# Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications

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## Abstract

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## 1. Introduction

## 2. Option Pricing

#### 2.1. The Risk-Neutral Approach

#### 2.2. Black–Scholes Model

#### 2.3. Finite-Moment Lévy-Stable Model

#### 2.4. Space-Time Option Pricing Model

## 3. Series Representation of the Pricing Formulas under the Space-Time Fractional Diffusion

#### 3.1. Risk-Neutral Parameter

#### 3.1.1. Mellin–Barnes Representation of the Risk-Neutral Parameter

#### 3.1.2. Series Representation of the Risk-Neutral Parameter

#### 3.2. Option Price

#### 3.2.1. Mellin–Barnes Representation of the Option Price

#### 3.2.2. Series Representation of the Option Price

## 4. Applications

#### 4.1. Call Price

#### 4.2. Implied Volatility

#### 4.2.1. At-the-Money Volatility

#### 4.2.2. Volatility Smile

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**In the first graph, we plot the evolution of ${\mu}_{\gamma}$ in function of $\gamma $ for different stability parameters $\alpha \in [1.6,1]$ and market volatility $\sigma =20\%$. We only consider $\gamma >0.38$ so that the condition $\gamma >1-\frac{1}{\alpha}$ is fulfilled for any of the chosen stabilities. In graph 2 and 3, we plot the evolution of ${\mu}_{\gamma}$ in function of the market volatility and the stability parameter resp., for different values of the fractionality $\gamma $.

**Figure 2.**Residues contributing to the evaluation of the double Mellin–Barnes integral (49).

**Figure 3.**Convergence of the m and n partial sums of the double-fractional call price double series formula (55).

**Figure 4.**Evolution of the double-fractional call price in function of various parameters (time-fractionality parameter $\gamma $, stability parameter $\alpha $, asset (spot) price S and market volatility $\sigma $.

**Figure 5.**The at-the-money implied volatility for the double-fractional Black–Scholes model, as a function of time fractionarity $\gamma $.

**Table 1.**Estimated values of option pricing parameters based on Black–Scholes model, FMLS model and Space-time fractional model. The estimation was done for all options and separately for call options and put options, respectively. We see that for this case is $\gamma $ very close to one, which does not have to be true for illiquid markets or during the abnormal periods. $AE$ denotes the aggregated error, which is defined as the sum of absolute differences between estimated price and market price. Table was taken from Ref. [7] and it is possible to find more details about the estimation there.

All Options | |||

Parameter | Black–Scholes | Lévy Stable | Double-Fractional |

$\alpha $ | - | 1.493(0.028) | 1.503(0.037) |

$\gamma $ | - | - | 1.017(0.019) |

$\sigma $ | 0.1696(0.027) | 0.140(0.021) | 0.143(0.030) |

AE | 8240(638) | 6994(545) | 6931(553) |

Call Options | |||

Parameter | Black–Scholes | Lévy Stable | Double-Fractional |

$\alpha $ | - | 1.563(0.041) | 1.585(0.038) |

$\gamma $ | - | - | 1.034(0.024) |

$\sigma $ | 0.140(0.021) | 0.118(0.026) | 0.137(0.020) |

AE | 3882(807) | 3610(812) | 3550(828) |

Put Options | |||

Parameter | Black–Scholes | Lévy Stable | Double-Fractional |

$\alpha $ | - | 1.493(0.031) | 1.508(0.036) |

$\gamma $ | - | - | 1.047(0.017) |

$\sigma $ | 0.193(0.039) | 0.163(0.034) | 0.163(0.037) |

AE | 3741(711) | 3114(591) | 2968(594) |

Strike | Call Price | BS Vol | F-BS Vol ($\mathit{\gamma}=0.8$) | F-BS Vol ($\mathit{\gamma}=0.9$) | F-BS Vol ($\mathit{\gamma}=1.1$) |
---|---|---|---|---|---|

900 | 118.9 | 0.4708 | 0.3163 | 0.3827 | 0.5900 |

940 | 92.7 | 0.4462 | 0.3066 | 0.3670 | 0.5330 |

980 | 69.5 | 0.4232 | 0.2929 | 0.3493 | 0.5210 |

1020 | 49.2 | 0.3976 | 0.2754 | 0.3284 | 0.4891 |

1060 | 32.3 | 0.3711 | 0.2557 | 0.3058 | 0.4574 |

1100 | 19.5 | 0.3475 | 0.2380 | 0.2857 | 0.4186 |

1150 | 8.9 | 0.3279 | 0.2269 | 0.2727 | 0.3938 |

1180 | 5.1 | 0.3301 | 0.2324 | 0.2789 | 0.3764 |

1220 | 2 | 0.3514 | 0.2514 | 0.3015 | 0.3692 |

1280 | 0.25 | 0.4110 | 0.2949 | 0.3544 | 0.4166 |

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**MDPI and ACS Style**

Aguilar, J.-P.; Korbel, J.
Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications. *Fractal Fract.* **2018**, *2*, 15.
https://doi.org/10.3390/fractalfract2010015

**AMA Style**

Aguilar J-P, Korbel J.
Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications. *Fractal and Fractional*. 2018; 2(1):15.
https://doi.org/10.3390/fractalfract2010015

**Chicago/Turabian Style**

Aguilar, Jean-Philippe, and Jan Korbel.
2018. "Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications" *Fractal and Fractional* 2, no. 1: 15.
https://doi.org/10.3390/fractalfract2010015