Nonparametric implied lévy densities

Likuan Qin, Viktor Todorov

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


This paper develops a nonparametric estimator for the Lévy density of an asset price, following an Itô semimartingale, implied by short-maturity options. The asymptotic setup is one in which the time to maturity of the available options decreases, the mesh of the available strike grid shrinks and the strike range expands. The estimation is based on aggregating the observed option data into nonparametric estimates of the conditional characteristic function of the return distribution, the derivatives of which allow to infer the Fourier transform of a known transform of the Lévy density in a way which is robust to the level of the unknown diffusive volatility of the asset price. The Lévy density estimate is then constructed via Fourier inversion. We derive an asymptotic bound for the integrated squared error of the estimator in the general case as well as its probability limit in the special Lévy case. We further show rate optimality of our Lévy density estimator in a minimax sense. An empirical application to market index options reveals relative stability of the left tail decay during high and low volatility periods.

Original languageEnglish (US)
Pages (from-to)1025-1060
Number of pages36
JournalAnnals of Statistics
Issue number2
StatePublished - Apr 2019


  • Fourier inversion
  • Itô semimartingale
  • Lévy density
  • Nonparametric density estimation
  • Options
  • Stochastic volatility

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'Nonparametric implied lévy densities'. Together they form a unique fingerprint.

Cite this