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#1
05-13-2008, 05:30 PM
 enirehtac07 Join Date: Nov 2007 Posts: 23
SOA Sample #18

I'm not sure if a thread already exists for this problem, but I'll post it anyway.
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A market-maker sells 1,000 1-year European gap call options, and delta-hedges the position with shares. You are given:

Each gap call option is written on 1 share of a nondividend-paying stock.
The current price of the stock is 100.
The stock's volatility is 100%.
Each gap call option has a strike price of 130.
Each gap call option has a payment trigger of 100.
The risk-free interest rate is 0%.

Under the Black-Scholes framework, determine the initial number of shares in the delta-hedge.
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The solution uses:
delta_gap = delta_regular call - 30*N'(d2)/(S*sigma*sqrt(T))

I understand this formula except for the last part - why is the second term being divided by S*sigma*sqrt(T)?

Thanks!!

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ANS: 586
#2
05-13-2008, 05:38 PM
 its_me Member Join Date: Jan 2003 Location: AO Posts: 1,772

Search for the post where jraven derives it step-by-step.
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#3
05-13-2008, 05:40 PM
 blahbla Member Join Date: May 2008 Posts: 1,408

Quote:
 Originally Posted by enirehtac07 I'm not sure if a thread already exists for this problem, but I'll post it anyway. ----------------------- A market-maker sells 1,000 1-year European gap call options, and delta-hedges the position with shares. You are given: Each gap call option is written on 1 share of a nondividend-paying stock. The current price of the stock is 100. The stock's volatility is 100%. Each gap call option has a strike price of 130. Each gap call option has a payment trigger of 100. The risk-free interest rate is 0%. Under the Black-Scholes framework, determine the initial number of shares in the delta-hedge. ----------------------- The solution uses: delta_gap = delta_regular call - 30*N'(d2)/(S*sigma*sqrt(T)) I understand this formula except for the last part - why is the second term being divided by S*sigma*sqrt(T)? Thanks!! ----------------------- ANS: 586
lol i actually spent about an hour deriving delta for a regular black scholes call the long way....that was awesome..JK!

the partial derivative with respect to S of 30N(d2) is equal to

30*N'(d2)* d (d2)/dS that 1/S*sigma*sqrt(T) is d/dS(d2)
#4
05-14-2008, 10:15 AM
 badmaj5 Member CAS Join Date: Sep 2005 Location: Chicago, IL Posts: 454

anyone know how the end result would look for delta_gap if we were looking at a put instead of a call in this problem?
#5
05-14-2008, 11:12 AM
 blahbla Member Join Date: May 2008 Posts: 1,408

Quote:
 Originally Posted by badmaj5 anyone know how the end result would look for delta_gap if we were looking at a put instead of a call in this problem?
it would come out to be 1 - delta of the gap call
#6
05-14-2008, 11:19 AM
 The Spocker Guest Posts: n/a

Quote:
 Originally Posted by its_me Search for the post where jraven derives it step-by-step.

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