# A Quasi-Closed-Form Solution for the Valuation of American Put Options

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## Abstract

**:**

## 1. Introduction

## 2. The American Put Option Valuation Model

#### 2.1. Analytical Solution for the Value of an American Put Option

- ${\alpha}_{n}={\left(2r\tau +\tau {\sigma}^{2}\right)}^{2}+4\delta {\tau}^{2}\left(-2r+\delta +{\sigma}^{2}\right)+8n{\sigma}^{2}$ for $n=1,2,3,4$
- $\beta =\tau {\sigma}^{2}-2r\tau $
- ${a}_{m-i}$ is a parameter with ${a}_{m-i}=0$ for m − i ≤ 0 and i = 1,2,3.

- ${d}_{m-i}$ is a parameter with ${d}_{m-i}=0\text{}\mathrm{for}\text{}m-i\le 0\text{}\mathrm{and}\text{}i=1,2,3$.

#### 2.2. The Application of the Richardson Extrapolation in the Valuation Model of an American Put Option

## 3. Results of the American Put Option Valuation Model

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**American put option as a function of S and τ. The values of the parameters underlying the construction of this figure are as follows: n, the number of time intervals is 2; r, the interest rate is 0.06; σ, the volatility of the stock price is 0.4; K, the strike price is 100; and ϕ, the value of constant dividends is 1.6.

**Figure 2.**American put option as a function of S and σ. The values of the parameters underlying the construction of this figure are as follows: n, the number of time intervals is 2; r, the interest rate is 0.06; τ, the term remaining to maturity is 3 years; K, the strike price is 100; and ϕ, the value of constant dividends is 1.6.

**Figure 3.**American put option as a function of $S$. The values of the parameters underlying the construction of this figure are as follows: $n$, the number of time intervals is 4; $\tau $, the term remaining to maturity is 3 years; $r$, the interest rate is 0.08; $\sigma $, the volatility of the stock price is 0.20; $K$, the strike price is 100; $\underset{\_}{{S}_{4}}$, the critical price is 79.66; $\delta $, the rate of dividend distribution is 0.02 and 0.14; and, $\varphi $, the value of constant dividends is 0.0.

(1) (S; δ) | (2) TRUE | (3) RAN4 | (4) EXP3 | (5) QD+ | (6) QD* | (7) PA | (8) MLE (n = 4) |
---|---|---|---|---|---|---|---|

(80;0.12) | 25.658 | 25.654 | 25.657 | 25.753 | 25.715 | 25.648 | 25.654 |

(90;0.12) | 20.083 | 20.084 | 20.082 | 20.202 | 20.176 | 20.079 | 20.084 |

(100;0.12) | 15.498 | 15.498 | 15.497 | 15.618 | 15.601 | 15.509 | 15.498 |

(110;0.12) | 11.803 | 11.805 | 11.802 | 11.912 | 11.900 | 11.834 | 11.805 |

(120;0.12) | 8.886 | 8.885 | 8.885 | 8.978 | 8.970 | 8.941 | 8.885 |

(80;0.08) | 22.205 | 22.197 | 22.208 | 22.162 | 22.149 | 22.196 | 22.197 |

(90;0.08) | 16.207 | 16.197 | 16.211 | 16.181 | 16.17 | 16.194 | 16.197 |

(100;0.08) | 11.704 | 11.688 | 11.707 | 11.708 | 11.7 | 11.692 | 11.688 |

(110;0.08) | 8.367 | 8.354 | 8.37 | 8.395 | 8.39 | 8.36 | 8.354 |

(120;0.08) | 5.93 | 5.918 | 5.932 | 5.971 | 5.967 | 5.929 | 5.918 |

(80;0.04) | 20.35 | 20.35 | 20.351 | 20.334 | 20.337 | 20.342 | 20.35 |

(90;0.04) | 13.497 | 13.496 | 13.5 | 13.465 | 13.471 | 13.473 | 13.496 |

(100;0.04) | 8.944 | 8.936 | 8.947 | 8.926 | 8.931 | 8.912 | 8.936 |

(110;0.04) | 5.912 | 5.902 | 5.915 | 5.916 | 5.92 | 5.876 | 5.902 |

(120;0.04) | 3.898 | 3.887 | 3.9 | 3.92 | 3.922 | 3.86 | 3.887 |

(80;0.00) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

(90;0.00) | 11.697 | 11.701 | 11.699 | 11.703 | 11.709 | 11.701 | 11.701 |

(100;0.00) | 6.932 | 6.932 | 6.935 | 6.95 | 6.958 | 6.914 | 6.932 |

(110;0.00) | 4.155 | 4.153 | 4.157 | 4.185 | 4.191 | 4.132 | 4.153 |

(120;0.00) | 2.51 | 2.507 | 2.512 | 2.548 | 2.551 | 2.484 | 2.507 |

$RMSRE$ | - | 0.00105 | 0.00032 | 0.00590 | 0.00587 | 0.00413 | 0.00105 |

(1) (S,τ,σ,r,δ) | (2) BENCH | (3) BJST | (4) IB | (5) CEAM | (6) MLE (n = 4) |
---|---|---|---|---|---|

(80;3.0;0.40;0.06;0.02) | 29.26 | 29.1 | 29.1 | 29.33 | 29.24 |

(85;3.0;0.40;0.06;0.02) | 26.92 | 26.77 | 26.77 | 26.94 | 26.9 |

(90;3.0;0.40;0.06;0.02) | 24.8 | 24.65 | 24.65 | 24.82 | 24.77 |

(95;3.0;0.40;0.06;0.02) | 22.88 | 22.73 | 22.73 | 22.89 | 22.85 |

(100;3.0;0.40;0.06;0.02) | 21.13 | 20.98 | 20.98 | 21.11 | 21.1 |

(105;3.0;0.40;0.06;0.02) | 19.54 | 19.4 | 19.39 | 19.49 | 19.5 |

(110;3.0;0.40;0.06;0.02) | 18.08 | 17.95 | 17.94 | 18.03 | 18.05 |

(115;3.0;0.40;0.06;0.02) | 16.76 | 16.63 | 16.62 | 16.71 | 16.72 |

(120;3.0;0.40;0.06;0.02) | 15.54 | 15.42 | 15.4 | 15.5 | 15.51 |

(100;3.0;0.40;0.02;0.02) | 25.89 | 25.78 | 25.82 | 25.89 | 25.86 |

(100;3.0;0.40;0.04;0.02) | 23.3 | 23.17 | 23.17 | 23.3 | 23.26 |

(100;3.0;0.40;0.06;0.02) | 21.13 | 20.98 | 20.99 | 21.11 | 21.1 |

(100;3.0;0.40;0.08;0.02) | 19.27 | 19.13 | 19.14 | 19.25 | 19.25 |

(100;3.0;0.40;0.10;0.02) | 17.66 | 17.54 | 17.55 | 17.63 | 17.65 |

(100;3.0;0.30;0.06;0.02) | 15.17 | 15.04 | 15.06 | 15.19 | 15.15 |

(100;3.0;0.35;0.06;0.02) | 18.16 | 18.02 | 18.03 | 18.19 | 18.13 |

(100;3.0;0.40;0.06;0.02) | 21.13 | 20.98 | 20.99 | 21.11 | 21.1 |

(100;3.0;0.45;0.06;0.02) | 24.07 | 23.91 | 23.91 | 24.06 | 24.03 |

(100;3.0;0.50;0.06;0.02) | 26.98 | 26.8 | 26.81 | 27.03 | 26.93 |

(100;0.5;0.40;0.06;0.02) | 10.27 | 10.21 | 10.23 | 10.26 | 10.26 |

(100;1.0;0.40;0.06;0.02) | 13.88 | 13.78 | 13.8 | 13.81 | 13.85 |

(100;1.5;0.40;0.06;0.02) | 16.37 | 16.25 | 16.26 | 16.35 | 16.34 |

(100;2.0;0.40;0.06;0.02) | 18.28 | 18.15 | 18.16 | 18.26 | 18.25 |

(100;2.5;0.40;0.06;0.02) | 19.83 | 19.69 | 19.7 | 19.81 | 19.8 |

(100;3.0;0.40;0.06;0.02) | 21.13 | 20.98 | 20.99 | 21.11 | 21.1 |

(100;3.5;0.40;0.06;0.02) | 22.24 | 22.08 | 22.08 | 22.22 | 22.2 |

(100;4.0;0.40;0.06;0.02) | 23.19 | 23.03 | 23.04 | 23.23 | 23.16 |

(100;4.5;0.40;0.06;0.02) | 24.02 | 23.87 | 23.87 | 24.06 | 23.99 |

(100;5.0;0.40;0.06;0.02) | 24.76 | 24.61 | 24.61 | 24.85 | 24.73 |

(100;5.5;0.40;0.06;0.02) | 25.41 | 25.26 | 25.26 | 25.5 | 25.39 |

(100;3.0;0.40;0.06;0.00) | 19.85 | 19.69 | 19.71 | 19.85 | 19.83 |

(100;3.0;0.40;0.06;0.02) | 21.13 | 20.98 | 20.99 | 21.11 | 21.1 |

(100;3.0;0.40;0.06;0.04) | 22.49 | 22.36 | 22.36 | 22.47 | 22.45 |

$RMSRE$ | - | 0.0069 | 0.0064 | 0.0019 | 0.0015 |

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**MDPI and ACS Style**

Viegas, C.; Azevedo-Pereira, J.
A Quasi-Closed-Form Solution for the Valuation of American Put Options. *Int. J. Financial Stud.* **2020**, *8*, 62.
https://doi.org/10.3390/ijfs8040062

**AMA Style**

Viegas C, Azevedo-Pereira J.
A Quasi-Closed-Form Solution for the Valuation of American Put Options. *International Journal of Financial Studies*. 2020; 8(4):62.
https://doi.org/10.3390/ijfs8040062

**Chicago/Turabian Style**

Viegas, Cristina, and José Azevedo-Pereira.
2020. "A Quasi-Closed-Form Solution for the Valuation of American Put Options" *International Journal of Financial Studies* 8, no. 4: 62.
https://doi.org/10.3390/ijfs8040062