Minimizing shortfall risk for multiple assets derivatives

The risk minimizing problem $\mathbf{E}[l((H-X_T^{x,\pi})^{+})]\overset{\pi}{\longrightarrow}\min$ in the Black-Scholes framework with correlation is studied. General formulas for the minimal risk function and the cost reduction function for the option $H$ depending on multiple underlying are derived. The case of a linear and a strictly convex loss function $l$ are examined. Explicit computation for $l(x)=x$ and $l(x)=x^p$, with $p>1$ for digital, quantos, outperformance and spread options are presented. The method is based on the quantile hedging approach presented in \cite{FL1}, \cite{FL2} and developed for the multidimensional options in \cite{Barski}.