Buy-Back Contracts as a Special Type of Options Contracts in a Two-Party Supply Chain

In a buy-back contract, the manufacturer agrees to buy back any unsold inventory from the retailer at a predetermined price. This can be viewed as a call option, where the retailer has the right (but not the obligation) to sell the inventory back to the manufacturer at a predetermined price.

To show that a buy-back contract is a special type of options contract, we can construct an option contract with special parameters that replicates the same sales and costs as the buy-back contract.

Let's assume that the manufacturer sells a product to the retailer at a wholesale price of $W per unit, and the retailer sells the product to the end consumer at a retail price of $R per unit. The manufacturer offers a buy-back contract to the retailer, agreeing to buy back any unsold inventory at a price of $B per unit.

To construct an equivalent option contract, we can set the strike price of the option equal to the buy-back price ($K=B$). We can also set the option premium equal to the buy-back price ($P=B$), since this is the amount the manufacturer would have to pay to buy back the inventory.

Now, let's consider two scenarios:

1. The retailer sells all the inventory to end consumers. In this case, the retailer pays the wholesale price of $W per unit to the manufacturer, and sells the product to end consumers at the retail price of $R per unit. The retailer makes a profit of $R-W per unit.

Under the buy-back contract, the manufacturer does not have to buy back any inventory, so there is no additional cost to the manufacturer.

Under the option contract, the retailer does not exercise the option, since the market price ($R$) is higher than the strike price ($B$). The retailer keeps the inventory and sells it to end consumers at the retail price of $R per unit, making a profit of $R-W per unit. The manufacturer receives the option premium of $P=B$ per unit, which is the same as the buy-back price.

Thus, in this scenario, both the buy-back contract and the option contract result in the same sales and costs for both the manufacturer and the retailer.

2. The retailer has unsold inventory. In this case, the retailer pays the wholesale price of $W per unit to the manufacturer, but is unable to sell all the inventory to end consumers. The remaining inventory is sold back to the manufacturer at the buy-back price of $B per unit.

Under the buy-back contract, the manufacturer buys back the unsold inventory at a cost of $B per unit, resulting in a loss of $B-W per unit.

Under the option contract, the retailer exercises the option and sells the unsold inventory back to the manufacturer at the strike price of $K=B$ per unit. The manufacturer pays the strike price of $K=B$ per unit, resulting in a loss of $B-W per unit. The retailer receives the option payoff of $K-B=0$ per unit, which is the same as the buy-back price.

Thus, in this scenario, both the buy-back contract and the option contract result in the same sales and costs for both the manufacturer and the retailer.

Therefore, a buy-back contract can be viewed as a special type of options contract, where the strike price and option premium are set to the buy-back price, and the option is a call option to sell the inventory back to the manufacturer.