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The Black, Scholes and Merton option pricing model assumes the stock price changes are lognormally distributed.

Then, How to show graphically

  1. How this distribution changes when

1a)an investor or/and trader is long the stock and long the put?

1b)an investor or/and trader is long the stock and short the call?

How to answer this question? I know the stock price changes are distributed lognormally.

Any hint or even correct answer to both these questions will be accepted.

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  • You probably want Quantitative Finance, but I don't know that site's policy on homework questions
    – AakashM
    Commented Feb 8, 2022 at 8:58
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    I don;t see how a trader holding an option and/or the stock changes the distribution of stock prices at all. Maybe you're reading the question wrong? Or it's a trick question?
    – D Stanley
    Commented Feb 8, 2022 at 14:33
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    1a) an investor or/and trader is long the stock and long the put? (1a) is equivalent to a long call. 1b) an investor or/and trader is long the stock and short the call? (1b) is equivalent to a short put. If you are asking what the P&L diagrams for these strategies are, they are easily seen by googling for "option strategy graphs". If not, then what are you after? Commented Feb 8, 2022 at 16:04

1 Answer 1

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If you have a distribution of stock prices then the lower limit is 0. However, if the position is the stock plus a (long) put position then the value of the position cannot be less then the strike price on the put. Hence at the strike price, there is a sharp vertical cut off.

When the investor is long the stock and short calls against it, the maximum value of the position is the strike price.

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  • I agree with your answers to both the questions. But for more clarity , would you give me a graphical pictogram? I didn't understand meaning of 'sharp vertical cut off at the strike price' in your answer to first question. Commented Feb 8, 2022 at 15:21
  • @DhamnekarWinod If you look at this URL: optionsmile.com/researches/… you will see on the right hand side a chart of possible values for a covered calls. You will notice that there is a vertical line. That is, the value of the position cannot be more than the strike price.
    – Bob
    Commented Feb 8, 2022 at 20:21

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