Integral representations of risk functions for basket derivatives

The risk minimizing problem $\mathbf{E}[l((H-X_T^{x,\pi }{)}^{+})]\min$ 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 $l(x)=x^p$, with $p>1$ for digital, quantos, outperformance and spread options are derived.