# A Threshold Type Policy for Trading a Mean-Reverting Asset with Fixed Transaction Costs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Method

#### 2.1. Formulation of the Optimal Trading Problem

**Remark**

**1.**

**Remark**

**2.**

#### 2.2. Bounds of the Value Functions

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

#### 2.3. The HJB Equations

#### 2.4. Solutions of the HJB Equations

#### 2.5. A Verification Theorem

**Theorem**

**1.**

**Lemma**

**2.**

**Proof of Lemma**

**2.**

**Lemma**

**3.**

**Proof**

**of**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof**

**of**

**Lemma**

**4.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**3.**

**Remark**

**4.**

## 3. Numerical Results and Discussion

**Remark**

**5.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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b | 1 | 1.5 | 2 | 2.5 | 3 |
---|---|---|---|---|---|

${x}_{0}$ | −4.36 | −4.46 | −4.58 | −4.66 | −4.74 |

${x}_{1}$ | 0 | 0.64 | 1.22 | 1.78 | 2.34 |

${x}_{2}$ | 0.74 | 1.22 | 1.7 | 2.18 | 2.66 |

a | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
---|---|---|---|---|---|

${x}_{0}$ | −4.2 | −4.4 | −4.58 | −4.72 | −4.86 |

${x}_{1}$ | 0.98 | 1.14 | 1.22 | 1.3 | 1.36 |

${x}_{2}$ | 1.58 | 1.62 | 1.7 | 1.74 | 1.78 |

$\mathit{\sigma}$ | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 |
---|---|---|---|---|---|

${x}_{0}$ | −4.56 | −4.56 | −4.58 | −4.58 | −4.6 |

${x}_{1}$ | 1.12 | 1.16 | 1.22 | 1.28 | 1.36 |

${x}_{2}$ | 1.54 | 1.62 | 1.7 | 1.8 | 1.9 |

$\mathit{\rho}$ | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 |
---|---|---|---|---|---|

${x}_{0}$ | −5 | −4.86 | −4.58 | −4.32 | −4.12 |

${x}_{1}$ | 1.56 | 1.36 | 1.22 | 1.06 | 0.9 |

${x}_{2}$ | 1.94 | 1.84 | 1.7 | 1.56 | 1.42 |

K | 0.01 | 0.05 | 0.1 | 0.5 | 1 |
---|---|---|---|---|---|

${x}_{0}$ | −5 | −5 | −4.58 | −2.56 | −1.6 |

${x}_{1}$ | 1.46 | 1.3 | 1.22 | 0.7 | −0.46 |

${x}_{2}$ | 1.56 | 1.68 | 1.7 | 1.78 | 1.86 |

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## Share and Cite

**MDPI and ACS Style**

Luu, P.; Tie, J.; Zhang, Q.
A Threshold Type Policy for Trading a Mean-Reverting Asset with Fixed Transaction Costs. *Risks* **2018**, *6*, 107.
https://doi.org/10.3390/risks6040107

**AMA Style**

Luu P, Tie J, Zhang Q.
A Threshold Type Policy for Trading a Mean-Reverting Asset with Fixed Transaction Costs. *Risks*. 2018; 6(4):107.
https://doi.org/10.3390/risks6040107

**Chicago/Turabian Style**

Luu, Phong, Jingzhi Tie, and Qing Zhang.
2018. "A Threshold Type Policy for Trading a Mean-Reverting Asset with Fixed Transaction Costs" *Risks* 6, no. 4: 107.
https://doi.org/10.3390/risks6040107