A BSM question

[Anir]

Hi Team BT,

Request help with this question.

A non-dividend-paying stock is currently trading at USD 40 and has an expected return of 12% per year. Using the Black-Scholes-Merton (BSM) model, a 1-year, European-style call option on the stock is valued at USD 1.78.
The parameters used in the model are:
N(d1) = 0.29123 N(d2) = 0.20333
The next day, the company announces that it will pay a dividend of USD 0.5 per share to holders of the stock
on an ex-dividend date 1 month from now and has no further dividend payout plans for at least 1 year. This
new information does not affect the current stock price, but the BSM model inputs change, so that:
N(d1) = 0.29928 N(d2) = 0.20333
If the risk-free rate is 3% per year, what is the new BSM call price?
a. USD 1.61
b. USD 1.78
c. USD 1.95
d. USD 2.11

Answer as per GARP is (C).

But how can a call price increase when there is a dividend announcement?

The dividend need to be discounted and reduced from the current stock price:
New S0* N(d1)= 40- (0.5*EXP(-3%*1/12)*0.299828= 11.8219
Before the dividend the So*N(d1)= 40*0.29123= 11.6492.
Hence the change 11.8219-11.6492= 0.1727 should be reduced from existing call price of 1.78.

Please help me understand where am I going wrong with this.

Thanks,
Anir.

 

[ShaktiRathore]

Hi
So which is stock price does not change after day 1 as per question and rest of inputs changes as per question which accordingly changes N(d1) and N(d2). Here we are not considering the effect of dividends only but also other inputs as volatility,thats why there is net increase in option price(may be volatility estimate increase after day 1 etc.)
Thanks

 

[David Harper CFA FRM]

I agree with @Anir : this question makes no sense to me. I am submitting it to GARP. (The expected return of 12% is a red herring; it should play no role)

In BSM, the introduction of a dividend:

• Decreases S(0), as shown in the answer correctly
• Decreases d1 and therefore N(d1), unequivocally, as it reduces the numerator in d1
• Decreases d2, similarly
  
• And overall must decrease the option value, as discussed in Hull.

To leave N(d2) unchanged in just plain confusing. But to increase N(d1) is wrong. You can't introduce unrealistic changes and expect the reader to assume them. Thanks,

 

[ShaktiRathore]

Yes there is errors in the new values of N(d1) and N(d2) with the dividend the resulting value of d1/N(d1) is increasing?? Which should be decreasing due to dividends. d2 shall also change with dividends, assumming a constant d2 is absolutely absurd then.how can the question assume unchanged N(d2) value is something questionable.
Assuming something questionable as above is not what expected from Garp. Just assuming anything is not something expected.yes question can be answered using above assumptions and answer shall also come but answer itself becomes faulty when assumptions themselves are faulty defeats the purpose of the question.
Thanks

 

[td]

I had another question on this one.

Conceptually, does the BSM assume:
1. dividends are a risk free asset,
2. can be discounted using risk free return,
3. dividends will be instantaneously incorporated into a stock price
4. and that the stock price will subsequently grow at rate determined by 1+ ND1?

Since the dividend was a one time event and was quoted per share, I calculated the present value of the dividend separately without multiplying times ND1. How can you discount the dividend by the risk free rate and still project the net share price forward using ND1. This seems to be a contradiction.

 

[td]

would a dividend payment impact the price of a European put?

 

[ShaktiRathore]

Hi
Yes Dividend will impact price of european put by increasing its value. Yes dividend are certain and known in advance therefore should be discounted by risk free rate,dividends lowers value of share by there present value.
Thanks

 

[sdonahue]

So, to clarify, BSM for a European Call should read:

C = Se^(-q(e^-r*t)*t)*N(d1) - K(e(-r*t))*N(d2)

Correct?

Thanks,

 

[David Harper CFA FRM]

Hi @sdonahue Here is simple spreadsheet for BSM with dividend, so you can see it in action https://www.dropbox.com/s/jwh1ort66gyh9wy/bsm-0514.xlsx?dl=0
This assumes continuous dividend yield (q) rather than lump sum, such that:

• c = S*exp(-qT)*N(d1) - K*exp(-rT)*N(d2), where d1 = [ln(S/K) + (r - q + σ^2/2)*T]/[σ*sqrt(T)]

i.e., mathematically the dividend reduces the stock price, effectively, in both the "outer" BSM and the "inner" d1. If instead we compute the lump sum, then it simply subtracts directly from the (S) and the (q) is omitted. Thanks,

 

[sdonahue]

Thank you, David. I understand now. Sorry for the confusion.

 

[David Harper CFA FRM]

Yes @sdonahue the BSM is a risk-neutral valuation such that the expected return given in the question is a red herring: the risk-free rate is used for drift and to discount the dividend. Discounting the lump sum of $0.5 is analogous to the continuous discounting given by exp(-qT). So the answer here is correct to both reduce d1 and to replace S(0) = 40 with S(0) = $39.50 = 40 - 0.5*exp(-3%*1/12); i.e., stock price reduced by the present value of a $0.5 dividend received in one month