# Pricing Various Types of Power Options under Stochastic Volatility

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Specification and Pricing Formula for Power Options

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Application to Various Power Payoffs

#### 3.1. Symmetric Power Option

#### 3.2. Polynomial Options

#### 3.3. Soft Strike Options

## 4. Numerical Experiments

#### 4.1. The General Case

#### 4.2. The Symmetric Case

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The relation between $\kappa $ and prices. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\theta =0.2$, $\xi =0.4$, and $\rho =-0.5$.

$\mathit{\alpha}$ | $1.00$ | $1.02$ | $1.04$ | $1.06$ | $1.08$ | $1.10$ | |
---|---|---|---|---|---|---|---|

$\mathit{\kappa}$ | |||||||

$0.5$ | $14.9602$ | $20.2509$ | $26.1164$ | $32.5757$ | $39.6907$ | $47.4996$ | |

$1.0$ | $15.2305$ | $20.6047$ | $26.5185$ | $32.0335$ | $40.1910$ | $48.0786$ | |

$1.5$ | $15.1380$ | $20.5057$ | $26.4116$ | $32.9027$ | $40.0499$ | $47.9059$ | |

$2.0$ | $15.2287$ | $20.6531$ | $26.5154$ | $32.0269$ | $40.1845$ | $48.0589$ | |

$2.5$ | $15.2286$ | $20.6070$ | $26.5239$ | $32.0315$ | $40.1913$ | $48.0525$ | |

$3.0$ | $15.3656$ | $20.7493$ | $26.6874$ | $32.2116$ | $40.4029$ | $48.2944$ |

**Table 2.**The relation between $\theta $ and prices. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\kappa =2$, $\xi =0.4$, and $\rho =-0.5$.

$\mathit{\alpha}$ | $1.00$ | $1.02$ | $1.04$ | $1.06$ | $1.08$ | $1.10$ | |
---|---|---|---|---|---|---|---|

$\mathit{\theta}$ | |||||||

$0.0$ | $10.6090$ | $15.4146$ | $20.6931$ | $26.4904$ | $32.8560$ | $39.8447$ | |

$0.1$ | $13.0813$ | $18.1847$ | $23.7941$ | $29.9698$ | $36.7552$ | $44.2054$ | |

$0.2$ | $15.2287$ | $20.6031$ | $26.5154$ | $33.0269$ | $40.1845$ | $48.0589$ | |

$0.3$ | $16.9669$ | $22.5634$ | $28.7321$ | $35.5366$ | $42.9980$ | $51.2359$ | |

$0.4$ | $18.0210$ | $23.7736$ | $30.1102$ | $37.0876$ | $44.7763$ | $53.2598$ |

**Table 3.**The relation between $\xi $ and prices. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\kappa =2$, $\theta =0.2$, and $\rho =-0.5$.

$\mathit{\alpha}$ | $1.00$ | $1.02$ | $1.04$ | $1.06$ | $1.08$ | $1.10$ | |
---|---|---|---|---|---|---|---|

$\mathit{\xi}$ | |||||||

$0.1$ | $33.3792$ | $41.0389$ | $49.4924$ | $58.8494$ | $69.1927$ | $80.6479$ | |

$0.2$ | $20.6700$ | $26.7056$ | $33.3712$ | $40.7167$ | $48.8059$ | $57.7199$ | |

$0.3$ | $16.9545$ | $22.5310$ | $28.6808$ | $35.4542$ | $42.8946$ | $51.0943$ | |

$0.4$ | $15.2287$ | $20.6031$ | $26.5154$ | $33.0269$ | $40.1845$ | $48.0589$ | |

$0.5$ | $14.2011$ | $19.4552$ | $25.2250$ | $31.5801$ | $38.5712$ | $46.2667$ |

**Table 4.**The relation between $\rho $ and prices. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\kappa =2$, $\theta =0.2$, and $\xi =0.4$.

$\mathit{\alpha}$ | $1.00$ | $1.02$ | $1.04$ | $1.06$ | $1.08$ | $1.10$ | |
---|---|---|---|---|---|---|---|

$\mathit{\rho}$ | |||||||

$-1.00$ | $20.6539$ | $26.7006$ | $33.3639$ | $40.7141$ | $48.8088$ | $57.7294$ | |

$-0.75$ | $17.8747$ | $23.5758$ | $29.8529$ | $36.7716$ | $44.3839$ | $52.7656$ | |

$-0.50$ | $15.2287$ | $20.6031$ | $26.5154$ | $33.0269$ | $40.1845$ | $48.0589$ | |

$-0.25$ | $12.7100$ | $17.7755$ | $23.3432$ | $29.4704$ | $36.1996$ | $43.5962$ | |

$0.00$ | $10.3129$ | $15.0862$ | $20.3285$ | $26.0932$ | $32.4186$ | $39.3655$ | |

$0.25$ | $8.0320$ | $12.5291$ | $17.4640$ | $22.8867$ | $28.8315$ | $35.3550$ | |

$0.50$ | $5.8622$ | $10.0981$ | $14.7427$ | $19.8428$ | $25.4289$ | $31.5538$ | |

$0.75$ | $3.7989$ | $7.7878$ | $12.1582$ | $16.9537$ | $22.2018$ | $27.9514$ | |

$1.00$ | $1.8374$ | $5.5927$ | $9.7040$ | $14.2123$ | $19.1418$ | $24.5381$ |

**Table 5.**The relation between $\kappa $ and prices when $n=2$. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\theta =0.4$, $\xi =0.14$, and $\rho =-0.52$.

$\mathit{\kappa}$ | $2.00$ | $2.25$ | $2.50$ | $2.75$ | $3.00$ |
---|---|---|---|---|---|

Prices | $41.8056$ | $59.1852$ | $83.1452$ | $95.6851$ | $107.2798$ |

**Table 6.**The relation between $\theta $ and prices when $n=2$. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\kappa =2$, $\xi =0.14$, and $\rho =-0.53$.

$\mathit{\theta}$ | $0.40$ | $0.41$ | $0.42$ | $0.43$ | $0.44$ |
---|---|---|---|---|---|

Prices | $147.1643$ | $218.7923$ | $286.2040$ | $343.9156$ | $399.0800$ |

**Table 7.**The relation between $\xi $ and prices when $n=2$. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\kappa =2$, $\theta =0.4$, and $\rho =-0.52$.

$\mathit{\xi}$ | $0.130$ | $0.132$ | $0.134$ | $0.136$ | $0.138$ | $0.140$ |
---|---|---|---|---|---|---|

Prices | $469.1018$ | $365.7494$ | $282.3005$ | $206.7740$ | $124.4067$ | $41.8056$ |

**Table 8.**The relation between $\rho $ and prices when $n=2$. $S=100$, $K=80$, $T-t=0.5$, $r=0.05$, $q=0.02$, $\kappa =2$, $\theta =0.4$, and $\xi =0.14$.

$\mathit{\rho}$ | $-0.550$ | $-0.545$ | $-0.540$ | $-0.535$ | $-0.530$ | $-0.525$ | $-0.520$ |
---|---|---|---|---|---|---|---|

Prices | $378.1934$ | $319.2700$ | $261.1295$ | $203.7638$ | $147.1643$ | $91.3230$ | $41.8056$ |

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**MDPI and ACS Style**

Lee, Y.; Kim, Y.; Lee, J.
Pricing Various Types of Power Options under Stochastic Volatility. *Symmetry* **2020**, *12*, 1911.
https://doi.org/10.3390/sym12111911

**AMA Style**

Lee Y, Kim Y, Lee J.
Pricing Various Types of Power Options under Stochastic Volatility. *Symmetry*. 2020; 12(11):1911.
https://doi.org/10.3390/sym12111911

**Chicago/Turabian Style**

Lee, Youngrok, Yehun Kim, and Jaesung Lee.
2020. "Pricing Various Types of Power Options under Stochastic Volatility" *Symmetry* 12, no. 11: 1911.
https://doi.org/10.3390/sym12111911