Optimal Risk Management Using Options

This paper provides an analytical solution to the problem of how an institution might optimally manage the market risk of a given exposure, under the assumption that the institution wishes to minimize its Value at Risk (VaR) using options. The solution specifies the VaR-minimizing level of moneyness of the options as a function of the underlying parameters. We show that the optimal hedge consists of a position in a single option whose strike price is independent of the level of expense the institution is willing to incur for its hedging program. The optimal strike price is increasing in the asset's drift, decreasing in its volatility for most reasonable parameter, decreasing in the risk-free interest rate, nonmonotonic in the horizon of the hedge, and increasing in the level of protection desired by the institution (i.e., the percentile of the distribution relevant for the VaR). Finally, we also show that the costs associated with a suboptimal choice of exercise price are economically significant.