The $k$ period log return is defined as $$r_{t}(k)=log(S_{t}/S_{t-k}),$$ Where $S_{t}$ is the stock closing price at time $t$. For argument sake, assume that by time I mean a stock trading day and historical stock closing prices are known and denoted by $S_{1}, S_{2}, S_{3}, ...., S_{T-1}$. Here are my questions:

  1. I have seen people trying to predict the change in stock price direction e.g. one period log return i.e. $r_{T}(1)=log(S_{T}/S_{T-1})$ is positive or negative based on past historical data. The argument is that they buy if they expect $r_{T}(1)>0$ and they sell if $r_{T}(1)<0$.

    • (a) Could anyone please tell me in this case that at which point of time during trading day $T$ exactly we should buy the stock and at which point of time we should sell?
    • (b) Should we buy in the morning of day $T$ and then wait until the end of day $T$ to sell it?
    • (c) Can we put a sell order exactly at closing time on day T?
    • (d) Given $k=1$, does this sort of trading strategy fall in the form of "day trading"?
  2. Suppose one predicts $r_{T}(5)=log(S_{T}/S_{T-5})>0$.

    • (a) Does this mean that investor should by the stock at time $T-5$ i.e. 5 days earlier and hold on to it for the next five days and the sell it in trading day $T$?

you should be buying and selling over whatever the prediction time interval was. So, if you're return prediction was positive and was defined as being over the period wednesday from 10:00 am to 4:00 pm , then you should be buying on wednesday at 10:00 am and selling at 4:00 pm.

The prediction time interval is defined by the model one is using to predict the direction so it's up to the modeller.

This is more of a generic answer but maybe it answers both of your questions ? If not, maybe someone else will say more. I've never tried direction prediction.

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