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  #1  
Old 03-06-2018, 11:07 PM
Actuary2017 Actuary2017 is offline
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Hi all

I am trying to solve the following problem, but I could not get any of the available choices.

Three month European call and put options on ABC stock sell at an exercise price (strike price) of $70. The stock price volatility is 35%, and the risk-free interest rate is 7.2% per annum, compounded annually. On January 1, an investor buys 100 put options and sells 100 call options. The options expire on April 1. On February 1, the stock price is the same as it was on January 1, the risk-free interest rate has not changed, and the stock price volatility has not changed. The only difference is that the options have two months left to expiration instead of three months. What is the gain (positive) or loss (negative) on the investor’s portfolio during January?

A. –$60 B. –$40 C. –$20 D. +$40 E. +$60
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  #2  
Old 03-07-2018, 04:08 AM
lylovehk1989 lylovehk1989 is offline
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Is that answer D?
Using put-call parity will get.
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  #3  
Old 03-07-2018, 04:29 AM
lylovehk1989 lylovehk1989 is offline
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Is that answer D?
Using put-call parity will get.
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  #4  
Old 03-07-2018, 07:56 AM
SweepingRocks SweepingRocks is offline
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I got B, here's how:

So the investor is buying Puts and selling Calls. He's receiving the amount for calls (C) and paying the amount for Puts (-P). So his portfolio could be expressed as 100(C-P). By Put Call Parity, we know this is equal to 100([Prepaid Forward Price of Stock]-[Prepaid Forward Price of Strike]). The thing about this question is that it says nothing about dividends, so for my sanity, I'm going to assume there are no dividends. This makes the prepaid forward stock price S. The Prepaid forward of strike with 3 months to expiration is 70(1.072^-.25), which is about 68.793807 and the prepaid forward of strike with 2 months to expiration is 70(1.072^-.1666), which is about 69.1935441.

So in January, the portfolio is 100S-6879.3807 and the portfolio in February is 100S-6919.35441. If we subtract the January portfolio from the February portfolio, we'll get -39.97371 as our answer.

Let me know if that makes any sense or if I did it wrong :P
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  #5  
Old 03-07-2018, 08:19 AM
crcosme88 crcosme88 is offline
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Quote:
Originally Posted by SweepingRocks View Post
I got B, here's how:

So the investor is buying Puts and selling Calls. He's receiving the amount for calls (C) and paying the amount for Puts (-P). So his portfolio could be expressed as 100(C-P). By Put Call Parity, we know this is equal to 100([Prepaid Forward Price of Stock]-[Prepaid Forward Price of Strike]). The thing about this question is that it says nothing about dividends, so for my sanity, I'm going to assume there are no dividends. This makes the prepaid forward stock price S. The Prepaid forward of strike with 3 months to expiration is 70(1.072^-.25), which is about 68.793807 and the prepaid forward of strike with 2 months to expiration is 70(1.072^-.1666), which is about 69.1935441.

So in January, the portfolio is 100S-6879.3807 and the portfolio in February is 100S-6919.35441. If we subtract the January portfolio from the February portfolio, we'll get -39.97371 as our answer.

Let me know if that makes any sense or if I did it wrong :P
Let G be the gain and Vt be the value of the investor's portfolio at time t. Then

G = Vt - AccumulatedValue(V0) =100S - 6919.35441 - (100S - 6879.3807)*(1.072^.1666) = -0.58107S.

So I think the gain is a function of the stock price unless we ignore interest...

Also, why is the stock volatility given?
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Last edited by crcosme88; 03-07-2018 at 08:26 AM.. Reason: Got the February and January values mixed up
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  #6  
Old 03-07-2018, 08:37 AM
SweepingRocks SweepingRocks is offline
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Quote:
Originally Posted by crcosme88 View Post
Let G be the gain and Vt be the value of the investor's portfolio at time t. Then

G = Vt - AccumulatedValue(V0) =100S - 6919.35441 - (100S - 6879.3807)*(1.072^.1666) = -0.58107S.

So I think the gain is a function of the stock price unless we ignore interest...
If that were the case, wouldn't the problem be unsolvable since we don't have the stock price?

Quote:
Originally Posted by crcosme88 View Post
Also, why is the stock volatility given?
Could be thrown in extraneously to throw you off
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Old 03-07-2018, 09:07 AM
crcosme88 crcosme88 is offline
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Quote:
Originally Posted by SweepingRocks View Post
If that were the case, wouldn't the problem be unsolvable since we don't have the stock price?



Could be thrown in extraneously to throw you off
I wouldn't say it's unsolvable, but maybe we're forgetting about some obscure formula. When interest should be ignored, it's usually stated explicitly in the problem.
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Old 03-07-2018, 09:36 AM
SweepingRocks SweepingRocks is offline
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Originally Posted by crcosme88 View Post
I wouldn't say it's unsolvable, but maybe we're forgetting about some obscure formula. When interest should be ignored, it's usually stated explicitly in the problem.
I apologize, but I don't see why my solution isn't right
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Old 03-07-2018, 09:53 AM
crcosme88 crcosme88 is offline
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I apologize, but I don't see why my solution isn't right
I think you are right. The problem is asking you to calculate the gain, not the profit, so there's no need to accumulate the value of the portfolio in January with interest.
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