There is a long tradition of scholars seeking to understand the distributional
regularities of financial returns. Research traces back to the turn of the 19th
century. Since then, it underwent a lot of drastic changes, which are to be shown
in this paper.
The aim of this paper is to show theoretical models that account for the distributional
regularities in financial returns as well as to illustrate the empirical
analysis. It is necessary to understand the evolution of research on this topic
because it came about in a consecutive manner. Thus, this paper will document
over one hundred years of research on distributional properties of financial returns.
The second chapter will start with the results of Louis Bachelier and his normal
distribution hypothesis. Then it will describe Benoît Mandelbrot's groundbreaking
results, which rejected Bachelier's normal hypothesis and introduced the
Lévy-stable distributions. Mandelbrot's work had such an impact that it will be
described in greater detail.
The third chapter will present the results of research that followed after Mandelbrot's
findings. It will also display and explain the results of recent research.
Table of Contents
1 Introduction
2 Mandelbrot and the Lévy-stable Distributions
2.1 The Lévy-stable distributions
2.1.1 Stability and Power-law Tail Behavior
2.1.2 Parameters
2.2 The Generalization of the Central Limit Theorem
2.3 Empirical Evidence of Mandelbrot’s Hypothesis
2.3.1 The Data
2.3.2 Frequency Distributions
2.3.3 Normal Probability Graphs
3 Research in the post-Mandelbrot Era
3.1 Recent Contributions: Power-law Behavior Outside the Lévy-Regime
3.2 Explanation for the Power-law Tail Behavior
4 Summary
Objectives and Topics
This paper documents the evolution of research regarding the distributional properties of financial returns over the last century. It focuses on the transition from the traditional normal distribution hypothesis to the development and subsequent refinement of models accounting for heavy-tailed phenomena, such as Lévy-stable distributions and power-law behavior.
- Evolution of statistical models for financial return distributions.
- Evaluation of the normal distribution hypothesis and its limitations.
- Analysis of Mandelbrot’s Lévy-stable distributions and empirical evidence provided by Fama.
- Exploration of post-Mandelbrot research, specifically power-law tail behavior and the inverse cubic law.
Excerpt from the Book
2.3.2 Frequency Distributions
An obvious way to examine distributions of price changes are frequency distributions. Fama observed the development of prices of the thirty Dow Jones stocks and computed the results. He compared the observations within 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0 and greater than 5 standard deviations of the mean change with the theoretical data of the normal distribution.
The results are shown in Tables 1 and 2. Table 1 shows the frequency distributions of the thirty Dow Jones stocks and the normal distribution, which is on top of the figure. Table 2 compares the frequency distributions by subtracting the theoretical value of the normal distribution from the entry of the particular stock.
The columns with positive numbers in Table 2 indicate that the normal distribution has too little probability mass in the center, i.e. in the range of 0.5 to 1.5 standard deviations, and the tails, i.e. in the > 5.0 standard deviations spectrum. The most striking feature of the data is the following: Although the difference in the >5.0 standard deviation column between the average frequency distribution and the normal distribution seems to be very small (0.12 percent), the difference is 2,000 times larger than the expected frequency. Thus, the distribution to describe security price changes had to have a lepokurtotic shape.
Summary of Chapters
1 Introduction: Introduces the historical context of distributional regularities in financial returns and outlines the scope of the paper, covering models from Bachelier to contemporary research.
2 Mandelbrot and the Lévy-stable Distributions: Discusses the rejection of the normal distribution hypothesis in favor of Mandelbrot’s Lévy-stable distributions, detailing their properties and empirical validation by Fama.
3 Research in the post-Mandelbrot Era: Examines subsequent challenges to the Lévy-stable hypothesis and the emergence of power-law behaviors as a more accurate description of market fluctuations.
4 Summary: Recaps the transition between theories and emphasizes the lasting importance of recognizing heavy-tailed behavior for practical applications like risk management.
Keywords
Financial Returns, Normal Distribution, Lévy-stable Distributions, Mandelbrot, Central Limit Theorem, Leptokurtotic, Power-law, Tail Behavior, Efficient Market Hypothesis, Volatility, Security Prices, Inverse Cubic Law, Fama, Stock Market, Statistical Distributions.
Frequently Asked Questions
What is the core subject of this paper?
The paper examines the history and development of mathematical models used to describe the statistical properties, particularly the distributions, of financial market returns over the past 100 years.
What are the primary themes discussed?
The main themes include the transition from Bachelier’s normal distribution hypothesis to Mandelbrot’s Lévy-stable distributions, empirical testing by Fama, and modern research on power-law tail behaviors.
What is the main objective of the research?
The goal is to show theoretical models that account for the observed regularities in financial returns and to document the evolution of this research field over time.
Which scientific methodology is utilized?
The work utilizes a literature-based historical review combined with the interpretation of empirical statistical analyses (such as frequency distribution tables and probability graphs) performed by scholars like Fama and Mandelbrot.
What is covered in the main body of the text?
The main body covers the theoretical foundations of the normal distribution versus Lévy-stable distributions, empirical evidence regarding leptokurtosis, and post-Mandelbrot advancements like power-law explanations for market volatility.
Which keywords characterize this work?
Key terms include financial returns, Lévy-stable distributions, power-law behavior, leptokurtosis, Mandelbrot’s hypothesis, and volatility modeling.
Why did Mandelbrot reject the normal distribution hypothesis?
Mandelbrot observed that financial data contained too many outliers (leptokurtotic behavior) that the normal distribution, due to its thin tails, failed to capture accurately.
How does the "inverse cubic law" relate to financial markets?
It is a power-law relation used to describe the probability of large returns, which helps explain significant market volatility and historical events like major market crashes.
- Arbeit zitieren
- Jakob Blatz (Autor:in), 2008, Distributional Regularities of Financial Returns, München, GRIN Verlag, https://www.grin.com/document/128878
