Do financial returns have finite or infinite variance? A paradox and an explanation

Abstract

One of the major points of contention in studying and modelling financial returns is whether or not the variance of the returns is finite or infinite (sometimes referred to as the Bachelier–Samuelson Gaussian world versus the Mandelbrot stable world). A different formulation of the question asks how heavy the tails of the financial returns are. 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. Inspired by the prelimit theorems of Klebanov, Rachev and Szekely (1999 Klebanov, L, Rachev, S and Szekely, G. 1999. Pre-limit theorems and their applications. Acta Applicandae Mathematicae, 58: 159–174. [Crossref], [Web of Science ®] , [Google Scholar]) and Klebanov, Rachev and Safarian (2000 Klebanov, L, Rachev, S and Safarian, M. 2000. Local prelimit theorems and their applications to finance. Appl. Math. Lett., 13: 73–78. [Crossref], [Web of Science ®] , [Google Scholar]), we provide an explanation of this paradox. We show that, for financial returns, a natural family 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. Important examples demonstrate the existence of a natural scale associated with the model at which such an apparent shift in the tails occurs.

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