Pricing and hedging derivative securities in markets with uncertain volatilities

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σminand σmax. These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the ‘pricing’ volatility is selected dynamically from the two extreme values, σmin, σmax, according to the convexity of the value-function. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model captures the importance of diversification in managing derivatives positions. It can be used systematically to construct efficient hedges using other derivatives in conjunction with the underlying asset.