# A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation

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## Abstract

**:**

## 1. Introduction

#### 1.1. PDE Formulation

#### 1.2. Initial Conditions

#### 1.3. Boundary Conditions

#### 1.4. Caputo Fractional Derivative

#### 1.5. Time Discretization

#### 1.6. Background on Numerical Methods

#### 1.7. Paper’s Outline

## 2. Spatial Discretization Nodes

## 3. A Fast High-Order Discretization

## 4. Construction of the Solver

## 5. Numerical Tests

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Computational illustrations in Example 1 for PMBS.

**Top Left**: The point plot of the computational solution for $m=n=40$.

**Top Right**: The plot of the computational solution for $m=n=40$.

**Bottom Left**: The point plot of the computational solution for $m=n=80$.

**Bottom Right**: The plot of the computational solution for $m=n=80$.

**Figure 2.**Computational illustrations in Example 2 for PMBS.

**Top Left**: The point plot of the computational solution for $m=n=21$.

**Top Right**: The plot of the computational solution for $m=n=21$.

**Bottom Left**: The point plot of the computational solution for $m=n=41$.

**Bottom Right**: The plot of the computational solution for $m=n=41$.

m,n | ${\mathit{u}}_{\mathit{FDM}}$ | ${\mathit{\epsilon}}_{\mathit{FDM}}$ | T${}_{\mathbf{FDM}}$ | ${\mathit{u}}_{\mathit{SM}}$ | ${\mathit{\epsilon}}_{\mathit{SM}}$ | T${}_{\mathbf{SM}}$ | ${\mathit{u}}_{\mathit{PMBS}}$ | ${\mathit{\epsilon}}_{\mathit{PMBS}}$ | T${}_{\mathbf{PMBS}}$ |
---|---|---|---|---|---|---|---|---|---|

10 | 16.070 | 2.11 × 10${}^{0}$ | 0.01 | 17.378 | 8.1 × 10${}^{-1}$ | 0.01 | 17.381 | 8.0 × 10${}^{-1}$ | 0.01 |

20 | 17.889 | 2.98 × 10${}^{-1}$ | 0.02 | 17.694 | 4.94 × 10${}^{-1}$ | 0.02 | 17.701 | 4.8 × 10${}^{-1}$ | 0.02 |

40 | 17.913 | 2.75 × 10${}^{-1}$ | 0.16 | 18.003 | 1.85 × 10${}^{-1}$ | 0.18 | 18.091 | 9.7 × 10${}^{-2}$ | 0.16 |

80 | 18.039 | 1.48 × 10${}^{-1}$ | 3.96 | 18.213 | 2.46 × 10${}^{-2}$ | 3.84 | 18.205 | 1.6 × 10${}^{-2}$ | 3.63 |

120 | 18.055 | 1.33 × 10${}^{-1}$ | 20.02 | 18.190 | 1.64 × 10${}^{-3}$ | 20.96 | 18.187 | 1.3 × 10${}^{-3}$ | 19.17 |

m,n | ${\mathit{u}}_{\mathit{FDM}}$ | ${\mathit{\epsilon}}_{\mathit{FDM}}$ | T${}_{\mathbf{FDM}}$ | ${\mathit{u}}_{\mathit{SM}}$ | ${\mathit{\epsilon}}_{\mathit{SM}}$ | T${}_{\mathbf{SM}}$ | ${\mathit{u}}_{\mathit{PMBS}}$ | ${\mathit{\epsilon}}_{\mathit{PMBS}}$ | T${}_{\mathbf{PMBS}}$ |
---|---|---|---|---|---|---|---|---|---|

11 | 23.914 | 7.1 × 10${}^{-1}$ | 0.01 | 23.631 | 9.9 × 10${}^{-1}$ | 0.03 | 23.712 | 9.1 × 10${}^{-1}$ | 0.02 |

21 | 24.316 | 3.1 × 10${}^{-1}$ | 0.03 | 24.117 | 5.1 × 10${}^{-1}$ | 0.05 | 24.239 | 3.9 × 10${}^{-1}$ | 0.05 |

41 | 24.466 | 1.6 × 10${}^{-1}$ | 0.15 | 24.356 | 2.7 × 10${}^{-1}$ | 0.21 | 24.546 | 8.3 × 10${}^{-2}$ | 0.19 |

81 | 24.545 | 8.4 × 10${}^{-2}$ | 4.14 | 24.565 | 6.4 × 10${}^{-2}$ | 4.69 | 24.601 | 2.8 × 10${}^{-2}$ | 4.37 |

161 | 24.580 | 4.9 × 10${}^{-2}$ | 63.97 | 24.636 | 6.1 × 10${}^{-3}$ | 64.01 | 24.624 | 5.9 × 10${}^{-3}$ | 62.64 |

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**MDPI and ACS Style**

Ghabaei, R.; Lotfi, T.; Ullah, M.Z.; Shateyi, S.
A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation. *Fractal Fract.* **2023**, *7*, 323.
https://doi.org/10.3390/fractalfract7040323

**AMA Style**

Ghabaei R, Lotfi T, Ullah MZ, Shateyi S.
A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation. *Fractal and Fractional*. 2023; 7(4):323.
https://doi.org/10.3390/fractalfract7040323

**Chicago/Turabian Style**

Ghabaei, Rouhollah, Taher Lotfi, Malik Zaka Ullah, and Stanford Shateyi.
2023. "A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation" *Fractal and Fractional* 7, no. 4: 323.
https://doi.org/10.3390/fractalfract7040323