American Option and European Option

Call

Put call parity (for non-dividend paying stock):
C = P + S - K/(1+rt)
= P + S - K + K - K/(1+rt)
= (S - K) + P + K * (1 - 1/(1+rt)) > S - K

(S - K): intrinsic value
P + K * (1 - 1/(1+rt)): time value (>0 before maturity)

Prior to maturity, selling a call option itself rather than exercising it would be profitable due to the positive time value.

C_A = C_E
C_A: American call option value
C_E: European call option value


Put

Put call parity (for non-dividend paying stock):
C = P + S - K/(1+rt)
P = C - S + K/(1+rt)
= C - K + K - S + K/(1+rt)
= (K - S) + C - K * (1 - 1/(1+rt))

(K - S): intrinsic value
C - K * (1 - 1/(1+rt)): time value (>0, =0, or <0 before maturity) When C - K * (1 - 1/(1+rt)) < 0, C < K * (1 - 1/(1+rt)) (deep in-the-money put) P < (K - S) Prior to maturity, selling a deep-in-the-money put option itself would be less profitable than its intrinsic value due to the negative time value. So an American put option value should be higher than European put option's one considering the exercise option.

P_A > P_E
P_A: American put option value
P_E: European put option value