# An Entropic Approach for Pair Trading

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Main Results

**Theorem**

**1.**

#### Numerical Example

## 4. Proof of Theorem 1

**Proof**

**of**

**Theorem**

**1.**

## 5. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

- Elliott, R.; Van Der Hoek, J.; Malcolm, W. Pairs trading. Quant. Financ.
**2005**, 5, 271–276. [Google Scholar] [CrossRef] - Gatev, E.; Goetzman, W.; Rouwenhorst, K. Pairs trading: Performance of a relative-value arbitrage rule. Rev. Financ. Stud.
**2006**, 19, 787–827. [Google Scholar] [CrossRef] - Avellaneda, M.; Lee, J. Statistical arbitrage in the US equities market. Quant. Financ.
**2010**, 10, 761–782. [Google Scholar] [CrossRef] - Kakushadze, Z. Mean-reversion and optimization. J. Asset Manag.
**2015**, 16, 14–40. [Google Scholar] [CrossRef] - Caporale, G.; Gil-Alana, L.; Plastun, A. Searching for inefficiencies in exchange rate dynamics. Comput. Econ.
**2017**, 49, 405–432. [Google Scholar] [CrossRef] - Chen, C.; Lin, T. Nonparametric tolerance limits for pair trading. Financ. Res. Lett.
**2017**, 21, 1–9. [Google Scholar] [CrossRef] - Chen, C.; Wang, Z.; Sriboonchitta, S.; Lee, S. Pair trading based on quantile forecasting of smooth transition GARCH models. N. Am. J. Econ. Financ.
**2017**, 39, 38–55. [Google Scholar] [CrossRef] - Ekström, E.; Lindberg, C.; Tysk, J. Optimal liquidation of a pairs trade. In Advanced Mathematical Methods for Finance; Nunno, G., Øksendal, B., Eds.; Springer: Berlin, Germany, 2011. [Google Scholar]
- Riedel, F. Optimal stopping with multiple priors. Econometrica
**2009**, 77, 857–908. [Google Scholar] - Krätschmer, V.; Schoenmakers, J. Representations for optimal stopping under dynamic monetary utility functionals. SIAM J. Financ. Math.
**2010**, 1, 811–832. [Google Scholar] [CrossRef] - Krätschmer, V.; Ladkau, M.; Laeven, R.; Schoenmakers, J.; Stadje, M. Robust Optimal Stopping. Available online: http://www.uni-ulm.de/fileadmin/websiteuniulm/mawi.inst.140/Team/MStadje/KLLSS-StoppingAmbiguity-041715-a.pdf (accessed on 30 June 2017).
- Zadeh, L. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Serguieva, A.; Hunter, J. Fuzzy interval methods in investment risk appraisal. Fuzzy Sets Syst.
**2004**, 142, 443–466. [Google Scholar] - Zhou, X.; Dong, M. Can fuzzy logic make technical analysis 20/20? Financ. Anal. J.
**2004**, 60, 54–75. [Google Scholar] [CrossRef] - Gradojevic, N.; Gencay, R. Fuzzy logic, trading uncertainty and technical trading. J. Bank. Financ.
**2013**, 37, 578–586. [Google Scholar] [CrossRef] - Bowden, R. Directional entropy and tail uncertainty, with applications to financial hazard. Quant. Financ.
**2011**, 11, 437–446. [Google Scholar] [CrossRef] - Gradojevic, N.; Gencay, R. Overnight interest rates and aggregate market expectations. Econ. Lett.
**2008**, 100, 27–30. [Google Scholar] [CrossRef] - Gencay, R.; Gradojevic, N. Crash of ’87—was it expected? Aggregate market fears and long range dependence. J. Empir. Financ.
**2010**, 17, 270–282. [Google Scholar] [CrossRef] - Gradojevic, N.; Caric, M. Predicting systemic risk with entropic indicators. J. Forecast.
**2017**, 36, 16–25. [Google Scholar] [CrossRef] - Yang, J.; Qiu, W. A measure of risk and a decision-making model based on expected utility and entropy. Eur. J. Oper. Res.
**2005**, 164, 792–799. [Google Scholar] [CrossRef] - Stutzer, M.J. Simple entropic derivation of a generalized Black-Scholes option pricing model. Entropy
**2000**, 2, 70–77. [Google Scholar] [CrossRef] - Kitamura, Y.; Stutzer, M.J. Connections between entropic and linear projections in asset pricing estimation. J. Econ.
**2002**, 107, 159–174. [Google Scholar] [CrossRef] - Bekiros, S. Timescale analysis with an entropy-based shift-invariant discrete wavelet transform. Comput. Econ.
**2014**, 44, 231–251. [Google Scholar] [CrossRef] - Peskir, G.; Shiryaev, A. Optimal Stopping and Free-Boundary Problems; Birkhäuser Verlag: Basel, Switzerland, 2006. [Google Scholar]
- Detlefsen, K.; Scandolo, G. Conditional and dynamic convex risk measures. Financ. Stoch.
**2005**, 9, 539–561. [Google Scholar] [CrossRef] - Föllmer, H.; Penner, I. Convex risk measures and the dynamics of their penalty functions. Stat. Decis.
**2006**, 24, 61–96. [Google Scholar] [CrossRef]

**Figure 1.**Pair values, means and boundaries. (

**a**) shows the pair of Daiichi Commodities Co., Ltd. (8746) and Asahi Industries Co., Ltd. (5456), (

**b**) is Fuji Oil Co., Ltd. (5017) and Sado Steam Ship Co., Ltd. (9176), (

**c**) is Fuji Oil Co., Ltd. (5017) and Takata Corporation (7312), (

**d**) is PADO Corporation (4833) and Oi Electric Co., Ltd. (6822), (

**e**) is PADO Corporation (4833) and Seiwa Electric MFG. Co., Ltd. (6748), and (

**f**) is Sado Steam Ship Co., Ltd. (9176), Daiko Denshi Tsushin Ltd. (8023).

**Table 1.**The rate of return for different $\lambda $. Pair 1 is Daiichi Commodities Co., Ltd. (8746) and Asahi Industries Co., Ltd. (5456); Pair 2 is Fuji Oil Co., Ltd. (5017) and Sado Steam Ship Co., Ltd. (9176); Pair 3 is Fuji Oil Co., Ltd. (5017) and Takata Corporation (7312); Pair 4 is PADO Corporation (4833) and Oi Electric Co., Ltd. (6822); Pair 5 is PADO Corporation (4833) and Seiwa Electric MFG. Co., Ltd. (6748); Pair 6 is Sado Steam Ship Co., Ltd. (9176), Daiko Denshi Tsushin Ltd. (8023).

$\mathit{\lambda}=0.001$ | $\mathit{\lambda}=0.01$ | $\mathit{\lambda}=0.1$ | $\mathit{\lambda}=+\mathbf{\infty}$ | |
---|---|---|---|---|

Pair 1 | 0.152 | 0.152 | 0.165 | 0.165 |

Pair 2 | 0.321 | 0.170 | 0.170 | 0.170 |

Pair 3 | 0.071 | 0.028 | 0.028 | 0.028 |

Pair 4 | 0.189 | 0.076 | 0.076 | 0.076 |

Pair 5 | 0.097 | 0.088 | 0.088 | 0.088 |

Pair 6 | 0.093 | 0.133 | 0.133 | 0.133 |

Names | Return |
---|---|

Fullcast Holdings Co., Ltd. (code: 4848) | 0.099 |

Daiichi Commodities Co., Ltd. (8746) | −0.113 |

Fuji Oil Co., Ltd. (5017) | −0.091 |

FIDEA Holdings Co., Ltd. (8713) | −0.125 |

Yoshicon Co., Ltd. (5280) | 0.016 |

PADO Corporation (4833) | −0.157 |

Sado Steam Ship Co., Ltd. (9176) | −0.016 |

Joban Kaihatsu Co., Ltd. (1782) | −0.047 |

Meiwa Estate Company Limited (8869) | −0.026 |

Oi Electric Co., Ltd. (6822) | −0.059 |

Takata Corporation (7312) | −0.034 |

Toei Reefer Line Ltd. (9133) | −0.147 |

Nihon House Holdings Co., Ltd. (1873) | −0.076 |

Sanei Architecture Planning Co., Ltd. (3228) | 0.521 |

Shinhoku Steel Corporation (5542) | −0.275 |

Daiko Denshi Tsushin Ltd. (8023) | −0.226 |

Shinnihon Corporation (1879) | 0.193 |

Asahi Industries Co., Ltd. (5456) | −0.121 |

Seiwa Electric MFG. Co., Ltd. (6748) | −0.050 |

Daisue Construction Co., Ltd. (1814) | −0.129 |

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yoshikawa, D. An Entropic Approach for Pair Trading. *Entropy* **2017**, *19*, 320.
https://doi.org/10.3390/e19070320

**AMA Style**

Yoshikawa D. An Entropic Approach for Pair Trading. *Entropy*. 2017; 19(7):320.
https://doi.org/10.3390/e19070320

**Chicago/Turabian Style**

Yoshikawa, Daisuke. 2017. "An Entropic Approach for Pair Trading" *Entropy* 19, no. 7: 320.
https://doi.org/10.3390/e19070320