# Multivariate General Compound Point Processes in Limit Order Books

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Definition of Multivariate General Compound Point Process (MGCPP)

**Definition**

**1**

#### 2.1. Assumptions for Multivariate Point Processes

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**1.**

**Remark**

**2.**

_{i}is in the form of

#### 2.2. Definition for MGCPP

**Remark**

**3.**

**Remark**

**4.**

## 3. LLNs and Diffusion Limits for MGCPP

#### 3.1. LLN for MGCPP

**Theorem**

**1**

**Proof**

**of Theorem 1.**

#### 3.2. Diffusion Limits for MGCPP: Stochastic Centralization

**Theorem**

**2**

**Proof**

**of Theorem 2.**

**Corollary**

**1**

**Corollary**

**2**

**Proof**

**of Corollarys 1 and 2.**

**Remark**

**5.**

**Remark**

**6.**

#### 3.3. Numerical Examples for FCLT: Stochastic Centralization

#### 3.3.1. Data Description and Parameter Estimations

#### 3.3.2. Comparison with MGCHP with Two Dependent Orders

**Remark**

**7.**

**Remark**

**8.**

#### 3.3.3. MGCPP with $\mathcal{N}$-State Dependent Orders

**Remark**

**9.**

## 4. Diffusion Limit for the MGCPP: Deterministic Centralization

#### 4.1. FCLT for MGCPP: Deterministic Centralization

**Theorem**

**3.**

**Proof**

**of Theorem 3.**

**Remark**

**10.**

**Remark**

**11.**

#### 4.2. Numerical Examples for FCLT: Deterministic Centralization

#### 4.3. Rolling Cross-Validation

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Standard deviation comparisons for 2 stocks by FCLT I for multivariate general compound point process (MGCPP) and multivariate general compound Hawkes process (MGCHP).

**Figure 2.**Standard deviation comparisons for MGCPP with 2-state Markov chain and 7-state Markov chain simulated by Apple’s stock data.

**Figure 3.**Standard deviation comparisons for MGCPP with 2-state Markov chain and 7-state Markov chain simulated by Google’s stock data.

**Figure 4.**Standard deviation comparisons for MGCPP with 2-state Markov chain and 7-state Markov chain simulated by Amazon’s stock data.

**Figure 5.**Standard deviation comparisons for 5 stocks by FCLT II for the MGCPP. INTC and MSFT are simulated with 2-state Markov chain while AAPL, AMZN, and GOOG are using 7-state Markov chain.

Ticker | # of Orders in 1 Day | Avg # of Orders/s | # of Price Changes in 1 Day | Avg # of Price Changes/s |
---|---|---|---|---|

INTC | 404,986 | 17.3071 | 3218 | 0.1375 |

MSFT | 411,409 | 5.0640 | 4016 | 0.1716 |

AAPL | 118,497 | 5.0640 | 64,351 | 2.7500 |

AMZN | 57,515 | 2.4579 | 27,558 | 1.1777 |

GOOG | 49,482 | 2.1146 | 24,085 | 1.0293 |

**Table 2.**Estimated parameters of 5 stocks via the law of large numbers (LLN) and functional central limit theorem (FCLT) assumptions.

Ticker | $\overline{\mathit{\lambda}}$ |
---|---|

INTC | 0.1366 |

MSFT | 0.1729 |

AAPL | 2.2938 |

AMZN | 1.0374 |

GOOG | 0.8178 |

Ticker | INTC | MSFT | AAPL | AMZN | GOOG |
---|---|---|---|---|---|

INTC | 1.0000 | 0.3870 | 0.2948 | 0.2932 | 0.2389 |

MSFT | 0.3870 | 1.0000 | 0.4373 | 0.3984 | 0.3474 |

AAPL | 0.2948 | 0.4373 | 1.0000 | 0.3697 | 0.3322 |

AMZN | 0.2932 | 0.3984 | 0.3697 | 1.0000 | 0.3251 |

GOOG | 0.2389 | 0.3474 | 0.3322 | 0.3251 | 1.0000 |

**Table 4.**Transition matrix and constant parameters for two-state MGCPP. ${\alpha}^{*}$ and ${\sigma}^{*}$ were calculated by Equation (19).

Ticker | p_{uu} | p_{dd} | ${\mathit{\sigma}}^{*}$ | ${\mathit{a}}^{*}$ |
---|---|---|---|---|

INTC | 0.5373 | 0.5814 | 0.0057 | −2.5023 × ${10}^{-4}$ |

MSFT | 0.5711 | 0.6044 | 0.0060 | −2.0145 × ${10}^{-4}$ |

AAPL | 0.4954 | 0.4955 | 0.0050 | −2.1529 × ${10}^{-7}$ |

AMZN | 0.4511 | 0.4590 | 0.0046 | −3.6077 × ${10}^{-5}$ |

GOOG | 0.4536 | 0.4886 | 0.0047 | −1.6584 × ${10}^{-4}$ |

**Table 5.**The mean square error (MSE) of the real standard deviation and theoretical standard deviations from MGCHP and MGCPP.

Ticker | MGCHP MSE | MGCPP MSE |
---|---|---|

INTC | $3.4039\times {10}^{-8}$ | $3.9858\times {10}^{-8}$ |

MSFT | $9.6454\times {10}^{-8}$ | $8.6189\times {10}^{-8}$ |

**Table 6.**Coefficients calculated by MGCHP and MGCPP models. Benchmark coefficients are coefficients of the least-square regression curves. Percentage errors are differences between two stochastic models with the benchmark.

Ticker | MGCHP Coefficient | MGCPP Coefficient | Benchmark Coefficient | MGCHP % Error | MGCPP % Error |
---|---|---|---|---|---|

INTC | 0.002086 | 0.002089 | 0.002162 | $3.515\%$ | $3.377\%$ |

MSFT | 0.002494 | 0.002487 | 0.002609 | $4.408\%$ | $4.676\%$ |

**Table 7.**The MSE and coefficients computed by MGCPP with 2-state and 7-state Markov chain for different tickers. The regression coefficients were derived by fitting the real standard deviations with square root curve. The MGCPP coefficients were computed by Equation (23).

Ticker | MSE | Regression Ceofficient | MGCPP Ceofficient | Percentage Error |
---|---|---|---|---|

AAPL 2-state | 0.2467 | 0.0278 | 0.0076 | 72.66% |

AAPL 7-state | 0.0064 | 0.0311 | 0.0288 | 7.40% |

GOOG 2-state | 0.4161 | 0.0307 | 0.0044 | 85.67% |

GOOG 7-state | 0.0081 | 0.0307 | 0.0287 | 6.51% |

AMZN 2-state | 0.1233 | 0.0189 | 0.0048 | 74.60% |

AMZN 7-state | 0.0225 | 0.0205 | 0.0147 | 28.29% |

Ticker | INTC | MSFT | AAPL | AMZN | GOOG |
---|---|---|---|---|---|

INTC | 0.4844 | 0.1719 | 1.2393 | 0.5317 | 0.5312 |

MSFT | 0.1719 | 0.5634 | 1.8361 | 0.7834 | 0.7162 |

AAPL | 1.2393 | 1.8361 | 62.3800 | 6.7811 | 6.4331 |

AMZN | 0.5317 | 0.7834 | 6.7811 | 19.2883 | 1.9617 |

GOOG | 0.5312 | 0.7162 | 6.4331 | 1.9617 | 22.7980 |

Ticker | MSE | Benchmark Ceofficient | MGCPP Ceofficient | Percentage Error |
---|---|---|---|---|

INTC 2-state | 1.4452 × ${10}^{-5}$ | 2.2361 × ${10}^{-3}$ | 2.0958 × ${10}^{-3}$ | 6.27% |

MSFT 2-state | 6.6227 × ${10}^{-5}$ | 2.5157 × ${10}^{-3}$ | 2.4919 × ${10}^{-3}$ | 0.94% |

AAPL 7-state | 6.1382 × ${10}^{-3}$ | 2.7799 × ${10}^{-2}$ | 2.8788 × ${10}^{-2}$ | 3.56% |

GOOG 7-state | 8.0981 × ${10}^{-3}$ | 3.0736 × ${10}^{-2}$ | 2.8686 × ${10}^{-2}$ | 6.67% |

AMZN 7-state | 1.1156 × ${10}^{-2}$ | 1.8940 × ${10}^{-2}$ | 1.4747 × ${10}^{-2}$ | 22.14% |

Overall Percentage Error | 7.92% |

**Table 10.**Test Errors for different tickers by applying 5-fold cross-validation. The errors are percentage errors between benchmark coefficients and the MGCPP coefficients.

Ticker | Fold 1 | Fold 2 | Fold 3 | Fold 4 | Fold 5 | Mean Error |
---|---|---|---|---|---|---|

INTC | 6.75% | 0.39% | 3.16% | 14.32% | 16.60% | 8.24% |

MSFT | 20.33% | 31.35% | 16.96% | 8.33% | 22.61% | 19.92% |

AAPL | 8.22% | 0.51% | 22.53% | 21.34% | 23.33% | 15.01% |

GOOG | 19.60% | 20.41% | 16.41% | 6.13% | 12.51% | 15.19% |

AMZN | 20.78% | 4.87% | 7.98% | 18.81% | 42.15% | 18.92% |

Overall Test Error | ${E}_{test}=15.46\%$ |

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**MDPI and ACS Style**

Guo, Q.; Remillard, B.; Swishchuk, A.
Multivariate General Compound Point Processes in Limit Order Books. *Risks* **2020**, *8*, 98.
https://doi.org/10.3390/risks8030098

**AMA Style**

Guo Q, Remillard B, Swishchuk A.
Multivariate General Compound Point Processes in Limit Order Books. *Risks*. 2020; 8(3):98.
https://doi.org/10.3390/risks8030098

**Chicago/Turabian Style**

Guo, Qi, Bruno Remillard, and Anatoliy Swishchuk.
2020. "Multivariate General Compound Point Processes in Limit Order Books" *Risks* 8, no. 3: 98.
https://doi.org/10.3390/risks8030098