Buy-Back Contracts: A Special Case of Options Contracts in Supply Chains

A buy-back contract in a two-party supply chain specifies that the manufacturer agrees to buy back a certain quantity of unsold products from the retailer at a pre-determined price. This contract can be viewed as a special type of options contract, where the manufacturer has the option to purchase the unsold products from the retailer at a pre-determined price.

To prove this, we can construct an options contract with special parameters that replicates the sales and costs under the buy-back contract. Let us consider the following parameters:

• Strike price (K): The pre-determined price at which the manufacturer agrees to buy back the unsold products from the retailer.
• Spot price (S): The actual price at which the retailer sells the products to the end customers.
• Quantity (Q): The total quantity of products that the retailer purchases from the manufacturer.

Under the buy-back contract, the retailer purchases Q units of products from the manufacturer at a wholesale price of W per unit. The retailer then sells these products to the end customers at a spot price of S per unit. If there are any unsold products at the end of the selling season, the manufacturer agrees to buy them back from the retailer at a price of K per unit.

Let us now construct an options contract with the following parameters:

• Call option: The manufacturer has the option to purchase Q units of products from the retailer at a pre-determined price of K per unit.
• Premium (P): The price that the manufacturer pays to the retailer for the option contract.

Under this options contract, if the spot price S is greater than the strike price K, the manufacturer will exercise the option and purchase Q units of products from the retailer at the strike price K. If the spot price S is less than or equal to the strike price K, the manufacturer will not exercise the option and the retailer will keep the premium P.

To replicate the sales and costs under the buy-back contract, we can set the premium P equal to the wholesale price W that the retailer pays to the manufacturer for each unit of product. In this case, if the manufacturer exercises the option, it will pay the strike price K per unit to the retailer, which is the same as the buy-back price. If the manufacturer does not exercise the option, the retailer will keep the premium P, which is the same as the wholesale price W that it paid to the manufacturer for each unit of product.

Therefore, we have shown that a buy-back contract can be viewed as a special type of options contract with special parameters, where the premium is equal to the wholesale price and the strike price is equal to the buy-back price.