Comparison of Two Methods for Superreplication

We compare two methods for superreplication of options with convex pay-off functions. One method entails the overestimation of an unknown covariance matrix in the sense of quadratic forms. With this method the value of the superreplicating portfolio is given as the solution of a linear Black--Scholes BS-type equation. In the second method, the choice of quadratic form is made pointwise. This leads to a fully non-linear equation, the so-called Black--Scholes--Barenblatt (BSB) equation, for the value of the superreplicating portfolio. In general, this value is smaller for the second method than for the first method. We derive estimates for the difference between the initial values of the superreplicating strategies obtained using the two methods.