Studies on the application of the alpha-stable distribution in economics - Frain, John C. (Trinity College (Dublin, Ireland). Department of Economics, 2009)
Bubbles, booms and busts in asset prices give rise to a considerable misallocation of resources when they are growing and the subsequent adjustment can be very long and painful. Yet, there is no accepted diagnosis of a bubble. In effect, there is a sense in which a bubble and a bust cannot occur in the usual econometric models. These models, almost always, depend on the normal or Gaussian distribution. Yet when one looks at data for asset prices the number and size of extreme losses and gains are orders of magnitude greater than a normal distribution would predict. The very existence of these extreme values must lead one to question the validity of the normality assumption and to look for an alternative. From time to time several alternatives have been proposed. A common proposal is to use mixtures of normal distributions. The simplest such solution is to have a mixture of two normal distributions — the first, with low volatility, represents the fundamental state with no bubble and the second, with high volatility, the bubble. The price of the asset in question is seen as switching from one state to the other with the switching being determined by some form of deterministic or stochastic process. Other solutions involve what are, in effect, infinite mixtures of normal distributions. Chief amongst these are the various GARCH proceses and the t-distribution. Various other “fat-tailed” distributions have been proposed but these have not received universal acceptance and probably never will. While such distributions often fit the data well, We have not seen any convincing theoretical arguments why they should. The purpose of this thesis is to examine the use of the a-stable distribution in this context and to determine some of the consequences of its use. The a-stable distribution is a generalisation of the normal distribution. It was first proposed as a distribution for asset returns and commodity prices by Mandelbrot in the early 1960s. It attracted alot of attention up to the early 1970s and then interest faded. There were two reasons for the waning interest. First the advances made at the time in portfolio and option pricing theory were dependent on the normal distribution. At the time almost all of this work could not have been replicated without the normality assumption. Secondly for actual application the computer power available at the time was simply not sufficient to properly use the a-stable distribution. Thus astable analysis was primitive relative to the corresponding normal analysis.
Keywords: Economics, Ph.D.; Ph.D. Trinity College Dublin