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A stock XYZ is trading at $\$11.70$ today.

Consider an American $put$ option on 1 share of XYZ with strike $K$ = 12 and expiration in $T$ = .25 years.

The put is selling today for $\$1.46$. Assume the risk-free rate is $1\%$.

Times goes by and 1 month later, the put option is trading at $\$0.85$. You decide to sell your put option. What is your profit or loss?

The trouble I am having with this problem is that I am not sure if there is a difference when you $sell$ or $exercise$ a put. I believe they are different. If so, then when I sell my put option, would I just calculate the future value of my put $\$1.46$ and subtract what it is worth it now($\$.85$)?

I am just confused on how to go about this problem.

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    There is a big difference between selling and exercising! If the put has time value left then it has positive value, even if it is out of the money at the time. – 2017-02-21
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    So when you sell, you are selling someone the option of exercising the option at any time. But what is the person buying paying, since the option won't pay off until the future? – 2017-02-21
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    In this case, the person is paying $.85$. I know that because you said so. More abstractly, the person pays the expected value of the exercise in the future. Note: as it is an American put you can't just look at the expiry date, it might have more value if exercised early. – 2017-02-21
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    So in this case, the profit/loss is the difference of premium paid? – 2017-02-21
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    Well, not quite. Your man bought the premium on one day and sold on a different day. To reckon $P\& L$ you need to align the pricing dates. Specifically, you should compute the value of the initial premium as of the sale date. – 2017-02-21
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    I understand it now. thx! – 2017-02-21

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