Topics in Applied Probability:Tempered Stable Distributions, Ad Placement, and Turing's Formula. Lecture 1

Lecture 1: Do financial returns have a finite or infinite variance? A paradox and an explanation. Abstract: One of the major points of contention in studying and modeling financial returns is whether or not the variance of the returns is finite or infinite. The available empirical evidence can be, and has been, interpreted in more than one way. The apparent paradox, which has puzzled many a researcher, is that the tails appear to become less heavy for less frequent (e.g. monthly) returns than for more frequent (e.g. daily) returns, a phenomenon not easily explainable by the standard models. We provide an explanation of this paradox and show that, for financial returns, natural families of models are those with tempered heavy tails. These models can generate observations that appear heavy tailed for a wide range of aggregation levels before becoming clearly light tailed at even larger aggregation scales.