Production Planning with Shortfall Hedging, under Partial Information and Budget Constraint

Abstract

We study production planning integrated with risk hedging by considering shortfall as the risk measure. In addition to the one-time production quantity decision, there is a real-time hedging strategy throughout the horizon; and the goal is to minimize the gap between a pre-specified target and the total terminal wealth achieved by both production and hedging. We assume partial information --- hedging is executed based on information from the financial market only, and impose a budget constraint to cap any loss from hedging. To find the optimal hedging strategy, we construct a dual problem, which provides a lower bound to the original problem. Solving the lower-bound problem yields the optimal terminal wealth from hedging; the real-time hedging strategy is then mapped out via martingale representation theorem. Interestingly, the optimal hedging strategy takes the form of a portfolio of two options, a digital option and a put option. With the hedging strategy optimized, we show that optimizing production quantity is a convex minimization problem. With both production and hedging optimized, we provide a complete characterization of the efficient frontier: the minimized shortfall as an increasing function of the target. We also derive an explicit quantification of the shortfall reduction achieved by hedging. Asymptotic analysis on several key parameters (such as the target, the budget, and the production quantity) generates additional insights to the hedging strategy and its impact.