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in the notes about arbitrage arguments I am reading, I notice the statement

We can also see that $$C^E_t>(S_t-K\mathrm{e}^{-r(T-t)})^+$$ Notice that the inequality holds STRICTLY!

I don't particularly understand why the inequality must be strict. What arbitrage can occur when equality occurs? What exactly should I be containing in my portfolio to replicate this?

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It is because to show the existence of arbitrage, it suffices to show that there is no chance of losing money,and a positive chance of making money. Arbitrage does not imply you are certain to make money. Thus, equality in your equation implies that we can create for zero cost a portfolio long the option/ short the intrinsic, which will have a positive chance of making money as long as the underlying stock has a chance of crossing the strike.

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