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# option on a dividend paying stock

#### Plirts

##### New Member
Hi!
I hope we will not need this formula on the exam, but still... Would it be enough to reduce stock value by the present value of dividends (i.e. S = S0-PV of all dividends) and use this new stock price in d1 OR will the dividend rate q also appear in d1 formula in part (r-q+sigma^2/2)?
Thank you!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Plirts,
I think you are correct to infer a low testability but: the dividend reduces the stock price in BOTH (with consistency) the outer BSM [i.e., c = S(reduced by dividend)*N(d1) - K*exp(-rT)*N(d2)] function and the "inner" d1/d2.

The one confusion that arises is that the dividend in d1, like in the outer, can reduce either continuously or as a PV[of future lumpy dividends] such that d1 can be either:
d1 = LN[(S - PV dividends)/K] + (r + variance/2)T]/(volatility*SQRT[T]), or if q = continuous dividend yield:
d1 = LN[([S/K] + (r - q + variance/2)T]/(volatility*SQRT[T]). But not both as that would double reduce!
These are identical as we can re-express S - PV[divs] = S*exp(-qT), and then LN(exp(-qT) = -qT.

Thanks,

#### Plirts

##### New Member
Thanks, it is clear now. My main point was in case of cont. dividend if "ln(S/K)" remains as it is but I did not think about double reduce.
Thanks,
Plirts

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