# Quantitative and Comparative Analyses of Limit Order Books with General Compound Hawkes Processes

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. One-Dimensional Hawkes Process

#### 2.1.1. Definition

#### 2.1.2. Calibration

#### 2.2. General Compound Hawkes Process

#### 2.2.1. Definition

#### 2.2.2. Diffusive Limit

## 3. Application

#### 3.1. Limit Order Book

#### 3.2. Data

#### 3.3. Descriptive Data Analysis

#### 3.3.1. QQ-Plot

#### 3.3.2. Autocorrelation

#### 3.3.3. Clustering Feature

## 4. Hawkes Process and Models Calibrations

#### 4.1. Hawkes Process’ Parameters Calibration

#### 4.2. Mid Price Modelling and Calibration

#### 4.2.1. GCHPDO

#### 4.2.2. GCHP2SDO

#### 4.2.3. GCHPnSDO

## 5. Error Measurement

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bacry, Emmanuel, Sylvain Delattre, Marc Hoffmann, and Jean-Francois Muzy. 2013. Some limit theorems for hawkes processes and application to financial statistics. Stochastic Processes and their Applications 123: 2475–99. [Google Scholar] [CrossRef]
- Bacry, Emmanuel, Iacopo Mastromatteo, and Jean-François Muzy. 2015. Hawkes processes in finance. arXiv. [Google Scholar]
- Cartea, Álvaro, Sebastian Jaimungal, and José Penalva. 2015. Algorithmic and High-Frequency Trading. Cambridge: Cambridge University Press. [Google Scholar]
- Chavez-Casillas, Jonathan, Robert J. Elliott, Bruno Remillard, and Anatoliy V. Swishchuk. 2019. A level-1 limit order book with time dependent arrival rates. Methodology and Computing in Applied Probability 21: 699–719. [Google Scholar] [CrossRef]
- Cont, Rama, and Adrien De Larrard. 2013. Price dynamics in a Markovian limit order markets. SIAM Journal on Financial Mathematics 4: 1–25. [Google Scholar] [CrossRef]
- Da Fonseca, José, and Riadh Zaatour. 2013. Hawkes process: Fast calibration, application to trade clustering, and diffusive limit. Journal of Futures Markets 34: 548–79. [Google Scholar] [CrossRef]
- Embrechts, Paul, Thomas Liniger, and Lu Lin. 2011. Multivariate hawkes processes: An application to financial data. Journal of Applied Probability 48: 367–78. [Google Scholar] [CrossRef]
- Gould, Martin D., Mason A. Porter, Stacy Williams, Mark McDonald, Daniel J. Fenn, and Sam D. Howison. 2013. Limit order books. Quantitative Finance 13: 1709–42. [Google Scholar] [CrossRef]
- Hawkes, Alan G. 1971. Spectra of some self-exciting and mutually exciting point processes. Biometrika 58: 83–90. [Google Scholar] [CrossRef]
- Laub, Patrick J., Thomas Taimre, and Philip K. Pollett. 2015. Hawkes Processes. arXiv. [Google Scholar]
- Lorenzen, Florian. 2012. Analysis of Order Clustering Using High Frequency Data: A Point Process Approach. Ph.D. thesis, Swiss Federal Institute of Technology Zurich (ETH Zurich), Zurich, Switzerland. [Google Scholar]
- Swishchuk, Anatoliy, and Aiden Huffman. 2018. General Compound Hawkes Process in Limit Order Books. arXiv. [Google Scholar]
- Swishchuk, Anatoliy, and Nelson Vadori. 2017. A semi-Markovian modeling of limit order markets. SIAM Journal on Financial Mathematics 8: 240–73. [Google Scholar] [CrossRef]
- Swishchuk, Anatoliy, Tyler Hofmeister, Katharina Cera, and Julia Schmidt. 2017. General semi-Markov model for limit order books. International Journal of Theoretical and Applied Finance 20: 1750019. [Google Scholar] [CrossRef]
- Swishchuk, Anatoliy, Bruno Remillard, Robert Elliott, and Jonathan Chavez-Casillas. 2019. Compound Hawkes processes in limit order books. In Financial Mathematics, Volatility and Covariance Modelling, 1st ed. Edited by Julien Chevallier, Goutte Stéphane, Guerreiro David, Saglio Sophie and Bilel Sanhaji. Abingdon: Routledge, vol. 2. [Google Scholar]
- Zhou, Ke, Hongyuan Zha, and Le Song. 2013. Learning triggering kernels for multi-dimensional hawkes processes. Paper presented at the 30th International Conference on Machine Learning, PMLR, Atlanta, GA, USA, June 16–21; vol. 29, pp. 1301–9. [Google Scholar]

**Figure 3.**Autocorrelation under different $\tau $. (

**a**) $\tau =20$ s. (

**b**) $\tau =30$ s. (

**c**) $\tau =60$ s. (

**d**) $\tau =90$ s.

Stock | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\mathit{E}\left[\mathit{N}\right([0,1]\left)\right]$ | $\frac{\widehat{\mathit{\lambda}}}{1-\frac{\widehat{\mathit{\alpha}}}{\widehat{\mathit{\beta}}}}$ |
---|---|---|---|---|---|

EBAY | 0.05142964 | 24.16091427 | 208.361411 | 0.058169396 | 0.05817548 |

FB | 0.1460757 | 7.0368564 | 33.6354758 | 0.184708815 | 0.184721079 |

MU | 0.06685776 | 8.49267842 | 47.38655865 | 0.081446361 | 0.081456495 |

PCAR | 0.04105567 | 28.42153021 | 234.1878409 | 0.046720101 | 0.046726496 |

SMH | 0.0157736 | 10.6972559 | 102.6908889 | 0.017602398 | 0.017607795 |

CSCO | 0.0223269 | 642.6391460 | 1311.8684573 | 0.043657407 | 0.043766646 |

Stock | Tick-Size Rate |
---|---|

EBAY | 21.35% |

FB | 18.18% |

MU | 9.85% |

PCAR | 22.22% |

SMH | 12.96% |

CSCO | 100% |

Stock | $\mathit{a}\left(1\right)$ | $\mathit{a}\left(2\right)$ |
---|---|---|

EBAY | 0.0091730474 | −0.0092358804 |

FB | 0.0095449949 | −0.0093280239 |

MU | 0.0098359729 | −0.0096846330 |

PCAR | 0.0116390041 | −0.0108269962 |

SMH | 0.0152972973 | −0.0134196891 |

n | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|

MSE | 0.00046701 | 0.000418703 | 0.000399311 | 0.000397898 | 0.000393683 |

n | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|

MSE | 0.000317633 | 0.000330058 | 0.000344656 | 0.000349087 | 0.000361214 |

n | 4 | 6 | 8 |
---|---|---|---|

MSE | 0.000263588 | 0.000272237 | 0.000275675 |

n | 6 | 9 | 12 | 15 |
---|---|---|---|---|

MSE | 0.000238185 | 0.000132224 | 0.000129238 | 0.000124427 |

n | 6 | 7 | 9 | 10 |
---|---|---|---|---|

MSE | 0.00003853463 | 0.00003436168 | 0.00002395992 | 0.00002475711 |

Stock | GCHPDO (MSE) | GCHP2SDO (MSE) | GCHPnSDO (MSE) | Best Model |
---|---|---|---|---|

EBAY | 0.002200996 | 0.000566305 | 0.000393683 | GCHP8SDO |

FB | 0.00173283 | 0.000241354 | 0.000317633 | GCHP2SDO |

MU | 0.000450242 | 0.000227604 | 0.000263588 | GCHP2SDO |

PCAR | 0.003129217 | 0.000468104 | 0.000124427 | GCHP15SDO |

SMH | 0.001477308 | 0.000053474 | 0.00002395992 | GCHP9SDO |

Stock | Error Rate |
---|---|

EBAY | 26.05% |

FB | 10.95% |

MU | 24.42% |

PCAR | 12.18% |

SMH | 0.67% |

CSCO | 7.99% |

Stock | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\mathit{E}\left[\mathit{N}\right([0,1]\left)\right]$ | $\frac{\widehat{\mathit{\lambda}}}{1-\frac{\widehat{\mathit{\alpha}}}{\widehat{\mathit{\beta}}}}$ |
---|---|---|---|---|---|

EBAY | 0.037386079 | 11.06718675 | 77.58564954 | 0.04356481 | 0.043606288 |

FB | 0.16361531 | 9.49962096 | 42.48237019 | 0.210416667 | 0.210739442 |

MU | 0.059890034 | 9.767710137 | 65.90938964 | 0.070277777 | 0.070309895 |

PCAR | 0.03032757 | 10.88944631 | 82.44806627 | 0.034861111 | 0.034942673 |

SMH | 0.01580581 | 41.96535566 | 351.0221707 | 0.01787037 | 0.017952006 |

CSCO | 0.022357326 | 677.5504008 | 1385.215404 | 0.04337963 | 0.04376324 |

n | 4 | 5 | 6 | 7 |
---|---|---|---|---|

MSE | 0.000154926 | 0.000151477 | 0.000182662 | 0.000193251 |

n | 4 | 5 | 6 | 7 |
---|---|---|---|---|

MSE | 0.000204729 | 0.000211706 | 0.000241339 | 0.000249355 |

n | 4 | 6 | 8 |
---|---|---|---|

MSE | 0.000031205 | 0.000032304 | 0.000038158 |

n | 4 | 6 | 8 | 10 |
---|---|---|---|---|

MSE | 0.000143116 | 0.000154922 | 0.000162016 | 0.00032631 |

n | 4 | 6 | 7 | 9 | 10 |
---|---|---|---|---|---|

MSE | 0.000063641 | 0.000051274 | 0.000077954 | 0.000102804 | 0.00010632 |

Stock | GCHPDO (MSE) | GCHP2SDO (MSE) | GCHPnSDO (MSE) | Best Model | Error Rate |
---|---|---|---|---|---|

EBAY | 0.000395775 | 0.000070119 | 0.000151477 | GCHP2SDO | 15.67% |

FB | 0.002979562 | 0.000164628 | 0.000204729 | GCHP2SDO | 4.15% |

MU | 0.000691422 | 0.000027226 | 0.000031205 | GCHP2SDO | 1.52% |

PCAR | 0.001220103 | 0.000075708 | 0.000143116 | GCHP2SDO | 6.75% |

SMH | 0.001058399 | 0.000076867 | 0.000051274 | GCHP6SDO | 4.21% |

CSCO | 0.000020788 | N/A | N/A | GCHPDO | 13.29% |

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## Share and Cite

**MDPI and ACS Style**

He, Q.; Swishchuk, A.
Quantitative and Comparative Analyses of Limit Order Books with General Compound Hawkes Processes. *Risks* **2019**, *7*, 110.
https://doi.org/10.3390/risks7040110

**AMA Style**

He Q, Swishchuk A.
Quantitative and Comparative Analyses of Limit Order Books with General Compound Hawkes Processes. *Risks*. 2019; 7(4):110.
https://doi.org/10.3390/risks7040110

**Chicago/Turabian Style**

He, Qiyue, and Anatoliy Swishchuk.
2019. "Quantitative and Comparative Analyses of Limit Order Books with General Compound Hawkes Processes" *Risks* 7, no. 4: 110.
https://doi.org/10.3390/risks7040110