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#1
11-01-2008, 01:18 PM
 badmaj5 Member CAS Join Date: Sep 2005 Location: Chicago, IL Posts: 454
ASM Practice Exam 9, #3

A market maker writes a call option on a stock and purchases half the amount of stock needed to delta-hedge the option. You are given:
i) the stock price is 35
ii) the stock pays no dividends
iii) the risk-free rate is .1
iv) the option premium is 3
v) delta for the option is .45
vi) gamma for the option is .12
vii) theta for the option is -.02 per day

Estimate the profit to the market maker one day later if the stock price declines to 33.

Why does this solutions NOT work??

-[.5*.12*(-2)^2 - .02 + (.1/365)*(.5(.45)(35) - 3)]

All I'm doing is using the regular formula for market maker profit, but saying they bought half the stock necessary to do so by using .5(.45)(35) instead of just (.45)(35). This gets me -.22, but the correct answer is .23 by not using this formula but instead breaking the profit into each separate piece. I don't know why what I did doesn't work??
#2
11-01-2008, 01:39 PM
 Bruin086 Member Join Date: Jun 2008 Posts: 550

Quote:
 Originally Posted by badmaj5 A market maker writes a call option on a stock and purchases half the amount of stock needed to delta-hedge the option. You are given: i) the stock price is 35 ii) the stock pays no dividends iii) the risk-free rate is .1 iv) the option premium is 3 v) delta for the option is .45 vi) gamma for the option is .12 vii) theta for the option is -.02 per day Estimate the profit to the market maker one day later if the stock price declines to 33. Why does this solutions NOT work?? -[.5*.12*(-2)^2 - .02 + (.1/365)*(.5(.45)(35) - 3)] All I'm doing is using the regular formula for market maker profit, but saying they bought half the stock necessary to do so by using .5(.45)(35) instead of just (.45)(35). This gets me -.22, but the correct answer is .23 by not using this formula but instead breaking the profit into each separate piece. I don't know why what I did doesn't work??
You used the Market Maker DH formula for profit but he's NOT exactly delta hedged so that's why it doesn't apply.

Rather, use the MM profit formula where profit = change in call price + delta * change in stock price + interest lost

So first you need to find the price of the Call after 1 day and with the stock dropping by 2.

C(epsilon = -2, h=1) = 3 + .45*-2 + .5*.12*(-2)^2 + (-.02)(1)
this equals 2.32

Now we have the necessary information for our profit formula. Let's take it piece by piece.

Profit on calls: because the MM wrote the call, he sold it for 3 at time 0 and can sell it one day later for 2.32. This means the profit = (3-2.32) = .68

Profit on stocks: The market maker delta hedges by buying stocks at time 0 and can sell it one day later for 33. Because it's a HALF delta hedge, we divide delta by 2. Profit = .45/2*(33-35) = -.45

Finally we need our interest piece. this equals -(rh)(delta/2 * S(0) - C(0)) = (.10/365)(.45/2*35-3) = -.001336

so our total profit is profit from calls + profit from stock - interest lost = .68 - .45 - .001336 = .22866 = .23
#3
11-01-2008, 01:42 PM
 cscottie31 Member Join Date: May 2007 Posts: 163

I screwed this up, too. Here's what I grasped from the solution:

Item (v) in your list shows that delta for the option is 0.45. You have sold one option, so the market-maker's profit on the call alone depends on the full 0.45. In addition to selling the option, to hedge this position the market-maker purchased half of the amount of shares of stock needed to delta hedge. In order to fully delta-hedge, 0.45 shares should've been purchased. However, only 0.225 shares have been purchased. Since the stock dropped in value, this is a loss to the market-maker.

Also, notice that this solution does not use the formula for market-maker profit, but the one for the overnight profit from a delta-hedged portfolio. It just used the approximation for interest. From formula (11.1) on page 195, the actual interest is [exp(.1/365)-1]*4.875=0.001335799, which is the approximation rounded.
#4
11-01-2008, 01:46 PM
 Bruin086 Member Join Date: Jun 2008 Posts: 550

you can use the actual interest calculation, not the approximate interest, and you still get the same answer. like you said you just need to use the overnight profit formula.
#5
11-01-2008, 02:04 PM
 merrycherub Member Join Date: Aug 2007 Posts: 112

excellent!!!

Quote:
 Originally Posted by silva086 You used the Market Maker DH formula for profit but he's NOT exactly delta hedged so that's why it doesn't apply. Rather, use the MM profit formula where profit = change in call price + delta * change in stock price + interest lost So first you need to find the price of the Call after 1 day and with the stock dropping by 2. C(epsilon = -2, h=1) = 3 + .45*-2 + .5*.12*(-2)^2 + (-.02)(1) this equals 2.32 Now we have the necessary information for our profit formula. Let's take it piece by piece. Profit on calls: because the MM wrote the call, he sold it for 3 at time 0 and can sell it one day later for 2.32. This means the profit = (3-2.32) = .68 Profit on stocks: The market maker delta hedges by buying stocks at time 0 and can sell it one day later for 33. Because it's a HALF delta hedge, we divide delta by 2. Profit = .45/2*(33-35) = -.45 Finally we need our interest piece. this equals -(rh)(delta/2 * S(0) - C(0)) = (.10/365)(.45/2*35-3) = -.001336 so our total profit is profit from calls + profit from stock - interest lost = .68 - .45 - .001336 = .22866 = .23
#6
11-01-2008, 02:21 PM
 billvp Member Join Date: Nov 2007 Posts: 303

you should always do these piece by piece, they're fairly obvious how to do that way
#7
11-01-2008, 07:19 PM
 badmaj5 Member CAS Join Date: Sep 2005 Location: Chicago, IL Posts: 454

Thanks everybody!

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