# A Front-Fixing Implicit Finite Difference Method for the American Put Options Model

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## Abstract

**:**

## 1. Introduction

## 2. The American Put Options Model

## 3. The Front-Fixing Method

## 4. A New Implicit Finite Difference Scheme

## 5. Consistency and Stability

#### 5.1. Consistency

#### 5.2. Stability

## 6. A Posteriori Error Estimator

## 7. Numerical Results

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Implicit method: amplification factor module $\left|\lambda \right|$ for different values of n with $N=320$, $\mu =20$ and ${x}_{\infty}=1$.

**Figure 2.**Implicit method: Amplification factor module $\left|\lambda \right|$ for different values of $\mu $ at time $\tau =T$. The implicit method results in stability for any value of $\mu $.

**Figure 3.**Explicit method: Amplification factor module $\left|\lambda \right|$ for different values of $\mu $ at time $\tau =T$. The explicit method becomes unstable for $\mu \ge 26$.

**Figure 4.**Numerical results obtained by the implicit scheme (8): on the

**left**${p}_{j}^{N}$ versus ${x}_{j}$, on the

**right**, ${S}_{f}^{n}$ versus ${\tau}^{n}$, obtained with $J=80$, $N=320$ and $\mu =20$.

**Figure 5.**Numerical results obtained by implicit scheme (8), ${p}_{j}^{n}$ with $J=80$, $N=320$ and $\mu =20$.

**Figure 6.**Numerical estimated errors ${e}_{r}\left({p}^{n}\right)$ and ${e}_{r}\left({S}_{f}^{n}\right)$ versus ${\tau}^{n}$ for the explicit (

**left**) and implicit method (

**right**), setting $\u03f5=0.005$, $\mu =20$, ${J}_{5}=160$ and ${N}_{5}=1280$.

**Table 1.**Free boundary value ${S}_{f}^{N}$ at $\tau =T$ for different truncated boundary locations ${x}_{\infty}$.

${\mathit{x}}_{\mathit{\infty}}$ | $\mathit{N}=10$ | $\mathit{N}=20$ | $\mathit{N}=40$ |
---|---|---|---|

$0.7$ | $0.8710513251234$ | $0.8661003470130$ | $0.86351373607045$ |

1 | $0.8710513685210$ | $0.8661003514438$ | $0.86351373597835$ |

2 | $0.8710513685385$ | $0.8661003514444$ | $0.86351373597828$ |

4 | $0.8710513685388$ | $0.8661003514438$ | $0.86351373597789$ |

$\mathit{\delta}\mathit{r}$ | $\frac{|\mathit{\delta}{\mathit{S}}_{\mathit{f}}|}{\mathit{\delta}\mathit{r}}$ | $\frac{|{\mathit{S}}_{\mathit{f}}^{\mathit{n}+1}+\mathit{\delta}{\mathit{S}}_{\mathit{f}}|}{\mathit{r}+\mathit{\delta}\mathit{r}}$ | $\frac{|{\mathit{S}}_{\mathit{f}}^{\mathit{n}+1}+\mathit{\delta}{\mathit{S}}_{\mathit{f}}|}{|{\mathit{S}}_{\mathit{f}}^{\mathit{n}+1}|}$ | $\frac{{\left|\right|\mathit{\delta}\mathit{p}\left|\right|}_{\mathit{\infty}}}{\mathit{\delta}\mathit{r}}$ | $\frac{{\left|\right|\mathit{\delta}\mathit{p}\left|\right|}_{\mathit{\infty}}}{\mathit{r}+\mathit{\delta}\mathit{r}}$ | $\frac{\left|\right|{\mathit{p}}^{\mathit{n}+1}+\mathit{\delta}\mathit{p}{\left|\right|}_{\mathit{\infty}}}{\left|\right|{\mathit{p}}^{\mathit{n}+1}{\left|\right|}_{\mathit{\infty}}}$ |
---|---|---|---|---|---|---|

$\delta r=0.1$ | 0.740531 | 7.909584 | 1.008584 | 0.745939 | 1.181324 | 0.946084 |

$\delta r=0.05$ | 0.759815 | 8.251885 | 1.004403 | 0.765465 | 1.271923 | 0.972340 |

$\delta r=0.01$ | 0.776018 | 8.548762 | 1.000899 | 0.781872 | 1.352228 | 0.994350 |

$\mathbf{\delta}\mathbf{\sigma}$ | $\frac{|\mathbf{\delta}{\mathbf{S}}_{\mathbf{f}}|}{\mathbf{\delta}\mathbf{\sigma}}$ | $\frac{|{\mathbf{S}}_{\mathbf{f}}^{\mathbf{n}+\mathbf{1}}+\mathbf{\delta}{\mathbf{S}}_{\mathbf{f}}|}{\mathbf{\sigma}+\mathbf{\delta}\mathbf{\sigma}}$ | $\frac{|{\mathbf{S}}_{\mathbf{f}}^{\mathbf{n}+\mathbf{1}}+\mathbf{\delta}{\mathbf{S}}_{\mathbf{f}}|}{|{\mathbf{S}}_{\mathbf{f}}^{\mathbf{n}+\mathbf{1}}|}$ | $\frac{{\left|\right|\mathbf{\delta}\mathbf{p}\left|\right|}_{\infty}}{\mathbf{\delta}\mathbf{\sigma}}$ | $\frac{\left|\right|{\mathbf{p}}^{\mathbf{n}+\mathbf{1}}+\mathbf{\delta}\mathbf{p}{\left|\right|}_{\infty}}{\mathbf{\sigma}+\mathbf{\delta}\mathbf{\sigma}}$ | $\frac{\left|\right|{\mathbf{p}}^{\mathbf{n}+\mathbf{1}}+\mathbf{\delta}\mathbf{p}{\left|\right|}_{\infty}}{\left|\right|{\mathbf{p}}^{\mathbf{n}+\mathbf{1}}{\left|\right|}_{\infty}}$ |

$\delta \sigma =0.1$ | 1.013264 | 3.829016 | 0.9765080 | 1.025923 | 0.716437 | 1.147543 |

$\delta \sigma =0.05$ | 1.010532 | 4.059731 | 0.9882857 | 1.022231 | 0.702173 | 1.073572 |

$\delta \sigma =0.01$ | 1.007935 | 4.260559 | 0.9976631 | 1.018774 | 0.689935 | 1.014676 |

**Table 3.**Implicit method: Richardson’s repeated extrapolations for the free boundary value ${S}_{f}^{N}$ at $\tau =T$.

N | ${\mathit{U}}_{\mathit{g},0}$ | ${\mathit{U}}_{\mathit{g},1}$ | ${\mathit{U}}_{\mathit{g},2}$ | ${\mathit{U}}_{\mathit{g},3}$ | ${\mathit{U}}_{\mathit{g},4}$ | ${\mathit{U}}_{\mathit{g},5}$ |
---|---|---|---|---|---|---|

5 | $0.884069$ | |||||

20 | $0.866100$ | $0.860111$ | ||||

80 | $0.863100$ | $0.862100$ | $0.862232$ | |||

320 | $0.862719$ | $0.862592$ | $0.862625$ | $0.862631$ | ||

1280 | $0.862717$ | $0.862716$ | $0.862724$ | $0.862726$ | $0.862726$ | |

5120 | $0.862738$ | $0.862746$ | $0.862748$ | $0.862748$ | $0.862748$ | $0.862748$ |

**Table 4.**Comparison of American put option price $P(S,T)$ with parameters (33). PM: penalty method; EM: explicit method; EMR: explicit method with Richardson’s extrapolation; IM: implicit method; IMR: implicit method with Richardson’s extrapolation.

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**MDPI and ACS Style**

Fazio, R.; Insana, A.; Jannelli, A.
A Front-Fixing Implicit Finite Difference Method for the American Put Options Model. *Math. Comput. Appl.* **2021**, *26*, 30.
https://doi.org/10.3390/mca26020030

**AMA Style**

Fazio R, Insana A, Jannelli A.
A Front-Fixing Implicit Finite Difference Method for the American Put Options Model. *Mathematical and Computational Applications*. 2021; 26(2):30.
https://doi.org/10.3390/mca26020030

**Chicago/Turabian Style**

Fazio, Riccardo, Alessandra Insana, and Alessandra Jannelli.
2021. "A Front-Fixing Implicit Finite Difference Method for the American Put Options Model" *Mathematical and Computational Applications* 26, no. 2: 30.
https://doi.org/10.3390/mca26020030