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### Stable Distributions 1. Introduction

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1.1          Stable distributions are a class of probability distributions that have interesting theoretical and practical properties that make them potentially useful for modelling financial data. In a sense that we will explore further below, they generalise the Normal distribution. They also allow fat tails and skewness, characteristics that are also frequently observed in financial data. Traditionally they have been perceived to be subject to the practical disadvantage that they have infinite variances (apart from the special case of the Normal distribution) and thus are not particularly easy to manipulate mathematically. However, more recently, mathematical tools and programs have been developed that simplify such manipulations.

1.2          Whether stable distributions are actually good at modelling financial data is not something that we explore in depth in these pages. Longuin (1993), when analysing the distribution of U.S. equity returns, concluded that their distribution was not sufficiently fat-tailed to be adequately modelled by Levy stable distributions, even if it was fatter tailed than implied by the normal distribution. Moreover, implicit within the theoretical justification for (non-Normal) stable distributions in such a context is an assumption that aggregate returns arise from the combined impact of a large number of smaller independent innovations, so that a generalisation of the Central Limit Theorem applies, see Section 4. Fat-tailed behaviour in the distribution of aggregate returns in line with stable laws can then be expected to arise if it is assumed that each of these smaller innovations is also (suitably) fat-tailed. The challenge is that this is not necessarily how fat tails arise in aggregate return data. Fat tails may instead arise partly or wholly due to distributional mixtures, e.g. regime shifts or time-varying volatility, or from one-off (systemic) ‘shocks’ that cannot be conceptually decomposed in to lots of smaller independent elements, see e.g. Kemp (2009). The latter might include the impact of an aggregate loss of risk appetite (and feedback effects that might then arise because of changed perceptions amongst market participants regarding the views of others).

1.3          The implicit assumption underlying stable distributions referred to in the previous paragraph is revealed by their defining characteristic, and the reason for the term stable, which is that they retain their shape (suitably scaled and shifted) under addition. The definition of a stable distribution is that if  are independent, identically distributed random variables coming from such a distribution, then for every  we have the following relationship for some constants  and :

Here  means equality in distributional form, i.e. the left and right hand sides have the same probability distribution. The distribution is called strictly stable if  for all . Some authors use the term sum stable to differentiate from other types of stability that might apply.

1.4          Normal distributions satisfy this property, indeed they are the only distributions with finite variance that do so. Other probability distributions that exhibit the stability property described above include the Cauchy distribution and the Levy distribution.

1.5          The class of all distributions that satisfy the above property is described by four parameters, . In general there are no simple closed form formulae for the probability densities, , and cumulative distribution functions, , applicable to these distributional forms (exceptions are the normal, Cauchy and Levy distributions), but there are now reliable computer algorithms for working with them.