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From volatility smiles to the volatility of volatility

Author

Listed:
  • Bernard Dumas

    (INSEAD
    University of Torino AXA Chair)

  • Elisa Luciano

    (University of Torino
    Collegio Carlo Alberto)

Abstract

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.

Suggested Citation

  • Bernard Dumas & Elisa Luciano, 2019. "From volatility smiles to the volatility of volatility," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 387-406, December.
    • Handle: RePEc:spr:decfin:v:42:y:2019:i:2:d:10.1007_s10203-019-00263-w
    DOI: 10.1007/s10203-019-00263-w
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    References listed on IDEAS

    as
    1. Mark Britten‐Jones & Anthony Neuberger, 2000. "Option Prices, Implied Price Processes, and Stochastic Volatility," Journal of Finance, American Finance Association, vol. 55(2), pages 839-866, April.
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    3. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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    Cited by:

    1. Elisa Alòs & Maria Elvira Mancino & Tai-Ho Wang, 2019. "Volatility and volatility-linked derivatives: estimation, modeling, and pricing," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 321-349, December.
    2. Kim, Sangkwon & Kim, Junseok, 2021. "Robust and accurate construction of the local volatility surface using the Black–Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).

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    More about this item

    Keywords

    Stochastic volatility; Implicit volatility; Forward volatility; VIX;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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