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

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2983
Author(s):  
Vasile Brătian ◽  
Ana-Maria Acu ◽  
Camelia Oprean-Stan ◽  
Emil Dinga ◽  
Gabriela-Mariana Ionescu
Keyword(s):  
Brownian Motion ◽  
Stock Market ◽  
Fractional Brownian Motion ◽  
Dynamic Equation ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Market Indexes ◽  
Stock Indexes ◽  
Fractal Market Hypothesis ◽  
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.

Bounds for the American perpetual put on a stock index

2001 ◽  
Vol 38 (01) ◽  
pp. 55-66 ◽  
Author(s):  
V. Paulsen
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Decomposition Technique ◽  
Early Exercise

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

scholarly journals Factor analysis of the Lithuanian equity market indices

2009 ◽  
Vol 50 ◽  
Author(s):  
Svetlana Danilenko
Keyword(s):  
Factor Analysis ◽  
Stock Market ◽  
Stock Prices ◽  
Equity Market ◽  
Stock Index ◽  
Change Dynamics ◽  
Statistical Measures ◽  
Components Method ◽  
Main Components ◽  
Stock Indexes

Statistical measures that can reproduce the state of the stock market and the tendencies of its change dynamics are the stock indexes. Having in mind the more complicated state of the finance system it is important to answer the question of what impacts the fluctuations of the stock prices. The article discusses various factors that impact the fluctuations of the Lithuanian stock index OMXV ; also stock index factor analysis is performed. Factors are determined using the main components method.

scholarly journals Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[

2019 ◽  
Vol 11 (1) ◽  
pp. 76
Author(s):  
Eric Djeutcha ◽  
Didier Alain Njamen Njomen ◽  
Louis-Aimé Fono
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Existence And Uniqueness ◽  
Hurst Parameter ◽  
Existence And Uniqueness Theorem ◽  
Martingale Approximation ◽  
Arbitrage Opportunities ◽  
Mixed Fractional Brownian Motion ◽  
Geometric Fractional Brownian Motion

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Hong Fan ◽  
Lingli Feng ◽  
Ruoyu Zhou
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Systemic Risk ◽  
Banking System ◽  
Risk Measurement ◽  
Geometric Brownian Motion ◽  
Hurst Index ◽  
Measurement Framework ◽  
Asset Value

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.

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