Stock Prices Follow a Brownian Motion

2017 ◽  
pp. 23-35
Author(s):  
Azar Karimov
Keyword(s):  
Brownian Motion ◽  
Stock Prices

Stochastic Processes for the Risk Management

2015 ◽  
pp. 188-200
Author(s):  
Gamze Özel
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Financial Markets ◽  
Interest Rates ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Stock Prices ◽  
Financial Risk ◽  
Future Events ◽  
Software Codes

The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.

scholarly journals The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Wawan Hafid Syaifudin ◽  
Endah R. M. Putri
Keyword(s):  
Brownian Motion ◽  
Kalman Filter ◽  
Model Predictive Control ◽  
Portfolio Optimization ◽  
Stock Returns ◽  
Predictive Control ◽  
Stock Prices ◽  
Geometric Brownian Motion ◽  
The Future ◽  
Stock Portfolio

<p style='text-indent:20px;'>A stock portfolio is a collection of assets owned by investors, such as companies or individuals. The determination of the optimal stock portfolio is an important issue for the investors. Management of investors' capital in a portfolio can be regarded as a dynamic optimal control problem. At the same time, the investors should also consider about the prediction of stock prices in the future time. Therefore, in this research, we propose Geometric Brownian Motion-Kalman Filter (GBM-KF) method to predict the future stock prices. Subsequently, the stock returns will be calculated based on the forecasting results of stock prices. Furthermore, Model Predictive Control (MPC) will be used to solve the portfolio optimization problem. It is noticeable that the management strategy of stock portfolio in this research considers the constraints on assets in the portfolio and the cost of transactions. Finally, a practical application of the solution is implemented on 3 company's stocks. The simulation results show that the performance of the proposed controller satisfies the state's and the control's constraints. In addition, the amount of capital owned by the investor as the output of system shows a significant increase.</p>

scholarly journals Comparison of Stochastic Forecasting Models

2021 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Brownian Motion ◽  
Stock Prices ◽  
Absolute Error ◽  
Geometric Brownian Motion ◽  
Time Dependent ◽  
Confidence Bounds ◽  
History Of ◽  
Brownian Motion Model ◽  
Stochastic Forecasting ◽  
Two Parameters

In this work, we compare several stochastic forecasting techniques like Stochastic Differential Equations (SDE), ARIMA, the Bayesian filter, Geometric Brownian motion (GBM), and the Kalman filter. We use historical daily stock prices of Microsoft (MSFT), Target (TGT) and Tesla (TSLA) and apply all algorithms to try to predict 54 days ahead. We find that there are instances in which all algorithms do well, or do poorly. We find that all three stocks have a strong auto-correlation and a high Hurst factor which shows that it is possible to predict future prices based on a short history of past prices. In our geometric Brownian motion model, we have two parameters for drift and diffusion which are not time dependent. In our more general SDE model (TDNGBM), we have time-dependent drift and time-dependent diffusion terms which makes it more effective than GBM. We measure all algorithms on the correlation between the predicted and actual values, the mean absolute error (MAE) and also the confidence bounds generated by the methods. Confidence intervals are more important than point forecasts, and we see that TDNGBM and ARIMA produce good bounds.

scholarly journals Forecasting Nestle Stock Price by using Brownian Motion Model during Pandemic Covid-19

2021 ◽  
Vol 7 (2) ◽  
pp. 58-64
Author(s):  
Siti Raihana Hamzah ◽  
Hazirah Halul ◽  
Assan Jeng ◽  
Umul Ain’syah Sha’ari
Keyword(s):  
Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Investment Decision ◽  
Primary Objective ◽  
Percentage Error ◽  
Economic Downturn ◽  
Term Prediction ◽  
Investment Decision Making ◽  
Short Term Prediction

In the modern financial market, investors have to make quick and efficient investment decisions. The problem arises when the investor does not know the right tools to use in investment decision making. Different tools can be implemented in trading strategies to predict future stock prices. Therefore, the primary objective of this paper is to analyse the performance of the Geometric Brownian Motion (GBM) model in forecasting Nestle stock price by assessing the performance evaluation indicators. To analyse the stocks, two software were used, namely Microsoft Excel and Python.  The model is trained for 16 weeks (4 months) of data from May to August 2019 and 2020. The simulated sample is for four weeks (1 month) which is for September 2019 and 2020. The findings show that during the Pandemic Covid-19, short-term prediction using GBM is more efficient than long-term prediction as the lowest Mean Square Error (MSE) value is at one week period.  In addition, the Mean Absolute Percentage Error (MAPE) for all GBM simulations is highly accurate as it shows that MAPE values are less than 10%, indicating that the GBM method can be used to predict Nestle stock price during an economic downturn.

scholarly journals The voice of Physics in finance: A glance on the theoretical application of heat equation to stock price diffusions

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Leonard Mushunje
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Stock Prices ◽  
Stock Price ◽  
Price Volatility ◽  
Heat Diffusion ◽  
Motion Model ◽  
Asset Valuation ◽  
Brownian Motion Model ◽  
Theoretical Application

Stock price volatility is considered the main matter of concern within the investment grounds. However, the diffusivity of these prices should as well be considered. As such, proper modelling should be done for investors to stay healthy-informed. This paper suggest to model stock price diffusions using the heat equation from physics. We hypothetically state that, our model captures and model the diffusion bubbles of stock prices with a better precision of reality. We compared our model with the standard geometric Brownian motion model which is the wide commonly used stochastic differential equation in asset valuation. Interestingly, the models proved to agree as evidenced by a bijective relation between the volatility coefficients of the Brownian motion model and the diffusion coefficients of our heat diffusion model as well as the corresponding drift components. Consequently, a short proof for the martingale of our model is done which happen to hold. 

Stochastic Processes for the Risk Management

2018 ◽  
pp. 473-488
Author(s):  
Gamze Özel
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Financial Markets ◽  
Interest Rates ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Stock Prices ◽  
Financial Risk ◽  
Future Events ◽  
Software Codes

The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.

Stochastic Processes for the Risk Management

2018 ◽  
pp. 174-189
Author(s):  
Gamze Özel
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Financial Markets ◽  
Interest Rates ◽  
Stochastic Processes ◽  
Stochastic Models ◽  
Stock Prices ◽  
Financial Risk ◽  
Future Events ◽  
Software Codes

The financial markets use stochastic models to represent the seemingly random behavior of assets such as stocks, commodities, relative currency prices such as the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro, and interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates. This chapter gives an overview of the stochastic models and methods used in financial risk management. Given the random nature of future events on financial markets, the field of stochastic processes obviously plays an important role in quantitative risk management. Random walk, Brownian motion and geometric Brownian motion processes in risk management are explained. Simulations of these processes are provided with some software codes.

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.

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