Using stable distributions to model local financial data

Abstract

Stable distributions are a family of probability distributions which have many properties that are useful in applications in many fields such as finance. One of their main characteristics is that they allow for skewness and heaviness of tails. The normal distribution, which is a member of this family, is a special case, since it is the only distribution that has a finite variance. Due to this characteristic, there are several works in literature that show that non-normal stable distributions are a better alternative to the normal distribution (which is also a stable distribution). In this dissertation we will review some of the main concepts of stable distributions. There are over thirteen methods of parameter estimation of stable distributions available nowadays, we shall describe two of these methods which are, the Integrated Squared Error Estimation method (ISEE) adopted by Heathcote (1977) and the Maximum Likelihood Estimation (MLE) method adopted by Nolan (2001). We will prove that both methods lead to estimators that are asymptotically normal and consistent. Then we shall apply the two methods on local financial data which includes exchange rates and stock returns. The parameter estimates obtained from the two methods are compared graphically and tested for their goodness of fit. Both methods allow us to accept the stable distribution as a model of the local exchange rates but force us to reject the stable distribution as a model of the local stock returns.