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

MICHAEL GRABCHAKy and GENNADY SAMORODNITSKY
Department of Statistics, Cornell University, Ithaca, NY 14853, USA
School of Operations Research and Information Engineering, and Department of Statistical Science,
Cornell University, Ithaca, NY 14853, USA
2 December 2009

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) and Klebanov, Rachev and Safarian (2000), 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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