Decisions in Economics and Finance

, Volume 42, Issue 2, pp 387–406 | Cite as

From volatility smiles to the volatility of volatility

  • Bernard Dumas
  • Elisa LucianoEmail author


The paper reviews models of the option surface and reduced-form models for stochastic volatility in continuous time, under the risk-neutral measure. It defines “forward volatilities,” analogous to forward interest rates in the theory of the term structure, and provides a proof that the forward volatility is a conditional expected value, under the risk-neutral measure, of the future spot volatility. The theory developed here is the analog of Heath–Jarrow–Morton bond-pricing theory. The link is established between forward volatilities and so-called “model-free” volatility measures such as the VIX.


Stochastic volatility Implicit volatility Forward volatility VIX 

JEL Classification

G13 G17 



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Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2019

Authors and Affiliations

  1. 1.INSEADFontainebleauFrance
  2. 2.University of Torino AXA ChairTurinItaly
  3. 3.ESOMAS DepartmentUniversity of TorinoTurinItaly
  4. 4.Collegio Carlo AlbertoTurinItaly

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