Stable distributions

In spite of wide-spread awareness that most risk factor distributions are heavy-tailed, to date, risk management systems have essentially relied either on historical, or on univariate and multi-variate Normal (or Gaussian) distributions for Monte Carlo scenario generation. Unfortunately, historical scenarios only capture conditions actually observed in the past, and in effect use empirical probabilities that are zero outside the range of the observed data, a clearly undesirable feature. On the other hand Gaussian Monte Carlo scenarios have probability densities that converge to zero too quickly (exponentially fast) to accurately model real-world risk factor distributions that generate extreme losses. When such large returns occur separately from the bulk of the data they are often called outliers.

Figure 1 shows quantile-quantile (qq)-plots of daily returns versus the best-fit Normal distribution of nine randomly selected Micro-cap stocks for the two-year period 2000-2001. If the returns were Normally distributed, the quantile points in the qq-plots would all fall close to a straight line. Instead they all deviate significantly from a straight line (particularly in the tails), reflecting a higher probability of occurrence of extreme values than predicted by the Normal distribution, and showing several outliers.

rns 0.2 tur et

-3 -1 12 3 Quantiles of Standard Normal ns 0.2 tur

-3 -1 12 3 Quantiles of Standard Normal s rn tur

-3 -1 12 3 Quantiles of Standard Normal

-3 -1 12 3 Quantiles of Standard Normal ns 0.2 tur

-3 -1 12 3 Quantiles of Standard Normal s rn tur

-3 -1 12 3 Quantiles of Standard Normal et 0.0

0.10

-3 -1 12 3 Quantiles of Standard Normal

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-3 -1 12 3 Quantiles of Standard Normal

-3 -1 12 3 Quantiles of Standard Normal

-3 -1 12 3 Quantiles of Standard Normal

Quantiles of Standard Normal

Quantiles of Standard Normal

Quantiles of Standard Normal

Quantiles of Standard Normal

0.10

Quantiles of Standard Normal

0.10

Quantiles of Standard Normal

Quantiles of Standard Normal

Quantiles of Standard Normal

Figure 1

Such behavior occurs in many asset and risk factor classes, including well-known indices such as the S&P 500, and corporate bond prices. The latter are well known to have quite non-Gaussian distributions that have substantial negative skews to reflect down-grading and default events. For such returns, non-Normal distribution models are required to accurately model the tail behavior and compute probabilities of extreme returns.

Various non-Normal distributions have been proposed for modeling extreme events, including:

• Mixtures of two or more Normal distributions.

• t-distributions, hyperbolic distributions, and other scale mixtures of normal distributions.

• Gamma distributions.

• Extreme Value distributions.

• Stable non-Gaussian distributions (also known as Levy-Stable and Pareto-Stable distributions).

Among the above, only Stable distributions have attractive enough mathematical properties to be a viable alternative to Normal distributions in trading, optimization and risk management systems. A major drawback of all alternative models is their lack of stability. Benoit Mandelbrot (1963) demonstrated that the stability property is highly desirable for asset returns. These advantages are particularly evident in the context of portfolio analysis and risk management.

An attractive feature of Stable models, not shared by other distribution models, is that they allow generation of Gaussian-based financial theories and, thus allow construction of a coherent and general framework for financial modeling. These generalizations are possible only because of specific probabilistic properties that are unique to (Gaussian and non-Gaussian) Stable laws, namely; the Stability property, the Central Limit Theorem, and the Invariance Principle for Stable processes.

Benoit Mandelbrot (1963), then Eugene Fama (1963), provided seminal evidence that Stable distributions are good models for capturing the heavy-tailed (leptokurtic) returns of securities. Many follow-on studies came to the same conclusion, and the overall Stable distributions theory for finance is provided in the definitive work of Rachev and Mittnik (2000).

But in spite the convincing evidence, Stable distributions have seen virtually no use in capital markets. There have been several barriers to the application of stable models, both conceptual and technical:

• Except for three special cases, described below, Stable distributions have no closed form expressions for their probability densities.

• Except for Normal distributions, which are a limiting case of Stable distributions (with a = 2 and ft = 0), Stable distributions have infinite variance and only a mean value for a > 1.

• Without a general expression for stable probability densities, one cannot directly implement maximum likelihood methods for fitting these densities, even in the case of a single (univariate) set of returns.

The availability of practical techniques for fitting univariate and multivariate stable distributions to asset and risk factor returns has been the barrier to the progress of Stable distributions in finance. Only the recent development of advanced numerical methods has removed this obstacle. These patented methods form the foundation of the Cognity™ market & credit risk management and portfolio optimization solution (see further comments in the concluding section).

Univariate Stable distributions A Stable distribution for a random risk factor X is defined by its characteristic function:

where:

is any probability density function in a location-scale family for X:

is any probability density function in a location-scale family for X:

A stable distribution is therefore determined by the four key parameters:

A stable distribution is therefore determined by the four key parameters:

1. a determines density's kurtosis with 0 < a < 2 (e.g. tail weight).

2. 3 determines density's skewness with —1 < 3 < 1.

3. a is a scale parameter (in the Gaussian case, a = 2 and 2a2 is the variance).

4. f is a location parameter (f is the mean if 1 < a < 2).

Stable distributions for risk factors allow for skewed distributions when p = 0 and fat tails relative to the Gaussian distribution when a < 2. Figure 2 shows the effect of a on tail thickness of the density as well as peakedness at the origin relative to the Normal distribution (collectively the "kurtosis" of the density), for the case of p = 0, f = 0, and a = 1. As the values of a decrease the distribution exhibits fatter tails and more peakedness at the origin.

Symmetric PDFs

Symmetric PDFs

Figure 2

a=

0.5

a=

1

a=

1.5

a=

2

The case of a = 2 and p = 0 and with the reparameterization in scale, a = \fla, yields the Gaussian distribution, whose density is given by:

The case a = 1 and p = 0 yields the Cauchy distribution with much fatter tails than the Gaussian, and is given by:

Figure 3 illustrates the influence of p on the skewness of the density for a = 1.5, f = 0 and a = 1. Increasing (decreasing) values of p result in skewness to the right (left).

Fitting Stable and Normal distributions: DJIA example. Aside from the Gaussian, Cauchy, and one other special case of stable distribution for a positive random variable with a = 0.5, there is no closed form expression for the probability density of a Stable random variable.

Thus one is not able to directly estimate the parameters of a Stable distribution by the method of maximum likelihood. To estimate the four parameters of the stable laws, the Cognity™

Figure 3

solution uses a special patent-pending version of the FFT (Fast Fourier Transform) approach to numerically calculate the densities with high accuracy, and then applies MLE (Maximum Likelihood Estimation) to estimate the parameters.

The results from applying the Cognity™ Stable distribution modeling to the DJIA daily returns from 1 January 1990 to 14 February 2003 is displayed in Figure 4. In both cases a GARCH model has been used to account for the clustering of volatility.

Figure 4 shows the left-hand tail detail of the resulting stable density, along with that of a Normal density fitted using the sample mean and sample standard deviation, and that of a non-parametric kernel density estimate (labeled "Empirical" in the plot legend). The parameter estimates are:

• Stable parameters a = 1.699, f = -0.120, jA = 0.0002, and a = 0.006.

• Normal density parameter estimates fi = 0.0003, and a = 0.010.

DJIA DAILY RETURNS

Note that the Stable density tail behavior is reasonably consistent with the Empirical non-parametric density estimate, indicating the existence of some extreme returns. At the same time it is clear from the figure that the tail of the Normal density is much too thin, and will provide inaccurate estimates of tail probabilities for the DJIA returns. Table 1 shows just how bad the Normal tail probabilities are for several negative returns values.

TABLE 1: PROBABILITY (DJIA RETURN < X)

X

-0.04

-0.05

-0.06

-0.07

Stable Fit Normal Fit

0.0066 0.000056

0.0043 0.0000007

0.0031 3.68E-09

0.0023 7.86E-12

A daily return smaller than —0.04 with the Stable distribution occurs with probability 0.0066, or roughly seven times every four years, whereas such a return with the Normal fit occurs on the order of once every four years.

Similarly, a return smaller than —0.05 with the Stable occurs about once per year and with the Normal fit about once every forty years. Clearly the Normal distribution fit is an exceedingly optimistic predictor of DJIA tail return values.

Figure 5 displays the central portion of the fitted densities as well as the tails, and shows that the Normal fit is not nearly peaked enough near the origin as compared with the empirical density estimate (even though the GARCH model was applied), while the stable distribution matches the empirical estimate quite well in the center as well as in the tails.

DJIA

DJIA

Fitting Stable distributions: micro-caps example. Noting that micro-cap stock returns are consistently strongly non-normal (see sample of normal qq-plots at the beginning of this section), we fit stable distributions to a random sample of 182 micro-cap daily returns for the two-year period 2000-2001. The results of the 95% confidence interval for the estimation of the tail weight parameter alpha are displayed in the boxplot in Figure 6.

ESTIMATED ALPHAS OF 182 MICRO-CAP STOCKS

ESTIMATED STABLE PARAMETER ALPHAS

Figure 6

The median of the estimated alphas is 1.57, and the upper and lower quartiles are 1.69 and 1.46 respectively. Somewhat surprisingly, the distribution of the estimated tail weight parameter alpha turns out to be quite Normal.

Multivariate Stable distribution modeling Multivariate Stable distribution modeling involves univariate Stable distributions for each risk factor, each with its own parameter estimates oii, j3i, (i,&i,i = 1, 2, ■■■, K, where K is the number of risk factors, along with a dependency structure.

One way to produce the dependency structure is through a subordinated process approach as follows. First compute a robust mean vector and covariance matrix estimate of the risk factors by trimming a small percentage of the observations on a coordinate-wise basis (to get rid of the outliers, and have a good covariance estimate for the central bulk of the data). Next you generate multivariate normal scenarios with this mean vector and covariance matrix. Then you multiply each random variable component of the scenarios by a Stable subordinator which is a strictly positive Stable random variable with index ai/2, i = 1, 2, ■■■ ,K. The vector of subordinators is usually independent of the normal scenario vectors, but it can also be dependent. See, for example, Rachev and Mittnik (2000), and Rachev, Schwartz and Khindanova (2003).

Another very promising approach to building the cross-sectional dependence model is through the use of copulas, an approach that is quite attractive because it allows for modeling higher correlations during extreme market movements, thereby accurately reflecting lower portfolio diversification at such times. The next section briefly discussion copulas.

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