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Path-conditional forward volatility


  • Carey, Alexander


In derivatives modelling, it has often been necessary to make assumptions about the volatility of the underlying variable over the life of the contract. This can involve specifying an exact trajectory, as in the Black and Scholes (1973), Merton (1973) or Black (1976) models; one that depends on the level of the underlying variable as in the local volatility models of Dupire (1994), Derman and Kani (1994) and Rubinstein (1994); or fixing the parameters of a more general stochastic volatility process as in Hull and White (1987) or Heston (1993). These forward-looking assumptions are by their very nature destined to be disproved, and what is more are at odds with the frequent model recalibration that (rightly) takes place in practice. In Carey (2005), the Black-Scholes analytical framework is extended, via the definition of higher-order volatilities and the derivation of moment formulae for the case where they are deterministic. In this paper, we show that the same formulae can be obtained under markedly weaker assumptions, which leave the future volatilities unspecified. Instead, we impose constraints on new, related quantities, which we term "path-conditional forward volatilities." Under this scheme, the model inputs are no longer the future spot volatilities, but rather their forward counterparts. One consequence, we show, is that contrary to conventional wisdom, the Black-Scholes formula can in principle be used without any reference to future volatility.

Suggested Citation

  • Carey, Alexander, 2006. "Path-conditional forward volatility," MPRA Paper 4964, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:4964

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    References listed on IDEAS

    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Carey, Alexander, 2005. "Higher-order volatility," MPRA Paper 4993, University Library of Munich, Germany.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    6. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    7. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    8. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    More about this item


    higher-order volatility; higher-order moments; forward volatility; option pricing;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates


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