The problem of completeness of the forward rate based bond market model driven by a Lévy process under the physical measure is examined. The incompleteness of market in the case when the Lévy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump measure under a martingale measure are presented and the corresponding integral representation of local martingales is proven.

The risk minimizing problem $E\left[l\left((H-X_T^{x,\pi })⁺\right)\right]\stackrel{\pi }{\to }min$ in the multidimensional Black-Scholes framework is studied. Specific formulas for the minimal risk function and the cost reduction function for basket derivatives are shown. Explicit integral representations for the risk functions for l(x) = x and $l\left(x\right)=x^p$, with p > 1 for digital, quantos, outperformance and spread options are derived.