This paper analyzes an optimal risk management problem using put options when there
are different underlying exposures. For simplicity, we assume that there are two different
underlying assets and that each underlying asset is associated with several put options
of different exercise prices. The objective of this study is to find an optimal solution to
minimize the Value-at-Risk (VaR). Because it is difficult to solve the optimization problem
that minimizes the exact VaR, we first solve an suboptimal problem that provides an upper
bound of the VaR. It is proved that a suboptimal solution can be attained by choosing two
put options, except for some extreme cases. Usually, one exercise price option for each
underlying asset is chosen to solve the suboptimal problem. Sometimes, two options have
to be chosen for one underlying asset (and none for the other underlying asset) to solve the
suboptimal problem. In this case the exercise prices for the options have to be adjacent. In
a numerical example, we compare the suboptimal solution with an (approximate) optimal
solution that is obtained by taking minimum of the exact VaR varying hedge ratios of the
put options about ten thousand times. The result shows that the suboptimal solution is a
good approximation for the optimal solution. The suboptimal solution is also used for the
sensitivity analysis of the hedging problem.
