It's reasonable,
For in the money option, the time value is affected by two sources:
1) time value caused by volatility
2) time value caused by interest rate
for put option because you are selling the underlying asset (S) get the cash (K) you tend to want to exercise it early since the earlier you get the cash, the more interest you will get.
In this sense, interest rate tend to have a negative effect on the time value for in the money put option. For deep in the money put, the positive effect on the time value by volatility is very low. When the negative effect outweighs the positive effect, the time value can be negative. So that the put price is less than K-S.
This is the main reason we focus much on the early exercise of the American put (no dividend). Negative time value means you will get hurt if you wait, in this case, exercise is a wise decision. But for European option, you have to wait, so that the time value of money will hurt you.
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by Chemist_MZ
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