# Univariate and Multivariate GARCH Models Applied to Bitcoin Futures Option Pricing

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Theoretical Framework

#### 3.1. Heston–Nandi Futures Option Pricing Model

#### 3.2. Multivariate GARCH Futures Option Pricing Model

**Proposition**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

#### 3.3. Multivariate GARCH Models

## 4. Empirical Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1. | The dataset was obtained from the Thomson Reuters Datastream databank. |

2. | The market prices were obtained from CME Group. |

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April | May | June | |
---|---|---|---|

Mean | 0.0132 | 0.0139 | 0.0126 |

Median | 0.0057 | 0.0038 | 0.0112 |

Maximum | 0.1631 | 0.1626 | 0.1578 |

Minimum | −0.1557 | −0.2118 | −0.1605 |

Std. Dev. | 0.0517 | 0.0572 | 0.0563 |

Skewness | −0.2318 | −0.5091 | −0.4326 |

Kurtosis | 4.6279 | 5.3923 | 3.9755 |

Jarque-Bera | 13.0122 | 25.0679 | 4.9587 |

Probability | 0.0015 | 0.0000 | 0.0838 |

Observations | 109 | 89 | 70 |

April | May | June | |
---|---|---|---|

$\lambda $ | 5.2100 | 4.2970 | 4.0490 |

${\alpha}_{0}$ | 0.0024 | 0.0030 | 0.0028 |

${\alpha}_{1}$ | 2.0 × 10${}^{-4}$ | 3.5 × 10${}^{-10}$ | 3.2 × 10${}^{-11}$ |

${\beta}_{1}$ | 2.3 × 10${}^{-7}$ | 0.0685 | 0.0930 |

AIC | −4.1841 | −3.7271 | −3.2558 |

April | May | June | |
---|---|---|---|

$\lambda $ | 5.0020 | 4.2970 | 4.0490 |

${\alpha}_{0}$ | 0.0023 | 0.0030 | 0.0028 |

${\alpha}_{1}$ | 2.3 × 10 ${}^{-4}$ | 3.5 × 10${}^{-10}$ | 3.5 × 10${}^{-11}$ |

${\beta}_{1}$ | 3.4 × 10${}^{-8}$ | 0.0671 | 0.0927 |

$\gamma $ | 16.1600 | 0.3934 | 0.0986 |

AIC | −2.1870 | −1.7271 | −1.2558 |

Expiry | Test Statistic |
---|---|

April | 1.2856 |

May | 2.8 × 10${}^{-8}$ |

June | −2.1× 10${}^{-8}$ |

Symmetric HN | Asymmetric HN | |
---|---|---|

RMSE | 669.6865 | 724.2534 |

MAE | 496.0976 | 533.5693 |

Symmetric HN | Asymmetric HN | |
---|---|---|

RMSE | 1302.6712 | 1302.6727 |

MAE | 1092.2221 | 1092.2235 |

Symmetric HN | Asymmetric HN | |
---|---|---|

RMSE | 1149.2917 | 1149.2875 |

MAE | 1018.2780 | 1018.2741 |

Expiry | Test Statistic |
---|---|

April | −2.5072 * |

May | −3.3034 * |

June | 4.0599 |

Estimated Parameter | |
---|---|

${\theta}_{1}$ | 1.4 × 10${}^{-9}$ |

${\theta}_{2}$ | 0.5657 |

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**MDPI and ACS Style**

Venter, P.J.; Maré, E.
Univariate and Multivariate GARCH Models Applied to Bitcoin Futures Option Pricing. *J. Risk Financial Manag.* **2021**, *14*, 261.
https://doi.org/10.3390/jrfm14060261

**AMA Style**

Venter PJ, Maré E.
Univariate and Multivariate GARCH Models Applied to Bitcoin Futures Option Pricing. *Journal of Risk and Financial Management*. 2021; 14(6):261.
https://doi.org/10.3390/jrfm14060261

**Chicago/Turabian Style**

Venter, Pierre J., and Eben Maré.
2021. "Univariate and Multivariate GARCH Models Applied to Bitcoin Futures Option Pricing" *Journal of Risk and Financial Management* 14, no. 6: 261.
https://doi.org/10.3390/jrfm14060261