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.