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Tempered stable distributions and processes

Uwe Küchler, Stefan Tappe

TL;DR

Tempered stable distributions form a flexible six-parameter framework $TS(\alpha^+,\beta^+,\lambda^+;\alpha^-,\beta^-,\lambda^-)$ that unifies several known families such as CGMY and bilateral Gamma. The authors develop a comprehensive theory including limit theorems with explicit convergence rates, weak convergence closures, density analysis and density-transformations under measure changes, and practical parameter estimation from moments and sample paths. They further apply the framework to finance by constructing exponential stock models and deriving option pricing tools under tempered stable measures, highlighting the boundary role of the bilateral Gamma case. Collectively, the results illuminate asymptotic behavior, provide rate bounds for normal approximation, and furnish inference and pricing methods for risk management with tempered stable dynamics.

Abstract

We investigate the class of tempered stable distributions and their associated processes. Our analysis of tempered stable distributions includes limit distributions, parameter estimation and the study of their densities. Regarding tempered stable processes, we deal with density transformations and compute their $p$-variation indices. Exponential stock models driven by tempered stable processes are discussed as well.

Tempered stable distributions and processes

TL;DR

Tempered stable distributions form a flexible six-parameter framework that unifies several known families such as CGMY and bilateral Gamma. The authors develop a comprehensive theory including limit theorems with explicit convergence rates, weak convergence closures, density analysis and density-transformations under measure changes, and practical parameter estimation from moments and sample paths. They further apply the framework to finance by constructing exponential stock models and deriving option pricing tools under tempered stable measures, highlighting the boundary role of the bilateral Gamma case. Collectively, the results illuminate asymptotic behavior, provide rate bounds for normal approximation, and furnish inference and pricing methods for risk management with tempered stable dynamics.

Abstract

We investigate the class of tempered stable distributions and their associated processes. Our analysis of tempered stable distributions includes limit distributions, parameter estimation and the study of their densities. Regarding tempered stable processes, we deal with density transformations and compute their -variation indices. Exponential stock models driven by tempered stable processes are discussed as well.

Paper Structure

This paper contains 10 sections, 30 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Tempered stable distributions and processes
  3. Closure properties of tempered stable distributions
  4. Convergence of tempered stable distributions to a normal distribution
  5. Laws of large numbers for tempered stable distributions
  6. Statistics of tempered stable distributions
  7. Analysis of the densities of tempered stable distributions
  8. Density transformations of tempered stable processes
  9. The $p$-variation index of a tempered stable process
  10. Exponential stock models driven by tempered stable processes

Key Result

Lemma 2.5

Suppose that $\beta \in (0,1)$. The one-sided tempered stable distribution has the characteristic function where the power stems from the main branch of the complex logarithm.

Theorems & Definitions (78)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 68 more