TABLE OF CONTENTS In this chapter, we look at fat tails in distributions of underlying variable moves from a practical perspective. We are especially concerned with obtaining some sort of volatility for fat tails. We introduce the idea using some examples, and deal with practical questions at the end.
It has been known for many years that the probability distribution of underlying changes of financial variables d tx( t) = x( t + dt) ? x( t) over time interval dt is at best only approximately Gaussian. Practically all production models in finance use Gaussian assumptions, either associating x( t) directly with a financial variable or using some simple transformation like x( t) = In r( t) which leads to the lognormal model [1]. The description of deviations from Gaussian behavior forms a large part of this book. In this chapter, we will focus on jump outliers, giving rise to "fat tails".
Gaussian behavior means we assume d tx( t) = [ ?( t)+ ?( x,t) ?( t)] dt. The stochastic variable ?( t) satisfies ?? 2( t) ? = 1/ dt, and has probability distribution 
. In addition, for different times we have the delta-function normalization ??( t) ?( t ?) ? = ?( t ? t ?
