Google Scholar

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M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

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M. G. Crandall and P. L. Lions, Two apprximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, 43 (1984), 1-19. doi: 10.1090/S0025-5718-1984-0744921-8.  Google Scholar

[8]

P. Gassiat, F. Gozzi and H. Pham, Investment/consumption problem in illiquid markets with regimes switching, SIAM J Control Optimization, 52 (2014), 1761-1786. doi: 10.1137/120876976.  Google Scholar

[9]

E. Jouini, Price functionals with bid-ask spreads: An axiomatic approach, J. Mathematical Economics, 34 (2000), 547-558. doi: 10.1016/S0304-4068(99)00023-3.  Google Scholar

[10]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.  Google Scholar

[11]

H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar

[12]

M. Pemy and Q. Zhang, Optimal stock liquidation in a regime switching model with finite time horizon, J. Mathematical Analysis & Applications, 321 (2006), 537-552. doi: 10.1016/j.jmaa.2005.08.034.  Google Scholar

[13]

M. Pemy, Q. Zhang and G. Yin, Liquidation of a large block of stock, J. Banking & Finance, 31 (2007), 1295-1305. doi: 10.1016/j.jbankfin.2006.10.014.  Google Scholar

[14]

M. Pemy, Q. Zhang and G. Yin, Liquidation of a large block of stock with regime switching, Mathematical Finance, 18 (2008), 629-648. doi: 10.1111/j.1467-9965.2008.00351.x.  Google Scholar

[15]

A. Schied and T. Schöneborn, Optimal portfolio liquidation for CARA investors, SSRN Working Paper, (2007), 1-11. doi: 10.2139/ssrn.1018088.  Google Scholar

show all references

References:
[1]

P. Bank and D. Baum, Hedging and portfolio optimization in financial markets with a large trader, Mathematical Finance, 14 (2004), 1-18. doi: 10.1111/j.0960-1627.2004.00179.x.  Google Scholar

[2]

F. Black, Towards a fully automated exchange: Part 1, Financial Analyst Journal, 27 (1971), 29-34. Google Scholar

[3]

U. Çetin, R. A. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 8 (2004), 311-341. doi: 10.1007/s00780-004-0123-x.  Google Scholar

[4]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2006.  Google Scholar

[5]

J. Cvitanic and I. Karatzas, Hedging and portfolio optimization under transaction costs: A martingale approach, Mathematical Finance, 6 (1996), 370-398. doi: 10.1214/aoap/1034968136.  Google Scholar

[6]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin American Mathematical Society, 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[7]

M. G. Crandall and P. L. Lions, Two apprximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, 43 (1984), 1-19. doi: 10.1090/S0025-5718-1984-0744921-8.  Google Scholar

[8]

P. Gassiat, F. Gozzi and H. Pham, Investment/consumption problem in illiquid markets with regimes switching, SIAM J Control Optimization, 52 (2014), 1761-1786. doi: 10.1137/120876976.  Google Scholar

[9]

E. Jouini, Price functionals with bid-ask spreads: An axiomatic approach, J. Mathematical Economics, 34 (2000), 547-558. doi: 10.1016/S0304-4068(99)00023-3.  Google Scholar

[10]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.  Google Scholar

[11]

H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar

[12]

M. Pemy and Q. Zhang, Optimal stock liquidation in a regime switching model with finite time horizon, J. Mathematical Analysis & Applications, 321 (2006), 537-552. doi: 10.1016/j.jmaa.2005.08.034.  Google Scholar

[13]

M. Pemy, Q. Zhang and G. Yin, Liquidation of a large block of stock, J. Banking & Finance, 31 (2007), 1295-1305. doi: 10.1016/j.jbankfin.2006.10.014.  Google Scholar

[14]

M. Pemy, Q. Zhang and G. Yin, Liquidation of a large block of stock with regime switching, Mathematical Finance, 18 (2008), 629-648. doi: 10.1111/j.1467-9965.2008.00351.x.  Google Scholar

[15]

A. Schied and T. Schöneborn, Optimal portfolio liquidation for CARA investors, SSRN Working Paper, (2007), 1-11. doi: 10.2139/ssrn.1018088.  Google Scholar

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