Path-conditional forward volatility
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.
|Date of creation:||28 Feb 2006|
|Date of revision:|
|Contact details of provider:|| Postal: |
Web page: https://mpra.ub.uni-muenchen.de
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Carey, Alexander, 2005. "Higher-order volatility," MPRA Paper 4993, University Library of Munich, Germany.
- 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 327-43.
- Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
- Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
- Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
- 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.
- 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 637-54, May-June.
- Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:4964. See general information about how to correct material in RePEc.
If references are entirely missing, you can add them using this form.