General Black-Scholes models accounting for increased market volatility from hedging strategies

Abstract

Increases in market volatility of asset prices have been observed and analyzed in recent y ears and their cause has generally been attributed to the popularity of portfolio insurance strategies for derivative securities. The basis of derivative pricing is the Black-Scholes model and its use is so extensive that it is likely to in uence the market itself. In particular it has been suggested that this is a factor in the rise in volatilities. In this work we present a class of pricing models that account for the feedback e ect from the Black-Scholes dynamic hedging strategies on the price of the asset, and from there back o n to the price of the derivative. These models do predict increased implied volatilities with minimal assumptions beyond those of the Black-Scholes theory. They are characterized by a nonlinear partial di erential equation that reduces to the Black-Scholes equation when the feedback is removed.