A research on optimal option prices in supply contract

This paper considers a supply contract in a two-stage supply chain consisting of one supplier and one buyer. Market demand is stochastic, and there are two opportunities for demand forecasts. With the first forecast at the beginning of production, the buyer puts a firm fixed order Q to the supplier, purchasing a quantity q_o of options at the same time. With the second forecast at the beginning of the sales season, the buyer determines execution quantity q_e of the option within the purchased quantity of options (q_ϵ ≤ q_o). The supplier determines option price p_o and option execution price p_e, while the buyer determines Q, q_o, q_ϵ under the given p_o and p_ϵ. Both buyer and supplier try to maximize their own profit; however, the buyer's decision variables Q, q_o, q_e are dependent on the supplier's decision variables p_o and p_ϵ. The purpose of this research is to determine the optimal p_o and p_ϵ for the supplier. The solution process is two-fold. In the first step, we solve the buyer's optimization problem to determine optimal Q, q_o and q_ϵ, which can maximize the buyer's profit. The optimal order quantity Q∗, option quantity q_o∗ and execution quantity q_ϵ∗ are functions of p_o and p_ϵ. Therefore, in the second step, we solve the supplier's optimization problem to determine the optimal p_o and p_ϵ so as to maximize the supplier's profit. We construct the supplier's profit model and derive an analytical function to determine the optimal option price p∗_o and option execution price p∗_ϵ. Since the analytical function consists of seven order implicit system equations, we apply a numerical solution procedure using Mathematica instead of deriving the closed analytical form of p∗_o and p∗_ϵ. Through numerical experiments, we show that the optimal option price p∗_o and option execution price p∗_e always provide maximum profit to the supplier, while the buyer keeps maximum profit at the same time.