# 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 & Related Fields, 2012, 2 (1) : 81-100. doi: 10.3934/mcrf.2012.2.81
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