# Market Graph Clustering via QUBO and Digital Annealing

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## Abstract

**:**

## 1. Introduction

## 2. Previous Work

#### 2.1. Market Graph

#### 2.2. Cardinality-Constrained Index-Tracking

#### 2.3. Clustering for Index-Tracking

## 3. Methods

#### 3.1. Market Graph

#### 3.2. Data Sets

#### 3.3. QUBO Model

#### 3.4. Parameters

- z is a solution containing $(k-1)$ exemplars. It is infeasible.
- ${z}^{*}$ is a solution containing k exemplars. It is feasible.
- Both these vectors are in ${\mathbb{R}}^{n}$.
- Therefore, we have the equality ${\sum}_{i}{z}_{i}^{*}=\left({\sum}_{i}{z}_{i}\right)+1=k$.

#### 3.5. The Fujitsu DA: Purpose-Built Architecture

#### 3.6. Asset Weights

## 4. Numerical Experiments

#### 4.1. Test Data

#### 4.2. Index-Tracking Performance

#### 4.3. Computation Times, Objective Function and $\gamma $

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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1. | More specifically, these DA computations were done using an environment built exclusively for the University of Toronto’s research. |

Index | Num Stocks |
---|---|

DAX 100 | 84 |

FTSE 100 | 88 |

Hang Seng | 30 |

Nikkei 225 | 224 |

Russell 2000 | 1317 |

Russell 3000 | 2150 |

Standard and Poors 100 | 97 |

Standard and Poors 500 | 456 |

Index | $\tilde{\mathit{\gamma}}$ |
---|---|

DAX 100 | 0.026 |

FTSE 100 | 0.025 |

Hang Seng | 0.019 |

Nikkei 225 | 0.022 |

Russell 2000 | 0.004 |

Russell 3000 | 0.003 |

Standard and Poors 100 | 0.027 |

Standard and Poors 500 | 0.007 |

Index | Cardinality Constraint $\left(\mathit{k}\right)$ |
---|---|

DAX 100 | 10 |

FTSE 100 | 10 |

Hang Seng | 10 |

Nikkei 225 | 10 |

Russell 2000 | 90 |

Russell 3000 | 70 |

Standard and Poors 100 | 10 |

Standard and Poors 500 | 40 |

Index | Tracking-Error |
---|---|

DAX 100 | 0.0118 |

FTSE 100 | 0.0096 |

Hang Seng | 0.0089 |

Nikkei I225 | 0.0223 |

Russell 2000 | 0.0187 |

Russell 3000 | 0.0123 |

Standard and Poors 100 | 0.0097 |

Standard and Poors 500 | 0.0137 |

Index | DA Run Time | DA Obj Fn | Gi Run Time | Gi Obj Fn |
---|---|---|---|---|

DAX100 | 520 | 1.388 | 10,712 | 1.389 |

FTSE100 | 522 | 1.351 | 10,712 | 1.352 |

Hang Seng | 520 | 1.050 | 70 | 1.050 |

Nikkei 225 | 522 | 1.165 | 10,701 | 1.167 |

Russell 2000 | 532 | 9.892 | 10,700 | 9.893 |

Russell 3000 | 589 | 12.585 | 10,703 | 12.586 |

SP 100 | 522 | 1.473 | 10,700 | 1.474 |

SP 500 | 523 | 5.426 | 10715 | 5.427 |

Index | Best $\mathit{\gamma}$ |
---|---|

DAX 100 | 0.022 |

FTSE 100 | 0.021 |

Hang Seng | 0.019 |

Nikkei 225 | 0.021 |

Russell 2000 | 0.003 |

Russell 3000 | 0.003 |

Standard and Poors 100 | 0.023 |

Standard and Poors 500 | 0.006 |

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**MDPI and ACS Style**

Hong, S.W.; Miasnikof, P.; Kwon, R.; Lawryshyn, Y. Market Graph Clustering via QUBO and Digital Annealing. *J. Risk Financial Manag.* **2021**, *14*, 34.
https://doi.org/10.3390/jrfm14010034

**AMA Style**

Hong SW, Miasnikof P, Kwon R, Lawryshyn Y. Market Graph Clustering via QUBO and Digital Annealing. *Journal of Risk and Financial Management*. 2021; 14(1):34.
https://doi.org/10.3390/jrfm14010034

**Chicago/Turabian Style**

Hong, Seo Woo, Pierre Miasnikof, Roy Kwon, and Yuri Lawryshyn. 2021. "Market Graph Clustering via QUBO and Digital Annealing" *Journal of Risk and Financial Management* 14, no. 1: 34.
https://doi.org/10.3390/jrfm14010034