Pricing and Hedging Derivative Securities in Markets with Uncertain Volatility
Marco Avellaneda, Arnon Levy and Antonio Paras

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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 , sigma_min and sigma_max.  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 derivatve 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'' volaitlity is selected dynamically from the two extreme values sigma_min and sigma_max,  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 toconstruct efficient hedges using other derivatives in conjunction with the underlying asset.

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