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QFI Core Exam Old Advanced Portfolio Management Forum

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  #1  
Old 02-26-2016, 05:08 PM
komorgan komorgan is offline
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Default Mistake in Paul Wilmott's Intro. Quant. Finance Solution Manual?

Hello,


http://www.wiley.com/legacy/wileychi...2/supp/c08.pdf

Had a question about the answer to Question 5 in the attached link. Specifically, in the scenario where the stock increases until expiration (i.e. Situation A), I'm confused about the part where it says "The cost
of buying the share in this fashion should be equal to the initial
premium for the option plus the exercise price, E"

I did an experiment (see attached workbook) where the stock increases over 10 years, and the call option writer rehedges his/her position at the end of each year by purchasing more (or less) stock, according to the change in the value of delta (N[d1]).

Ultimately, the total net cost of purchasing stocks (100.28) was about equal to ONLY the exercise price (100), NOT equal to exercise price PLUS premium (which I assume was the initial value of the option, 58.24).

Is that quoted statement incorrect, or did I make a mistake in my math.

Thanks.
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  #2  
Old 02-26-2016, 05:20 PM
komorgan komorgan is offline
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NOTE: I reposted the above post under "Quantitative Finance and Investment (QFI) track", as I thought this was a more relevant topic for this question.
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Old 02-26-2016, 06:21 PM
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JVP3122 JVP3122 is offline
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I think there's something wrong with your math in the spreadsheet, but I didn't spend enough time looking through it to figure out what it is.

What your spreadsheet suggests is ultimately that you could write a call and collect premium, then delta hedge and ultimately at expiration own one share (at a cost approximately equal to the strike price) and then deliver that to the person who bought your call. This would imply that there is some arbitrage opportunity for the call writer who then collected that premium and was able to buy the stock over time at essentially the strike price.

Just looking again now I noticed something weird going on in row 7 (volatility). I might suggest you look there to correct the spreadsheet to see if it matches up. If you fix it could you re-upload it? I'm at the end of my day so I don't feel like sitting and trying to fix it.
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Old 02-26-2016, 07:11 PM
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Zakfischer Zakfischer is offline
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You're close!

The first mistake is to fix your vol -- it should be constant.

The second is that the total cost of stock purchases needs to take care of the time value of money properly. For example, if you need to buy half a share at time 0, then that cost will accumulate at the risk free rate to the end of the projection. You had to borrow money to buy this stock, and the loan would grow at the risk free rate. Therefore the cost increases.

I tried more refined time steps to get closer.

Also, just as a sanity check, you know that the cost has to be greater than the exercise price. You will buy some at time 0 at the exercise price, but the rest will be at a higher price. So your conclusion cannot be right!

Lastly, look at put call parity. If you solve for SN(d1), you can see why their conclusion works.

Hopefully that helps!
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Last edited by Zakfischer; 02-26-2016 at 07:16 PM..
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Old 02-28-2016, 02:33 PM
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There is no 'apparent' problem in the proposed solution.

the discounted net cost of hedging shall be equal to the call option value (under the BS model).

Under scenario (a): The net cost of hedging = (The cost of purchasing the shares to remain delta neutral during the period)-(The exercise price).

Under scenario (b): The net cost of hedging=The total loss incurred when shorting shares to remain delta neutral.

As indicated in earlier: Why do you need to change the volatility of the option during the option life? Under BSM, the vol stays constant....

Any 'calculated' difference between the discounted net cost of hedging and the B/S price may be due to (i) frequency of re-balancing and more practically, to (ii) transaction costs.
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Old 02-28-2016, 03:17 PM
komorgan komorgan is offline
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Okay, I think I made all the necessary corrections (can't believe I overlooked the volatility problem!!). Let me know if any further corrections are needed.

So now the premium is 57.08 while the Present Value of Stock Purchases Minus Exercise Price is 49.92. So I'm assuming the difference is exclusively attributed (since we're assuming no transaction costs) to the fact that I'm not re-hedging continuously.

Thanks everyone for all the help!
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Last edited by komorgan; 02-28-2016 at 03:49 PM..
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Old 02-28-2016, 10:35 PM
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It looks good.

Note how the calculation for the cost of hedging for scenario (b) is different from that of scenario (a). Under (b), you need not deliver the asset since the option is not exercised.

Good job!!
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