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| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | Rokiah@Rozita Ahmad, Prof. Dr. | |
| dc.contributor.author | Zieneb Ali Elshegmani (P50038) | |
| dc.date.accessioned | 2023-10-13T09:32:08Z | - |
| dc.date.available | 2023-10-13T09:32:08Z | - |
| dc.date.issued | 2013-10-06 | |
| dc.identifier.other | ukmvital:74767 | |
| dc.identifier.uri | https://ptsldigital.ukm.my/jspui/handle/123456789/499459 | - |
| dc.description | Asian options are path dependent options whose payoff functions depend on the average of the stock price over life of the option. There are two types of Asian options depending on the way the average is computed; namely arithmetic and geometric, with each one includes two types of options, call option, which gives the holder of the option the right to buy the underlying security and put option which gives the right to sell. For both options, the holder has no obligation to fulfill the right. In the case of the geometric Asian options, there is a closed-form solution to value these types of options. However this is not the same for the arithmetic Asian option, because the arithmetic average of a set of lognormal random variables is not log-normally distributed. The main objective of this research is to solve a second order partial differential equation (PDE) that arises in the valuation of the continuous arithmetic Asian options. Firstly, we present derivation of this partial differential equation, and we provide several methods for the analytical solution using three different integral transformations. By using Fourier transform in a stock price we transform a second order PDE to the first order. Besides Fourier, Mellin transform is also utilized in the share price to transform the PDE to a simpler first order PDE with constant coefficients, which can be solved analytically. Likewise, Laplace transform in time is used to transform the PDE to an ordinary differential equation (ODE), and provide its analytical solution. Based on the general stochastic differential equations we derive a modified partial differential equation for the arithmetic Asian option PDE.We provide two different modifications, and we present four different cases of the modified PDE, which can be transformed to the classical Black-Scholes PDE and then to a heat equation with constant coefficients, which can be solved analytically using general transformation methods. Furthermore, the PDE is reduced to a simplified parabolic equation with constant coefficients. Previous studies showed that, the PDE of the arithmetic Asian options cannot be reduced to the simplest parabolic equation with constant coefficients using the multiplication of two transformation functions. Product of two transformation functions would results in a more complicated equation. However, this study has shown that using the multiplication of three functions in this transformation has successfully produced the simplest parabolic equation with constant coefficients. Apart from a constant factor, this study also extended to the PDE with time dependent parameters. Comparison of the results for the solution of both forms is done. Some important results with the arithmetic Asian option PDE are rationalized to the general form of the partial differential equations.,Ph.D | |
| dc.language.iso | eng | |
| dc.publisher | UKM, Bangi | |
| dc.relation | Faculty of Science and Technology / Fakulti Sains dan Teknologi | |
| dc.rights | UKM | |
| dc.subject | Differential equation | |
| dc.subject | Differential equations | |
| dc.subject | Partial | |
| dc.title | Analytical solution of the arithmetic Asian option partial differential equation using several transformation methods | |
| dc.type | Theses | |
| dc.format.pages | 108 | |
| dc.identifier.callno | QA377.E449 2013 | |
| dc.identifier.barcode | 000312 | |
| Appears in Collections: | Faculty of Science and Technology / Fakulti Sains dan Teknologi | |
Files in This Item:
| File | Description | Size | Format | |
|---|---|---|---|---|
| ukmvital_74767+Source01+Source010.PDF Restricted Access | 335.14 kB | Adobe PDF |  View/Open |
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