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Black Scholes: Question 9 Sept2023 Exam

B

Bernadette Pieterse

Active Member
Hello,

For Q9 in the Sept2023 exam:
Why do they add 0.1*exp(-0.02*10) at the end when calculating the price?
Also, how does this question differ from the first practice question in the chapter that suggests a long put and short call? The wording seems almost identical.

Thanks in advance!
 
This bit is cash accumulating to 10% to make sure that the floor is 10%. The principle of the hedge can be achieved in many different ways:
long call (10), short call (50)
long put (10), short put (50)
underlying, short call (50) and long put (10)

However, the "heights" of these strategies would be different on the position diagram. So, if we want the "height" to be a particular number (10 in this case) then we need to use cash.

Challenge for you - create the same diagram in Excel using all 3 of these strategies and use cash to correct the height of your diagrams.
John
 
I think the difference is that the studying material is asking to hedge against the risk but not replicating the payoff at maturity. But the past paper is asking to price the investment product which involved replication of payoff at maturity. I do not quite get the why 0.1*exp(-0.02*10) being the floor.
I believe the answer has assumed the initial price of stock index to be 1 so payoff at maturity according to my methodology are as follows:
1.1 for 0<=S_T<=1.1
S_T for 1.1<S_T<=1.5
1.5 for S_T>1.5


"height" in this case would be 1.1
underlying, short call (50) and long put (10) would lead to an answer for the investment product price to be
long put (return @10%) +short call (return @50%) + stock index= 0.07611 + 0.0541 + 1 = 1.022

If I replicate the payoff using call options only, the replicating portfolio would be:
long call (10), short call (50), cash of 1.1exp(-0.02*10)
long call (return @10%) +short call (return @50%) + cash of 1.1exp(-0.02*10)= 0.17551 + 0.0541 + 0.90060 = 1.022

However if I draw the payoff diagram according to the answer, the payoff would be:

0.1 for 0<=S_T<=1.1
S_T-1 for 1.1<S_T<=1.5
0.5 for S_T>1.5

why the payoff at maturity is represented as the percentage of return of the initial investment but not the total amount (initial investment + return). Why 10% is sensible to be the floor but not 110%?
 
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