Modelling tail risk with tempered stable distributions: an overview

  • Hasan FallahgoulEmail author
  • Gregoire Loeper
S.I.: Recent Developments in Financial Modeling and Risk Management


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.


Lévy process Stable distribution Tail risk Tempered stable distribution 

JEL classification

C5 G12 



We are grateful to Frank Fabozzi, Young (Aaron) Kim, and Stoyan Stoyanov for helpful comments. We also thank the editor and two anonymous referees for insightful remarks. The Centre for Quantitative Finance and Investment Strategies has been supported by BNP Paribas.


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Authors and Affiliations

  1. 1.School of Mathematical Sciences and Centre of Quantitative Finance and Investment StrategiesMonash UniversityMelbourneAustralia

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