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It a simple school problem with five different parts. I understand a), b), c), and e); however I am very confused on what d) means. The problem goes like this:

"Suppose that each day the price of a stock moves up 1/8 of a point, moves down 1/8 of a point, or remains unchanged. For i>=1, let U_i and D_i be the events that the price of the Stock moves up and down on the ith trading day, respectively. In terms of U_i's and D_i's, find an expression for the event that the price of the stock: d) is the same as today after three trading days;"

What I do not understand about d) is that is it saying the price of stock has not changed and has remained the same across all three days (or possibly four because of the 'today' part)?Could it possibly mean that the stock has been steadily rising or falling for the past I don't know how many days. Is it saying that it is the same trend as of three days ago? Could someone please help clarify this for me?

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+1 for a nicely phrased question. Because this is a homework/self-study problem, could you add the self-study tag to your question? See stats.stackexchange.com/tags/self-study/info –  Patrick Coulombe Aug 29 at 19:52

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up vote 3 down vote accepted

The word order in the question might be a bit confusing. You are asked to find an expression for the event "that the price of the stock is the same as today after three trading days". A more clear wording might have been "the price of the stock after three trading days is the same as [the price of the stock] today." That is, the event in question is defined by stating equality of two values . These values are "the price today" and "the price after three days". What the price is between these days, or before today, or after the third trading day, does not matter.

Some numeric examples (3 made up histories for the price of the stock): \begin{equation} \begin{array}{cccc|c} \textrm{Today} & \textrm{After 1st day} & \textrm{After 2nd day} & \textrm{After 3rd day} & \textrm{Comparison} \\ 1 & 1 \frac{1}{8} & 1 \frac{1}{4} & 1 \frac{3}{8} & 1\neq 1\frac{3}{8} \\ 1 & 1 \frac{1}{8} & 1 \frac{1}{4} & 1 & 1 = 1 \\ 1 & 1 & 1 & 5 & 1 \neq 5 \end{array} \end{equation}

The event defined in part d of the question occurs if the stock price behaves according to the second row, but does not occur if the stock price behaves according to the first/third row. (Note that my example price histories in the 2nd and 3rd rows are not compatible with the process defined in the question. This is deliberate to highlight that the problem here seems to be about the interpretation of the wording, which is unrelated to the details of the process specification)

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Thanks this helped a lot! –  user2544603 Aug 29 at 20:27

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