# A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. The Solution Method

#### 3.1. Spatial Discretization

#### 3.2. Temporal Discretization

## 4. Numerical Results and Analysis

^{®}2019. The validity of the scheme has been verified through the produced results.

## 5. Conclusions and Recommendations

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Comparison between exact and numerical solutions of the linear and nonlinear European put option model for various stock price values at $t=0$.

S | SSP-RK3 + FD6 | Dura and Moşneagu [41] | Exact | ||
---|---|---|---|---|---|

Linear Model | Nonlinear Model | Solution | |||

a = 0.02 | a = 0.05 | (Linear Model) | |||

4.0 | 5.753096 | 5.753096 | 5.753096 | 5.753102 | 5.753100 |

8.0 | 1.902102 | 1.904440 | 1.915559 | 1.902102 | 1.902434 |

10.0 | 0.668906 | 0.673788 | 0.696650 | 0.668360 | 0.669390 |

16.0 | 0.005324 | 0.005498 | 0.006353 | 0.005419 | 0.005386 |

20.0 | 1.091776 × 10${}^{-4}$ | 1.139578 × 10${}^{-4}$ | 1.375594 × 10${}^{-4}$ | 1.170806 × 10${}^{-4}$ | 1.129336 × 10${}^{-4}$ |

N | M | ${\u2225.\u2225}_{\mathit{ds},\mathit{\infty}}$ | Ratio ${\u2225.\u2225}_{\mathit{ds},\mathit{\infty}}$ | ${\u2225.\u2225}_{\mathit{ds},2}$ | Ratio ${\u2225.\u2225}_{\mathit{ds},2}$ |
---|---|---|---|---|---|

51 | 101 | 0.018822 | 0.007153 | ||

101 | 1001 | 0.018035 | 1.043695 | 0.002887 | 2.477658 |

201 | 2001 | 0.002668 | 6.759370 | 6.547562 × 10${}^{-4}$ | 4.409275 |

401 | 4001 | 4.036584 × 10${}^{-4}$ | 6.609549 | 1.392947 × 10${}^{-4}$ | 4.700510 |

N | M | Error (Linear Model) | Error (Nonlinear Model) |
---|---|---|---|

51 | 101 | 0.013100 | 0.012081 |

101 | 1001 | 0.004132 | 0.003932 |

201 | 2001 | 8.054754 × 10${}^{-4}$ | 9.081600 × 10${}^{-4}$ |

N | M | CPU (Linear Model) | CPU (Nonlinear Model) | CPU (Nonlinear Model) |
---|---|---|---|---|

a = 0.02 | a = 0.05 | |||

51 | 101 | 0.136891 | 0.166548 | 0.171630 |

101 | 2001 | 3.223077 | 3.807224 | 3.830958 |

201 | 4001 | 12.112536 | 14.122904 | 14.242410 |

401 | 8001 | 63.284679 | 72.833058 | 72.978609 |

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**MDPI and ACS Style**

Gulen, S.; Popescu, C.; Sari, M. A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities. *Mathematics* **2019**, *7*, 760.
https://doi.org/10.3390/math7080760

**AMA Style**

Gulen S, Popescu C, Sari M. A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities. *Mathematics*. 2019; 7(8):760.
https://doi.org/10.3390/math7080760

**Chicago/Turabian Style**

Gulen, Seda, Catalin Popescu, and Murat Sari. 2019. "A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities" *Mathematics* 7, no. 8: 760.
https://doi.org/10.3390/math7080760