Integrated Commodity Inventory Management and Financial Hedging: A Dynamic Mean-Variance Analysis

We consider a firm purchasing a storable raw material commodity from a spot market with volatile commodity prices and the access to an associated financial derivatives market. The purchased commodity is processed into an end product with uncertain demand and lost sales. The firm aims to integrate the inventory replenishment and financial hedging decisions to maximize the mean-variance of terminal wealth over a finite horizon. Recognizing time-inconsistency of mean-variance criteria, we employ the dynamic programming approach to obtain a time-consistent policy. Assuming no arbitrage in financial market, we show that the mean-variance utility functions under the time-consistent policy have a recursive representation which enables us to readily characterize the structure of the time-consistent policy. We analyze two types of hedging instruments, vanilla hedges and exotic hedges, and show that inventory and financial hedging decisions can be separated in the presence of forward contracts and a myopic state-dependent base stock policy is optimal. The optimal hedging policy can be obtained by minimizing the variance of the hedging portfolio, the value of excess inventory and the profit-to-go as a function of future price. In the presence of a continuum of option strikes, we show how to construct custom exotic derivatives using forwards and options of all strikes to replicate the profit-to-go function. We then show the optimality of the time-consistent policy under exotic hedge for the initial mean-variance objective. We further investigate the dynamic interplay of inventories and financial hedge and show that they can be substitutes in a dynamic environment. Finally, we compare the performances in different hedging environments to discuss how financial hedges add value and provide a numerical study.