# American Institute of Mathematical Sciences

March  2012, 2(1): 81-100. doi: 10.3934/mcrf.2012.2.81

## Optimal trend-following trading rules under a three-state regime switching model

 1 Department of Mathematics and Actuarial Science, Roosevelt University, Chicago, IL 60605, United States 2 Department of Mathematics, University of Georgia, Athens, GA 30602

Received  November 2010 Revised  November 2011 Published  January 2012

Momentum (or trend-following) trading strategies are widely used in the investment world. To better understand the nature of trend-following trading strategies and discover the corresponding optimality conditions, we consider the cases when the market trends are fully observable. In this paper, the market follows a regime switching model with three states (bull, sideways, and bear). Under this model, a set of sufficient conditions are developed to guarantee the optimality of trend-following trading strategies. A dynamic programming approach is used to verify these optimality conditions. The value functions are characterized by the associated HJB equations and are shown to be either linear functions or infinity depending on the parameter values. The results in this paper will help an investor to identify market conditions and to avoid trades which might be unprofitable even under the best market information. Finally, the corresponding value functions will provide an upper bound for trading performance which can be used as a general guide to rule out unrealistic expectations.
Citation: Jie Yu, Qing Zhang. Optimal trend-following trading rules under a three-state regime switching model. Mathematical Control and Related Fields, 2012, 2 (1) : 81-100. doi: 10.3934/mcrf.2012.2.81
##### References:
 [1] N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49. doi: 10.3905/jod.1998.408011. [2] J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497-514. doi: 10.1142/S0219024902001523. [3] A. Cadenillas and S. R. Pliska, Optimal trading of a security when there are taxes and transaction costs, Finance & Stochastics, 3 (1999), 137-165. doi: 10.1007/s007800050055. [4] G. M. Constantinides, Capital market equilibrium with personal tax, Econometrica, 51 (1983), 611-636. doi: 10.2307/1912150. [5] M. Dai, Q. Zhang and Q. J. Zhu, Trend following trading under a regime switching model, SIAM J. Financial Math, 1 (2010), 780-810. doi: 10.1137/090770552. [6] R. M. Dammon and C. S. Spatt, The optimal trading and pricing of securities with asymmetric capital gains taxes and transaction costs, Rev. Financial Studies, 9 (1996), 921-952. doi: 10.1093/rfs/9.3.921. [7] R. J. Elliott, "Stochastic Calculus and Applications," Springer-Verlag, New York, 1982. [8] X. Guo, "Inside Information and Stock Fluctuations," Ph.D thesis, Rutgers University, 1999. [9] X. Guo and Q. Zhang, Optimal selling rules in a regime switching model, IEEE Trans. Automatic Control, 50 (2005), 1450-1455. doi: 10.1109/TAC.2005.854657. [10] J. D. Hamilton, A new approach to the economic analysis of non-stationary time series, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559. [11] K. Helmes, Computing optimal selling rules for stocks using linear programming, in "Mathematics of Finance" (eds. G. Yin and Q. Zhang), Contemporary Mathematics, 351, American Mathematical Society, Providence, RI, (2004), 187-198. [12] T. C. Johnson and M. Zervos, The optimal timing of investment decisions, work in progress, 2006. [13] I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Springer, New York, 1998. [14] H. T. Kong and Q. Zhang, An optimal trading rule of a mean-reverting asset, Discrete and Continuous Dynamical System Series B, 14 (2010), 1403-1417. doi: 10.3934/dcdsb.2010.14.1403. [15] H. T. Kong, Q. Zhang and G. Yin, A trend-following strategy: Conditions for optimality, Automatica, 47 (2011), 661-667. doi: 10.1016/j.automatica.2011.01.039. [16] R. Liu, G. Yin and Q. Zhang, Option pricing in a regime switching model using the fast Fourier transform, Applied Mathematics and Stochastic Analysis, 2006, Art. ID 18109, 22 pp. [17] A. Løkka and M. Zervos, Long-term optimal real investment strategies in the presence of adjustment costs, work in progress, 2007. [18] G. B. Di Masi, Y. M. Kabanov and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probability and Applications, 39 (1994), 173-181. doi: 10.1137/1139008. [19] A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control Optim., 46 (2007), 839-876. doi: 10.1137/050640758. [20] B. Øksendal, "Stochastic Differential Equations," 6th edition, Springer-Verlag, New York, 2003. [21] D. D. Yao, Q. Zhang and X. Y. Zhou, A regime-switching model for European options, in "Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems" (eds. H. Yan, G. Yin and Q. Zhang), International Series in Operations Research and Management Sciences, 94, Springer, New York, (2006), 281-300. [22] G. Yin, R. H. Liu and Q. Zhang, Recursive algorithms for stock liquidation: A stochastic optimization approach, SIAM J. Optim., 13 (2002), 240-263. doi: 10.1137/S1052623401392901. [23] G. Yin and C. Zhu, "Hybrid Switching Diffusions: Properties and Applications," Springer, New York, 2010. [24] H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica J. IFAC, 44 (2008), 1511-1518. doi: 10.1016/j.automatica.2007.11.003. [25] Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 64-87. doi: 10.1137/S0363012999356325. [26] Q. Zhang and G. Yin, Nearly optimal asset allocation in hybrid stock-investment models, J. Optim. Theory Appl., 121 (2004), 419-444. doi: 10.1023/B:JOTA.0000037412.23243.6c. [27] X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.

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##### References:
 [1] N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49. doi: 10.3905/jod.1998.408011. [2] J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497-514. doi: 10.1142/S0219024902001523. [3] A. Cadenillas and S. R. Pliska, Optimal trading of a security when there are taxes and transaction costs, Finance & Stochastics, 3 (1999), 137-165. doi: 10.1007/s007800050055. [4] G. M. Constantinides, Capital market equilibrium with personal tax, Econometrica, 51 (1983), 611-636. doi: 10.2307/1912150. [5] M. Dai, Q. Zhang and Q. J. Zhu, Trend following trading under a regime switching model, SIAM J. Financial Math, 1 (2010), 780-810. doi: 10.1137/090770552. [6] R. M. Dammon and C. S. Spatt, The optimal trading and pricing of securities with asymmetric capital gains taxes and transaction costs, Rev. Financial Studies, 9 (1996), 921-952. doi: 10.1093/rfs/9.3.921. [7] R. J. Elliott, "Stochastic Calculus and Applications," Springer-Verlag, New York, 1982. [8] X. Guo, "Inside Information and Stock Fluctuations," Ph.D thesis, Rutgers University, 1999. [9] X. Guo and Q. Zhang, Optimal selling rules in a regime switching model, IEEE Trans. Automatic Control, 50 (2005), 1450-1455. doi: 10.1109/TAC.2005.854657. [10] J. D. Hamilton, A new approach to the economic analysis of non-stationary time series, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559. [11] K. Helmes, Computing optimal selling rules for stocks using linear programming, in "Mathematics of Finance" (eds. G. Yin and Q. Zhang), Contemporary Mathematics, 351, American Mathematical Society, Providence, RI, (2004), 187-198. [12] T. C. Johnson and M. Zervos, The optimal timing of investment decisions, work in progress, 2006. [13] I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Springer, New York, 1998. [14] H. T. Kong and Q. Zhang, An optimal trading rule of a mean-reverting asset, Discrete and Continuous Dynamical System Series B, 14 (2010), 1403-1417. doi: 10.3934/dcdsb.2010.14.1403. [15] H. T. Kong, Q. Zhang and G. Yin, A trend-following strategy: Conditions for optimality, Automatica, 47 (2011), 661-667. doi: 10.1016/j.automatica.2011.01.039. [16] R. Liu, G. Yin and Q. Zhang, Option pricing in a regime switching model using the fast Fourier transform, Applied Mathematics and Stochastic Analysis, 2006, Art. ID 18109, 22 pp. [17] A. Løkka and M. Zervos, Long-term optimal real investment strategies in the presence of adjustment costs, work in progress, 2007. [18] G. B. Di Masi, Y. M. Kabanov and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probability and Applications, 39 (1994), 173-181. doi: 10.1137/1139008. [19] A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control Optim., 46 (2007), 839-876. doi: 10.1137/050640758. [20] B. Øksendal, "Stochastic Differential Equations," 6th edition, Springer-Verlag, New York, 2003. [21] D. D. Yao, Q. Zhang and X. Y. Zhou, A regime-switching model for European options, in "Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems" (eds. H. Yan, G. Yin and Q. Zhang), International Series in Operations Research and Management Sciences, 94, Springer, New York, (2006), 281-300. [22] G. Yin, R. H. Liu and Q. Zhang, Recursive algorithms for stock liquidation: A stochastic optimization approach, SIAM J. Optim., 13 (2002), 240-263. doi: 10.1137/S1052623401392901. [23] G. Yin and C. Zhu, "Hybrid Switching Diffusions: Properties and Applications," Springer, New York, 2010. [24] H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica J. IFAC, 44 (2008), 1511-1518. doi: 10.1016/j.automatica.2007.11.003. [25] Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 64-87. doi: 10.1137/S0363012999356325. [26] Q. Zhang and G. Yin, Nearly optimal asset allocation in hybrid stock-investment models, J. Optim. Theory Appl., 121 (2004), 419-444. doi: 10.1023/B:JOTA.0000037412.23243.6c. [27] X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.
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