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Local volatility changes in the black-scholes model

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  • Hans Peter Bermin
  • Arturo Kohatsu

Abstract

In this paper we address a problem arising in risk management; namely the study of price variations of different contingent claims in the Black-Scholes model due to anticipating future events. The method we propose to use is an extension of the classical Vega index, i.e. the price derivative with respect to the constant volatility, in the sense that we perturb the volatility in different directions. This directional derivative, which we denote the local Vega index, will serve as the main object in the paper and one of the purposes is to relate it to the classical Vega index. We show that for all contingent claims studied in this paper the local Vega index can be expressed as a weighted average of the perturbation in volatility. In the particular case where the interest rate and the volatility are constant and the perturbation is deterministic, the local Vega index is an average of this perturbation multiplied by the classical Vega index. We also study the well-known goal problem of maximizing the probability of a perfect hedge and show that the speed of convergence is in fact dependent of the local Vega index.

Suggested Citation

  • Hans Peter Bermin & Arturo Kohatsu, 1999. "Local volatility changes in the black-scholes model," Economics Working Papers 416, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:416
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    File URL: https://econ-papers.upf.edu/papers/416.pdf
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    References listed on IDEAS

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    1. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
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    Cited by:

    1. Hans‐Peter Bermin & Arturo Kohatsu‐Higa & Miquel Montero, 2003. "Local Vega Index and Variance Reduction Methods," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 85-97, January.

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    More about this item

    Keywords

    Contingent claims; hedging; local Vega index; Malliavin calculus; stochastic flows;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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