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Local volatility changes in the black-scholes model

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  • Hans Peter Bermin
  • Arturo Kohatsu
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    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.

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    File URL: http://www.econ.upf.edu/docs/papers/downloads/416.pdf
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    Bibliographic Info

    Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 416.

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    Date of creation: Sep 1999
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    Handle: RePEc:upf:upfgen:416

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    Web page: http://www.econ.upf.edu/

    Related research

    Keywords: Contingent claims; hedging; local Vega index; Malliavin calculus; stochastic flows;

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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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