Production with Risk Hedging—Optimal Policy and Efficient Frontier

Abstract

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. The online appendix is available at https://doi.org/10.1287/opre.2017.1597

Applications of Stochastic Orders and Stochastic Functions in Inventory and Pricing Problems

“…A commonly studied risk measure is the variance of the profit (see, e.g., Chiu and Choi 2016, Ridder et al 1998, Rubio-Herreroa et al 2015, Wang and Yao 2017, that is M 2 ðyÞ. Zhang et al (2020b) analyze the skewness (i.e., the third moment M 3 ðyÞ) and the kurtosis (i.e., the fourth moment M 4 ðyÞ).…”

Mean–Variance Hedging for Production Planning with Multiple Products

“…Next, we review papers that relate to ours in that demand uncertainty is partly linked to financial assets and thus induces risk hedging strategy. In a recent work, Wang and Yao (2017), production planning with risk hedging is studied in a single-product setting, with the demand depending on a single financial asset. The need to extend to multiple products and multiple financial assets has been motivated above through practical business cases.…”

“…The following lemma is essentially the highdimensional version of Lemma 4 in Wang and Yao (2017) and can be easily verified by applying It ô's Lemma and using the admissibility conditions in Equations (A2) and (A15); hence, the proof is omitted. LEMMA A1.…”

“…Properties of Itô's integral ( in our case), guarantee that the total hedging wealth from this discretized implementation converges (in L 2 ) to its theoretical value in continuous time (i.e., χ T ) as the implementation frequency increases. Wang and Yao (2017) numerically show that not so frequent update (e.g., monthly) of the hedging position generates a performance quite close to that from daily updates. Of course, how frequent the updates of hedging need depends on specific applications and is limited by available demand update frequency, and the firm should conduct extensive analysis (by simulation, for instance) to see if our model can achieve a performance close to the theoretical value.…”

“…This extension also leads to significant technical difficulties and several new insights as we shall illustrate in the sections below. The work of Wang and Yao (2017) was motivated by Caldentey and Haugh (2006), the latter also considers a single product with a more general production model but with a less explicit solution, and it makes no attempt to characterize the efficient frontier.…”