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

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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

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

Received February 2015 Revised July 2015 Published August 2016

Abstract
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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 & 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, (2012), 3693. doi: 10.1109/ACC.2012.6315392. Google Scholar
[2]	C. Blanco and D. Soronow, Mean reverting processes - Energy price processes used for derivatives pricing and risk management,, Commodities Now, 5 (2001), 68. Google Scholar
[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. doi: 10.1088/1469-7688/2/5/302. Google Scholar
[4]	T. J. I'A. Bromwich, An introduction to the Theory of Infinite Series,, American Mathematical Society, (1991). Google Scholar
[5]	A. Cowles and H. Jones, Some posteriori probabilities in stock market action,, Econometrica, 5 (1937), 280. doi: 10.2307/1905515. Google Scholar
[6]	J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates,, Econometrica, 53 (1985), 385. doi: 10.2307/1911242. Google Scholar
[7]	M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model,, SIAM Journal on Financial Mathematics, 1 (2010), 780. doi: 10.1137/090770552. Google Scholar
[8]	R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets,, Second edition. Springer Finance. Springer-Verlag, (2005). Google Scholar
[9]	E. Fama and K. R. French, Permanent and temporary components of stock prices,, J. Political Economy, 96 (1988), 246. doi: 10.1086/261535. Google Scholar
[10]	J. P. Fouque, G. Papanicolaou and R. K. Sircar, Derivatives in Financial Markets with Stochastic Volatility,, Cambridge University Press, (2000). Google Scholar
[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. doi: 10.2307/1061676. Google Scholar
[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. Google Scholar
[13]	X. Guo and Q. Zhang, Optimal selling rules in a regime switching model,, IEEE Trans. Automatic Control, 50 (2005), 1450. doi: 10.1109/TAC.2005.854657. Google Scholar
[14]	C. M. Hafner and H. Herwartz, Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis,, J. Empirical Finance, 8 (2001), 1. doi: 10.1016/S0927-5398(00)00024-4. Google Scholar
[15]	J. C. Hull, Options, Futures, and Other Derivatives,, 3rd Ed., (1997). Google Scholar
[16]	S. Iwarere and B. R. Barmish, A confidence interval triggering method for stock trading via feedback control,, Proc. American Control Conference, (2010), 6910. doi: 10.1109/ACC.2010.5531311. Google Scholar
[17]	I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer, (1998). doi: 10.1007/b98840. Google Scholar
[18]	A. Merhi and M. Zervos, A model for reversible investment capacity expansion,, SIAM J. Control Optim., 46 (2007), 839. doi: 10.1137/050640758. Google Scholar
[19]	W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics,, 3rd Edition, (1966). Google Scholar
[20]	M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling,, Springer, (1997). doi: 10.1007/978-3-662-22132-7. Google Scholar
[21]	R. Norberg, The Markov chain market,, ASTIN Bulletin, 33 (2003), 265. doi: 10.2143/AST.33.2.503693. Google Scholar
[22]	Q. S. Song and Q. Zhang, An optimal pairs-trading rule,, Automatica, 49 (2013), 3007. doi: 10.1016/j.automatica.2013.07.012. Google Scholar
[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. doi: 10.1002/asmb.893. Google Scholar
[24]	O. A. Vasicek, An equilibrium characterization of the term structure,, Journal of Financial Economics, 5 (1977), 177. Google Scholar
[25]	Z. X. Wang and D. R. Guo, Special Functions,, World Scientific Publishing Co Pte Ltd, (1989). doi: 10.1142/0653. Google Scholar
[26]	G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications, A Two-Time-Scale Approach,, 2nd Ed, (2013). doi: 10.1007/978-1-4614-4346-9. Google Scholar
[27]	H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high,, Automatica, 44 (2008), 1511. doi: 10.1016/j.automatica.2007.11.003. Google Scholar
[28]	Q. Zhang, Stock trading: An optimal selling rule,, SIAM J. Control Optim., 40 (2001), 64. doi: 10.1137/S0363012999356325. Google Scholar
[29]	Q. Zhang, Explicit solutions for an optimal stock selling problem under a Markov chain model,, J. Mathematical Analysis and Applications, 420 (2014), 1210. doi: 10.1016/j.jmaa.2014.06.049. Google Scholar

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References:

[1]	B. R. Barmish and J. A. Primbs, On market-neutral stock trading arbitrage via linear feedback,, Proc. American Control Conference, (2012), 3693. doi: 10.1109/ACC.2012.6315392. Google Scholar
[2]	C. Blanco and D. Soronow, Mean reverting processes - Energy price processes used for derivatives pricing and risk management,, Commodities Now, 5 (2001), 68. Google Scholar
[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. doi: 10.1088/1469-7688/2/5/302. Google Scholar
[4]	T. J. I'A. Bromwich, An introduction to the Theory of Infinite Series,, American Mathematical Society, (1991). Google Scholar
[5]	A. Cowles and H. Jones, Some posteriori probabilities in stock market action,, Econometrica, 5 (1937), 280. doi: 10.2307/1905515. Google Scholar
[6]	J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates,, Econometrica, 53 (1985), 385. doi: 10.2307/1911242. Google Scholar
[7]	M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model,, SIAM Journal on Financial Mathematics, 1 (2010), 780. doi: 10.1137/090770552. Google Scholar
[8]	R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets,, Second edition. Springer Finance. Springer-Verlag, (2005). Google Scholar
[9]	E. Fama and K. R. French, Permanent and temporary components of stock prices,, J. Political Economy, 96 (1988), 246. doi: 10.1086/261535. Google Scholar
[10]	J. P. Fouque, G. Papanicolaou and R. K. Sircar, Derivatives in Financial Markets with Stochastic Volatility,, Cambridge University Press, (2000). Google Scholar
[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. doi: 10.2307/1061676. Google Scholar
[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. Google Scholar
[13]	X. Guo and Q. Zhang, Optimal selling rules in a regime switching model,, IEEE Trans. Automatic Control, 50 (2005), 1450. doi: 10.1109/TAC.2005.854657. Google Scholar
[14]	C. M. Hafner and H. Herwartz, Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis,, J. Empirical Finance, 8 (2001), 1. doi: 10.1016/S0927-5398(00)00024-4. Google Scholar
[15]	J. C. Hull, Options, Futures, and Other Derivatives,, 3rd Ed., (1997). Google Scholar
[16]	S. Iwarere and B. R. Barmish, A confidence interval triggering method for stock trading via feedback control,, Proc. American Control Conference, (2010), 6910. doi: 10.1109/ACC.2010.5531311. Google Scholar
[17]	I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer, (1998). doi: 10.1007/b98840. Google Scholar
[18]	A. Merhi and M. Zervos, A model for reversible investment capacity expansion,, SIAM J. Control Optim., 46 (2007), 839. doi: 10.1137/050640758. Google Scholar
[19]	W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics,, 3rd Edition, (1966). Google Scholar
[20]	M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling,, Springer, (1997). doi: 10.1007/978-3-662-22132-7. Google Scholar
[21]	R. Norberg, The Markov chain market,, ASTIN Bulletin, 33 (2003), 265. doi: 10.2143/AST.33.2.503693. Google Scholar
[22]	Q. S. Song and Q. Zhang, An optimal pairs-trading rule,, Automatica, 49 (2013), 3007. doi: 10.1016/j.automatica.2013.07.012. Google Scholar
[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. doi: 10.1002/asmb.893. Google Scholar
[24]	O. A. Vasicek, An equilibrium characterization of the term structure,, Journal of Financial Economics, 5 (1977), 177. Google Scholar
[25]	Z. X. Wang and D. R. Guo, Special Functions,, World Scientific Publishing Co Pte Ltd, (1989). doi: 10.1142/0653. Google Scholar
[26]	G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications, A Two-Time-Scale Approach,, 2nd Ed, (2013). doi: 10.1007/978-1-4614-4346-9. Google Scholar
[27]	H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high,, Automatica, 44 (2008), 1511. doi: 10.1016/j.automatica.2007.11.003. Google Scholar
[28]	Q. Zhang, Stock trading: An optimal selling rule,, SIAM J. Control Optim., 40 (2001), 64. doi: 10.1137/S0363012999356325. Google Scholar
[29]	Q. Zhang, Explicit solutions for an optimal stock selling problem under a Markov chain model,, J. Mathematical Analysis and Applications, 420 (2014), 1210. doi: 10.1016/j.jmaa.2014.06.049. Google Scholar

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