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September  2016, 6(3): 467-488. doi: 10.3934/mcrf.2016012

An optimal mean-reversion trading rule under a Markov chain model

Jingzhi Tie 1, and  Qing Zhang 1,

1. 

Department of Mathematics, University of Georgia, Athens, GA 30602, United States, United States

Received  February 2015 Revised  July 2015 Published  August 2016

This paper is concerned with a mean-reversion trading rule. In contrast to most market models treated in the literature, the underlying market is solely determined by a two-state Markov chain. The major advantage of such Markov chain model is its striking simplicity and yet its capability of capturing various market movements. The purpose of this paper is to study an optimal trading rule under such a model. The objective of the problem under consideration is to find a sequence stopping (buying and selling) times so as to maximize an expected return. Under some suitable conditions, explicit solutions to the associated HJ equations (variational inequalities) are obtained. The optimal stopping times are given in terms of a set of threshold levels. A verification theorem is provided to justify their optimality. Finally, a numerical example is provided to illustrate the results.
Keywords: optimal stopping, Mean-reversion Markovian asset, variational inequalities..
Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 49L2.
Citation: Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control and Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012
References:
[1]

B. R. Barmish and J. A. Primbs, On market-neutral stock trading arbitrage via linear feedback, Proc. American Control Conference, Montreal, (2012), 3693-3698. doi: 10.1109/ACC.2012.6315392.

[2]

C. Blanco and D. Soronow, Mean reverting processes - Energy price processes used for derivatives pricing and risk management, Commodities Now, 5 (2001), 68-72.

[3]

L. P. Bos, A. F. Ware and B. S. Pavlov, On a semi-spectral method for pricing an option on a mean-reverting asset, Quantitative Finance, 2 (2002), 337-345. doi: 10.1088/1469-7688/2/5/302.

[4]

T. J. I'A. Bromwich, An introduction to the Theory of Infinite Series, American Mathematical Society, AMS Chelsea Publishing, Providence, RI, 1991.

[5]

A. Cowles and H. Jones, Some posteriori probabilities in stock market action, Econometrica, 5 (1937), 280-294. doi: 10.2307/1905515.

[6]

J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.

[7]

M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model, SIAM Journal on Financial Mathematics, 1 (2010), 780-810. doi: 10.1137/090770552.

[8]

R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Second edition. Springer Finance. Springer-Verlag, New York, 2005.

[9]

E. Fama and K. R. French, Permanent and temporary components of stock prices, J. Political Economy, 96 (1988), 246-273. doi: 10.1086/261535.

[10]

J. P. Fouque, G. Papanicolaou and R. K. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2000.

[11]

L. A. Gallagher and M. P. Taylor, Permanent and temporary components of stock prices: Evidence from assessing macroeconomic shocks, Southern Economic Journal, 69 (2002), 345-362. doi: 10.2307/1061676.

[12]

E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative-value arbitrage rule, Review of Financial Studies, 19 (2006), 797-827.

[13]

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.

[14]

C. M. Hafner and H. Herwartz, Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis, J. Empirical Finance, 8 (2001), 1-34. doi: 10.1016/S0927-5398(00)00024-4.

[15]

J. C. Hull, Options, Futures, and Other Derivatives, 3rd Ed., Prentice Hall, Upper Saddle River, NJ, 1997.

[16]

S. Iwarere and B. R. Barmish, A confidence interval triggering method for stock trading via feedback control, Proc. American Control Conference, Baltimore, MD, (2010), 6910-6916. doi: 10.1109/ACC.2010.5531311.

[17]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998. doi: 10.1007/b98840.

[18]

A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control Optim., 46 (2007), 839-876. doi: 10.1137/050640758.

[19]

W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Edition, Springer-Verlag, Berlin $\cdot$ Heidelberg $\cdot$ GmbH, 1966.

[20]

M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, Springer, New York, 1997. doi: 10.1007/978-3-662-22132-7.

[21]

R. Norberg, The Markov chain market, ASTIN Bulletin, 33 (2003), 265-287. doi: 10.2143/AST.33.2.503693.

[22]

Q. S. Song and Q. Zhang, An optimal pairs-trading rule, Automatica, 49 (2013), 3007-3014. doi: 10.1016/j.automatica.2013.07.012.

[23]

J. Van der Hoek and R. J. Elliott, American option prices in a Markov chain market model, Applied Stochastic Models in Business and Industry, 28 (2012), 35-59. doi: 10.1002/asmb.893.

[24]

O. A. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188.

[25]

Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing Co Pte Ltd, Singapore, 1989. doi: 10.1142/0653.

[26]

G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications, A Two-Time-Scale Approach, 2nd Ed, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9.

[27]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica, 44 (2008), 1511-1518. doi: 10.1016/j.automatica.2007.11.003.

[28]

Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 64-87. doi: 10.1137/S0363012999356325.

[29]

Q. Zhang, Explicit solutions for an optimal stock selling problem under a Markov chain model, J. Mathematical Analysis and Applications, 420 (2014), 1210-1227. doi: 10.1016/j.jmaa.2014.06.049.

show all references

References:
[1]

B. R. Barmish and J. A. Primbs, On market-neutral stock trading arbitrage via linear feedback, Proc. American Control Conference, Montreal, (2012), 3693-3698. doi: 10.1109/ACC.2012.6315392.

[2]

C. Blanco and D. Soronow, Mean reverting processes - Energy price processes used for derivatives pricing and risk management, Commodities Now, 5 (2001), 68-72.

[3]

L. P. Bos, A. F. Ware and B. S. Pavlov, On a semi-spectral method for pricing an option on a mean-reverting asset, Quantitative Finance, 2 (2002), 337-345. doi: 10.1088/1469-7688/2/5/302.

[4]

T. J. I'A. Bromwich, An introduction to the Theory of Infinite Series, American Mathematical Society, AMS Chelsea Publishing, Providence, RI, 1991.

[5]

A. Cowles and H. Jones, Some posteriori probabilities in stock market action, Econometrica, 5 (1937), 280-294. doi: 10.2307/1905515.

[6]

J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.

[7]

M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model, SIAM Journal on Financial Mathematics, 1 (2010), 780-810. doi: 10.1137/090770552.

[8]

R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Second edition. Springer Finance. Springer-Verlag, New York, 2005.

[9]

E. Fama and K. R. French, Permanent and temporary components of stock prices, J. Political Economy, 96 (1988), 246-273. doi: 10.1086/261535.

[10]

J. P. Fouque, G. Papanicolaou and R. K. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2000.

[11]

L. A. Gallagher and M. P. Taylor, Permanent and temporary components of stock prices: Evidence from assessing macroeconomic shocks, Southern Economic Journal, 69 (2002), 345-362. doi: 10.2307/1061676.

[12]

E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative-value arbitrage rule, Review of Financial Studies, 19 (2006), 797-827.

[13]

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.

[14]

C. M. Hafner and H. Herwartz, Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis, J. Empirical Finance, 8 (2001), 1-34. doi: 10.1016/S0927-5398(00)00024-4.

[15]

J. C. Hull, Options, Futures, and Other Derivatives, 3rd Ed., Prentice Hall, Upper Saddle River, NJ, 1997.

[16]

S. Iwarere and B. R. Barmish, A confidence interval triggering method for stock trading via feedback control, Proc. American Control Conference, Baltimore, MD, (2010), 6910-6916. doi: 10.1109/ACC.2010.5531311.

[17]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998. doi: 10.1007/b98840.

[18]

A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control Optim., 46 (2007), 839-876. doi: 10.1137/050640758.

[19]

W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Edition, Springer-Verlag, Berlin $\cdot$ Heidelberg $\cdot$ GmbH, 1966.

[20]

M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, Springer, New York, 1997. doi: 10.1007/978-3-662-22132-7.

[21]

R. Norberg, The Markov chain market, ASTIN Bulletin, 33 (2003), 265-287. doi: 10.2143/AST.33.2.503693.

[22]

Q. S. Song and Q. Zhang, An optimal pairs-trading rule, Automatica, 49 (2013), 3007-3014. doi: 10.1016/j.automatica.2013.07.012.

[23]

J. Van der Hoek and R. J. Elliott, American option prices in a Markov chain market model, Applied Stochastic Models in Business and Industry, 28 (2012), 35-59. doi: 10.1002/asmb.893.

[24]

O. A. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188.

[25]

Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing Co Pte Ltd, Singapore, 1989. doi: 10.1142/0653.

[26]

G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications, A Two-Time-Scale Approach, 2nd Ed, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9.

[27]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica, 44 (2008), 1511-1518. doi: 10.1016/j.automatica.2007.11.003.

[28]

Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 64-87. doi: 10.1137/S0363012999356325.

[29]

Q. Zhang, Explicit solutions for an optimal stock selling problem under a Markov chain model, J. Mathematical Analysis and Applications, 420 (2014), 1210-1227. doi: 10.1016/j.jmaa.2014.06.049.

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