The Effect of Time to Expiration

The Effect of Time to Expiration

Consider two American calls that differ only in their times to expiration. One has a time to expiration of T₁ and a price of

   C_a(S₀, T₁, X),

the other has a time to expiration of T₂ and a price of

   C_a(S0, T₂, X),

where T₁ < T₂. Which of these options will have a greater value?

Suppose today is the expiration day of the shorter-lived option. The stock price is S_T. The value of the expiring option is

   Max(0, S_T1 -X).

The second option still has time to expiration T₂ - T₁. Its minimum value is

   Max(0, S_T - X).

Thus, when the shorter-lived option expires, its value is the minimum value of the longer-lived option. Therefore:

                                    C_a(S0, T₂, X) ≥ C_a(S₀, T₁, X)

The time value of a call varies with the time to expiration and the proximity of the stock price to the exercise price. Investors pay for the time value of the call based on the uncertainty of the future stock price. If the stock price is very high, the call is said to be deep-in-the-money and the time value will be low. If the stock price is very low, the call is said to be deep-out-of-the-money and the time value likewise will be low. The time value will be low because at these extremes, the uncertainty about the call expiring in or out-of-the-money is lower. The uncertainty is greater when the stock price is near the exercise price, and it is at this point that the time value is higher.

These properties of the time value and our previous results enable us to get a general idea of what the call price looks like relative to the stock price. The following figure illustrates this point for American calls. The curved line is the price of the call, which lies above its intrinsic value, Max(0, S₀ - X).

 

As expiration approaches, the call price loses its time value, a process called time value decay, and the curve moves gradually toward the intrinsic value. At expiration, the call price collapses onto the intrinsic value. The relationship between time to expiration and option price also hold for European calls.