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Machine Learning to Compute Implied Volatility from European/American Options Considering Dividend Yield^{ †}

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## Abstract

**:**

## 1. Problem Formulation

## 2. Methodology

#### 2.1. ANN for European Implied Volatility

#### 2.2. ANN for American Implied Volatility

## 3. Numerical Results

## References

- Liu, S.; Leitao, A.; Borovykh, A.; Oosterlee, C.W. On Calibration Neural Networks for extracting implied information from American options. arXiv
**2020**, arXiv:2001.11786. [Google Scholar] - Fang, F.; Oosterlee, C.W. Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions. Numer. Math.
**2009**, 114, 27. [Google Scholar] [CrossRef] - Liu, S.; Oosterlee, C.W.; Bohte, S.M. Pricing Options and Computing Implied Volatilities using Neural Networks. Risks
**2019**, 7, 16. [Google Scholar] [CrossRef]

**Figure 1.**The Vega for American options: One tail is near zero, and the other tail is flat (zero value).

Parameters | Values |
---|---|

Hidden layers | 4 |

Neurons(each layer) | 400 |

Activation | ReLU |

Initialization | Glorot |

Optimizer | Adam |

Batch size | 1024 |

Parameters | Range | |
---|---|---|

Inputs | Stock price ($K/{S}_{0}$) | [0.3, 1.8] |

Time to maturity ($\tau $) | [0.08, 2.5] | |

Risk-free rate (r) | [0.0, 0.25] | |

Dividend yield (q) | [0.0, 0.25] | |

Scaled time value ($\widehat{V}$) | - | |

Output | Volatility ($\sigma $) | (0.01, 1.05) |

Phase | European Options | American Options | ||||||
---|---|---|---|---|---|---|---|---|

MSE | MAE | MAPE | ${\mathrm{R}}^{\mathbf{2}}$ | MSE | MAE | MAPE | ${\mathrm{R}}^{\mathbf{2}}$ | |

Training | 1.72 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-7}$ | 3.17 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 6.99 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 0.9999976 | 7.12 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-7}$ | 5.66 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 1.42 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 0.999990 |

Testing | 1.94 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-7}$ | 3.35 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 7.39 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 0.9999972 | 1.93 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ | 6.52 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | 2.35 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ | 0.999974 |

Method | GPU (s) | CPU (s) | Robustness |
---|---|---|---|

Newton-Raphson | 19.68 | 23.06 | No |

Brent | 52.08 | 60.67 | Yes |

Bi-section | 337.94 | 390.91 | Yes |

IV-ANN | 0.20 | 1.90 | Yes |

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## Share and Cite

**MDPI and ACS Style**

Liu, S.; Leitao, Á.; Borovykh, A.; Oosterlee, C.W.
Machine Learning to Compute Implied Volatility from European/American Options Considering Dividend Yield. *Proceedings* **2020**, *54*, 61.
https://doi.org/10.3390/proceedings2020054061

**AMA Style**

Liu S, Leitao Á, Borovykh A, Oosterlee CW.
Machine Learning to Compute Implied Volatility from European/American Options Considering Dividend Yield. *Proceedings*. 2020; 54(1):61.
https://doi.org/10.3390/proceedings2020054061

**Chicago/Turabian Style**

Liu, Shuaiqiang, Álvaro Leitao, Anastasia Borovykh, and Cornelis W. Oosterlee.
2020. "Machine Learning to Compute Implied Volatility from European/American Options Considering Dividend Yield" *Proceedings* 54, no. 1: 61.
https://doi.org/10.3390/proceedings2020054061