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I am currently working on American options.

I saw that we can derive a PDE for American style options in the same way as with BS for European options.

In a textbook, I found that the PDE leads to an inequality because "we can earn no more than the risk-free rate on our portfolio". Can someone explain why ?

Thanks

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    $\begingroup$ If you could, you could construct a hedged portfolio which earns more than the risk-free rate. Going long this portfolio and financing this with a short position in a RF bond would be an arbitrage then. $\endgroup$
    – freistil90
    Jan 31 at 14:23
  • $\begingroup$ OTOH you could earn less than RF in case of early exercise, as you will forfeit the "intrinsic value" of the option, which the counterparty will capture as an early exercise premium from his point of view. He makes, you lose. $\endgroup$
    – nbbo2
    Feb 1 at 16:39

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