Explain why the arguments leading to putcall parity for European options cannot be used to give a similar result for American options.

I want to ask that do my explaination for this question good enough and understandable? bc I will have the presentation of this question in this week but I don't know whether my solution is correct or not. Please give me feedbacks. :(

According to chapter 1, we have learned that:

A European option can be exercised only on the maturity date; an American option can be exercised at any time during its life.

And because they are European, the options cannot be exercised prior to time T. Since the portfolios have identical values at time T, they must have identical values today. If this were not the case, an arbitrageur could buy the less expensive portfolio and sell the more expensive one. Because the portfolios are guaranteed to cancel each other out at time T, this trading strategy would lock in an arbitrage profit equal to the difference in the values of the two portfolios.

Regard to american option, this options are known for early exercise ( bc the option can be exercised any time during its life maturity as well as these American options (call and put) does not have the exact expiration date so they dont worth at the same value) they will cause departure in the present value of two portfolios and hence the put call parity would not hold.

Due to that difference, suppose we have P + S >C +Ke-rT

Arbitrage opportunity appears when theres a difference in value between the two portfolios but these two have the same expiration date (T). But for the American option, this situation does not lead to an arbitrage opportunity. Because of this assumption, although theres a difference in value between these portfolios, the American option has the early exercise so we cannot determine when we can gain the arbitrage opportunity. In specific, If we buy the call, short the put, and short the stock, we cannot be sure of the result because we do not know when the put will be exercised.

Therefore, the arguments leading to putcall parity for European options cannot be used to give a similar result for American options

So:

American options need to have expiry on maturity in order to sustain with the theory of put call parity.

European options only allow exercise at the time of maturity so they will truly reflect the put call parity.