Production Planning with Risk Hedging

We study production planning integrated with risk hedging in a continuous-time stochastic setting. The (cumulative) demand process is modeled as a sum of two components: the demand rate is a general function in a tradable financial asset (which follows another stochastic process), and the noise component follows an independent Brownian motion. There are two decisions: a production quantity decision at the beginning of the planning horizon, and a dynamic hedging strategy throughout the horizon. Thus, the total terminal wealth has two components: production payoff, and profit/loss from the hedging strategy. The production quantity and hedging strategy are jointly optimized under the mean-variance and the shortfall criteria. For each risk objective, we derive the optimal hedging strategy in closed form and express the associated minimum risk as a function of the production quantity, the latter is then further optimized. With both production and hedging (jointly) optimized, we provide a complete characterization of the efficient frontier. By quantifying the risk reduction contributed by the hedging strategy, we demonstrate its substantial improvement over a production-only decision. To derive the mean-variance hedging strategy, we use a numeraire-based approach, and the derived optimal strategy consists of a risk mitigation component and an investment component. For the shortfall hedging, a convex duality method is used, and the optimal strategy takes the form of a put option and a digital option, which combine to close the gap from the target left by production (only). Furthermore, we extend the models and results by allowing multiple products, with demand rates depending on multiple assets. We also make extension by allowing the asset price to follow various stochastic processes (other than the geometric Brownian motion).