Production with Risk Hedging : Optimal Policy and Efficient Frontier

Demand for many products may depend on the price of a tradable asset or on the economy in general. For example, demand for equipment that plants or harvests corn correlates with the corn price on the commodity market, and discount stores experienced increased sales revenue during the last recession. Thus, we model demand as a stochastic process with two components: in addition to the usual Gaussian component reflecting demand volatility, there is a drift component taking the form of a function of a tradable asset price. (In the case of dependence on the general economy, the asset price can be a broad market index, such as the S&P 500 Index.) With this demand model, we study the one-time production quantity decision along with a real-time risk-hedging strategy over a given planning horizon (the production cycle). Pursuing a mean-variance formulation, we derive the optimal solution to both production and hedging decisions. We give a complete characterization of the efficient frontier and quantify the improvement in risk-return trade-off achieved by the hedging strategy. Furthermore, we show that the hedging strategy is self-financing in the sense that the expected total wealth from both production and hedging stays nonnegative at all times.