# Options with Extreme Strikes

## Abstract

**:**

## 1. Introduction

## 2. Main Results

#### 2.1. Asian Options

**Proposition 1**. (i)

#### 2.2. American Options

**Proposition 2**. (i)

#### 2.3. Parisian Down-And-Out Options

**Proposition 3**. (i)

#### 2.4. Parisian Up-And-In Options

**Proposition 4**. (i)

#### 2.5. Parisian Up-And-Out Options

**Proposition 5**. (i)

#### 2.6. Parisian Down-And-In Options

**Proposition 6**. (i)

#### 2.7. Perpetual Options

**Proposition 7**. (i)

## 3. Conclusions and Future Directions

**Table 1.**Summary of four categories for call option prices at extreme strikes, where T is the maturity, D is the option window and ${\gamma}_{0}>1$ is a constant depending on $r,q,\sigma $.

Call Option $C\left(K\right)$ | Option Types |
---|---|

${e}^{-\frac{1}{2{\sigma}^{2}T}{(logK)}^{2}+o\left({(logK)}^{2}\right)}$ | European, American, Asian, |

Parisian Down-And-Out, Up-And-In | |

${e}^{-\frac{1}{2{\sigma}^{2}D}{(logK)}^{2}+o\left({(logK)}^{2}\right)}$ | Parisian Up-And-Out |

${e}^{-\frac{1}{2{\sigma}^{2}(T-D)}{(logK)}^{2}+o\left({(logK)}^{2}\right)}$ | Parisian Down-And-In |

${e}^{(1-{\gamma}_{0})logK+o(logK)}$ | Perpetual American |

**Table 2.**Summary of four categories for put option prices at extreme strikes, where T is the maturity, D is the option window and ${\gamma}_{1}<0$ is a constant depending on $r,q,\sigma $.

Put Option $P\left(K\right)$ | Option Types |
---|---|

${e}^{-\frac{1}{2{\sigma}^{2}T}{(logK)}^{2}+o\left({(logK)}^{2}\right)}$ | European, American, Asian, |

Parisian Up-And-Out, Down-And-In | |

${e}^{-\frac{1}{2{\sigma}^{2}D}{(logK)}^{2}+o\left({(logK)}^{2}\right)}$ | Parisian Down-And-Out |

${e}^{-\frac{1}{2{\sigma}^{2}(T-D)}{(logK)}^{2}+o\left({(logK)}^{2}\right)}$ | Parisian Up-And-Out |

${e}^{(1-{\gamma}_{1})logK+o(logK)}$ | Perpetual American |

## 4. Appendix: Proofs

**Proof of Proposition 1**. (i) Note that under the risk-neutral measure,

**Proof of Proposition 2**. (i) The price of an American call option is at least as much as the European counterpart:

**Proof of Proposition 3**. (i) It is clear that the price of the Parisian down-and-out option is bounded above by the price of the vanilla European option, and it is shown in [28] that we have the lower bound (see Equation (2) in Section 3 of [28]):

**Proof of Proposition 4**. (i) It is clear that the option price ${C}_{UI}\left(K\right)$ is less or equal to its vanilla European counterpart. For the lower bound, notice that:

**Proof of Proposition 5**. (i) For any $\delta >0$, we have the upper bound:

**Proof of Proposition 6**. (i) We have the following lower bound, for sufficiently small $\delta >0$,

**Proof of Proposition 7**. It is well known that there is an explicit formula (see, e.g., page 259 in [30]) for the price of a perpetual American call option (for sufficiently large K):

## Acknowledgments

## Conflicts of Interest

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Zhu, L.
Options with Extreme Strikes. *Risks* **2015**, *3*, 234-249.
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Zhu L.
Options with Extreme Strikes. *Risks*. 2015; 3(3):234-249.
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**Chicago/Turabian Style**

Zhu, Lingjiong.
2015. "Options with Extreme Strikes" *Risks* 3, no. 3: 234-249.
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