# Fractional Partial Differential Equations Associated with Lêvy Stable Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractional Diffusion Model and Option Pricing

**Definition**

**1.**

**Definition**

**2.**

## 3. The Model

#### 3.1. L$\widehat{e}$vy Process

**Theorem**

**1**(L$\widehat{e}$vy-Khintchine presentation theorem)

**.**

#### 3.2. L$\widehat{e}$vy Stable Processes

## 4. Main Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Merton, R. Continuous-Time Finance, 1st ed.; Blackwell: Oxford, UK, 1990. [Google Scholar]
- Bertoin, J. Lêvy Processes; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Sato, K.I. Lêvy Processes and Infinitely Divisible Distributions; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Schoutens, W. Lêvy Processes in Finance: Pricing Financial Derivativese, 1st ed.; Wiley Series in Probability and Statistics; Wiley: Chichester, UK, 2003. [Google Scholar]
- Shokrollahi, F.; Kılıçman, A.; Ibrahim, N.A. Greeks and Partial Differential Equations for some Pricing Currency Option Models. Malaysian J. Math. Sci.
**2015**, 9, 417–442. [Google Scholar] - Cont, R.; Voltchkova, E. Finite difference methods for option pricing in jump diffusion and exponential Lêvy models. SIAM J. Numer. Anal.
**2005**, 43, 1596–1626. [Google Scholar] [CrossRef] - Lewis, A.L. A Simple Option Formula for General Jump-Diffusion and Other Exponential Lêvy Processes; Working paper; Envision Financial Systems and Option City: Newport Beach, CA, USA, 2001. [Google Scholar]
- Carr, P.; Madan, D. Option valuation using the fast Fourier transform. J. Comput. Financ.
**1999**, 2, 61–73. [Google Scholar] [CrossRef] [Green Version] - Cartea, A.; del-Castillo-Negrete, D. Fractional Diffusion Models of Option Prices in Markets with Jumps. Phys. A Stat. Mech. Its Appl.
**2006**, 374, 749–763. [Google Scholar] [CrossRef] [Green Version] - Chen, W.; Xu, X.; Zhu, S.P. Analytical pricing European-style option under the modified Black-Scholes equation with a partial-fractional derivative. Q. Appl. Math.
**2014**, 72, 597–611. [Google Scholar] [CrossRef] - Demirci, E.; Ozalp, N. A method for solving differential equations of fractional order. J. Comput. Appl. Math.
**2012**, 236, 2754–2762. [Google Scholar] [CrossRef] [Green Version] - Jumarie, G. Derivation and solutions of some farctional Black-Scholes equations in space and time. J. Comput. Math. Appl.
**2010**, 59, 1142–1164. [Google Scholar] [CrossRef] [Green Version] - Du, M. Analytically Pricing European Options under the CGMY Model; University of Wollongong Thesis Collection 1954–2016; University of Wollongong: Dubai, United Arab Emirates.
- De Olivera, F.; Mordecki, E. Computing Greeks for Lêvy Models: The Fourier Transform Approach. arXiv
**2014**, arXiv:1407.1343v1. [Google Scholar]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aljedhi, R.A.; Kılıçman, A.
Fractional Partial Differential Equations Associated with L*ê*vy Stable Process. *Mathematics* **2020**, *8*, 508.
https://doi.org/10.3390/math8040508

**AMA Style**

Aljedhi RA, Kılıçman A.
Fractional Partial Differential Equations Associated with L*ê*vy Stable Process. *Mathematics*. 2020; 8(4):508.
https://doi.org/10.3390/math8040508

**Chicago/Turabian Style**

Aljedhi, Reem Abdullah, and Adem Kılıçman.
2020. "Fractional Partial Differential Equations Associated with L*ê*vy Stable Process" *Mathematics* 8, no. 4: 508.
https://doi.org/10.3390/math8040508