Valuing the Option to Purchase an Asset at a Proportional Discount

I analyze the value of a nonstandard call option that allows the holder to purchase an underlying asset at a discount proportional to the asset's market price. Several applications for this type of option exist, including its use in employee compensation contracts. I derive the value of this option for a dividend-paying asset and for an option whose exercise price reflects a time-varying discount factor. The derived value incorporates the optimal time at which the option should be exercised. One application of this option relates to a residential real estate program in China.