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Options on a Stock with Market-Dependent Volatility

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  • J. Chalupa

Abstract

A market is considered whose index has strongly price-dependent local volatility. A tractable parametrization of the volatility is formulated, and option valuation of a stock with two-factor dynamics is investigated. One factor is the market index; when the second factor is uncorrelated with the first, the option valuation equation can separate. A formal solution is given for a European call. The call value depends on both the stock price and the market index. Even if the prices of a set of calls were fitted with a one-factor implied volatility, the calls could not be hedged solely with an offsetting position in the stock. For example, delta-hedging involves two deltas, one corresponding to the stock and the other to the market index. In a numerical example, the magnitude of the market delta is found to be significant. The CAPM is used as an example to explore how market-dependent volatilities could be implemented in multifactor models. In the process, the Black-Scholes equation with standard boundary conditions is reduced to quadrature for squared volatilities proportional to (1+an*sm^n)/(1+ad*sm^n); sm is the market index, and n, an, and ad are constants.

Suggested Citation

  • J. Chalupa, 1997. "Options on a Stock with Market-Dependent Volatility," Finance 9710005, EconWPA, revised 07 Jan 1998.
  • Handle: RePEc:wpa:wuwpfi:9710005
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    More about this item

    Keywords

    equity options; implied volatility; hedging;

    JEL classification:

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

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