How dividends reduce stock price (S) in Black-Scholes

[David Harper CFA FRM]

A good question came up today about dividends in BSM, as a proposed correction to one of my practice questions @ http://www.bionicturtle.com/how-to/question/t4.hull-chapters-13-14

Shyamasonti is correct to want to reduce the stock price in d1. Basically a dividend yield reduces the stock price in the "outer" BSM:
c = S(0)*exp(-qT)*N(d1) - ...

and the dividend also (additionally) reduces the stock price "inside" the d1:
d1 = ( LN[S(0)/K] + ( r - q + sigma^2/2)*T ) / ....

Inside the d1, alternatively, we can use the equivalent:
d1 = ( LN[S(0)*exp(-qT)/K] + ( r + sigma^2/2)*T ) / ....

Because, here's why it's great to get facile with LN() and EXP():
LN[S(0)*exp(-qT)/K] = LN[S(0)] + LN[exp(-qT)] - LN(K) = LN[S(0)/K] - qT

In summary:

• Dividends reduce the stock price in both the "outer" BSM and the "inner" d1 (therefore d2, too)
• But with respect to d1, see how we can either (i) discount the stock price with exp(-qT) or (ii) parse out the subtraction term, -qT, but we should not do both as both double-counts the reduction?

Further, this refers to continuous dividend yields, where are convenient but not realistic. If we have lumpy dividends, we must reduce the stock price directly (and there will be no -qT term)

 

[trabala38]

Thanks a lot for this comment.

I had seen in the questions the two different approaches and I now understand where it comes from !

So obvious when you analyze the equation of d1 !

Thanks again,

trabala38

 

[David Harper CFA FRM]

Trabala38, thanks, glad you liked! To be honest, for years I've sort of glossed over the treatment of LUMPY dividends b/c it's so common to run it with assumption of continuous dividends, I don't think i was crystal clear on why, in the case of lumpy dividends (which are more realistic), i didn't see the -qT, so finally I feel like i have clarity on the consitency of the approach w.r.t. lumpy dividends.

 

[mide]

hello David,

I intend to write the FRM Level 1 and 2 together in november 2012.do you have any practice questions to gloss at pending the release of the AIMS?

 

[David Harper CFA FRM]

mide,

do we have practice questions? um, we have thousands ... the lower boards are practice questions that get updated to PDFS

Lower boards (for paid members, of course) @ http://forum.bionicturtle.com/#financial-risk-manager-frm-practice-questions.7

thanks, David

 

[Mark W]

David,

Great post, I was just wondering the same myself but had a search and found this! Like trabala38 says, obvious when you analyse!

However, I think there's a typo in your equation:

Because, here's why it's great to get facile with LN() and EXP():
LN[S(0)*exp(-qT)/K] = LN[S(0)] * LN[exp(-qT)] - LN(K) = LN[S(0)/K] - qT

Should be:

Because, here's why it's great to get facile with LN() and EXP():
LN[S(0)*exp(-qT)/K] = LN[S(0)] + LN[exp(-qT)] - LN(K) = LN[S(0)/K] - qT

Thanks,

Mark

 

[Jhoony]

Hello David,
I found this Q&A very useful. However, could you please clarify one more thing, because I decided to use only one approach no matter if there is cont. dividend or dividend in absolute values. So, whenever there are dividends (cont. or abs. value) I can:

1. calculate the S(0) minus dividend to get the dividend free S(0) or S(0)*; either by S(0)exp(-div*T) or S(0) - PV(dividends)
2. calculate the d1 as [LN(S(0)*/K) + (rf + o^2/2)*T] / [o*SQRT(T)] - here I do not need to incorporate the dividends, am I right, because they are already included in S(0)*
3. for the call = S(0)* x N(d1) - Kexp(-rf*T)N(d2)

I prefer this method, because the input "spot price is always clean" as in the case there are no dividends at all.
Does this approach work every time?

Thank you!