# Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Model Set-Up and Path-Dependent Options

#### 2.1. Stochastic Volatility Model

#### 2.2. Path-Dependent Options

## 3. Asymptotic Expansions

**Theorem**

**1.**

## 4. Determining ${P}_{0}$ and ${P}_{1}$ for Down-and-Out Put Options

#### 4.1. ${P}_{0}$ Term for Down-and-Out Put Options

#### 4.2. ${P}_{1}$ Term for Down-and-Out Put Options

**Theorem**

**2.**

## 5. Determining ${P}_{0}$ and ${P}_{1}$ for Lookback Put Options

#### 5.1. ${P}_{0}$ Term for Lookback Put Options

#### 5.2. ${P}_{1}$ Term for Lookback Put Options

**Theorem**

**3.**

## 6. Numerical Results and Sensitivity Analysis

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mellin Transform

Function | Mellin Tansform |
---|---|

h | $\widehat{h}$ |

$s{h}^{\prime}$ | $-w\widehat{h}$ |

${s}^{2}{h}^{\u2033}$ | $w(w+1)\widehat{h}$ |

${s}^{3}{h}^{\left(3\right)}$ | $-w(w+1)(w+2)\widehat{h}$ |

$\frac{{e}^{\delta}{s}^{\eta}}{2\sqrt{\lambda \pi}}{e}^{-\frac{1}{4\lambda}{(lns)}^{2}}$ | ${e}^{\lambda {(w+\eta )}^{2}+\delta}$ |

$s{h}^{\prime}+{s}^{2}{h}^{\u2033}$ | ${w}^{2}\widehat{h}$ |

$-s{h}^{\prime}-3{s}^{2}{h}^{\u2033}-{s}^{3}{h}^{\left(3\right)}$ | ${w}^{3}\widehat{h}$ |

## Appendix B. Derivation of Formulas (20) and (27)

#### Appendix B.1. Derivation of Formula (20)

#### Appendix B.2. Derivation of Formulas (27)

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**Figure 1.**Plots of $\sqrt{\u03f5}{P}_{1}$ and ${P}_{0}+\sqrt{\u03f5}{P}_{1}$ with different values of $\u03f5$ against the initial value of the underlying asset, for the down-and-out put option.

**Figure 2.**Plots of $\sqrt{\u03f5}{P}_{1}$ and ${P}_{0}+\sqrt{\u03f5}{P}_{1}$ with different values of $\u03f5$, against the initial value of the underlying asset, for floating strike put options.

**Figure 3.**Plots ${P}_{0}+\sqrt{\u03f5}{P}_{1}$ with different values of $\rho $, against the initial value of the underlying asset, for down-and-out put option and floating strike put option.

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**MDPI and ACS Style**

Cao, J.; Li, X.; Zhang, W.
Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform. *J. Risk Financial Manag.* **2023**, *16*, 456.
https://doi.org/10.3390/jrfm16100456

**AMA Style**

Cao J, Li X, Zhang W.
Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform. *Journal of Risk and Financial Management*. 2023; 16(10):456.
https://doi.org/10.3390/jrfm16100456

**Chicago/Turabian Style**

Cao, Jiling, Xi Li, and Wenjun Zhang.
2023. "Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform" *Journal of Risk and Financial Management* 16, no. 10: 456.
https://doi.org/10.3390/jrfm16100456