This paper considers exponential utility indifference pricing for a
multidimensional non-traded assets model and provides two approximations for
the utility indifference price: a linear approximation by Picard iteration and
a semigroup approximation by splitting techniques. The key tool is the
probabilistic representation for the utility indifference price by the solution
of fully coupled linear forward-backward stochastic differential equations. We
apply our methodology to study the counterparty risk of derivatives in
incomplete markets.