Illustration scheme of the classes of random processes discussed above. The solid circle denotes the stable Gaussian process. Mantegna and Stanley (2000)
Security returns are non-stationary, so we speak here of the asymptotic pdf. The distribution of returns is approximately symmetric and has high kurtosis (that is, fat tails and a peaked centre compared with the normal distribution). The distributions are increasingly fat-tailed as data frequency increases (smaller interval sizes).
A random process Y is infinitely divisible if, for every natural number n, it can be represented as the sum of n independent identically distributed (i.i.d.) random variables:
Y = X1 + X2+…+Xn
Consider the sum of n i.i.d. random variables:
Y = X1 + X2+…+Xn
when the functional form of Y is the same as the functional form of Xi, the stochastic process is said to be stable.
Special cases of stable distributions:
A power law relationship between two scalar quantities x and y is any such that the relationship can be written as
y = axk
where a and k are constants.
A random variable x with a Pareto distribution has a probability density function given by f(x) = akax-(a+1), x ≥ k where a and k are positive constants.
The Pareto distribution is a “power law” distribution.The Gaussian distribution, also called the normal distribution or the bell curve, is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent variables is approximately normal. The Gaussian distribution is the only stable distribution having all of its moments finite.
The central limit theorem states that the sum of a number of random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization of the central limit theorem states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | a + 1 (and therefore having infinite variance) will tend to a stable Levy distribution f(x;a,0,c,0) as the number of variables grows.
Illustration scheme of the classes of random processes discussed above. The solid circle denotes the stable Gaussian process. Mantegna and Stanley (2000)
"PDF of returns for the Shanghai market data with Δt = 1. This plot is compared to a stable symmetric Levy distribution using the value α = 1.44 determined from the slope [in a log-log plot of the central peak of the PDF as a function of the time increment]. The agreement is very good over the main central portion, with deviations for large z. Two attempts to fit a Gaussian are also shown. The wider Gaussian is chosen to have the same standard deviation as the empirical data. However, the peak in the data is much narrower and higher than this Gaussian, and the tails are fatter. The narrower Gaussian is chosen to fit the central portion, however the standard deviation is now too small. It can be seen that the data has tails which are much fatter and furthermore have a non-Gaussian functional dependence." Johnson, Jefferies and Hui (2003)
For how long have we known about the fat tails?
"Heavy tails: the (unconditional) distribution of returns seems to display a power-law or Pareto-like tail, with a tail index which is finite, higher than two and less than five for most data sets studied. In particular this excludes stable laws with infinite variance and the normal distribution. However the precise form of the tails is difficult to determine."
Cont (2001)
"The degree of leptokurtosis is much larger for high-frequency data (Fig. 8.2)."
Mantegna and Stanley (2000)
"...For example, researchers beginning with Black (1976) have found evidence that stock returns are negatively correlated with changes in returns volatility—i.e., volatility tends to rise in response to “bad news” (excess returns lower than expected) and to fall in response to “good news“ (excess returns higher than expected)."
Nelson (1991)
"Asset returns tend to be leptokurtic. The documentation of this empirical regularity by Mandelbrot (1963), Fama (1965) and others led to a large literature on modelling stock returns as i.i.d. draws from thick-tailed distributions; see e.g., Mandelbrot (1963), Fama (1963, 1965), Clark (1973), and Blattberg and Gonedes (1974)."
Bollersvlev, Engle and Nelson (1994)
[foreign exchange (FX) rates, interbank money market rates, and Eurofutures contracts]
"The distributions of returns are increasingly fat-tailed as data frequency increases (smaller interval sizes) and are hence distinctly unstable. The second moments of the distributions most probably exist while the fourth moments tend to diverge."
Dacorogna, et al. (2001)
We find that the distributions of realized daily variances, standard deviations and covariances are skewed to the right and leptokurtic, but that the distributions of logarithmic standard deviations and correlations are approximately Gaussian. Volatility movements, moreover, are highly correlated across the two exchange rates. We also find that the correlation between the exchange rates (as opposed to the correlation between their volatilities) increases with volatility. Finally, we confirm the wealth of existing evidence of strong volatility clustering effects in daily returns. However, in contrast to earlier work, which often indicates that volatility persistence decreases quickly with the horizon, we find that even monthly realized volatilities remain highly persistent. Nonetheless, realized volatilities do not have unit roots; instead, they appear fractionally integrated and therefore very slowly mean-reverting. This finding is strengthened by our analysis of temporally aggregated volatility series, whose properties adhere closely to the scaling laws implied by the structure of fractional integration."
Andersen et al. (2000) The Distribution of Realized Exchange Rate Volatility
FX: Exchange rate returns standardized by realized volatility are very nearly Gaussian.
Andersen, et al. (2000)
"Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks."
Carr, et al. (2002)
"We examine “realized” daily equity return volatilities and correlations obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones Industrial Average. We find that the unconditional distributions of realized variances and covariances are highly right-skewed, while the realized logarithmic standard deviations and correlations are approximately Gaussian, as are the distributions of the returns scaled by realized standard deviations. Realized volatilities and correlations show strong temporal dependence and appear to be well described be long-memory processes. Finally, there is strong evidence that realized volatilities and correlations move together in a manner broadly consistent with latent factor structure."
Andersen et al., (2001) The Distribution of Stock Return Volatility.
"Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades3±10. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countriesÐsuggesting that a generic theoretical basis may underlie these phenomena."
Gabaix, et al. (2003)
"[individual stocks] For time scales from 5 min up to approximately 16 days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent 2.5,},4, well outside the stable Le´vy regime 0,a,2. For time scales Dt @(Dt)3'16 days, we observe results consistent with a slow convergence to Gaussian behavior."
We also find that the distribution of returns of individual companies and the S&P 500 index have the same asymptotic behavior."
Plerou, et al. (1999)
"IT IS WIDELY DOCUMENTED that daily financial return series display strong conditional heteroskedasticity."
"On the empirical front, a sizable literature has documented a strong positive contemporaneous correlation between daily trading volume and return volatility."
"[IBM volume] The skewness is positive in all subsamples. [...] In addition, the kurtosis exceeds three but is smaller than for the returns series. However, this finding is not robust. In the majority of subsamples the volume series has a higher kurtosis than returns. [...] The autocorrelations display a regular and smooth decline from significantly positive values at small lags to about zero at lags above thirty. [...] In sum, the derived volume series appears stationary..."
Andersen (1996)
[FX]
"The variety of opinions about the distributions of FX price changes and their generating process is wide. some authors claim the distributions to be close to Paretian stable (McFarland et al. 1982), some to Student distributions (Boothe and Glassmann 1987), some reject any single distribution (Calderon-Rossel and Ben-Horim 1982)."
"Non stable, fat-tailed distribution."
"Finite variance."
"Symmetric distribution."
"Decreasing leptokurticity.
Guillaume, et al. (1997)
"We find an asymptotic power-law behavior for the cumulative distribution with an exponent 3, well outside the Levy regime (0 < < 2)."
To put these results in the context of previous work, we recall that proposals for P(g) have included (i) a Gaussian distribution [1], (ii) a Levy distribution [2,11,12], and (iii) a truncated Levy distribution, where the tails become \approximately exponential" [3]. The inverse cubic law di ers from all three proposals: Unlike (i) and (iii), it has diverging higher moments (larger than 3), and unlike (i) and (ii) it is not a stable distribution."
Gopikrishnan, et al. (1998)
"The probability distribution of stock price changes is studied by analyzing a database (the Trades and Quotes Database) documenting every trade for all stocks in three major US stock markets, for the two year period January 1994 - December 1995. A sample of 40 million data points is extracted, which is substantially larger than studied hitherto. We find an asymptotic power-law behavior for the cumulative distribution with an exponent [alpha approximately equal to] 3, well outside the Lévy regime (0 < [alpha] < 2)."
Gopikrishnan, et al. (1998)
"...the S & P 500 index is reasonably fitted by a truncated Levy distribution with ? 1.4 over a time scale which ranges from a minute to a day, with convergence to a Gaussian at approximately one month 2 (Mantegna and Stanley, 1995). Also, Arneodo et al (1996) observed a 1/f2 power spectrum at long time scales consistent with Gaussian behavior; while at short time scales truncated Levy behavior was observed. They analyzed the DEM-USD exchange rate from October 1991 - November 1994."
Bak, Paczuski and Shubik (1996)
fat-tailed PDF of price changes Johnson, Jefferies and Hui (2003), page 69
[daily returns for 30 German stocks forming the DAX share index as well as the DAX itself]
[stocks] "...suggests that the tail behaviour of the data deviates from that of stable Paretian distributions for all cases considered. These results ® t with the picture emerging from the recent literature indicating that empirical distribution shapes of stock returns, though they initially appear similar to the stable laws, are characterized by a higher rate of decay of observations in the tails than is consistent with the stable distributions. Hence, one may conclude other types of distributions like the Student t or ARCH processes are more appropriate for describing financial data."
[stocks] "...the empirical tail regions are thinner than expected under a stable distribution."
Lux (1996)
[daily observations of an index of the most traded stocks]
"I show empirically that the extreme returns obey a Fréchet distribution."
"The daily returns have a positive mean of 0.031% and a high standard deviation of 1.053% (in annual unit an average return of 8.70% and a volatility of 17.60%). The returns are slightly skewed (-0.506) and presents excess kurtosis (22.057) which suggests departure from the normal distribution.
Longin (1996)
[daily observations of an index of the most traded stocks]
"A characteristic of the extremes is their clustering: there are 28 years (from among 106) during which the minima occur in the same week. In general, the price decrease precedes the price increase."
Longin (1996)
"The distribution of price fluctuations is one of the most basic properties of markets. For some markets the historical data spans a century at a daily timescale, and for at least the last decade every transaction is recorded. Nonetheless, the price distribution’s functional form is still a topic of active debate. Naively, central-limit theorem arguments suggest a Gaussian (normal) distribution. If p(t) is the price at time t, the log-return rt(t) is defined as rt(t) = log p(t + t) – log p(t). Dividing t into N subintervals, the total log-return rt(t) is by definition the sum of the log-returns in each subinterval. If the price changes in each subinterval are independent and identically distributed (IID) with a well-defined second moment, under the central limit theorem the cumulative distribution function f(rt) should converge to a normal distribution for large t.
For real financial data, however, convergence is very slow. While the normal distribution provides a good approximation for the center of the distribution for large t, for smaller values of t— less than about a month—there are strong deviations from normality. This is surprising, given that the autocorrelation of log-returns is typically very close to zero for times longer than about 15 to 30 minutes.2,3 What is the nature of these deviations from normality and what is their cause?
The actual distribution of log-returns has fat tails. That is, there is a higher probability for extreme values than for a normal distribution. As one symptom of this, the fourth moment is larger than expected for a Gaussian. We can measure this deviation in a scale-independent manner by using the kurtosis k = á(r – árñ)4ñ/á(r – árñ)2ñ2 (á ñ in28 dicates a time average). In the early 1960s, Benoit Mandelbrot4 (now famous as the grandfather of fractals) and Eugene Fama5 (now famous as the high priest of efficient market theory) presented empirical evidence that f was a stable Levy distribution. The stable Levy distributions are a natural choice because they emerge from a generalization of the central limit theorem. For random variables that are so fat-tailed that their second moment doesn’t exist, the normal central limit theorem no longer applies. Under certain conditions, however, the sum of N such variables converges to a Levy distribution.3 The Levy distributions are characterized by a parameter 1 £ m £ 2, where m = 2 corresponds to the special case of a normal distribution. For m < 2, however, the stable Levy distributions are so fat-tailed that their standard deviation and all higher moments are infinite—that is, árqñ = ¥ for q ³ 2. In practice, this means that numerical estimates of any moment q = 2 or higher will not converge. Based on daily prices in different markets, Mandelbrot and Fama measured m » 1.7, a result that suggested that short-term price changes were indeed ill-behaved: if the variance doesn’t exist, most statistical properties are ill defined.
Subsequent studies demonstrated, however, that the behavior is more complicated than this.6–12 First, for larger values of t, the distribution becomes progressively closer to normal. Second, investigations of larger data sets (including work by economists in the late ’80s and early ’90s6–8) make it clear that large returns asymptotically follow a power law f(r) ~ |r|–a, with a > 2. This finding is incompatible with the Levy distribution. The difference in the value of a is very important: with a > 2, the second moment (the variance) is well defined. A value 2 < a < ¥ is incompatible with the stable Levy distribution and indicates that simply generalizing the central limit theorem with long tails is not the correct explanation.
Physicists have contributed to this problem by studying really large data sets and looking at the scalings in close detail. A group at Olsen and Associates, led by Michel Dacorogna, studied intraday price movements in foreign exchange markets.9 Another group at Boston University, led by Rosario Mantegna and Eugene Stanley, has studied the intraday movements of the S&P index.10,11 More recently, they studied the fiveminute returns of 1,000 individual stocks traded on the AMEX, NASDAQ, and NYSE exchanges, over a two-year period involving roughly 40 million records.12 In this case, they observed the power-law scaling over about 90 standard deviations (see Figure 1). For larger values of |r|, these results dramatically illustrate that f(r) is approximately a power law with a » 3. Thus, the mean and variance are well-defined, the kurtosis clearly diverges, and the behavior of the skewness is not so clear.
Power-law scaling is not new to economics. The power-law distribution of wealth discovered by Vilfredo Pareto (1848–1923) in the 19th century predates any power laws in physics.13 And indeed, since Pareto, the existence of power laws has been controversial. One underlying reason is that power-law probability distributions are necessarily approximations. An inverse powerlaw cumulative distribution f(r) ~ |r|–a with an exponent a > 0 is not integrable at zero, and similarly, with an exponent a £ 0, it is not integrable at infinity. Thus, a power-law probability distribution cannot be exactly true for a variable with an unbounded range. When they apply at all, power-law distributions are necessarily only part of a more complete description, valid within certain limits. (See the “Power law distribution of wealth” sidebar for more on this topic.14)
Another reason for skepticism about power laws in economics is that sloppy statistical analysis has led to mistakes in the past. In the 1980s, there was considerable interest in the possibility that price changes might be described by a lowdimensional chaotic attractor. Physics and biology have many examples where the existence of low-dimensional chaos is unambiguous. Why not economics? Based on a numerical computation of fractal dimension, several researchers claimed to observe low-dimensional chaos in price series. Such computations are done by measuring the coarse-grained size of a set, in this case a possible attractor of returns in a state space whose variables are lagged returns, as a function of the scale of the coarse-graining. If this behaves as a power law in the limit where the scale is small, it implies low-dimensional chaos. But it is very easy to be fooled when performing such calculations. It is critical to test against a carefully formulated null hypothesis.15 More careful statistical analysis by José Scheinkman and Blake LeBaron showed that the claims of low-dimensional chaos in price series were not well-justified.16 While nonlinearity is clearly present, there is no convincing evidence of lowdimensionality. The power-law scaling that people thought they saw was apparently just an artifact of the finite size of their data sets.
The power law for large price moves is a very different story. To detect a chaotic attractor based on its fractal dimension in state space requires a test of the distribution’s fine-grained, microscopic properties. Low-dimensional chaos is a very strong hypothesis, because it would imply deep structure and short-term predictability in prices. A power law in the tails of the returns, in contrast, is just a statement about the frequency of large events and is a much weaker hypothesis. This becomes clear in the context of extreme value theory. For simplicity, consider the positive tail r ® ¥. Under very general conditions, there are only three possible limiting behaviors, which we can classify based on the tail index a:
1. There is a maximum value for the variable. The distribution vanishes for values greater than this maximum, and a < 0.
2. The tails decay exponentially and 1/a = 0 (an example is a normal distribution).
3. There are fat tails that decay as a power law with a > 0.
Price returns must be in one of these three categories, and the data clearly points to choice 3 with a > 2.2,6–12 Surprisingly, this implies that the price-formation process cannot be fully understood in terms of central limit theorem arguments, even in a generalized form. Power-law tails do obey a sort of partial central limit theorem: For a random variable with tail exponent a, the sum of N variables will also have the same tail exponent a.17 This does not mean that the full distribution is stable, however, because the distribution’s central part, as well as the power law’s cutoff, will generally vary. The fact that the distribution’s shape changes with t makes it clear that the random process underlying prices must have nontrivial temporal structure, as I’ll discuss next. This complicates statistical analysis of prices, both for theoretical and practical purposes, and gives an important clue about the behavior of economic agents and the price-formation process. But unlike low-dimensional chaos, it does not imply that the direction of price movements is predictable. (Also see the “Powerlaw scaling” sidebar.18)"
Farmer (1999)
"...long tailedness relative to the normal for daily returns, with returns over longer periods approaching normality (Fama 1965)..."
"The literature on market returns includes a number of models. In addition to Brownian motion and the normal distribution, Mandelbrot (1963) put forward the symmetric stable distribution; Press (1967) introduced a compound events model combining normally distributed jumps at Poisson Jump times; and Praetz (1972) suggested the t distribution. More recently, Bookstaber and McDonald (1987) have proposed a generalized beta distribution."
Madan and Seneta (1990)
[daily stock returns] "Changing variance can also explain the high levels of kurtosis in return distributions. Variance changes are often related to the rate of information arrivals, level of trading activity, and corporate financial and operating decisions, which tend to affect the level of stock price. A natural way of modeling this phenomenon is to represent return distributions as mixtures of distributions, or as distributions with stochastic moments."
Akgiray (1989)
"Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian"
Andersen, et al (1999)
[tick by tick data of two Nasdaq technology stocks]
"Studies as early as, for example, Fama (1965), showed that daily returns are more long tailed than the normal density, with the distribution of returns approaching normality as the holding period is extended to one month."
"Mandelbrot (1963) introduced a class of stable processes to account for the deviations of returns from Brownian motion."
"Virtually all empirical studies establish a positive correlation between volatility—measured as absolute or squared changes–and volume (see Karpoff(1987), Gallant, Rossi and Tauchen (1992)). More revently, Jones, Kaul, and Lipson (1994) study daily prices of Nasdaq securities and conclude that it is the number of trades and not their size that generates volatility: “The average trade size has virtually no explanatory power when volatility is conditioned on the number of transactions.”"
"...Blume, Easley, and O’Hara (1994) observe that volume provides information on the quality of market information."
"It is shown that, to recover normality in asset returns, the number of trades is a better time change then the traditionally used trading volume."
Ane and Geman (2000)
Why the Fat Tails?
"The most common explanation of the fat tails is that information shows up in infrequent clumps, rather than in a smooth and continuous fashion. The market reaction to clumps of information results in the fat tails."
Peters [book]
In real market, the probability distribution function (pdf) of returns is known to have fat tails with exponent -4 on average [1].
Challet, Marsili and Zhang (2001)
"Basically, the argument about the distribution of price change runs like this. Assume first that the probability distribution of price changes on any one day is identical with that of changes on any other day and, further, assume that those changes are independent of one another. Now assume that probability distribution of the price change over any two days is of the same form as the distribution for either one of the days; that is, it is the same except for parameters. If we ask ourselves what probability distributions have this property, we find a class of distributions known only as the infinitely divisible distributions. The binomial, Poisson, and normal all fall in this class, along with others. Now if we further require that the distribution of price changes over any interval, however small or however large, be the same—that is, if we consider the limiting distribution as the number of subdivisions becomes infinites—the possible solutions are sharply reduced. The most widely known such solution is the Gaussian, but it is merely one of a family of distributions characterized in Fama’s notation by parameters α, β, γ, and δ. In the Mandelbrot hypothesis β = 0, δ, which is a location parameter, can easily be taken as zero and for most purposes γ can be assumed to be 1. Then α is the critical variable, which is 2 in the Gaussian case and less than 2 in Mandelbrot’s models. When α is less than 2 but greater than 1, the second moment of the distribution does not exist, but the mean does."
Cootner in Cootner (1964), pages 233-234
"A method like the detrended fluctuation analysis is recalled emphasizing its value in sorting out correlation ranges, thereby leading to predictability at short horizon. The (m,k)-Zipf method is presented for sorting out short-range correlations in the sign and amplitude of the fluctuations. A well-known financial analysis technique, the so-called moving average, is shown to raise questions to physicists about fractional Brownian motion properties. Among spectacular results, the possibility of crash predictions has been demonstrated through the log-periodicity of financial index oscillations."
Ausloos (2000)
"Finally, in contrast to the random walk literature, which focuses on the conditional distribution of security returns, another strand of the early nance literature has focused on the marginal distribution of returns, and specifically on the notion of “stability”, the preservation of the parametric form of the marginal distribution under addition. This is an especially important property for security returns, which are summed over various holding periods to yield cumulative investment returns. For example, if Pt denotes the end-of-month-t price of a security, then its monthly continuously compounded return xt is defined as log(Pt/Pt - 1), hence its annual return is log(Pt/Pt - 12) = xt + xt - 1 + … + xt - 11. The normal distribution is a member of the class of stable distributions, but the non-normal stable distributions possess a distinguishing feature not shared by the normal: they exhibit leptokurtosis or “fat tails”, which seems to accord well with higher frequency nancial data, e.g., daily and weekly stock returns. Indeed, the fact that the historical returns of most securities have many more outliers than predicted by the normal distribution has rekindled interest in this literature, which has recently become part of a much larger endeavor known as “risk management”.
Of course, stable distributions have played a prominent role in the early development of modern probability theory (see, for example, Levy (1937)), but their application to economic and nancial modeling is relatively recent. Mandelbrot (1960, 1963) pioneered such applications, using stable distributions to describe the cross-sectional distributions of personal income and of commodity prices. Fama (1965) and Samuelson (1967) developed the theory of portfolio selection for securities with stably distributed returns, and Fama and Roll (1971) estimated the parameters of the stable distribution using historical stock returns. Since then, many others have considered stable distributions in a variety of nancial applications|see McCulloch (1996) for an excellent and comprehensive survey."
Lo (2000)
"In recent years, however, several studies have found, what appears to be strong evidence against the stable model (Gopikrishnan et al., 1999; McCulloch, 1997). These studies have estimated the tail exponent directly from the tail observations and commonly have found ® that appears to be signi¯cantly greater than 2, well outside the stable domain."
"The fluctuations of the Standard & Poor 500 exhibit scaling behaviour, follow a power law distribution and have a fractal temporal signature suggesting that this is an example of a self-organized critical phenomenon."
Burbidge (2000)
"In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments."
Brachet, Taflin and Tcheou (1997)
"A satisfactory probability distribution for daily returns must have high kurtosis and be either exactly or approximately symmetric."
Taylor (2005)
[intraday returns]
"Intraday returns have a fat-tailed distribution, whose kurtosis increases as the frequency of price observations increases."
Taylor (2005)
[daily returns]
"First, the distribution of returns is approximately symmetric and has high kurtosis, fat tails and a peaked center compared with the normal distribution.
Taylor (2005), page 93
[daily returns]
"The distribution of returns is not normal."
Taylor (2005)
"It is very clear that the returns-generating process is not even approximately Gaussian. This is an old conclusion that may first have been established in Alexander (1961). It has since been shown for almost all series of daily and more frequent returns."
Taylor (2005) page 69
