# Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Lemma**

**1.**

**Proof.**

**Direct problem**: To solve the option pricing problem (9) and (10), we set the boundary conditions as in [28]:

**Calibration problem**: Assume that the option prices $\{{\widehat{V}}_{ij},i=1,2,\dots ,N,j=1,2,\dots ,{M}_{i}\}$ with maturities $\left\{{T}_{i}\right\}$ and strike prices $\left\{{K}_{ij}\right\}$ in market are given. Market calibration requires us to find a volatility function $\sigma \left(\tau \right)$ and a parameter $\gamma $ which make the recalculated option price $V({S}_{0},{r}_{0},{\tau}_{0},{K}_{ij},{T}_{i},\sigma \left(\tau \right),\gamma )$ satisfy

## 3. Problem Regularization

#### 3.1. Existence and Stability of the Solutions of the Minimization Problem

**Theorem**

**1**

**Proof.**

**Theorem**

**2**

#### 3.2. ADMM Algorithm for the Regularized Problem

## 4. Computational Experiments

#### 4.1. Numerical Simulation

**Example**

**1.**

**Example**

**2.**

#### 4.2. Empirical Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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K | 64 | 68 | 72 | 76 | 80 |
---|---|---|---|---|---|

${T}_{1}=0.5$ | 1.4032 | 0.4094 | 0.0917 | 0.0165 | 0.0025 |

${T}_{2}=1.0$ | 2.5149 | 1.1465 | 0.4555 | 0.1600 | 0.0506 |

$\mathit{max}|{\mathit{\sigma}}_{\mathit{d}}-{\mathit{\sigma}}_{\mathit{d}-1}|$ | RMSE | $\mathit{max}|\mathit{\sigma}-\widehat{\mathit{\sigma}}|$ | $\mathit{\gamma}$ | AE | |
---|---|---|---|---|---|

$\xi =0$ | 1.9305 × 10${}^{-7}$ | 5.7762 × 10${}^{-4}$ | 1.0623 × 10${}^{-3}$ | 1.327 | 0.0056 |

$\xi =1\%$ | 3.8220 × 10${}^{-7}$ | 3.0564 × 10${}^{-3}$ | 9.3685 × 10${}^{-3}$ | 1.171 | 0.1504 |

$\xi =3\%$ | 7.7861 × 10${}^{-7}$ | 8.1093 × 10${}^{-3}$ | 2.8106 × 10${}^{-3}$ | 1.028 | 0.2934 |

$\xi =5\%$ | 1.2144 × 10${}^{-6}$ | 1.1219 × 10${}^{-2}$ | 4.6803 × 10${}^{-2}$ | 0.9512 | 0.3702 |

$\mathit{max}|{\mathit{\sigma}}_{\mathit{d}}-{\mathit{\sigma}}_{\mathit{d}-1}|$ | RMSE | $\mathit{max}|\mathit{\sigma}-\widehat{\mathit{\sigma}}|$ | $\mathit{\gamma}$ | AE | |
---|---|---|---|---|---|

$\xi =0$ | 7.7267 × 10${}^{-7}$ | 7.7984 × 10${}^{-4}$ | 1.6680 × 10${}^{-3}$ | 0.802 | 0.002 |

$\xi =1\%$ | 7.6843 × 10${}^{-7}$ | 9.3685 × 10${}^{-3}$ | 2.1668 × 10${}^{-3}$ | 0.811 | 0.011 |

$\xi =3\%$ | 7.5211 × 10${}^{-7}$ | 5.2394 × 10${}^{-3}$ | 2.8106 × 10${}^{-3}$ | 0.754 | 0.046 |

$\xi =5\%$ | 7.4837 × 10${}^{-7}$ | 8.4450 × 10${}^{-3}$ | 4.6843 × 10${}^{-2}$ | 0.712 | 0.088 |

Strike Price K | $\mathit{V}\left(\mathit{K},{\mathit{T}}_{1}\right)$ | $\mathit{V}\left(\mathit{K},{\mathit{T}}_{2}\right)$ | $\mathit{V}\left(\mathit{K},{\mathit{T}}_{3}\right)$ | $\mathit{V}\left(\mathit{K},{\mathit{T}}_{4}\right)$ |
---|---|---|---|---|

2.85 | 0.3928 | 0.4013 | 0.4179 | 0.4401 |

2.90 | 0.3434 | 0.3548 | 0.3758 | 0.4019 |

2.95 | 0.2966 | 0.3082 | 0.3347 | 0.3659 |

3.00 | 0.2470 | 0.2647 | 0.2984 | 0.3334 |

3.10 | 0.1523 | 0.1837 | 0.2269 | 0.2695 |

3.20 | 0.0726 | 0.1192 | 0.1671 | 0.2167 |

3.30 | 0.0275 | 0.0707 | 0.1210 | 0.1727 |

3.40 | 0.0084 | 0.0393 | 0.0860 | 0.1343 |

3.50 | 0.0030 | 0.0207 | 0.0590 | 0.1038 |

3.60 | 0.0014 | 0.0107 | 0.0422 | 0.0787 |

3.70 | 0.0008 | 0.0059 | 0.0290 | 0.0591 |

$\mathit{max}|{\mathit{\sigma}}_{\mathit{d}}-{\mathit{\sigma}}_{\mathit{d}-1}|$ | RMSE | $\mathit{max}|\mathit{\sigma}-\widehat{\mathit{\sigma}}|$ | $\mathit{\gamma}$ | |
---|---|---|---|---|

Empirical results | 4.99 × 10${}^{-7}$ | 0.0151 | 0.0402 | 0.8451 |

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**MDPI and ACS Style**

Zhao, J.; Xu, Z.
Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model. *Fractal Fract.* **2022**, *6*, 344.
https://doi.org/10.3390/fractalfract6070344

**AMA Style**

Zhao J, Xu Z.
Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model. *Fractal and Fractional*. 2022; 6(7):344.
https://doi.org/10.3390/fractalfract6070344

**Chicago/Turabian Style**

Zhao, Jiajia, and Zuoliang Xu.
2022. "Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model" *Fractal and Fractional* 6, no. 7: 344.
https://doi.org/10.3390/fractalfract6070344