  We study the problem of optimal pricing and hedging of a European option
written on an illiquid asset $Z$ using a set of proxies: a liquid asset $S$,
and $N$ liquid European options $P_i$, each written on a liquid asset $Y_i,
i=1,N$. We assume that the $S$-hedge is dynamic while the multi-name $Y$-hedge
is static. Using the indifference pricing approach with an exponential utility,
we derive a HJB equation for the value function, and build an efficient
numerical algorithm. The latter is based on several changes of variables, a
splitting scheme, and a set of Fast Gauss Transforms (FGT), which turns out to
be more efficient in terms of complexity and lower local space error than a
finite-difference method. While in this paper we apply our framework to an
incomplete market version of the credit-equity Merton's model, the same
approach can be used for other asset classes (equity, commodity, FX, etc.),
e.g. for pricing and hedging options with illiquid strikes or illiquid exotic
options.
