Modelling tail risk with tempered stable distributions: an overview

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In this study, we investigate the performance of different parametric models with stable and tempered stable distributions for capturing the tail behaviour of log-returns (financial asset returns). First, we define and discuss the properties of stable and tempered stable random variables. We then show how to estimate their parameters and simulate them based on their characteristic functions. Finally, as an illustration, we conduct an empirical analysis to explore the performance of different models representing the distributions of log-returns for the S&P500 and DAX indexes.

Original languageEnglish
Pages (from-to)1253-1280
Number of pages28
JournalAnnals of Operations Research
Issue number1-2
Publication statusPublished - Apr 2021


  • Lévy process
  • Stable distribution
  • Tail risk
  • Tempered stable distribution

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