Actuarial Outpost Hypothetical Black-Scholes Scenario
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 Investment / Financial Markets Old Exam MFE Forum

#1
03-25-2010, 05:19 PM
 Veckatimest Member Join Date: Nov 2009 Posts: 144
Hypothetical Black-Scholes Scenario

If a question gives the following information, is it possible to price a call on the stock?

(i) The stock's price follows the Black-Scholes framework
(ii) The stock's 1-year forward price is \$75
(iii) The strike price is \$68
(iv) The prepaid forward annual volatility is 0.25
(v) It is a dividend paying stock.
(vi) The option expires in 6 mos.
(vii) The cc risk-free rate is 0.05.

If it is possible, which prepaid forward price is to be used in the formula of d1: The forward price discounted back one year, or 6 mos?

Thanks.
#2
03-25-2010, 05:29 PM
 Force of Interest Member SOA Join Date: Sep 2009 Posts: 5,119

Not positive, but I dont think you could price this. I don't see how you can get a 6 month prepaid fwd without knowing either the dividend rate or the current stock price.
__________________
Quote:
 Originally Posted by win diesel You have to take the third derivative of any of these stats to show Amoron is doing a good job.
#3
03-25-2010, 11:12 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,253

Quote:
 Originally Posted by Veckatimest If a question gives the following information, is it possible to price a call on the stock? (i) The stock's price follows the Black-Scholes framework (ii) The stock's 1-year forward price is \$75 (iii) The strike price is \$68 (iv) The prepaid forward annual volatility is 0.25 (v) It is a dividend paying stock. (vi) The option expires in 6 mos. (vii) The cc risk-free rate is 0.05. If it is possible, which prepaid forward price is to be used in the formula of d1: The forward price discounted back one year, or 6 mos? Thanks.
Yes, you have enough information.

1 year, of course.. The strike would be discounted half a year.

Retracted below.

Last edited by Abraham Weishaus; 03-27-2010 at 11:41 PM..
#4
03-26-2010, 01:39 PM
 home_alone Member Join Date: Nov 2008 Posts: 428

I don't see why Abe was saying that you should use 1 year to discount the forward price instead of 6 months.
Are you evaluating the call at t=0 or t=6 months? The evaluating date of the call would make a difference in how to solve the question.
Would you be clear on that.

Thanks.
#5
03-26-2010, 01:43 PM
 Hydraskull Member Join Date: Jun 2008 Location: Svalbard Studying for life Favorite beer: Six Point Resin Posts: 6,451

Quote:
 Originally Posted by home_alone I don't see why Abe was saying that you should use 1 year to discount the forward price instead of 6 months. Are you evaluating the call at t=0 or t=6 months? The evaluating date of the call would make a difference in how to solve the question. Would you be clear on that. Thanks.
The option expires in six months. You are pricing the call at t=0. Pricing the call at t=6 months would just be (S - K).

You use the prepaid forward price when pricing a call. You have the one year forward price. By definition, the pre-paid forward price is the discounted value of the forward price. So discount by one year.
__________________
Quote:
 Originally Posted by Abraham Weishaus Only Hydraskull knows what he's talking about. Retracted below.
#6
03-26-2010, 03:14 PM
 epeddy1 Member SOA AAA Join Date: Jan 2010 Location: East Coast Studying for None!!! College: U of Chief Illiniwek Posts: 8,860

You're given the 1 year forward price, which is current price minus PVof dividends. You're pricing the option for 6 months. But the dividend information isn't given (is it continuous? monthly? one-time dividend paid after 6 months?).

I'm pretty sure you'd use the 6-mo forward price in the BS formula, which can't be derived with the info given. If the question said "the dividend is paid next month," then you could say the 1-yr forward price is the same as 6-mo, and you could solve.

The key info missing is when the dividend is paid.

Disclaimer: I'm not extremely confident in this answer.
#7
03-26-2010, 03:33 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,253

Dividends are irrelevant to discounting forwards. To turn a forward into a prepaid forward, discount at the risk-free rate.

Retracted below.

Last edited by Abraham Weishaus; 03-27-2010 at 11:41 PM..
#8
03-26-2010, 04:31 PM
 Force of Interest Member SOA Join Date: Sep 2009 Posts: 5,119

Quote:
 Originally Posted by Abraham Weishaus Yes, you have enough information. 1 year, of course.. The strike would be discounted half a year.
not that i really need to know this anymore, but i know the BS formula for a 6 month call would use the natural log of the 6 month prepaid fwd divided by the strike discounted at the risk free rate for 6 months. So i'm curious as to why you can use the 1 year prepaid fwd since it would not necessarily be equal to the 6 month prepaid fwd.
__________________
Quote:
 Originally Posted by win diesel You have to take the third derivative of any of these stats to show Amoron is doing a good job.
#9
03-26-2010, 04:45 PM
 Hydraskull Member Join Date: Jun 2008 Location: Svalbard Studying for life Favorite beer: Six Point Resin Posts: 6,451

Quote:
 Originally Posted by Force of Interest not that i really need to know this anymore, but i know the BS formula for a 6 month call would use the natural log of the 6 month prepaid fwd divided by the strike discounted at the risk free rate for 6 months. So i'm curious as to why you can use the 1 year prepaid fwd since it would not necessarily be equal to the 6 month prepaid fwd.
It's irrelevant because the prepaid forward price is the prepaid forward price. If you use the 1 year forward price to determine the prepaid forward price, you discount by 1 year. If you use the 6 month forward price to determine the prepaid forward price, you discount by 6 months and you end up with the prepaid forward price.
__________________
Quote:
 Originally Posted by Abraham Weishaus Only Hydraskull knows what he's talking about. Retracted below.
#10
03-26-2010, 06:07 PM
 Force of Interest Member SOA Join Date: Sep 2009 Posts: 5,119

Quote:
 Originally Posted by Hydraskull It's irrelevant because the prepaid forward price is the prepaid forward price. If you use the 1 year forward price to determine the prepaid forward price, you discount by 1 year. If you use the 6 month forward price to determine the prepaid forward price, you discount by 6 months and you end up with the prepaid forward price.
but you would arrive at different answers for the prepaid fwd under each of those scenarios, which was what my question was.

One of those would discount dividends for a year, and one would do it for 6 months.

for example, 50e^-d would not equal 50e^-.5d, so if the original question gave you the 1 year fwd and the 6 month fwd, you would get different answers for d1 and d2, so i must be missing something else.
__________________
Quote:
 Originally Posted by win diesel You have to take the third derivative of any of these stats to show Amoron is doing a good job.