Modeling the volatity of cryptocurrency markets

The application of the model of geometric Brownian motion (GBM) for the problem of modeling and forecasting prices for cryptocurrencies is analyzed. For prediction the solution of the stochastic differential equation of the GBM model is used, which has a linear drift and diffusion coefficients. Different scenarios of price movement are considered. Keywords: geometric Brownian motion (GBM), modeling, forecasting, cryptocurrency.

Limit theorems for prices of options written on semi-Markov processes

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

Efficient or Fractal Market Hypothesis? A Stock Indexes Modelling Using Geometric Brownian Motion and Geometric Fractional Brownian Motion

In this article, we propose a test of the dynamics of stock market indexes typical of the US and EU capital markets in order to determine which of the two fundamental hypotheses, efficient market hypothesis (EMH) or fractal market hypothesis (FMH), best describes market behavior. The article’s major goal is to show how to appropriately model return distributions for financial market indexes, specifically which geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) dynamic equations best define the evolution of the S&P 500 and Stoxx Europe 600 stock indexes. Daily stock index data were acquired from the Thomson Reuters Eikon database during a ten-year period, from January 2011 to December 2020. The main contribution of this work is determining whether these markets are efficient (as defined by the EMH), in which case the appropriate stock indexes dynamic equation is the GBM, or fractal (as described by the FMH), in which case the appropriate stock indexes dynamic equation is the GFBM. In this paper, we consider two methods for calculating the Hurst exponent: the rescaled range method (RS) and the periodogram method (PE). To determine which of the dynamics (GBM, GFBM) is more appropriate, we employed the mean absolute percentage error (MAPE) method. The simulation results demonstrate that the GFBM is better suited for forecasting stock market indexes than the GBM when the analyzed markets display fractality. However, while these findings cannot be generalized, they are verisimilar.

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

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.

Research on Systemic Risk of the Turkish Banking Industry Based on a Systemic Risk Measurement Framework of the Fractional Brownian Motion

Since the 2008 financial crisis, it is an important issue to assess the systemic risk of banks, but there is a lack of research on the assessment of the systemic risk of Turkey’s financial system. In addition, geometric Brownian motion is used in most of the assessment frameworks of systemic risk under the normal financial market state, while the Turkish financial market has the situation of spike and thick tail. Therefore, this paper proposes a fractional Brownian motion measurement framework of systemic risk to study the systemic risk of the Turkish financial system. Firstly, this paper uses the data of 11 Turkish listed banks from 2014 to 2019 to conduct a normality test and demonstrate that its market has the characteristics of a fractal market; that is, there is a spike and thick tail distribution phenomenon in the stock price trend. Then, this paper proposes a fractional Brownian motion systemic risk measurement framework (fBSM). Based on the proposed theoretical framework and the actual data of Turkish listed banks from 2014 to 2019, a dynamically evolving Turkish banking network system is constructed to measure the systemic risk in the Turkish banking system. The research results find that the systemic risk is the highest in 2017, which then improved and gradually recovered. In addition, when analyzing the sensitivity of the Hurst index, it shows that with the increase in Hurst index, the Hurst index elasticity of Turkish banks’ asset value increases gradually and the asset value also increases continuously. Hence, the Hurst index has a greater impact on asset value. Therefore, the measurement framework of systemic risk based on the fBSM can better monitor the systemic risk than the traditional geometric Brownian motion in the Turkish banking system.

Forecasting Cyclical and Non-cyclical Stock Prices on the Stock Exchange of Thailand

Forecasting is an important role in organizations for decision making and planning. This research is to forecast the cyclical and non-cyclical weekly stock prices on the Stock Exchange of Thailand by using the models of Geometric Brownian motion, Fourier’s series, and Cauchy initial value problem. The accuracy and performance of the models are based on the minimum root mean squared percentage error which is the error between actual and forecasted stock prices. The results showed that Geometric Brownian motion is suitable for forecasting both cyclical and non-cyclical stock prices because of minimum error. Moreover, the confidence intervals of forecasted stock prices are demonstrated. Therefore, Geometric Brownian motion should be selected to describe the movement of stock prices in Thailand.