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BSM model and Continuous vs Discrete Dividends

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Hi, I have a question concerning the treatment of continuous dividends vs discrete dividends within the BSM option pricing model.

For continuous dividends, they will accounted for within the call formula as So * e-qt.....and conversely with the put formula as well.

However with discrete dividends, any expected dividends will be accounted for via d1, ie ln(s/k)...will be altered so that s = pv of dividends expected, taking into account the risk free rate and time.

So, why is there a different treatment, with both instances there are expectations of dividends and therefore the intrinsic value of the option?
 
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