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Lower bounds on dividend-paying options


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Call Options are Not Exercised Early.
So, Ct = [ St - K. e EXP( -r*T) ] for Both European & American Call Options

However, Call Options are EXERCISED-EARLY when DIVIDENDS are paid - they are exercised just before the Dividend-Paying-Date. So in the case where Dividends are paid, shouldn't the
( Ct = St - K ) for American Call Options...?

Much gratitude for insights on this :)

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @gargi.adhikari I am not certain I completely follow your logic jump from your first to the second assertion, sorry, but I do appreciate these bound relationships are harder than meets the eye. But I don't think I need to understand because I think I can appreciate your question, which I interpret to be: if we can early exercise an American option (when the stock pays dividends) early, why isn't its lower bound given by max(0, S - K); ie, at least its intrinsic value. I mean, if you are asking about lower bounds?

Please note the summary table below from our note (I think Deepa did a good job on this!). We don't indicate lower bound for an American style option because, I think this is true(?), Hull doesn't show that. However, I do want to point out the asymmetric effect of the dividend: it hurts (subtracts) the call option holder's lower bound who forgoes it, while it helps (adds) the put option's lower bound who--from an intuitive perspective--is helped by the dividend that effectively lowers the stock's growth rate (and who is in any case, selling not buying!).

So with respect to the American style option on a dividend paying stock, I think the challenge around the lower bound relates to the same reason that it may or may not be optimal to early exercise. It is not true that it always should be exercised! I don't think Hull anywhere goes into this specifically (unassigned McDonald does, however), but mathematically the choice reduces to whether the dividend received (if we early exercise) outweighs the savings earned by deferring the strike price (if we do not). This "choice" implies that lower bound is a conditional, not a straight rule (this part is my extrapolation). For example, just to illustrate, let's assume: S = $12.00, K = $10.00. Rf = 3%, and T = 1.0 year. The minimum value on a Euro call if there is no dividend = 12 - 10*exp(-3%*1) = $2.30. Now let's just add the present value of the dividend: then the minimum value of the Euro call is reduced to = 12 - 10*exp(-3%*1) - D = $2.30 - D. The issue now, and why I suspect Hull doesn't show the min value of a American call on div paying stock, is that depending on whether it is optimal to early exercise, the American lower bound may equal this value (if it not optimal) or it will be higher (if it is optimal). Specifically,
  • It is optimal to exercise if D > K - K*exp(-rt) or in this case if D > 10 - 10*exp(-3%*1) or D > $0.30 because 0.30 is the value of waiting to exercise (cool, right?).
  • So, to my thinking, the minimum value of this American call on dividend paying stock would be the greater of:
    • If D > K - K*exp(-rt)--i.e., this option should be exercised now!--then MV = S - K + D = 12 - 10 + D = 2.00 + D; or
    • If D < (K - K*exp(-rt))--i.e., better to wait--then MV = same as Euro = S - D - K*exp(-rT) = $2.30 - D
    • You can see how this is a conditional lower bound
Please don't quote me on the exact algebra because it's just my logic and I do not have an author reference. I just wanted to illustrate the dynamics involved, specifically, to show why the lower bound is maybe more involved than you suspected (?). Geez i hope i answered your question ;) as i went out on this exploration limb!

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David Harper CFA FRM

David Harper CFA FRM
Staff member
@gargi.adhikari or anyone else who might be interested, based on the above, I started this sheet (as an addition to our big T3. Hull learning spreadsheet) at https://www.dropbox.com/s/cmv7tteif40r76i/0826-early-exercise.xlsx?dl=0 (see below).
the only difference in the four scenarios (in columns) is the continuous dividend yield assumption, which is translated into PV lump sum, D.
Then at the bottom I show the lower bounds: European without dividend, Euro with dividend (which simply subtracts D); and finally American style with dividend. This American-with-dividend, as discussed above, is conditional and depends on whether or not it is optimal to exercise. Caveat: this is just a first draft (I am not claiming it is error-free). I hope that adds color to the conditionality of the american style call option, thanks!