Francesca Biagini, Bernt Øksendal
Because of its properties (persistence/antipersistence and self-similarity), the fractional Brownian motion (fBm) has been suggested as a useful mathematical tool in many applications, including finance.
In this paper, we discuss the extension to the multi-dimensional case of the Wick-Ito integral with respect to fractional Brownian motion and apply this approach to study the problem of minimal variance hedging in a (possibly incomplete) market driven by m-dimensional fBm. The mean-variance optimal strategy is obtained by projecting the option value on a suitable space of stochastic integrals with respect to the fBm, which represents the attainable claims.
Here we prove first a multi-dimensional Ito type isometry for such integrals, which is used in the proof of the multi-dimensional Ito formula. These results are then applied in order to provide a necessary and sufficient characterization of the optimal strategy as a solution of a differential equation involving Malliavin derivatives.