# On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Numerical Method under the Generalized Black–Scholes Model

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 4. Numerical Results

## 5. Final Remarks

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Optimal exercise boundary for $r=0.02$ (and also $\lambda =2.7206\times {10}^{-6}$). ($T=2,$$\sigma =0.3,$$K=100$).

**Figure 4.**American put option values for different values of r(and $\lambda $). ($T=1,\sigma =0.1,K=100$).

(S,T,K,$\mathit{\sigma}$) | $\mathit{r}=0.02$ | $\mathit{r}=0.05$ | $\mathit{r}=0.08$ | $\mathit{r}=0.11$ |
---|---|---|---|---|

$(\mathbf{\lambda}=\mathbf{2.7206}\times {\mathbf{10}}^{-\mathbf{6}})$ | $(\mathbf{\lambda}=\mathbf{4.3814}\times {\mathbf{10}}^{-\mathbf{5}})$ | $(\mathbf{\lambda}=\mathbf{1.8500}\times {\mathbf{10}}^{-\mathbf{4}}$) | $(\mathbf{\lambda}=\mathbf{4.9585}\times {\mathbf{10}}^{-\mathbf{4}}$) | |

$(95,1.0,100,0.1)$ | $6.0165$ | $5.2869$ | $5.0116$ | $5.0025$ |

$(100,1.0,100,0.1)$ | $3.2347$ | $2.4576$ | $1.9151$ | $1.5293$ |

$(110,1.0,100,0.1)$ | $0.6661$ | $0.3897$ | $0.2210$ | $0.1218$ |

$(95,0.5,100,0.1)$ | $5.4703$ | $5.0889$ | $4.9855$ | $5.0050$ |

$(100,0.5,100,0.1)$ | $2.4243$ | $1.9738$ | $1.6338$ | $1.3700$ |

$(110,0.5,100,0.1)$ | $0.2274$ | $0.1439$ | $0.0894$ | $0.0544$ |

$(95,2.0,100,0.1)$ | $6.8305$ | $5.5790$ | $5.0588$ | $4.9961$ |

$(100,2.0,100,0.1)$ | $4.2805$ | $2.9778$ | $2.1717$ | $1.6569$ |

$(110,2.0,100,0.1)$ | $1.4508$ | $0.7599$ | $0.3859$ | $0.1901$ |

(S,T,r,$\mathit{\sigma}$) | Binomial | Front-Fixing | MBM-FDM | Simple Method | ||

$(80,1.0,0.05,0.2)$ | $20.0000$ | $20.0000$ | $20.0000$ | $20.0000$ | ||

$(90,1.0,0.05,0.2)$ | $11.4928$ | $11.4924$ | $11.4857$ | $11.4929$ | ||

$(100,1.0,0.05,0.2)$ | $6.0903$ | $6.0893$ | $6.0829$ | $6.0905$ | ||

$(110,1.0,0.05,0.2)$ | $2.9866$ | $2.9856$ | $2.9854$ | $2.9868$ | ||

$(120,1.0,0.05,0.2)$ | $1.3672$ | $1.3654$ | $1.3643$ | $1.3674$ | ||

RMSE | $0.0010$ | $0.0048$ | $0.0002$ | |||

The Proposed Method | ||||||

(S,T,r,$\mathbf{\sigma}$) | Computational Discrete Mesh (N × M) | |||||

125×25 | 250×50 | 500×100 | 1000×200 | 2000×400 | 4000×800 | |

$(80,1.0,0.05,0.2)$ | $19.9396$ | $19.9708$ | $19.9964$ |
$$20.0009$$
| $20.0001$ | $20.0000$ |

$(90,1.0,0.05,0.2)$ | $10.6927$ | $11.4142$ | $11.4760$ | $11.4886$ | $11.4913$ | $11.4923$ |

$(100,1.0,0.05,0.2)$ | $5.4337$ | $6.0368$ | $6.0775$ | $6.0862$ | $6.0887$ | $6.0897$ |

$(110,1.0,0.05,0.2)$ | $2.6967$ | $2.9603$ | $2.9787$ | $2.9836$ | $2.9852$ | $2.9859$ |

$(120,1.0,0.05,0.2)$ | $1.2610$ | $1.3592$ | $1.3637$ | $1.3655$ | $1.3663$ | $1.3667$ |

RMSE | 0.4838 | 0.1032 | 0.0231 | 0.0069 | 0.0028 | 0.0012 |

(S,T,K,r,$\mathit{\sigma}$) | Closed-Form Formula | The Proposed Method: Computational Mesh (N × M) | |||
---|---|---|---|---|---|

125 × 25 | 250 × 50 | 500 × 100 | 1000 × 200 | ||

$(100,0.5,100,0.02,0.3)$ | $7.9363$ | $7.7735$ | $7.9945$ |
$$7.9987$$
| $8.0020$ |

$(100,0.5,100,0.05,0.3)$ | $7.2091$ | $7.1535$ | $7.4053$ | $7.4137$ | $7.4191$ |

$(100,0.5,100,0.08,0.3)$ | $6.5281$ | $6.6098$ | $6.8795$ | $6.8966$ | $6.9015$ |

$(100,0.5,100,0.11,0.3)$ | $5.8926$ | $6.0840$ | $6.4102$ | $6.4294$ | $6.4347$ |

$(100,1.0,100,0.02,0.3)$ | $10.9493$ | $10.9206$ | $11.0657$ |
$$11.0786$$
| $11.0845$ |

$(100,1.0,100,0.05,0.3)$ | $9.5848$ | $9.8175$ | $9.9927$ | $10.0070$ | $10.0127$ |

$(100,1.0,100,0.08,0.3)$ | $8.3363$ | $8.8532$ | $9.0639$ | $9.0811$ | $9.0869$ |

$(100,1.0,100,0.11,0.3)$ | $7.2023$ | $8.0312$ | $8.2522$ | $8.2709$ | $8.2770$ |

$(100,2.0,100,0.02,0.3)$ | $15.1768$ | $15.2689$ | $15.3577$ |
$$15.3733$$
| $15.3808$ |

$(100,2.0,100,0.05,0.3)$ | $12.9045$ | $13.4118$ | $13.5299$ | $13.5726$ | $13.5801$ |

$(100,2.0,100,0.08,0.3)$ | $10.8381$ | $11.6097$ | $12.0134$ | $12.0352$ | $12.0422$ |

$(100,2.0,100,0.11,0.3)$ | $8.9858$ | $10.5430$ | $10.6741$ | $10.7056$ | $10.7140$ |

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Lee, J.-K. On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model. *Mathematics* **2020**, *8*, 1563.
https://doi.org/10.3390/math8091563

**AMA Style**

Lee J-K. On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model. *Mathematics*. 2020; 8(9):1563.
https://doi.org/10.3390/math8091563

**Chicago/Turabian Style**

Lee, Jung-Kyung. 2020. "On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model" *Mathematics* 8, no. 9: 1563.
https://doi.org/10.3390/math8091563