scholarly journals Pricing European Option When the Stock Price Process Is Being Driven by Geometric Brownian Motion

OALib ◽  
2019 ◽  
Vol 06 (08) ◽  
pp. 1-19
Author(s):  
Kebareng I. Moalosi-Court
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Geometric Brownian Motion ◽  
European Option ◽  
Price Process

OPTIMAL TRADE EXECUTION UNDER GEOMETRIC BROWNIAN MOTION IN THE ALMGREN AND CHRISS FRAMEWORK

2011 ◽  
Vol 14 (03) ◽  
pp. 353-368 ◽  
Author(s):  
JIM GATHERAL ◽  
ALEXANDER SCHIED
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Brownian Motion ◽  
Price Process ◽  
Risk Criterion ◽  
Execution Strategy ◽  
Optimal Trade Execution ◽  
Alternative Choice

With an alternative choice of risk criterion, we solve the HJB equation explicitly to find a closed-form solution for the optimal trade execution strategy in the Almgren–Chriss framework assuming the underlying unaffected stock price process is geometric Brownian motion.

scholarly journals The Geometric Brownian Motion of Indosat Telecommunications Daily Stock Price During the Covid-19 Pandemic in Indonesia

2021 ◽  
Vol 2084 (1) ◽  
pp. 012012
Author(s):  
Tiara Shofi Edriani ◽  
Udjianna Sekteria Pasaribu ◽  
Yuli Sri Afrianti ◽  
Ni Nyoman Wahyu Astute
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Service Providers ◽  
Main Idea ◽  
Geometric Brownian Motion ◽  
Government Policies ◽  
Percentage Error ◽  
Price Movement ◽  
Data Movement ◽  
Total Data

Abstract One of the major telecommunication and network service providers in Indonesia is PT Indosat Tbk. During the coronavirus (COVID-19) pandemic, the daily stock price of that company was influenced by government policies. This study addresses stock data movement from February 5, 2020 to February 5, 2021, resulted in 243 data, using the Geometric Brownian motion (GBM). The stochastic process realization of this stock price fluctuates and increases exponentially, especially in the 40 latest data. Because of this situation, the realization is transformed into log 10 and calculated its return. As a result, weak stationary in variance is obtained. Furthermore, only data from December 7, 2020 to February 5, 2021 fulfill the GBM assumption of stock price return, as R t 1 * , t 1 * = 1 , 2 , 3 , … , 40 . The main idea of this study is adding datum one by one as much as 10% – 15% of the total data R t 1 * , starting from December 4, 2020 backwards. Following this procedure, and based on the 3% < p-value < 10%, the study shows that its datum can be included in R t 1 * , so t 1 * = − 4. − 3 , − 2 , … , 40 and form five other data groups, R t 2 * , … , R t 6 * . Considering Mean Absolute Percentage Error (MAPE) and amount of data from each group, R t 6 * is selected for modelling. Thus, GBM succeeded in representing the stock price movement of the second most popular Indonesian telecommunication company during COVID-19 pandemic.

scholarly journals An Interval of No-Arbitrage Prices in Financial Markets with Volatility Uncertainty

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Hanlei Hu ◽  
Zheng Yin ◽  
Weipeng Yuan
Keyword(s):  
Brownian Motion ◽  
Financial Markets ◽  
Stock Price ◽  
Contingent Claims ◽  
Price Dynamics ◽  
Complex Situation ◽  
No Arbitrage ◽  
Price Process ◽  
Arbitrage Opportunities

In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric G-Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

2017 ◽  
Vol 6 (3) ◽  
pp. 85
Author(s):  
ömer önalan
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Price ◽  
Long Range Dependence ◽  
Price Process ◽  
Inverse Gamma ◽  
Proposed Model ◽  
Mixed Fractional Brownian Motion ◽  
Time Periods

In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

Analytical Approximations of American Call Options with Discrete Dividends

2018 ◽  
Vol 26 (3) ◽  
pp. 283-310
Author(s):  
Kwangil Bae
Keyword(s):  
Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Computing Time ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Option Prices ◽  
Call Options ◽  
Analytical Approximations ◽  
Discrete Dividends

In this study, we assume that stock prices follow piecewise geometric Brownian motion, a variant of geometric Brownian motion except the ex-dividend date, and find pricing formulas of American call options. While piecewise geometric Brownian motion can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our work differs from other studies using the same assumption in at least three respects. First, we investigate the analytical approximations of American call options and examine European call options as a special case, while most analytical approximations in the literature cover only European options. Second, we provide both the upper and the lower bounds of option prices. Third, our solutions are equal to the exact price when the size of the dividend is proportional to the stock price, while binomial tree results never match the exact option price in any circumstance. The numerical analysis therefore demonstrates the efficiency of our method. Especially, the lower bound formula is accurate, and it can be further improved by considering second order approximations although it requires more computing time.

scholarly journals Stock price prediction using geometric Brownian motion

2018 ◽  
Vol 974 ◽  
pp. 012047 ◽  
Author(s):  
W Farida Agustini ◽  
Ika Restu Affianti ◽  
Endah RM Putri
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Geometric Brownian Motion ◽  
Stock Price Prediction ◽  
Price Prediction

scholarly journals MENENTUKAN PORTOFOLIO OPTIMAL PADA PASAR SAHAM YANG BERGERAK DENGAN MODEL GERAK BROWN GEOMETRI MULTIDIMENSI

2015 ◽  
Vol 4 (3) ◽  
pp. 127 ◽  
Author(s):  
RISKA YUNITA ◽  
KOMANG DHARMAWAN ◽  
LUH PUTU IDA HARINI
Keyword(s):  
Stochastic Process ◽  
Exact Solution ◽  
Brownian Motion ◽  
Stock Price ◽  
Stock Return ◽  
Geometric Brownian Motion ◽  
Optimal Portfolio ◽  
The Future ◽  
Model Determination ◽  
Stock Price Movements

Model of stock price movements that follow stochastic process can be formulated in Stochastic Diferential Equation (SDE). The exact solution of SDE model is called Geometric Brownian Motion (GBM) model. Determination the optimal portfolio of three asset that follows Multidimensional GBM model is to be carried out in this research.Multidimensional GBM model represents stock price in the future is affected by three parameter, there are expectation of stock return, risk stock, and correlation between stock return. Therefore, theory of portfolio Markowitz is used on formation of optimal portfolio. Portfolio Markowitz formulates three of same parameter that is calculated on Multidimensional GBM model. The result of this research are optimal portfolio reaches with the proportion of fund are 39,38% for stock BBCA, 59,82% for stock ICBP, and 0,80% for stock INTP. This proportion of fund represents value of parameters that is calculated on modelling stock price.

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