On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.Comment: 14 pages, 0 figure

On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.

Penerapan Metode Perturbation Pada Perhitungan Nilai Opsi Asia

Metode perturbation merupakan salah satu cara untuk menyelesaikan persamaan diferensial yang tidak dapat diselesaikan dengan cara biasa, seperti persamaan diferensial Black-Scholes. Pendekatan solusi model Black-Scholes menggunakan metode perturbation menghasilkan nilai error yang kecil terhadap solusi eksaknya. Fokus utama dalam Tugas Akhir ini adalah penerapan metode perturbation pada perhitungan nilai opsi Asia tipe Eropa. Opsi Asia tipe Eropa adalah opsi dimana payo� bergantung pada rata-rata harga aset selama opsi tersebut berlaku dan hanya dapat dieksekusi pada saat jatuh tempo saja. Persamaan diferensial dalam Tugas Akhir ini menggunakan persamaan diferensial dari opsi Asia rata-rata geometrik tipe Eropa. Penerapan metode Perturbation pada persamaan diferensial opsi Asia rata-rata geometrik tipe Eropa melalui transformasi menjadi persamaan difusi yang selanjutnya diberikan parameter Perturbation sehingga didapatkan solusi. Hasil simulasi dengan menggunakan software MATLAB menunjukkan bahwa pengaruh parameter strike price (K), maturity date (T), risk free interest rate (r), dan volatilitas (�) pada model opsi Asia rata-rata geometrik tipe Eropa dengan metode Perturbation sesuai dengan trend nilai opsi Asia. ================================================================================================ Perturbation method is a method that can be used to solve di�erential equations that can not be solved in the usual way, such as the Black-Scholes di�erential equation. The solution approach of Black-Scholes model is using perturbation method yields a small error value against its exact solution. The main focus in this Final Project applied perturbation method in the calculation of value of Asian option of European type. The Asian option of the European type is an option where payo� depends on the average of asset price as long as the option is valid and can only be executed at maturity date only. The di�erential equation to be used in this Final Project is the di�erential equation of the European-Style Geometric Average Asian Option. The application of perturbation method to the di�erential equation of the European-Style Geometric Average Asian Option through transformation into a di�usion equation which is then given parameters of perturbation to obtain a solution. The results of simulation using MATLAB software shows that in uence of strike price (K), maturity date T, risk free interest rate (r), and volatility (�) in the European-Style Geometric Average Asian Option is same with trend value of Asian Option. Keywords: Perturbation Method, Asian Option, Europea