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An overwhelming part of the research in financial markets is based on the assumption that financial markets are driven by a gaussian evolution. This assumption has been widely discusssed, and it has often been shown than it is false for equity, forex, and commodities markets. Stable distributions have been proposed as a better model for market evolutions. However, stable distributions are NOT used a lot in the industry and quantitative research teams, because of a lack of proper understanding and available software. The lack of analytical formulas for the probability density and cumulative distribution functions is also an issue. Moreover, there is a conceptual resistance to change. Traders rely a lot on volatility (second moment of the distribution of the returns), which is undefined for stable distributions. In option trading desks, the volatility smile concept is used in order to overcome the difficulty, and risk teams have also developped various models in order to take into account "fat tails" in their risk models. This page is a little introduction on the non-normality of returns on the equity market, and provide some software available for research purposes. A little study on the S&P500, since 1950. We use here all the returns of the S&P500 indice (american stock market) from 1950. The data has been retrieved from Yahoo Quotes. This gives a total number of observations around 14000. Here are the parameters of this samples: sigma = 0.0089 mu = 3.48e-004 kurtosis =27.7 skewness = -0.9080 We notice that the kurtosis is very high. The distribution is left skewed. The histogram with the normal corresponding distribution (in red) obviously shows that the gaussian model is not appropriate.  Some crashes should not have occurred if the returns where normally distributed.  ... but these extremes events are not the only cause of the failure of the fit between the returns and a gaussian model. In green, a plot of the histogram and the corresponding gaussian model (green), when all returns > 3% have been eliminated.  Instead, let's try to fit the data to a stable distribution, using the quantile method by J. McCulloch (See the links below). We obtain the following parameters. alpha = 1.5150 (<2 : fat-tails. alpha=2 corresponds to the gaussian case) beta = -0.0515 (skewness at the left) c = 0.0046 (scale factor) delta = 0.0003 We use a bootstrap method in order to check than these parameter estimates are robust. 
A comparison of empirical versus normal and stable histogramms. 
A zoom shows that the fit with stable distribution seems far better. 
Here is a semi-log plot of the distribution density function. 
The three different cumulative functions, we notice that the fit of the empirical returns of the S&P with a stable distribution is very good. 
We notice that the fit is not perfect in the tails of the cumulative distributions. The normal model underestimates rare returns, and the stable model we have found overestimates them. This is partly due to the method we used in the model (quantile method). See "Links to other people and ressources" to know more on other methods. 
Here is a loglog plot of the cumulative distribution functions of a gaussian, empirical, stable distribution estimated with quantile method, and stable distribution estimated with maximum likelihood method. 
Zoom in the tails: 
As a short and temporary conclusion, stable distributions are a better model for equity returns. There are different numerical methods that can be used to estimates the parameters of the stable distribution. The goodness of estimation depends of two things : the model that is used to estimate the parameters, and, of course, the number of observations. The choice of the estimation model is a compromise between the quality of the estimates, and the speed of the algorithm. Even if stable distributions seem to better model asset returns, it seems that this model overestimates occurences of extreme events. Software used in this example: You will find below the modified versions that have been used for this work. stabrnd is used to generate numbers following a stable distribution, stabfit estimates the parameters of a stable distribution given a data sample, stabmlefit estimates the parameters (very slowly) with the maximum-likelihood method and dstable calculates the density of a given stable distribution. Download: - for scilab: stabfit.sci stabrnd.sci dstable.sci ("dstable" often fails because of the "optim" function in SCILAB which does not always converge) - for matlab: stabfit.m stabrnd.m stabmlefit.m dstable.m "stabfit" is a modified version of STAB by J. McCulloch. The website of J. Huston McCulloch contains several useful research papers and tools, including a option pricing software based on stable distributions. "dstable" and "stabmlefit" are derived works from the fBasics package of R CRAN, written by Diethelm Wurtz. fBasics package included in the R project (CRAN) contains Maximum Likelihood and Quantile methods for estimating parameters. Links to other people and ressources: John Nolan's homepage on stable distributions contains interesting papers and full packages to work with stable distributions in several languages : S-Plus, R, matlab and Mathematica. Mr Nolan has developed a software : STABLE.EXE, which let you estimate stable parameters by a Maximum Likelihood method. The model fits better the datas, especially for the tails of the distribution. For the S&P data, it gives : alpha = 1.67, beta = -0.084, delta = 0.0048 , gamma = 0.00048 Robert H. Rimmer's stable calculation homepage has several useful functions (pdf, cdf, stable data generation, and several fitting methods) for Mathematica that you can download freely, along with a Dow Jones Industrial study. XploRe, a statistical tool (an academic version (free) is available) containing functions for working with stable distributions (cdfstab, pdfstab, rndstab, and three different estimation methods: Kogon-Williams, Koutrouvelis, Moment, and Quantile (McCullogh) methods). You can have a look at downloadable e-books, especially the Handbook of Computational Statistics, which contains a section on stable distributions by Rafael Weron which compares the different estimate methods, also published on the page of publications of Hugo Steinhaus Center, that contains the research report "Performance of the estimators of stable law parameters", Research Report HSC/95/1. Benoit Mandelbrot, a continuous advocate of stable distributions and fractal models for financial markets. An early estimation giving an alpha of 1.7 on cotton prices (in french) was presented in 1962 at l'Académie des Sciences (Paris, France) Author: Jeremie Cosmao - jeremie.cosmao@hfri.org |