Finance > Stylized Facts


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), xk 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)

  • MANTEGNA, R.N. and H.E. STANLEY, 1999. Introduction to Econophysics: Correlations and Complexity in Finance. [Cited by 730] (106.16/year)
  • Possible distributions (in Mantegna and Stanley (2000)):

    "The degree of leptokurtosis is much larger for high-frequency data (Fig. 8.2)."
    Mantegna and Stanley (2000)

  • NELSON, D.B., 1991. Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica. [Cited by 1181] (80.05/year)
  • Abstract: "This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987."

    "...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)

  • SAMORODNITSKY, G. and G. SAMORODNITSKYM, 1994. Stable Non-Gaussian Random Processes. [Cited by 701] (59.02/year)
  • ERAKER, B., M. JOHANNES and N. POLSON, 2003. The Impact of Jumps in Volatility and Returns, The Journal of Finance. [Cited by 135] (49.02/year)
  • Abstract: "This paper examines continuous-time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood-based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998. Using formal and informal diagnostics, we find strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing."
  • BOLLERSLEV, T., R.F. ENGLE and D.B. NELSON, 1993. Arch Models. [Cited by 595] (46.21/year)
  • "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)

  • DACOROGNA, Michel M., et al., 2001. An introduction to high-frequency finance. [Cited by 222] (45.53/year)
  • [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)

  • ANDERSEN, T.G., et al., 2000. The Distribution of Stock Return Volatility. [Cited by 257] (44.66/year)
  • Abstract: "We exploit direct model-free measures of daily equity return volatility and correlation obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones Industrial Average over a five-year period to confirm, solidify and extend existing characterizations of stock return volatility and correlation. We find that the unconditional distributions of the variances and covariances for all thirty stocks are leptokurtic and highly skewed to the right, while the logarithmic standard deviations and correlations all appear approximately Gaussian. Moreover, the distributions of the returns scaled by the realized standard deviations are also Gaussian. Consistent with our documentation of remarkably precise scaling laws under temporal aggregation, the realized logarithmic standard deviations and correlations all show strong temporal dependence and appear to be well described by long-memory processes. Positive returns have less impact on future variances and correlations than negative returns of the same absolute magnitude, although the economic importance of this asymmetry is minor. Finally, there is strong evidence that equity volatilities and correlations move together, possibly reducing the benefits to portfolio diversification when the market is most volatile. Our findings are broadly consistent with a latent volatility fact or structure, and they set the stage for improved high-dimensional volatility modeling and out-of-sample forecasting, which in turn hold promise for the development of better decision making in practical situations of risk management, portfolio allocation, and asset pricing."

    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)

  • CARR, P., et al., 2002. The Fine Structure of Asset Returns: An Empirical Investigation, The Journal of Business. [Cited by 155] (41.29/year)
  • Abstract: "We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. 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. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation."

    "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)

  • ANDERSEN, T.G., et al., 2001. The Distribution of Realized Exchange Rate Volatility, Journal of the American Statistical Association, 96, 42--55. [Cited by 195] (41.02/year)
  • Abstract: "Using high-frequency data on Deutschemark and Yen returns against the dollar, we construct model-free estimates of daily exchange rate volatility and correlation, covering an entire decade. Our estimates, termed realized volatilities and correlations, are not only model-free, but also approximately free of measurement error under general conditions, which we discuss in detail. Hence, for practical purposes, we may treat the exchange rate volatilities and correlations as observed rather than latent. We do so, and we characterize their joint distribution, both unconditionally and conditionally. Noteworthy results include a simple normality-inducing volatility transformation, high contemporaneous correlation across volatilities, high correlation between correlation and volatilities, pronounced and persistent dynamics in volatilities and correlations, evidence of long-memory dynamics in volatilities and correlations, and remarkably precise scaling laws under temporal aggregation."

    "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.

  • BROWN, S.J. and J.B. WARNER, 1985. Using daily stock returns: The case of event studies. Journal of Financial Economics, Volume 14, Issue 1 , March 1985, Pages 3-31. [Cited by 823] (39.65/year)
  • Abstract: "This paper examines properties of daily stock returns and how the particular characteristics of these data affect event study methodologies. Daily data generally present few difficulties for event studies. Standard procedures are typically well-specified even when special daily data characteristics are ignored. However, recognition of autocorrelation in daily excess returns and changes in their variance conditional on an event can sometimes be advantageous. In addition, tests ignoring cross-sectional dependence can be well-specified and have higher power than tests which account for potential dependence."
  • GABAIX, X., et al., 2003. A theory of power-law distributions in financial market fluctuations.. Nature. [Cited by 113] (39.28/year)
  • Abstract: "Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades. 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 countries--suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades."

    "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)

  • MITZENMACHER, M., 2003. A brief history of generative models for power law and lognormal distributions. Internet Mathematics. [Cited by 108] (37.54/year)
  • GOPIKRISHNAN, P., et al., 1999. Scaling of the distribution of fluctuations of financial market indices. Physical Review E 60(5): 5305{5316. [Cited by 199] (28.94/year)
  • Abstract: "We study the distribution of fluctuations of the S&P 500 index over a time scale ?t by analyzing three distinct databases. Database (i) contains approximately 1 200 000 records, sampled at 1-min intervals, for the 13-year period 1984–1996, database (ii) contains 8686 daily records for the 35-year period 1962–1996, and database (iii) contains 852 monthly records for the 71-year period 1926–1996. We compute the probability distributions of returns over a time scale ?t, where ?t varies approximately over a factor of 104—from 1 min up to more than one month. We find that the distributions for ?t?~ 4 d (1560 min) are consistent with a power-law asymptotic behavior, characterized by an exponent a˜3, well outside the stable Lévy regime 0 "We find that the distributions for Dt< 4 d ~1560 min! are consistent with a power-law asymptotic behavior, characterized by an exponent a'3, well outside the stable Le´vy regime 0,a,2."
    "One possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility. For time scales longer than (Dt)3'4 d, our results are consistent with a slow convergence to Gaussian behavior."
    Gopikrishnan (1999)

  • MANDELBROT, B., 1963. The Variation of Certain Speculative Prices. The Journal of Business. [Cited by 1120] (26.12/year)
  • BROCK, W., J. LAKONISHOK and B. LEBARON, 1992. Simple Technical Trading Rules and the Stochastic Properties of Stock Returns. The Journal of Finance. [Cited by 336] (24.43/year)
  • Abstract: "This paper tests two of the simplest and most popular trading rules--moving average and trading range break--by utilizing the Dow Jones Index from 1897 to 1986. Standard statistical analysis is extended through the use of bootstrap techniques. Overall, their results provide strong support for the technical strategies. The returns obtained from these strategies are not consistent with four popular null models: the random walk, the AR(1), the GARCH-M, and the Exponential GARCH. Buy signals consistently generate higher returns than sell signals, and further, the returns following buy signals are less volatile than returns following sell signals. Moreover, returns following sell signals are negative, which is not easily explained by any of the currently existing equilibrium models."
  • LONGIN, F. and B. SOLNIK, 1995. Is the correlation in international equity returns constant: 1960-1990?. JOURNAL OF INTERNATIONAL MONEY AND FINANCE. [Cited by 258] (23.99/year)
  • Abstract: "We study the correlation of monthly excess returns for seven major countries over the period 1960-1990. We find that the international covariance and correlation matrices are unstable over time. A multivariate GARCH(1,1) modelling with constant conditional correlation helps capture some of the evolution in the conditional covariance structure. We include information variables in the mean and variance equations. The volatility of markets changed somewhat over the period 1960-1990 and the proposed GARCH modelling allows to capture this evolution in variances. However tests of specific deviations lead to a rejection of the hypothesis of a constant conditional correlation. An explicit modelling of the conditional correlation indicates an increase of the international correlation between markets over the past thirty years. We also find that the correlation rises in periods when the conditional volatility of markets is large. There is some preliminary evidence that economic variables such as the dividend yield and interest rates contain information about future volatility and correlation that is not contained in past returns alone. However, further theoretical work is required to provide a satisfactory model. The effects are not of great magnitude but are often statistically significant."
  • PLEROU, V., et al., 1999. Scaling of the distribution of price fluctuations of individual companies. Physical Review E. [Cited by 161] (23.41/year)
  • "[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)

  • CONT, R., 2001. Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance, Volume 1, Number 2 / February 01, 2001, pages 223 - 236. [Cited by 107] (22.51/year)
  • Abstract: "We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studied of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case."
  • ANDERSEN, Torben G., 1996. Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility, The Journal of Finance, Vol. 51, No. 1 (Mar., 1996) , pp. 169-204. [Cited by 206] (21.12/year)
  • Abstract: "This paper develops an empirical return volatility-trading volume model from a microstructure framework in which informational asymmetries and liquidity needs motivate trade in response to information arrivals. The resulting system modifies the so-called 'mixture of distribution hypothesis' (MDH). The dynamic features are governed by the information flow, modeled as a stochastic volatility process, and generalize standard autoregressive conditional heteroskedasticity specifications. Specification tests support the modified MDH representation and show that it vastly outperforms the standard MDH. The findings suggest that the model may be useful for analysis of the economic factors behind the observed volatility clustering in returns."

    "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)

  • PLEROU, V., et al., 1999. Scaling of the distribution of price fluctuations of individual companies. Physical Review E. [Cited by 142] (21.02/year)
  • Abstract: "We present a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major U.S. stock markets: (a) the New York Stock Exchange, (b) the American Stock Exchange, and (c) the National Association of Securities Dealers Automated Quotation stock market. Specifically, we consider (i) the trades and quotes database, for which we analyze 40 million records for 1000 U.S. companies for the 2-yr period 1994-95; and (ii) the Center for Research and Security Prices database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-yr period 1962-96. We study the probability distribution of returns over varying time scales Delta t, where Delta t varies by a factor of approximately 10(5), from 5 min up to approximately 4 yr. 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 < proportional to < 4, well outside the stable Levy regime 0 < alpha < 2. For time scales Delta t >> (Delta t)(x) approximately equal to 16 days, we observe results consistent with a slow convergence to Gaussian behavior. We also analyze the role of cross correlations between the returns of different companies and relate these correlations to the distribution of returns for market indices."
  • GUILLAUME, D.M.D., et al., 1997. From the bird's eye to the microscope: A survey of new stylized facts of the intra-daily foreign …. Finance and Stochastics. [Cited by 164] (18.48/year)
  • [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)

  • GOPIKRISHNAN, P.F., et al., 1998. Inverse cubic law for the distribution of stock price variations. The European Physical Journal B-Condensed Matter. [Cited by 143] (18.15/year)
  • Abstract: "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 , well outside the Lévy regime ."

    "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)

  • MANTEGNA, R.N. and H.E. STANLEY, 1994. Stochastic process with ultraslow convergence to a Gaussian: The truncated Levy flight. Physical Review Letters, 73, 2946-2949. [Cited by 211] (17.95/year)
  • Abstract: "We introduce a class of stochastic process, the truncated Lévy flight (TLF), in which the arbitrarily large steps of a Lévy flight are eliminated. We find that the convergence of the sum of n independent TLFs to a Gaussian process can require a remarkably large value of n—typically n˜104 in contrast to n˜10 for common distributions. We find a well-defined crossover between a Lévy and a Gaussian regime, and that the crossover carries information about the relevant parameters of the underlying stochastic process."
  • BAK, P., M. PACZUSKI and M. SHUBIK, 1996. Price Variations in a Stock Market With Many Agents. Arxiv preprint cond-mat/9609144. [Cited by 146] (14.78/year)
  • Abstract: "Large variations in stock prices happen with sufficient frequency to raise doubts about existing models, which all fail to account for non-Gaussian statistics. We construct simple models of a stock market, and argue that the large variations may be due to a crowd effect, where agents imitate each other’s behavior. The variations over different time scales can be related to each other in a systematic way, similar to the Levy stable distribution proposed by Mandelbrot to describe real market indices. In the simplest, least realistic case, exact results for the statistics of the variations are derived by mapping onto a model of diffusing and annihilating particles, which has been solved by quantum field theory methods. When the agents imitate each other and respond to recent market volatility, different scaling behavior is obtained. In this case the statistics of price variations is consistent with empirical observations. The interplay between “rational” traders whose behavior is derived from fundamental analysis of the stock, including dividends, and “noise traders”, whose behavior is governed solely by studying the market dynamics, is investigated. When the relative number of rational traders is small, “bubbles” often occur, where the market price moves outside the range justified by fundamental market analysis. When the number of rational traders is larger, the market price is generally locked within the price range they define."

    "...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)

  • LAMOUREUX, C.G. and W.D. LASTRAPES, 1990. Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects. The Journal of Finance. [Cited by 228] (14.47/year)
  • Abstract: "This paper provides empirical support for the notion that autoregressive conditional heteroskedasticity in daily stock return data reflects time dependence in the process generating information flow to the market. Daily trading volume, used as a proxy for information arrival time, is shown to have significant explanatory power regarding the variance of daily returns, which is an implication of the assumption that daily returns are subordinated to intraday equilibrium returns. Furthermore, autoregressive conditional heteroskedasticity effects tend to disappear when volume is included in the variance equation."
  • JOHNSON, N.F., et al., 2003. Financial Market Complexity: What Physics Can Tell Us about Market Behaviour. [Cited by 40] (13.90/year)
  • fat-tailed PDF of price changes Johnson, Jefferies and Hui (2003), page 69

  • SORNETTE, D., A. JOHANSEN and J.P. BOUCHAUD, 1995. Stock market crashes, Precursors and Replicas. Arxiv preprint cond-mat/9510036. [Cited by 147] (13.52/year)
  • LUX, T., 1996. The stable Paretian hypothesis and the frequency of large returns: an examination of major German …. Applied Financial Economics. [Cited by 132] (13.36/year)
  • Abstract: "A statistical analysis is provided of daily returns for 30 German stocks forming the DAX share index as well as the DAX itself during the period 1988-1994. Estimating the parameters of the stable laws and performing standard tests of fit, some evidence in favour of the stable Paretian hypothesis is found. However, application of a more recently developed semiparametric technique for analysis of the limiting behaviour in the tails of a distribution (Hill's tail index estimator) suggests that the empirical tail regions are thinner than expected under a stable distribution. Since the reliability of tail index estimation rests on the appropriateness of the chosen tail regions, it is also examined whether the tails indeed follow approximately the expected limiting distributions of extremes. It turns out that convergence to the limiting extreme value distributions cannot be rejected in the vast majority of cases for tails covering the most extreme 15% of observations or less. Furthermore, strong similarity in the extremal behaviour of the 30 series is found and the hypothesis of identical limit laws governing their extreme value distributions is not rejected."

    [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)

  • LONGIN, F.M., 1996. The Asymptotic Distribution of Extreme Stock Market Returns. The Journal of Business. [Cited by 126] (12.76/year)
  • [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)

  • FARMER, J.D., 1999. Physicists attempt to scale the ivory towers of finance. Computing in Science and Engineering. [Cited by 84] (12.22/year)
  • "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)

  • MADAN, D.B. and E. SENETA, 1990. The Variance Gamma (VG) Model for Share Market Returns. The Journal of Business. [Cited by 194] (12.22/year)
  • "...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)

  • CALVET, L. and A. FISHER, 2002. Multifractality in Asset Returns: Theory and Evidence, Review of Economics and Statistics. [Cited by 45] (11.99/year)
  • Abstract: "This paper investigates the multifractal model of asset returns (MMAR), a class of continuous-time processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the MMAR compounds a Brownian motion with a multifractal time-deformation. Prices follow a semi-martingale, which precludes arbitrage in a standard two-asset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than 2. The local variability of a sample path is highly heterogeneous and is usefully characterized by the local Hölder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The MMAR predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche mark/U.S. dollar exchange rates and several equity series. We develop an estimation procedure and infer a parsimonious generating mechanism for the exchange rate. In Monte Carlo simulations, the estimated multifractal process replicates the scaling properties of the data and compares favorably with some alternative specifications."
  • MADAN, D.B. and E. SENETA, 1990. The Variance Gamma (VG) Model for Share Market Returns, The Journal of Business, Volume (Year): 63 (1990), Issue (Month): 4 (October), Pages: 511-24. [Cited by 184] (11.68/year)
  • Abstract: "A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance that is distributed as a gamma variate. Its advantages include long tailedness, continuous-time specification, finite moments of all orders, elliptical multivariate unit period distributions, and good empirical fit. The process is pure jump, approximable by a compound Poisson process with high jump frequency and low jump magnitudes. Applications to option pricing show differential effects for options on the money, compared to in or out of the money."
  • AKGIRAY, V., 1989. Conditional Heteroscedasticity in Time Series of Stock Returns: Evidence and Forecasts. The Journal of Business. [Cited by 196] (11.61/year)
  • [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)

  • LONGIN, F.M., 1996. The Asymptotic Distribution of Extreme Stock Market Returns, The Journal of Business, Volume (Year): 69 (1996), Issue (Month): 3 (July), Pages: 383-408. [Cited by 110] (11.28/year)
  • Abstract: "This article presents a study of extreme stock market price movements. According to extreme value theory, the form of the distribution of extreme returns is precisely known and independent of the process generating returns. Using data for an index of the most traded stocks on the New York Stock Exchange for the period 1885-1990, the author shows empirically that the extreme returns obey a Frechet distribution."
  • ANDERSEN, T.G., et al., 2000. Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian. [Cited by 66] (11.23/year)
  • "Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian"
    Andersen, et al (1999)

  • AKGIRAY, V., 1989. Conditional Heteroscedasticity in Time Series of Stock Returns: Evidence and Forecasts. The Journal of Business, Volume (Year): 62 (1989), Issue (Month): 1 (January), Pages: 55-80. [Cited by 183] (10.92/year)
  • Abstract: "This article presents new evidence about the time-series behavior of stock prices. Daily return series exhibit significant levels of second-order dependence, and they cannot be modeled as linear white-noise processes. A reasonable return-generating process is empirically shown to be a first-order autoregressive process with conditionally heteroskedastic innovations. In particular, generalized autoregressive conditional heteroskedastic GARCH (1, 1) processes fit to data very satisfactorily. Various out-of-sample forecasts of monthly return variances are generated and compared statistically. Forecasts based on the GARCH model are found to be superior."
  • ANDERSEN, T.G., et al., 2000. Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian. [Cited by 62] (10.77/year)
  • "Introduction and Basic Ideas The prescriptions of modern financial risk management hinge critically on the associated characterization of the distribution of future returns (cf., Diebold, Gunther and Tay, 1998, and Diebold, Hahn and Tay, 1999). Because volatility persistence renders high-frequency returns temporally dependent (e.g., Bollerslev, Chou and Kroner, 1992), it is the conditional return distribution, and not the unconditional distribution, that is of relevance for risk management. This is especially true in high-frequency situations, such as monitoring and managing the risk associated with the day-to-day operations of a trading desk, where volatility clustering is omnipresent. Exchange rate returns are well-known to be unconditionally symmetric but highly leptokurtic. Standardized daily or weekly returns from ARCH and related stochastic volatility models also appear symmetric but leptokurtic; that is, the distributions are not only unconditionally, bu..."
  • ANÉ, T. and H. GEMAN, 2000. Order Flow, Transaction Clock, and Normality of Asset Returns. THE JOURNAL OF FINANCE. [Cited by 62] (10.55/year)
  • [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)

  • ANE, T. and H. GEMAN, 2000. Order Flow, Transaction Clock, and Normality of Asset Returns, The Journal of Finance, Volume 55, Number 5, October 2000, pp. 2259-2284. [Cited by 58] (10.08/year)
  • Abstract: "The goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change. Clark (1973) addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator. We extend Clark's results and find the following: (i) stochastic time changes are mathematically much less constraining than subordinators; (ii) the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns; (iii) this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions."
  • PETERS, E.E., 1996. Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility. [Cited by 54] (5.47/year)
  • 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]

  • IORI, G., 2002. A microsimulation of traders activity in the stock market: the role of heterogeneity, agents' …. Journal of Economic Behavior and Organization. [Cited by 36] (9.29/year)
  • TAKAYASU, H., A.H. SATO and M. TAKAYASU, 1997. Stable Infinite Variance Fluctuations in Randomly Amplified Langevin Systems. Physical Review Letters. [Cited by 82] (9.24/year)
  • CHATTERJEE, A., B.K. CHAKRABARTI and S.S. MANNA, 2003. Pareto Law in a Kinetic Model of Market with Random Saving Propensity. Arxiv preprint cond-mat/0301289. [Cited by 26] (9.04/year)
  • WEISBUCH, G., A. KIRMAN and D. HERREINER, 2000. Market Organisation and Trading Relationships. The Economic Journal. [Cited by 49] (8.34/year)
  • JOHANSEN, A.N. and D.N. SORNETTE, 1998. Stock market crashes are outliers. The European Physical Journal B-Condensed Matter. [Cited by 58] (7.36/year)
  • CHALLET, D., M. MARSILI and Y.C. ZHANG, 2001. Stylized facts of financial markets and market crashes in Minority Games. Arxiv preprint cond-mat/0101326. [Cited by 33] (6.77/year)
  • 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)

  • STAUFFER, D. and D. SORNETTE, 1999. Self-Organized Percolation Model for Stock Market Fluctuations. Arxiv preprint cond-mat/9906434. [Cited by 47] (6.83/year)
  • CONT, R., M. POTTERS and J.P. BOUCHAUD, 1997. Scaling in stock market data: stable laws and beyond. Scale invariance and beyond. [Cited by 56] (6.31/year)
  • CHAMBERS, J.M., C.L. MALLOWS and B.W. STUCK, 1976. A Method for Simulating Stable Random Variables. Journal of the American Statistical Association. [Cited by 182] (6.09/year)
  • BIHAM, O., et al., 1998. Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete …. Physical Review E. [Cited by 47] (5.97/year)
  • SOLOMON, S. and P. RICHMOND, 2001. Power Laws of Wealth, Market Order Volumes and Market Returns. Arxiv preprint cond-mat/0102423. [Cited by 29] (5.95/year)
  • MCCULLOCH, J.H., 1997. Measuring tail thickness to estimate the stable index alpha: A critique. Journal of Business and Economic Statistics 15: 74{81. [Cited by 51] (5.75/year)
  • Abstract: "A generalized Pareto or simple Pareto tail-index estimate above 2 has frequently been cited as evidence against infinite-variance stable distributions. It is demonstrated that this inference is invalid; tail index estimates greater than 2 are to be expected for stable distributions with a as low as 1.65. The nonregular distribution of the likelihood ratio statistic for a null of normality and an alternative of symmetric stability is tabulated by Monte Carlo methods and appropriately adjusted for sampling error in repeated tests. Real stock returns yield a stable a of 1.845 and reject iid normality at the .996 level."
  • WERON, R., 2001. Levy-stable distributions revisited: tail index> 2 does not exclude the Levy-stable regime. Arxiv preprint cond-mat/0103256. [Cited by 28] (5.74/year)
  • LILLO, F. and R.N. MANTEGNA, 2003. Power-law relaxation in a complex system: Omori law after a financial market crash. Physical Review E. [Cited by 15] (5.21/year)
  • MALCAI, O., O. BIHAM and S. SOLOMON, 1999. Power-law distributions and Lévy-stable intermittent fluctuations in stochastic systems of many …. Physical Review E. [Cited by 35] (5.09/year)
  • COOTNER…, P.H., 1967. The Random Character of Stock Market Prices. MIT Press. [Cited by 180] (4.63/year)
  • "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

  • AUSLOOS, M., 2000. Statistical physics in foreign exchange currency and stock markets. Physica A. [Cited by 23] (3.91/year)
  • "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)

  • PASQUINI, M. and M. SERVA, 1998. Multiscale behaviour of volatility autocorrelations in a financial market. Arxiv preprint cond-mat/9810232. [Cited by 27] (3.43/year)
  • WANG, B.H. and P.M. HUI, 2001. The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market. Eur. Phys. J. B. [Cited by 14] (2.87/year)
  • AKGIRAY, V. and G.G. BOOTH, 1988. The Stable-Law Model of Stock Returns. Journal of Business & Economic Statistics. [Cited by 51] (2.85/year)
  • FAMA, E.F., 1963. Mandelbrot and the Stable Paretian Hypothesis. The Journal of Business. [Cited by 120] (2.80/year)
  • HALL, J.A., W. BRORSEN and S.H. IRWIN, 1989. The Distribution of Futures Prices: A Test of the Stable Paretian and Mixture of Normals Hypotheses. The Journal of Financial and Quantitative Analysis. [Cited by 40] (2.37/year)
  • MANDELBROT, B. and H.M. TAYLOR, 1967. On the Distribution of Stock Price Differences. Operations Research. [Cited by 88] (2.26/year)
  • DUSAK, K., 1973. Futures Trading and Investor Returns: An Investigation of Commodity Market Risk Premiums. The Journal of Political Economy. [Cited by 65] (1.98/year)
  • MCFARLAND, J.W., R.R. PETTIT and S.K. SUNG, 1982. The Distribution of Foreign Exchange Price Changes: Trading Day Effects and Risk Measurement. The Journal of Finance. [Cited by 41] (1.72/year)
  • MANDELBROT, B., 1967. The Variation of Some Other Speculative Prices. The Journal of Business. [Cited by 61] (1.57/year)
  • CONT, R., 1999. Statistical properties of financial time series. Mathematical Finance: Theory and Practice. Lecture Series in …. [Cited by 9] (1.31/year)
  • LEVY, M. and S. SOLOMON, 1996. Dynamical Explanation for the Emergence of Power Law in a Stock Market Model. International Journal of Modern Physics. [Cited by 12] (1.21/year)
  • LO, A.W., 2000. Finance: A Selective Survey.. Journal of the American Statistical Association. [Cited by 7] (1.19/year)
  • "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)

  • HSU, D.A., R.B. MILLER and D.W. WICHERN, 1974. On the Stable Paretian Behavior of Stock-Market Prices. Journal of the American Statistical Association. [Cited by 37] (1.16/year)
  • SOLOMON, S., 1998. … Generically to Truncated Pareto Power Wealth Distribution, Truncated Levy Distribution of Market …. Arxiv preprint cond-mat/9803367. [Cited by 9] (1.14/year)
  • OFFICER, R.R., 1972. The Distribution of Stock Returns. Journal of the American Statistical Association, Vol. 67, No. 340. (Dec., 1972), pp. 807-812. [Cited by 36] (1.06/year)
    Abstract: "A detailed examination is made of the distribution of stock returns following reports that the distribution is best described by the symmetric stable class of distributions. The distributions are shown to be "fat-tailed" relative to the normal distribution but a number of properties inconsistent with the stable hypothesis are noted. In particular, the standard deviation appears to be a well behaved measure of scale."

  • BORAK, Szymon, Wolfgang HARDLE and Rafal WERON, 1 Stable Distributions. [not cited] (?/year)
  • "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."

  • BURBIDGE, R., MSci Project. [not cited] (?/year)
  • "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)

  • TAFLIN, M.E.B.E. and J.M. TCHEOU, Scaling transformation and probability distributions for financial time series. [not cited] (?/year)
  • "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)

  • TAYLOR, S.J., 2005. Asset price dynamics, volatility, and prediction. Princeton University Press. [not cited] (0/year)
  • "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