How would you resolve this apparent paradox the value max0, Finance Basics

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How would you resolve this apparent paradox the value max0

Call prices are directly related to the stock's volatility, yet higher volatility means that the stock price can go lower. How would you resolve this apparent paradox? The value Max[0, X(1 + r)-T - S0] was shown to be the lowest possible value of a European put. Why is this value irrelevant for an American put?

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