Excerpt

## Table of Contents

List of Figures

List of Tables

1 Introduction

2 Mandelbrot and the L´evy-stable Distributions

2.1 The L´evy-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 L´evy-Regime

3.2 Explanation for the Power-law Tail Behavior

4 Summary

5 Appendix

References

## List of Figures

1 Leptokurtotic Behavior of the Empirical Distribution of Wool Price Changes

2 Normal Probability Graphs

3 Normal Probability Graphs cont.

4 Comparison of Normal Distribution, L´evy Distribution and Data of S&P 500 Price Changes

5 Regimes in the Empirical Distribution of S&P 500 Returns

6 Empirical Cumulative Distributions of the Daily Returns of three Stock Indices

7 Empirical Cumulative Distributions of Different-Sized Companies

8 Daily Returns of the DAX

9 Daily Returns of the Deutsche Bank Stock

## List of Tables

1 Frequency Distributions of the Dow Jones Industrial Average Stocks

2 Comparison of Frequency Distributions and the Normal Distribution

## 1 Introduction

A very important part of the distributional properties of economic and ﬁnancial systems are the distributional regularities of ﬁnancial returns as they cover a big part of second ﬁeld - ﬁnancial systems.

Distribution in the sense of this paper is meant as the statistical distribution. 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 dis- tributional 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 re- turns.

The second chapter will start with the results of Louis Bachelier and his normal distribution hypothesis. Then it will describe Benoˆıt Mandelbrot’s groundbreak- ing results, which rejected Bachelier’s normal hypothesis and introduced the L´evy-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 Mandel- brot’s ﬁndings. It will also display and explain the results of recent research.

## 2 Mandelbrot and the L´evy-stable Distributions

In 1900 Louis Bachelier developed the ﬁrst theory to describe the behavior of security prices. In his thesis - *Th´eorie de la Sp´eculation* - he assumed that in each transaction price changes are independently, identically distributed variables (i.i.d.-variables in the following). It was further assumed that the transactions were evenly distributed across time and the distribution of price changes had ﬁnite variance.^{1} Thus, price changes over a period of time were the result of many single price changes caused by many single transactions.

Since the daily, weekly and monthly price changes were sums of many indepen- dent variables, their distribution - according to the Central Limit Theorem (CLT in the following) - had to be a normal distribution. That is so, because the CLT states that the sum of i.i.d.-variables converges towards the normal distribution regardless of the distribution of the single i.i.d.-variables.

Benoˆıt Mandelbrot questioned this hypothesis in his work “*The Variation of Certain Speculative Prices*”, objecting that there were too many outliers, which could not be simply neglected. The number of outliers contradicted the normal distribution hypothesis since the normal distribution has too little probability mass in its tails.

Mandelbrot examined the prices of cotton over a period of several years. For bet- ter suitability Mandelbrot analyzed log-returns instead of actual price changes.^{2} The data indicated a *leptokurtotic* behavior, i.e. the empirical distributions had more observations in the center and - most notably - in the tails. This can be seen in Fig. 1 and Figs. 8 and 9 in the appendix. There were far too many changes within the *±* 10 per cent range to maintain the normal distribution. The leptokurtosis of the data had to ﬁt the theoretical distribution. Or as Mandelbrot (1963) said: “There need not be any obvious discontinuity between ‘outliers’ and the rest of the distribution[...].”

Hence, he rejected the normal distribution hypothesis and proposed a new family of distributions of price changes - the L´evy-stable distributions. Let us now take a close look at this family of probability distributions.

illustration not visible in this excerpt

Figure 1: The leptokurtotic behavior of the empirical distribution of wool price changes. Source: Gerhard Tintner (1940), *The Variate-Difference Method*, Bloomington. Used in Mandelbrot, B. (1963), “The Variation of Certain Specu- lative Prices”, *The Journal of Business*, **36**, p. 395.

### 2.1 The L´evy-stable distributions

The term *L ´ evy-stable distributions* designates a class of probability distributions characterized by Paul L´evy in his studies in the late 1920’s.^{3} An often used synonym for the L´evy-stable distributions is *stable Paretian distributions*. This is due to two important facts of the L´evy-stable distributions:

1. They are stable distributions

2. Their tails show a power-law behavior like the Paretian distribution does Now these facts will be explained starting with the deﬁnition of stability.

#### 2.1.1 Stability and Power-law Tail Behavior

A distribution can be called stable if the sum of random variables of this dis- tribution follows itself the distribution of the summands. This is an important property of the normal distribution, i.e. if the random variables *X* 1 and *X* 2 are normal, then *Y* = *X* 1 + *X* 2 is also normal.^{4}

Power-law tail behavior means that the tails of the L´evy distributions asymptot- ically behave as

illustration not visible in this excerpt

which can also be said about the Pareto-distribution. This means that for very large and very small *x* the tails of the L´evy distributions behave like the tails of *x ^{− α}*.

So the L´evy-stable distributions are stable and its tails have asymptotic power- law behavior. Hence the synonym stable Paretian distributions.

Since it lacks a clear analytical representation, a L´evy-stable distribution can only be described by its characteristic function:^{5}

illustration not visible in this excerpt

Thus, L´evy-stable distributions are deﬁned by four parameters. They are now to be examined.

#### 2.1.2 Parameters

This section is mainly based on Fama (1963), pp. 421-422; Voit (2003), p. 126 and Nolan (2008), pp. 7-8.

In (2), * Β*, which is restricted to *Β ∈* [ *−* 1 *,* 1], deﬁnes the skewness. When *Β* = 0, the distribution is symmetric. *Β >* 0 gives a right skewed distribution, whereas *Β <* 0 gives a left skewed one - i.e. the left tail is the heavy one.

*α* - restricted to *α ∈* (0 *,* 2] - is the *index parameter* or *characteristic exponent* and corresponds to the power-law exponent in (1). When *α* = 2, the L´evy-stable distribution is a normal distribution. When *α ∈* (0 *,* 2), the tails are heavier than those of the normal distribution (i.e. when *α* = 2). They are becoming heavier as *α* moves towards 0. A ﬁnite variance exists only in the case of a normal distribution and therefore is inﬁnite when *α ∈* (0 *,* 2). However, the mean is ﬁnite when *α >* 1 *.*

*m* is the location parameter giving the position of the peak, and *α* is the scale factor, which distinguishes the width of the distribution. *m ∈* R and *α ≥* 0 is applied for *m* and *a*.

### 2.2 The Generalization of the Central Limit Theorem

This section is based on Fama (1965) pp. 42-44.

The consequence of these facts is that Mandelbrot did not fully reject the nor- mal distribution hypothesis but broadened it by the generalization of the CLT.^{6} Hence, both Bachelier’s normal distribution hypothesis and Mandelbrot’s L´evy- stable hypothesis assume a L´evy-stable distribution. They only differ in their assumption of the value of *α*.

As Gnedenko and Kolmogorov showed, the L´evy-stable distributions are the only possible limiting distributions for sums of i.i.d. random variables. A direct con- sequence of this is that any distribution that is stable has to be a L´evy-stable distribution.

So, the sum of many i.i.d. L´evy-stable variables is itself L´evy-stable distributed. Put in simple words, “stability means that the values of the parameters *α* and *Β* remain constant under addition.”^{7}

As mentioned earlier, the price change of a security is the sum of many single price changes caused by many single transactions. Thus, assuming that price changes are evenly spread over time and are i.i.d. L´evy-stable variables one comes to the conclusion that price changes over bigger time intervals are also L´evy-stable distributed with the same parameters *α* and *Β*.

If the variables of the single price changes have ﬁnite variance, the distribution of their sums will be a normal distribution, i.e. *α* will have the value 2. If, on the other hand, the variables of the single price changes have inﬁnite variance, the distribution of their sums will be a L´evy-stable distribution with *α ∈* (0 *,* 2).

According to Mandelbrot, the characteristic exponent *α* is *∈* (0 *,* 2)^{8}. This im- plies that the variances of price change from single transactions are allowed to be inﬁnite.

### 2.3 Empirical Evidence of Mandelbrot’s Hypothesis

Now we are going to take look at the examinations done by Eugene Fama in his thesis “The Behavior of Stock-Market Prices”^{9}.

#### 2.3.1 The Data

The data Fama analyzed in his work consisted of daily prices changes of the thirty stocks of the Dow Jones Industrial Average. Like Mandelbrot, Fama used log-returns. The observed time period ran from the end of 1957 to September 26, 1962. The data comprised roughly 1,200 - 1,700 observations per stock. This choice of stocks was quite conservative since blue chip stocks are more stable than other stocks. So if the data was to show a leptokurtotic behavior the nor- mal distribution hypothesis could easily be rejected.

**[...]**

^{1} cf. Fama (1965), pp. 41-42.

^{2} [illustration not visible in this excerpt]. One advantage of log-returns is that they are summable.

^{3} cf. L´evy (1965).

^{4} cf. Nolan (2008), p. 4.

^{5} cf. Voit (2003), p. 126.

^{6} for the generalization of the CLT, cf. Gnedenko and Kolmogorov (1954).

^{7} quoted Fama (1965), p. 43.

^{8} actually Mandelbrot came to the conclusion that *α ≈* 1 *.* 7 ﬁtted the data best.

^{9} cf. Fama (1965), pp. 45-60.

- Quote paper
- Jakob Blatz (Author), 2008, Distributional Regularities of Financial Returns, Munich, GRIN Verlag, https://www.grin.com/document/128878

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