Pg 69, Hull, Chapter 15: the BSM Model (Hull Text Q&A)

[Dr. Jayanthi Sankaran]

Hi David,

Given the following problem:

Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is six months, the volatility is 30% per annum, and the risk-free interest rate is 10% per annum. Two equal dividends are expected during the life of the option, with ex-dividend dates at the end of two months and five months. Assume the dividends are 40 cents. Use Black's approximation and the Derivagem software to value the option.

Answer

I get Hull's price of 0.7947, assuming that there is no early exercise. However, Hull goes on to value the option assuming that it is exercised at the five-month point just before the final dividend. The value of the option is then 0.7668. How do you get this value?

Thanks!
Jayanthi

 

[Dr. Jayanthi Sankaran]

In page 68 of the same reference as above, I notice that value of d2 calculated by using the BSM formula:

d2 = [ln(S(0)/K) + (r - sigma^2/2)*T]/sigma*SQRT(T) is different from
d2 = d1 - sigma*SQRT(T)

Is this because the stock pays dividends?

Thanks!
Jayanthi

Sorry - the data for the problem on page 68 is time to ex-dividend date = 0.125, D = $0.50,
S(0) = $30, K = $29, r = 5% per annum, sigma = 25% per annum and Time to maturity = 0.3333

 

[QuantMan2318]

I think Hull took it that it is better to exercise the American options just before the final Ex-dividend date. The option having dividends cannot be taken to be exercised at maturity as we have to first check if its possible to exercise it prior to maturity. Hull states that whenever Dn>K(1-e^(-r*(T-tn))), it is always advisable to exercise the option at just before the final Ex-dividend date tn. We can check for other ex-dividend dates also, but American options almost always are viable to be exercised just before the final ex-dividend date. Therefore, he took it at the 5 month point.

Now we can take the time to expiry as 5 months, and just input one dividend at the end of 2 months and then if we calculate using DerivaGem we get 0.7668.

In page 68 of the same reference as above, I notice that value of d2 calculated by using the BSM formula:

d2 = [ln(S(0)/K0) + (r - sigma^2/2)*T]/sigma*SQRT(T) is different from
d2 = d1 - sigma*SQRT(T)

Is this because the stock pays dividends?

Both are the same formulas, we can derive d2 from d1 which is the second formula or use the original formula, if we expand d1-sigma*SQRT(T), we get the first formula. Of course here S(o) should be S(o)-PV of all dividend.

 

[Dr. Jayanthi Sankaran]

Hi @QuantMan2318

Thanks for your detailed answer. I appreciate it. Conceptually, as far as the question on page 69 goes, I am clear about the value of the option being exercised just prior to the last ex-dividend date. However, I do not know how to use the DerivaGem software. I will calculate it manually, as will be required for the exam!

As regards the question on page 68, the reason for my query was that despite taking the present value of dividends in the d2 and d1 formulae, I get

d2 = [ln(S(0)/K) + (r - sigma^2/2)*T]/sigma*SQRT(T)

d2 = [ln($29.5031/$29) + (.05 - .25^2/2)*0.3333/.25*SQRT(.3333) = 0.1625

d2 = d1 - sigma*SQRT(T)
d2 = 0.3608 - .25*SQRT(.3333) = 0.2165 where

d1 = ln(S(0)/K) + (r + sigma^2/2)*T]/sigma*SQRT(T)
d1 = ln($29.5031/$29) + (.05 + .25^2/2)*0.3333/.25*SQRT(.3333) = 0.3608

I don't know where I am going wrong in my calculation as far as d2 is concerned!

Thanks
Jayanthi

 

[QuantMan2318]

Dear Ms. Sankaran

Thanks for your reply, can you tell us the question that is on pg. 68?

I seem to get 0.3068 as d1 based on your data and not 0.3608 and when that is input into d1-sigma*SQRT(T), I get 0.16246 which is the same as the original d2 formula. I took the following based on your data:
rf 5%
K $29
sigma 25%


Thanks
Manikandan V R

 

[Dr. Jayanthi Sankaran]

Dear Mr Manikandan,

Thanks for pointing out my typo. Yes, I just calculated the value for d1 and it turns out to be 0.3068. d2 then turns out to be d1 - sigma*SQRT(T) = .3068 - .25*SQRT(0.3333) = 0.16247 which is the same as the original d2 formula. Thanks for your clarification - appreciate it!

Jayanthi Sankaran

 

[Dr. Jayanthi Sankaran]

Dear Mr Manikandan,

As you rightly point out - Hull states that whenever Dn > K(1-e^(-r*(T-tn))), it is always advisable to exercise the option at just before the final Ex-dividend date tn. For the problem on page 69 above, this turns out to be

$0.40 > $20*[1 - exp^(-0.1*(0.5 - 0.4167)] = 0.16591 approx. 0.166

How do you manually calculate the value of the option assuming that it is optimal to exercise the option just before the five-month ex-dividend date?
Hull gets the value of the option using DerivaGem to be 0.7668. If you have the time could you please go through the steps in arriving at this value, manually?

Thank you very much!
Jayanthi Sankaran