Closed-form transformations from risk-neutral to real-world distributions

Xiaoquan Liu, Mark B. Shackleton, Stephen J. Taylor, Xinzhong Xu

Research output: Journal PublicationArticlepeer-review

60 Citations (Scopus)


Risk-neutral and real-world densities are derived from option prices and risk assumptions, and are compared with historical densities obtained from time series. Two parametric risk-transformations are used to convert risk-neutral densities into real-world densities. Both transformations are estimated by maximizing the likelihood of observed index levels, for two parametric density families. Results for the FTSE-100 index show that parametric densities derived from option prices have more explanatory power than historical densities and higher likelihoods than densities estimated by spline methods. A combination of parametric real-world and historical densities provides the preferred predictive densities.

Original languageEnglish
Pages (from-to)1501-1520
Number of pages20
JournalJournal of Banking and Finance
Issue number5
Publication statusPublished - May 2007
Externally publishedYes


  • Generalized beta
  • Lognormal mixture
  • Real-world density
  • Risk-neutral density

ASJC Scopus subject areas

  • Finance
  • Economics and Econometrics


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