# European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Some Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 2. Description of the Method Using the Caputo–Fabrizio Fractional Operator

## 3. Solution of the European Option Pricing Problem

#### 3.1. Fractional European Option Pricing Problem in the Sense of the Caputo–Fabrizio Derivative

#### 3.2. Fractional Generalized European Option Pricing Problem in the Sense of the Caputo–Fabrizio Derivative

## 4. Determining Stabilization and Convergence of Suggested Method

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FDE | Fractional differential equation |

LHAM | Laplace homotopy analysis method |

CF | Caputo–Fabrizio |

AB | Atangana–Baleanu |

LT | Laplace transform |

FC | Fractional calculus |

ENSO | El Niño–Southern Oscillation |

FBSE | Fractional Black–Scholes equation |

GFBSE | Generalized fractional Black–Scholes equation |

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**Figure 1.**The solution function of Equation (2) in the sense of Caputo–Fabrizio (CF) with respect to $\left(\alpha ,\tau \right)=\left[0,1\right]\times \left[0,1\right].$

**Figure 3.**Option prices $V\left(S,t\right)$ with respect to underlying asset S for different $\alpha $ values.

**Figure 4.**The solution function of (4) in the CF derivative sense with respect to $\left(\alpha ,\tau \right)=\left[0,1\right]\times \left[0,1\right].$

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**MDPI and ACS Style**

Yavuz, M.; Özdemir, N.
European Vanilla Option Pricing Model of Fractional Order without Singular Kernel. *Fractal Fract.* **2018**, *2*, 3.
https://doi.org/10.3390/fractalfract2010003

**AMA Style**

Yavuz M, Özdemir N.
European Vanilla Option Pricing Model of Fractional Order without Singular Kernel. *Fractal and Fractional*. 2018; 2(1):3.
https://doi.org/10.3390/fractalfract2010003

**Chicago/Turabian Style**

Yavuz, Mehmet, and Necati Özdemir.
2018. "European Vanilla Option Pricing Model of Fractional Order without Singular Kernel" *Fractal and Fractional* 2, no. 1: 3.
https://doi.org/10.3390/fractalfract2010003