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Penentuan Harga Opsi Dengan Volatilitas Stokastik Menggunakan Metode Monte Carlo Chalimatusadiah Chalimatusadiah; Donny Citra Lesmana; Retno Budiarti
Jambura Journal of Mathematics Vol 3, No 1: January 2021
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

ABSTRAKHal yang utama dalam perdagangan opsi adalah penentuan harga jual opsi yang optimal. Namun pada kenyataan sebenarnya fluktuasi harga aset yang terjadi di pasar menandakan bahwa volatilitas dari harga aset tidaklah konstan, hal ini menyebabkan investor mengalami kesulitan dalam menentukan harga opsi yang optimal. Artikel ini membahas tentang penentuan harga opsi tipe Eropa yang optimal dengan volatilitas stokastik menggunakan metode Monte Carlo dan pengaruh harga saham awal, harga strike, dan waktu jatuh tempo terhadap harga opsi Eropa. Adapun model volatilitas stokastik yang digunakan dalam penelitian ini adalah model Heston, yang mengasumsikan bahwa proses harga saham (St) mengikuti distribusi log-normal, dan proses volatilitas saham (Vt) mengikuti Proses Cox-Ingersoll-Ross. Hal pertama yang dilakukan dalam penelitian ini adalah mengestimasi parameter model Heston untuk mendapatkan harga saham dengan menggunakan metode ordinary least square dan metode numerik Euler-Maruyama. Langkah kedua adalah melakukan estimasi harga saham untuk mendapatkan harga opsi tipe Eropa menggunakan metode Monte Carlo. Hasil dari penelitian ini menunjukkan bahwa penggunaan metode Monte Carlo dalam penentuan harga opsi tipe Eropa dengan volatilitas stokastik model Heston menghasilkan solusi yang cukup baik karena memiliki nilai error yang kecil dan akan konvergen ke solusi eksaknya dengan semakin banyak simulasi. Selain itu, simulasi Monte Carlo memberikan kesimpulan bahwa parameter harga strike, harga saham awal dan waktu jatuh tempo memiliki pengaruh terhadap harga opsi yang konsisten dengan teori harga opsi. ABSTRACTWhat is important in options trading is determining the optimal selling price. However, in real market conditions, fluctuations in asset prices that occur in the market indicate that the volatility of asset prices is not constant, this causes investors to experience difficulty in determining the optimal option price. This article discusses the optimal determination of the European type option price with stochastic volatility using the Monte Carlo method and the effect of the initial stock price, strike price, and expiration date on European option prices. The stochastic volatility model used in this study is the Heston model, which assumes that the stock price process (S) follows the normal log distribution, and the stock volatility process (V) follows the Ingersoll-Ross Cox Process. The first thing to do in this study is to estimate the parameters of the Heston model to get stock prices using the ordinary least square method and the Euler-Maruyama numerical method. The second step is to estimate the share price to get the European type option price using a Monte Carlo Simulation. This study indicates that using the Monte Carlo method in determining the price of European type options with the Heston model of stochastic volatility produces a fairly good solution because it has a small error value and will converge to the exact solution with more simulations. Also, the Monte Carlo simulation concludes that the parameters of the strike price, initial stock price, and maturity date influence the option price, which is consistent with the option price theory.

Penghitungan Premi Asuransi Kendaraan Bermotor Menggunakan Generalized Linear Models dengan Distribusi Tweedie Tri Andika Julia Putra; Donny Citra Lesmana; I Gusti Putu Purnaba
Jambura Journal of Mathematics Vol 3, No 2: July 2021
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

ABSTRAKSeorang aktuaris mempunyai tugas penting dalam menentukan harga premi yang sesuai untuk setiap nasabah dengan risiko dan karakteristik yang berbeda. Banyak variabel yang dapat mempengaruhi harga premi. Oleh karena itu, aktuaris harus mengetahui variabel-variabel yang berpengaruh signifikan terhadap premi. Tujuan dari penelitian ini adalah untuk menentukan variabel yang dapat mempengaruhi besaran premi murni menggunakan distribusi campuran dalam menentukan besarnya premi melalui Generalized Linear Models (GLM) serta menentukan model harga premi yang sesuai berdasarkan variabel-variabel yang mempengaruhinya. Salah satu analisis statistik yang dapat digunakan untuk memodelkan premi asuransi adalah Generalized Linear Models. GLM merupakan perluasan dari model regresi klasik yang dapat mengakomodasi fleksibilitas untuk menggunakan beberapa distribusi data tetapi terbatas pada distribusi keluarga eksponensial. Dalam model GLM, premi diperoleh dengan mengalikan nilai ekspektasi bersyarat dari frekuensi klaim dan biaya klaim. Berdasarkan penelitian yang telah dilakukan diketahui bahwa frekuensi klaim dan besarnya klaim mengikuti distribusi Tweedie. Dari kedua model tersebut diketahui bahwa variabel yang mempengaruhi premi murni adalah jumlah anak, pendapatan per bulan, status pernikahan, pendidikan, pekerjaan, penggunaan kendaraan, besarnya bluebook yang dibayarkan, dan jenis kendaraan nasabah. Hal ini menunjukkan bahwa model GLM merupakan model yang representatif dan berguna bagi perusahaan asuransi. ABSTRACTIt is an important task for an actuary in determining the appropriate premium price for each customer with different risks and characteristics. Many variables can affect the premium price. Therefore, actuaries must determine the variables that significantly affect the premium. The purpose of this study is to determine the variables that can affect the amount of pure premium using a mixed distribution in determining the amount of premium through Generalized Linear Models (GLM) and determine the appropriate premium price model based on the variables that influence it. One of the statistical analyzes that can be used to model insurance premiums is the Generalized Linear Models. GLM is an extension of the classic regression model that can accommodate the flexibility of its users to use multiple data distributions but is limited to the exponential family distribution. In the GLM model, the premium is obtained by multiplying the conditional expected value of the frequency of claims and the cost of claims. Based on the research that has been done, it is known that the frequency of claims and the size of claims follow the Tweedie distribution. From the two models, it is known that the variables affecting the pure premium are the number of children, monthly income, marital status, education, occupation, vehicle use, the number of bluebooks paid, and the type of vehicle from the customer. This shows that the GLM model is a representative and useful model for the insurance company business.

PENENTUAN PREMI TAHUNAN BERSIH ASURANSI JIWA SEUMUR HIDUP JOINT LIFE DENGAN MODEL COPULA CLAYTON DAN COPULA GUMBEL Laila Qudrah Fikriyah; I Gusti Putu Purnaba; Windiani Erliana; Berlian Setiawaty; Donny Citra Lesmana
MILANG Journal of Mathematics and Its Applications Vol. 18 No. 1 (2022): MILANG Journal of Mathematics and Its Applications
Publisher : Dept. of Mathematics, IPB University

Asuransi jiwa seumur hidup adalah bentuk pengalihan risiko atas kerugian keuangan oleh tertanggung kepada penanggung yang disebabkan oleh hilangnya jiwa seseorang setelah polis disepakati. Pada status joint life pasangan suami istri Premi dibayarkan setiap tahun dan pembayaran manfaat dilakukan pada akhir tahun kematian pertama. Biasanya risiko kematian pasangan suami istri diasumsikan saling bebas, namun dalam kenyataannya kerap kali pasangan suami istri memiliki risiko bersama. Pada karya ilmiah ini, dilakukan penghitungan premi bersih tahunan dari asuransi jiwa seumur hidup joint life bagi pasangan suami istri menggunakan dua asumsi: (1) kebebasan mortalitas dan (2) ketidakbebasan mortalitas dengan model copula Clayton dan copula Gumbel. Berdasarkan hasil perhitungan untuk contoh kasus yang spesifik, premi tahunan yang dihitung menggunakan asumsi kebebasan mortalitas lebih besar jika dibandingkan dengan menggunakan asumsi ketidakbebasan mortalitas. Hasil ini berlaku juga untuk suku bunga yang bervariasi.

Determination of Term Life Insurance Premiums with Varying Interest Rates Following The CIR Model and Varying Benefits Value Dian Puspita; I Gusti Putu Purnaba; Donny Citra Lesmana
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.20542

Term life insurance is an insurance that provides protection for a certain period that has been agreed upon in the policy. The policy is an agreement that contains the participant's obligation to pay premiums contributions to the insurance company and the insurance company's obligation to pay benefits in the event of a risk to the insurance participant as agreed in the policy. Interest rates will influence the calculation of premium value and benefits in the long term. So we need a model of interest rates that will change by time. One of the models that can be used is the CIR model. This research purposes to simulate the CIR model that will be carried out to determine interest rates for calculating term life insurance premiums for five years, with premiums paid at the beginning of the 1/m interval or monthly premium payments and benefits paid at the end of the 1/m interval when the participant dies. The case that will be discussed is when the benefit various. The results of this study are the CIR model can be applied to calculate the term life insurance premiums for five years and the premium calculation results show that the amount of the premium increase every year with varying benefits.

Stock Hedging Using Strangle Strategy on Vanilla Options and Capped Options Donny Citra Lesmana; David Vijanarco Martal; Unika Nabila; Syifa Fauzia; Raymond Raymond; Zidni Kamal Hasan; M Ridwan Aprizky
Jurnal Akuntansi dan Keuangan Vol. 26 No. 1 (2024): MAY 2024
Publisher : Institute of Research and Community Outreach - Petra Christian University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.9744/jak.26.1.47-55

The financial market often experiences unexpected fluctuations that can impact stock values. Therefore, investors require hedging strategies to protect their investment values from unwanted price fluctuations. This study compares the hedging results using the strangle strategy on Vanilla options and Capped options on Micron Technology, Inc. (MU) stock. The methods used are Monte Carlo simulation and Black Scholes Merton to calculate the option prices. The research results indicate that the strangle strategy on Vanilla options has unlimited maximum profit potential, whereas on Capped options, the profit is capped above. However, the potential maximum loss on Capped options is lower than that on Vanilla options. Therefore, Capped options are preferred for hedging the MU stock. The research yields significant practical and theoretical benefits. Practically, it offers investors insights into more effective hedging choices for risk management and profit potential in the stock market. Opting for capped options allows investors to control risk better while preserving profit potential. Theoretically, the study enhances our understanding of cost efficiency and risk profiles across various options strategies, making a vital contribution to financial literature.

Perbandingan Harga Opsi Reset dengan Metode Monte Carlo Standar dan Control Variate Donny Citra Lesmana; Andyta Putri Pramesty; Yunita Nur’Afiyah; Akbar Cautsar Suryaputra; Anggi Setya Pratiwi; Wintang Dayinta Wiyanto; Yeremia Bertolla Gimanjar
Limits: Journal of Mathematics and Its Applications Vol 20, No 3 (2023)
Publisher : Institut Teknologi Sepuluh Nopember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12962/limits.v20i3.17053

Opsi bermanfaat untuk melindungi investor dari risiko kenaikan atau penurunan harga saham. Opsi yang digunakan dalam penelitian ini adalah opsi reset dengan tipe call dan put yang bersifat path-dependent. Dalam kebanyakan kasus, opsi yang bersifat path-dependent tidak memiliki solusi analitik sehingga perlu dilakukan pendekatan numerik untuk menghitung harga wajar opsi jenis ini. Dalam penelitian ini, metode numerik yang digunakan adalah metode Monte Carlo standar dan Monte Carlo control variate. Ditunjukkan pula bahwa harga opsi reset yang dihitung menggunakan kedua metode tersebut konvergen menuju solusi eksak. Hal ini menandakan bahwa kedua metode dapat digunakan untuk menentukan harga opsi reset. Hasil penghitungan kedua metode tersebut kemudian dibandingkan berdasarkan kecepatan kekonvergenannya yang dihitung menggunakan galat relatif yang dihasilkan dari masing-masing metode. Metode Monte Carlo control variate menghasilkan galat relatif yang lebih kecil dibandingkan dengan Monte Carlo standar pada kedua tipe opsi. Hal ini menunjukkan bahwa metode Monte Carlo control variate merupakan metode yang lebih akurat dibandingkan metode Monte Carlo standar. Pengaruh nilai parameter model juga dibandingkan untuk menentukan harga opsi reset. Nilai volatilitas yang semakin besar membuat harga opsi reset cenderung semakin mahal pada kedua tipe opsi reset. Waktu reset yang semakin dekat dengan tanggal terbit opsi ataupun waktu jatuh tempo opsi menyebabkan harga opsi reset cenderung semakin murah. Sedangkan tingkat bunga bebas risiko menyebabkan perubahan perilaku harga yang berbeda pada opsi put dan call. Dengan meningkatnya tingkat bunga bebas risiko, harga opsi call akan semakin meningkat, sedangkan harga opsi put semakin menurun.

Penentuan Harga Opsi Bermuda Menggunakan Simulasi Monte Carlo Reduksi Antithetic Variates Donny Citra Lesmana; Reinhart Hasudungan Sinaga; Cahya Rila Hadiva; Muthia Rasya Salsabilla; Hindana Dzulfa Pratiwi; Aufa Shahnazdiza Salsabila; Ravasya Nugraha
Limits: Journal of Mathematics and Its Applications Vol 20, No 3 (2023)
Publisher : Institut Teknologi Sepuluh Nopember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12962/limits.v20i3.17054

Opsi merupakan produk derivatif yang popular di kalangan investor pada industri keuangan. Hal ini terjadi karena perdagangan opsi menawarkan potensi profit yang tinggi disertai fleksibilitasnya dalam pengelolaan risiko keuangan. Salah satu jenis opsi yang menarik perhatian adalah opsi Bermuda. Opsi Bermuda adalah opsi yang memberikan keleluasaan bagi pemegang opsi untuk menggunakan haknya pada waktu tertentu selama opsi belum kadaluarsa. Penentuan harga opsi Bermuda menjadi suatu tantangan terutama karena melibatkan struktur payoff yang kompleks. Oleh karena itu, penelitian ini ditujukan untuk membandingkan dua metode simulasi yang umum digunakan, yakni simulasi Monte Carlo standar dan simulasi Monte Carlo dengan teknik reduksi variansi antithetic variates, sehingga didapatkan metode yang lebih efektif dalam menilai harga opsi Bermuda. Hasil penelitian menunjukkan bahwa metode simulasi Monte Carlo dengan antithetic variates memberikan hasil yang lebih akurat dan tingkat eror yang lebih rendah dibandingkan dengan metode Monte Carlo standar. Dengan demikian, metode ini terbukti lebih efektif dalam menilai harga opsi Bermuda.

Determination of Term Life Insurance Premiums with Varying Interest Rates Following The CIR Model and Varying Benefits Value Puspita, Dian; Purnaba, I Gusti Putu; Lesmana, Donny Citra
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.20542

Term life insurance is an insurance that provides protection for a certain period that has been agreed upon in the policy. The policy is an agreement that contains the participant's obligation to pay premiums contributions to the insurance company and the insurance company's obligation to pay benefits in the event of a risk to the insurance participant as agreed in the policy. Interest rates will influence the calculation of premium value and benefits in the long term. So we need a model of interest rates that will change by time. One of the models that can be used is the CIR model. This research purposes to simulate the CIR model that will be carried out to determine interest rates for calculating term life insurance premiums for five years, with premiums paid at the beginning of the 1/m interval or monthly premium payments and benefits paid at the end of the 1/m interval when the participant dies. The case that will be discussed is when the benefit various. The results of this study are the CIR model can be applied to calculate the term life insurance premiums for five years and the premium calculation results show that the amount of the premium increase every year with varying benefits.

PRICING EMPLOYEE STOCK OPTION USING TRINOMIAL TREE METHOD Lesmana, Donny Citra; Ramadhan, Reza Tri Ahmad; Nurjanah, Siti; Dharmawan, Vanaya Syahira
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 19 No 2 (2025): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol19iss2pp709-720

This study explores the Employee Stock Option (ESO) model proposed by Liao and Lyuu, which provides a robust framework for addressing critical factors such as dilution, early exercise, and employee forfeiture rates. The model is solved using the trinomial tree method, allowing for the consideration of three possible stock price movements: increase, unchanged, or decrease. This approach combines forward and backward calculations to accurately evaluate ESO values by accounting for the complex interactions of these parameters. Dilution effects are modeled by adjusting stock prices based on outstanding shares and strike prices, while early exercise probabilities are addressed using a modified Chi-Square distribution to represent employee behavior. Additionally, the forfeiture rate is dynamically adjusted based on ESO returns and the ratio of stock-to-strike prices. The analysis reveals that ESO price negatively correlates with strike price and forfeiture rate, whereas parameters such as vesting time, maturity date, risk-free rate, volatility, and the number of ESOs granted exhibit positive correlations. This comprehensive methodology demonstrates the practical applicability of the Liao and Lyuu model for real-world ESO valuation. By integrating these critical factors into a unified framework, the study contributes significantly to the literature on financial modeling and provides actionable insights for companies seeking to optimize their ESO programs.

IMPLEMENTATION OF MONTE CARLO MOMENT MATCHING METHOD FOR PRICING LOOKBACK FLOATING STRIKE OPTION Dewi, Komang Nonik Afsari; Lesmana, Donny Citra; Budiarti, Retno
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 16 No 4 (2022): BAREKENG: Journal of Mathematics and Its Applications
Publisher : PATTIMURA UNIVERSITY

Monte Carlo method was a numerical method that was popular in finance. This method had disadvantages at convergences, so the moment matching was used to improve the efficiency from Monte Carlo method. The research has discussed about pricing of the lookback floating strike option using the Monte Carlo moment matching method. The monthly stock price of PT TELKOM from 2004 to 2021 that used in this research. The results obtained by adding variance reduction moment matching in Monte Carlo method, which produces a relatively had smaller error when compared to the relative error of the standard Monte Carlo method. The orders of convergence from Monte Carlo method with variance reduction moment matching for call and put option are about 1.1 and 1.4. The conclusion that addition of the moment matching can increase the efficiency of the Monte Carlo method in determining the price of the lookback floating strike option.

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