Managing the Volatility Risk of Portfolios of Derivative Securities:
the Lagrangian Uncertain Volatility Model
Marco Avellaneda and Antonio Paras
Applied Mathematical Finance, 1996

We present  an  algorithm  for  hedging option portfolios and custom-tailored derivative securities which uses options to manage volatility risk.  The algorithm uses a  volatility band  to model heteroskedasticity  and a  non-linear partial differential equation to evaluate worst-case  volatility scenarios for any given forward liability structure. This equation gives sub-additive portfolio prices and hence provides a natural ordering of preferences in  terms of hedging with options. The second element of the algorithm consists of a portfolio optimization taking into account the prices of the options available in the market. Several examples are discussed,  including  possible applications to  market making in  equity and foreign-exchange derivatives.