Asymptotic analysis for hedging errors in models with respect to geometric fractional Brownian motion

Stochastics ◽  
2018 ◽  
Vol 91 (3) ◽  
pp. 407-432
Author(s):  
Wensheng Wang
Keyword(s):  
Brownian Motion ◽  
Asymptotic Analysis ◽  
Fractional Brownian Motion ◽  
Geometric Fractional Brownian Motion

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 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.

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