Regularly Varying Functions and Pareto-Type Distributions

Archil Gulisashvili
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Archil Gulisashvili: Ohio University

Chapter Chapter 7 in Analytically Tractable Stochastic Stock Price Models, 2012, pp 201-225 from Springer

Abstract: 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.

Keywords: Stock Price; Vary Function; Asymptotic Formula; Pareto Distribution; Implied Volatility (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-3-642-31214-4_7

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