Penentuan Persamaan Diferensial Black-Scholes Multi Aset dengan Menurunkan Backward Stochatic Differential Equations (BSDEs)

Alfajriyah, Aimmatul Ummah (2023) Penentuan Persamaan Diferensial Black-Scholes Multi Aset dengan Menurunkan Backward Stochatic Differential Equations (BSDEs). Masters thesis, Institit Teknologi Sepuluh Nopember.

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Abstract

Persamaan diferensial Black-Scholes banyak digunakan dalam penetapan harga opsi. Penentuan persamaan diferensial Black-Scholes umumnya dilakukan dengan menggunakan teori ∆-lindung nilai yang hanya dapat dilakukan di complete market. Metode penetapan harga opsi lain adalah dengan menurunkan backward stochastic differential equations (BSDEs). BSDEs memiliki aplikasi penting di bidang matematika keuangan, karena BSDEs dapat digunakan untuk menentukan harga produk keuangan di incomplete market. Tujuan utama dari penelitian ini adalah untuk menentukan persamaan diferensial Black-Scholes multi-aset menggunakan BSDEs. Penelitian ini diawali dengan membangun portofolio multi aset dalam bentuk BSDE. Hubungan antara BSDEs dan persamaan diferensial parsial diberikan oleh teori Feynman-Kac. Kemudian berdasarkan teorema keberadaan dan ketunggalan solusi BSDEs, didapatkan solusi BSDE portofolio multi aset ada dan tunggal. Solusi tersebut juga merupakan solusi dari persamaan diferensial Black-Scholes multi aset. Kemudian, dengan menurunkan BSDE, portofolio multi aset dapat diubah menjadi persamaan diferensial Black-Scholes multi aset. Pada penelitian ini juga didapatkan solusi eksak dari harga opsi multiaset pada opsi basket dengan mengubah persamaan diferensial Black-Scholes multi aset menjadi persamaan difusi. Sebagai aplikasi, beberapa simulasi harga opsi multi aset dilakukan.
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Multi-asset Black-Scholes differential equation is widely used in option pricing. The determination of Black-Scholes differential equation is generally carried out using the ∆-hedging theory, which can only be done on complete markets. Another option pricing method is to derive backward stochastic differential equations (BSDEs). BSDEs has important applications in the field of mathematical finance, since BSDEs can be used in pricing financial products on incomplete markets. A primary objective of this study is to determine the multi-asset Black-Scholes differential equation using the BSDEs. This research begins with constructing a multi-asset portfolio in the form of BSDE. Relationship between BSDEs and partial differential equations (PDEs) is given by Feynman-Kac theory. Then, based on the existence and uniqueness theorem of the BSDEs solution, we get the solution of BSDE of multi-asset portfolios is exist and unique. It is also a solution of multi asset Black-Scholes differential equation. Then, by deriving BSDE, the multi asset portfolio can be transformed into the multi-asset Black-Scholes differential equation. We also obtain the exact solution of the multi-asset option price on the basket option, by transforming multi-asset Black-Scholes differential equation into a diffusion equation. As an application, some simulations of multi-asset option prices are conducted.

Item Type:	Thesis (Masters)
Uncontrolled Keywords:	Black-Scholes, BSDEs, Feynman-Kac, Opsi Multi Aset, Persamaan Diferensial, Multi Asset Option, PDE.
Subjects:	H Social Sciences > HG Finance > HG4012 Mathematical models H Social Sciences > HG Finance > HG4529 Investment analysis H Social Sciences > HG Finance > HG4529.5 Portfolio management Q Science Q Science > Q Science (General) > Q180.55.M38 Mathematical models Q Science > QA Mathematics > QA274.2 Stochastic analysis Q Science > QA Mathematics > QA371 Differential equations--Numerical solutions Q Science > QA Mathematics > QA401 Mathematical models.
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44101-(S2) Master Thesis
Depositing User:	Aimmatul Ummah Alfajriyah
Date Deposited:	15 Feb 2023 04:15
Last Modified:	15 Feb 2023 04:15
URI:	http://repository.its.ac.id/id/eprint/97300

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