Calculating Dividends

[[email protected]]

Hi David,

I have a general question regarding the calculation of dividends for the BSM model (also relates to Put/Call parity, COC, etc).

In the BSM (for dividend-paying bonds)-- we have:
Value of a call = So(d1) - Ke^(-r-q)t(d2)
" " put = Ke^(-r-q)t(1-d2) - So(1-d1)

Where d1 = [ln1(S/k) + (r-q+(s.d.^2)/2)t] / s.d. (sq. root of t)
And d2 = s.d.(sq.root of t) - d1

I understand that this would be the case if you have a dividend in percentage (i.e. continuously at x%). But what if you have a dollar dividend? How would that be incorporated into the equation?

Also, how is this related to the extended put-call (per Ch.9 Hull), where you have:
p+So = C +D+Xe^(-rt)? Generally I've only seen dividends expressed in the (-r-q) format, not in the +D format -- can you please elaborate? Thank you!

 

[David Harper CFA FRM]

Hi shi,

Call option = S0*exp[-qT]*N(d1) - K*exp^[-rT]*N(d2)
With d1, d2 as you say
…. Please note: in ADDITION to yield "inside" d1, the divide yield (q) also appears in the "outer" Black-Scholes.

There is one KEY IDEA: dividends effectively reduce the stock price
(why? a stock has a total return, price appreciate + dividends. For a given total return, higher dividend implies lower price appreciation but the OPTION HOLDER does not receive the interim dividends)

(How do I remember this ?
it is not exactly the same but I "use" the cost of carrry:
F0 = S*EXP[+rate - dividend….]
… just as mnemonic device, okay, don't read anything else into that)

If the dividend instead is dollars/discrete, then we use the BSM:
(S0 - PV[all dividends during option life])*N(d1) - K*exp^[-rT]*N(d2)
… but in this case, reduction in the "inner" d1 is not required

Here is Hull on this one key idea (Ch 13)
" European options can be- analyzed by assuming that the stock price is the sum of two components: a riskless component that corresponds to the known dividends during the life of the option and a risky component. The riskless component, at any given time, is the present value of all the dividends during the life of the option discounted from the ex-dividend dates to the present at the risk-free rate. By the time the option matures, the dividends will have been paid and the riskless component will no longer exist. The Black-Scholes formula is therefore correct if So is equal to the risky component of the stock price and o- is the volatility of the process followed by the risky component. Operationally, this means that the Black-Scholes formula can be used provided that the stock price is reduced by the present value of all the dividends during the life of the option, the discounting being done from the ex-dividend dates at the" risk-free rate. As already mentioned, a dividend is counted as being during the life of the option only if its. ex-dividend date. occurs during the life of the option. "

Re: "how is this related to the extended put-call (per Ch.9 Hull), where you have: p+So = C +D+Xe^(-rt)?"
Same idea(!) but more natural (IMO) is:
p+So - D = C + Xe^(-rt)
… (-D) on the left, reducing the stock price, effectively

David

 

[[email protected]]

I see.

So if the BSM for a dividend call is:

So^(-qt)(d1) - Ke^(-rt)(d2),

should I still account for q in the d1 calc?
i.e. should d1 be:
d1 = ln(s/k)+(r-q+(s.d.^2/2))T/(s.d.*sigmaT) or should it simply be ln(s/k)+(r+(s.d.^2/2))T/(s.d.*sigmaT)?

In other words, if we account for q in the "outer shell" of the BSM -- i.e. in the stock price -- should we still account for it in the d1? I see the formula for d1 as including the q in hull... which is why I'm confused... do we do both? i.e. include (r-q) in the calc of d1 and e^(-qt) in the calc of the stock price?

Thank you!

 

[David Harper CFA FRM]

Hi shi,

Yes, if dividend (q) is constant %, it is included in both (it "deducts" in both inner and outer)
But if lump sum (D), it does not fit "inside" d1 and does not replace (-q); that is a continuous lognormal

David

 

[[email protected]]

That's very helpful, thank you!