BSM with dividends

[The Great Khan]

Hi All,

When I look at the BSM with dividends formulas in the study notes, I find something that is inconsistent with when I do the problems.

Specifically the part that trips me up is in the calculation of d1, we take ln(So/K), which when I was studying seemed to be specifically the current price, not discounting for dividends, since the BMS formula went out of it's way to specify So is discounted. However, when doing the problems it seems to use So discounted continuously at the dividend rate.

Am I correct in understanding that these two formulas use the notation So in a different way, with BSM formula explicitly discounting, and d1 assuming that So is already discounted?

 

[ami44]

In d1 the S0 in ln(S0/K) is the current price. The meaning is consistent in all formulas that you are citing.
What gave you the impression, that S0 is differently defined somewhere?

Also compare https://en.wikipedia.org/wiki/Black–Scholes_model

 

[The Great Khan]

P1.T4.5.3

5.3. A European call option has a time to maturity of six months on a stock with ex-dividend dates in two and five months. Each dividend pay $1 per share. The current stock and strike prices are both $50. The volatility of the stock is 18% per annum. The risk free rate is 4%. What is the price of the call option?

a) $2.00 b) $2.75 c) $3.08 d) $3.16


Answer:

The present value of the dividends = $1.00*exp(-2/12*4%) + $1.00*exp(-5/12*4%)
= $1.976827
Reduce stock price by PV of dividends such that S(0) = $50 - $1.976827 = $48.0232
d1 = [LN(48.0232/50) + ([4% + 18%^2/2]*0.5)] / [18%*SQRT(0.50)] = -0.0962;
d2 = d1 - 18%*SQRT(0.50) = -0.2234
N(d1) = 0.4617 and N(d2) = 0.4116

Then we apply the Black-Scholes, but please note the reduced stock price appears BOTH in the "outer" formula and in the "inner" d1 and d2:
c = S*N(d1) - K*exp(-rT)*N(d2) = $48.0232*0.4617 - $50*exp(-4%*6/12)*0.4116 = $2.000
​

As you can see, the current stock price is $50, but in calculating d1 he used the reduced stock price in both formulas.

 

[Dr. Jayanthi Sankaran]

Hi @The Great Khan,

As far as I can see, there seems to be no inconsistency as @ami44 has said!

 

[ami44]

You are right in the fact, that the price in d1 has to be discounted with the dividends too. This discounting is already included in your formula in the r - q term.
For better understanding, rewrite the numerator in d1 as: ln( S0/K * exp( (r - q + σ^2/2) * T) ) you see, the dividend is indeed subtracted from S0.

In the exercise that you cite is a discrete dividend instead of a continuous dividend yield used. If we call the NPV of this Dividend δ then d1 is calculated with the numerator ln((S0-δ)/K) + (r + σ^2/2) * T they could have also used ln( S0/K) + (r - ln(-(S0-δ)/S0)/T + σ^2/2) * T)
In this case ln(-(S0-δ)/S0)/T is taking the place of the continuous yield q

Did that help?

 

[The Great Khan]

Thanks ami,

That explains it. I was getting confused due to the discrete dividends.

 

[David Harper CFA FRM]

wonderful explanation @ami44 !!