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 and 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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[15]

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

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