Buy-Back Contract as a Special Type of Options Contract: A Proof with Parameter Analysis

A buy-back contract is an agreement between the manufacturer and the retailer where the manufacturer agrees to repurchase any unsold inventory from the retailer at a pre-agreed price. This type of contract is commonly used in industries with high demand uncertainty, such as fashion or electronics.

An options contract is a financial instrument that gives the holder the right, but not the obligation, to buy or sell an underlying asset at a pre-agreed price and time. There are two types of options contracts: call options and put options.

To prove that a buy-back contract can be viewed as a special type of options contract, we need to construct an options contract with special parameters that replicates the same sales and costs as a buy-back contract.

Let's assume that the manufacturer and the retailer agree on a buy-back contract where the manufacturer will repurchase any unsold inventory at a price of $X per unit. We can construct an options contract with the following parameters:

• Underlying asset: The inventory of the product.
• Strike price: $X per unit.
• Expiration date: The date when the buy-back contract expires.
• Option type: Put option.

Now, let's analyze what happens under both contracts in different scenarios:

1. If demand is high and all inventory is sold:

Under the buy-back contract, the retailer sells all inventory to customers and earns a profit of (selling price - cost) per unit. The manufacturer does not repurchase any inventory, so there is no cost to the manufacturer.

Under the options contract, the put option expires worthless because the inventory is sold, so there is no cost to the retailer. The manufacturer does not buy back any inventory, so there is no cost to the manufacturer.

In both cases, the sales and costs are the same under both contracts.

2. If demand is low and there is unsold inventory:

Under the buy-back contract, the retailer sells as much inventory as possible to customers at the selling price. Any unsold inventory is repurchased by the manufacturer at a cost of $X per unit.

Under the options contract, the retailer exercises the put option and sells the unsold inventory to the manufacturer at a price of $X per unit. The manufacturer buys the inventory at a cost of $X per unit.

In both cases, the sales and costs are the same under both contracts.

Therefore, we can conclude that a buy-back contract can be viewed as a special type of options contract where the option type is a put option and the strike price is the repurchase price. This equivalence allows us to use option pricing models to value buy-back contracts and to understand their risk characteristics.