Options on a Stock with Market-Dependent Volatility

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 rst, 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 tted with a one-factor implied volatility, the calls could not be hedged solely with an o setting 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 signi cant. 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 volatilities of the form 2 = 2 0(1+ aNS n m)=(1 + aDS n m); Sm is the market index, and n, 0, aN and aD are constants. Draft No. 2 of ewpn/9710005 on . A discussion has been added concerning how local volatilities could be implemented in the CAPM and related multifactor models.