Model-free backward and forward nonlinear PDEs for implied volatility

Peter Carr, Andrey Itkin, Sasha Stoikov

Research output: Contribution to journalArticlepeer-review


The authors derive backward and forward nonlinear partial differential equations that govern the implied volatility of a contingent claim whenever the latter is well defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. The authors also discuss suitable initial and boundary conditions for those partial differential equations. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.

Original languageEnglish (US)
Pages (from-to)51-78
Number of pages28
JournalJournal of Derivatives
Issue number1
StatePublished - Sep 2020


  • Options
  • Volatility measures

ASJC Scopus subject areas

  • Finance
  • Economics and Econometrics


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