scholarly journals Penentuan Harga Opsi Asia dengan Metode Monte Carlo

2017 ◽  
Vol 3 (1) ◽  
pp. 44-48
Author(s):  
Surya Amami Pramuditya
Keyword(s):  
Monte Carlo ◽  
Confidence Interval ◽  
Stock Price ◽  
Variance Reduction ◽  
Option Price ◽  
Numerical Procedure ◽  
Reduction Technique ◽  
Asian Options ◽  
Call Option ◽  
Strike Price

An option is a contract between a holder and a writer in which the writer grants the rights (not obligations) to the holder to buy or sell the assets of the writer at a certain price (strike price) at maturity time. Asian options are included in the dependent path option. This means that Asia's payoff option depends not only on the stock price at maturity time, but it is the average stock price during its maturity and symbolized A (average). Monte Carlo is basically used as a numerical procedure to estimate the expected value of pricing product derivatives. The techniques used are the standard Monte Carlo and variance reduction. The result obtained the Asia call option price and put for both techniques with 95% confidence interval. The variance reduction technique looks faster reducing 95% confidence interval than standard method.

scholarly journals PERBANDINGAN KEKONVERGENAN METODE CONDITIONAL MONTE CARLO DAN ANTITHETIC VARIATE DALAM MENENTUKAN HARGA OPSI CALL TIPE BARRIER

2018 ◽  
Vol 7 (3) ◽  
pp. 271
Author(s):  
NI LUH PUTU KARTIKA WATI ◽  
KOMANG DHARMAWAN ◽  
KARTIKA SARI
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Stock Price ◽  
Option Price ◽  
Life Time ◽  
Barrier Option ◽  
Strike Price ◽  
Stock Volatility ◽  
Volatility Risk ◽  
Conditional Monte Carlo

Barrier option is an option where the payoff price depends  on whether or not the stock price passes the barrier during its life time. The aim of the research is to compare the convergence between conditional Monte Carlo and antithetic variate methods in determining the call barrier option  price. The call barrier option price  is influenced by several factors: initial stock price, stock volatility, risk-free interest rate, maturity, strike price and barrier. The calculation of call barrier option price is obtained by simulating stock price movements with different simulation number. Based on the simulation result, it is obtained that the calculation of call barrier option price with conditional Monte Carlo method converge faster than the antithetic variate method.

Penentuan Harga Opsi Dengan Volatilitas Stokastik Menggunakan Metode Monte Carlo

2021 ◽  
Vol 3 (1) ◽  
pp. 80-92
Author(s):  
Chalimatusadiah Chalimatusadiah ◽  
Donny Citra Lesmana ◽  
Retno Budiarti
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Stochastic Volatility ◽  
Stock Price ◽  
Option Price ◽  
Least Square ◽  
Heston Model ◽  
Strike Price ◽  
Ordinary Least Square ◽  
The Monte Carlo Method

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.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

scholarly journals Penentuan Harga Opsi Barrier Menggunakan Metode Trinomial Kamrad-Ritchken Dengan Volatilitas Model Garch

2019 ◽  
Vol 10 (1) ◽  
pp. 83-92
Author(s):  
S Sulastri ◽  
Lienda Novieyanti ◽  
Sukono Sukono
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Barrier Options ◽  
Barrier Option ◽  
Market Conditions ◽  
Strike Price ◽  
Daily Data ◽  
Closing Price ◽  
The Right ◽  
Initial Price

Abstract. This study aims to minimize the violation of the assumptions of determining price options by taking into account the actual market conditions in order to obtain the right price that will provide high profits for investors. The method used to determine the option price in this study is the Kamrad Ritchken trinomial with volatility values that will be modeled first using GARCH. The data used in this study is daily data (5 working days per week) from the closing price of the stock price of PT. Bank Rakyat Indonesia, Tbk (BBRI. Based on the results of the research, the best model is GARCH (1,1). For the call up barrier option, increase the strike price with the initial price and barrier which causes the option price to call up the barrier "in" and "out" decreases, on the contrary to the put barrier option, an increase in strike price with the initial price and a barrier that causes the put barrier option price to both put up-in and put up-out. initial and barrier which still causes the call down barrier option price both in and out decreases, on the contrary in the put down barrier option, increasing strike price with the initial price and barrier which causes the put down barrier option price to increase in and out.Keywords: Barrier Options, Trinomial, Kamrad Ritchken, Volatility, GARCH  Abstrak. Penelitian ini bertujuan untuk meminimalkan pelanggaran asumsi-asumsi penentuan harga opsi dengan memperhatikan kondisi pasar yang sebenarnya sehingga diperoleh harga yang tepat yang akan memberikan keuntungan tinggi bagi investor. Metode yang digunakan untuk menentukan harga opsi dalam penelitian ini adalah trinomial Kamrad Ritchken dengan nilai volatilitas yang akan dimodelkan terlebih dahulu dengan menggunakan GARCH. Data yang digunakan dalam penelitian ini adalah data harian (5 hari kerja per minggu) dari harga penutupan harga saham PT. Bank Rakyat Indonesia, Tbk (BBRI). Berdasarkan hasil penelitian diperoleh model yang paling baik adalah GARCH (1,1). Untuk opsi call up barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi call up barrier baik "in" maupun "out" menurun, sebaliknya pada opsi put barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi put barrier baik put up-in maupun put up-out meningkat. Sedangkan untuk opsi call barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi call down barrier baik in maupun out menurun, sebaliknya pada opsi put down barrier, peningkatan strike price dengan harga awal dan barrier yang tetap menyebabkan harga opsi put down barrier baik in maupun out meningkat.Kata Kunci :  Opsi Barrier, Trinomial, Kamrad Ritchken, Volatilitas, GARCH

scholarly journals Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities

2019 ◽  
Vol 11 (3) ◽  
pp. 815 ◽  
Author(s):  
Yijuan Liang ◽  
Xiuchuan Xu
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Convergence Rates ◽  
Option Price ◽  
Correlation Coefficients ◽  
Control Variate ◽  
Stochastic Volatilities ◽  
Conditional Monte Carlo ◽  
Mc Simulation ◽  
Martingale Representation Theorem

Pricing multi-asset options has always been one of the key problems in financial engineering because of their high dimensionality and the low convergence rates of pricing algorithms. This paper studies a method to accelerate Monte Carlo (MC) simulations for pricing multi-asset options with stochastic volatilities. First, a conditional Monte Carlo (CMC) pricing formula is constructed to reduce the dimension and variance of the MC simulation. Then, an efficient martingale control variate (CV), based on the martingale representation theorem, is designed by selecting volatility parameters in the approximated option price for further variance reduction. Numerical tests illustrated the sensitivity of the CMC method to correlation coefficients and the effectiveness and robustness of our martingale CV method. The idea in this paper is also applicable for the valuation of other derivatives with stochastic volatility.

scholarly journals OPSI SAHAM PADA PASAR MODAL DI INDONESIA (STUDI PASAR OPSI SAAT PASAR OPSI MASIH BERLANGSUNG DI BURSA EFEK INDONESIA)

2019 ◽  
Vol 2 (2) ◽  
pp. 300
Author(s):  
Syanti Dewi ◽  
Ishak Ramli
Keyword(s):  
Interest Rate ◽  
Stock Prices ◽  
Stock Price ◽  
Stock Exchange ◽  
Option Price ◽  
Stock Option ◽  
Exchange Market ◽  
Strike Price ◽  
Option Trading ◽  
Put Option

Stock option exchange market is not working anymore in the Indonesian Stock Exchange, using the data option exchange market for the running period 2007-2008, we analyzed the effect of stock price, strike price, time to maturity, volatility and risk- free interest rate on the stock option’s price of listed stock call or put option trading at the Indonesian Stock Exchange during 2007-2008. The results found that the stock price, strike price, time to maturity, volatility and risk-free interest rate are positive significantly affecting the stock option price either the buying option price or the selling option price in Indonesia Stock Exchange 2007-2008 period. While there were no variables that significantly affected the call option during the periode 2007-2008, furthermore stock prices and strike prices significantly affected the put option prices. Time to maturity, Volatility, and risk free interest rate did not significantly affect the put option prices.That is why the stock option exchange market stop since the investor were not sure to the stock option price versus the risk of the volatility, time to maturity, and riskfree rate.

scholarly journals On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2930
Author(s):  
Siow Woon Jeng ◽  
Adem Kiliçman
Keyword(s):  
Monte Carlo ◽  
Stock Price ◽  
Variance Reduction ◽  
Volterra Equation ◽  
Price Volatility ◽  
Multilevel Methods ◽  
Heston Model ◽  
Control Variate ◽  
Stochastic Volterra Equation ◽  
And Control

The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient error[|Vt−Vtn|p] is finitely bounded by an exponential function of t. Furthermore, the weak error |error[Vt−Vtn]| and convergence for the stochastic Volterra equation are proven at the rate of O(n−H). In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.

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