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Thread: Please help for another ASM sample exam question

  1. #1
    Actuary.com - Level I Poster
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    Post Please help for another ASM sample exam question

    Thanks so much for last help...
    I have a question again from ASM samples.

    4-8. You are creating a synthetic 3-month 10,000 Treasury bill using options on a stock. You are given:
    i. The stock price is 40.
    ii. The options are European style 3-month options at-the-money
    iii. A call option has premium 2.50
    iv. A put option has premium 2.00
    v. The continuous dividend rate for the stock is 0.02
    Determine the number of shares of stock to purchase.

    a. 245.63
    b. 248.75
    c. 250.00
    d. 251.25
    e. 254.45

    The answer is B. Solution use delta= Bond Value / strike price * exp(-0.02/4).

    But what I did is using parity equation, Call-Put = S*exp(-0.02/4) - K*exp(-r/4), then Call-Put = 40*delta - 10000*exp(-r/4).
    I found exp(-r/4) by solving the equations of Call="Black Scholes equation" and Put="BS equation".
    Back to the Call-Put = 40*delta - 10000*exp(-r/4), I got delta=245.63---Answer A.

    Please someone point out my misunderstanding.....Thank you very much...

  2. #2
    Actuary.com - Level IV Poster
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    You solved for r correct? I have r~.0706.

    I solve these by using PCP myself. I know I want to create synthetic bond maturing at 10000 at time .25. So, today, I must invest 9825.05. Using PCP, I set it up as such:

    C-P-40e^(-.05*.25)=-40e^(-.0706*.25)

    The minus sign on the right hand side tells me I am investing, which is what we do when purchasing a bond. So, I need 40e^(-.0706*.25)*X=9825.05 to replicate the purchase of the bond. Therefore X=250.

    Now the trick is to realize that the prepaid forward on the stock is 40e^(-.02*.25), and we multiply the equation through by 250, so 250*40e^(-.02*.25) = 9950.1248, but each share only costs 40, so we divide by 40 to get the correct solution.

    I probably would have got this wrong on the exam, I would have made the mistake of simply choosing 250.

  3. #3
    Actuary.com - Level I Poster
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    I got the same r.
    I agree with your calculation. I first got the same as you.
    But I don't know why the solution is B-248.75.

    Urrrr...:skeptical:

    Quote Originally Posted by brandond View Post
    You solved for r correct? I have r~.0706.

    I solve these by using PCP myself. I know I want to create synthetic bond maturing at 10000 at time .25. So, today, I must invest 9825.05. Using PCP, I set it up as such:

    C-P-40e^(-.05*.25)=-40e^(-.0706*.25)

    The minus sign on the right hand side tells me I am investing, which is what we do when purchasing a bond. So, I need 40e^(-.0706*.25)*X=9825.05 to replicate the purchase of the bond. Therefore X=250.

    Now the trick is to realize that the prepaid forward on the stock is 40e^(-.02*.25), and we multiply the equation through by 250, so 250*40e^(-.02*.25) = 9950.1248, but each share only costs 40, so we divide by 40 to get the correct solution.

    I probably would have got this wrong on the exam, I would have made the mistake of simply choosing 250.

  4. #4
    Actuary.com - Level IV Poster
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    Each prepaid share is 40e^(-.05*.25). We are needing to buy 250*40e^(-.0706*.25) to replicate the price of the bond. We therefore spend 250*40e^(-.05*.25) total in buying stock. But at time 0, the stock is 40, so we divide 250*40e^(-.05*.25) by 40 to get the correct solution.

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