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Analytically Tractable Stochastic Stock Price Models pp 201–225Cite as

Regularly Varying Functions and Pareto-Type Distributions

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Abstract

Regularly varying functions have a deceptively simple structure. However, despite their apparent simplicity, they possess numerous interesting properties. Chapter 7 provides a short overview of the theory of regularly varying functions. In addition, the chapter discusses Pareto type distributions, which are distributions with regularly varying tails. A new notion of weak Pareto type functions is introduced and studied. A function is of a weak Pareto type if it can be squeezed between two regularly varying functions having the same index of regular variation. It is shown in Chap. 7 that the distributions of the stock price in the Hull-White, Stein-Stein, and Heston models are all of Pareto type. The proof of the previous statement is based on the asymptotic formulas for the stock price densities established in Chap. 6. Weak Pareto type functions will reappear in Sect. 10.6, devoted to the asymptotic equivalence in R. Lee’s moment formulas for the implied volatility.

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Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, USA

    Archil Gulisashvili

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  1. Archil Gulisashvili
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Gulisashvili, A. (2012). Regularly Varying Functions and Pareto-Type Distributions. In: Analytically Tractable Stochastic Stock Price Models. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31214-4_7

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