Model-Free Boundaries of Option Time Value and Early Exercise Premium

Based on the option put-call parity relation, we derive model-free boundary conditions of option time value and option early exercise premium with cash dividends on the underlying stock. The article produces four main results. (1) For European options, the difference in time value between a call option and a put option is the discount interest earned on the exercise price less the present value of cash dividends to be paid before the option expiration. (2) For American options, the difference in time value between a call option and a put option is bounded between the negative amount of the present value of dividends and the discount interest earned on the exercise price. (3) The early exercise premium of an American put option is bounded between zero and the discount interest earned on the exercise price. (4) The early exercise premium of an American call option is bounded between zero and the present value of cash dividends to be paid before the option expiration. From (3) and (4), the difference in early exercise premiums between an American put option and a call option is bounded below by the negative amount of the present value of cash dividends and bounded above by the discount interest earned on the exercise price. We numerically test these results in the Black-Scholes and binomial tree models. This article contributes to the finance literature. It extends the understanding of option time value and early exercise premium, provides boundary conditions for option-pricing model calibration, and indirectly helps enhance market efficiency and make optimal early exercise decisions, especially when the underlying stock pays cash dividends.