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