Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models

Søren Asmussen*, Mogens Bladt

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

2 Citations (Scopus)


The Gram–Charlier expansion of a target probability density, (Formula presented.), is an (Formula presented.) -convergent series (Formula presented.) in terms of a reference density (Formula presented.) and its orthonormal polynomials (Formula presented.). We implement this for the density of a regime-switching Lévy process at a given time horizon T. The main step is the evaluation of moments of all orders of (Formula presented.) in terms of model primitives, for which we give a matrix-exponential representation. A number of numerical examples, in part involving pricing of European options, are presented. The traditional choice of (Formula presented.) as normal with the same mean and variance as (Formula presented.) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, (Formula presented.) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.

Original languageEnglish
JournalQuantitative Finance
Pages (from-to)675-689
Number of pages15
Publication statusPublished - Apr 2022


  • Bell polynomials
  • CGMY process
  • Cumulants
  • European call option
  • Faà di Bruno's formula
  • Integrated CIR process
  • Markov additive process
  • Markov-modulation
  • Matrix-exponentials
  • Normal inverse Gaussian distribution
  • Risk neutrality
  • Tempered stable distribution


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