Managing the Volatility Risk of Portfolios of Derivative Securities: the Lagrangian Uncertain Volatility

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 diierential 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 options available in the market. Several examples are discussed, including possible applications to market-making in equity and foreign-exchange derivatives. for their useful comments and suggestions. All mistakes are, of course, ours.