# Logarithmic returns for realized variance?

I am wondering which method makes more sense when computing log returns. I am trying to compute log returns for realized variance, and I have the opening and closing prices for every minute.

Since the log return is defined as

$$r_{t+1} = \ln \left(\frac{p_{t+1}}{p_t} \right)$$

should I take the average of the open and closing prices at every point as use that as $p_t$?

Or should I find $r_{t+1}$ at every $t$ by assuming $p_{t+1}$ = closing price and $p_t$ = opening price?

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entirely depends on what you are trying to achieve. Similar question: "Should I buy a car today or in 20 years?" –  Matt Wolf Feb 28 '13 at 2:45

It depends on your investment strategy. The most common approach is to use the close price of $p_t$ and $p_{t+1}$. The volatility you measure using this method implies the "assumption" that your are able to trade at close every day.

If you choose to compute the daily returns from open to close, then you assume that you are selling your position every night and buying it back every morning: you have no overnight exposure which is unlikely unless you are fairly sophisticated.

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You should find r at every t by assuming p(t+1) = closing price and p(t) = Last period closing price.

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Could you explain why the user should do that? –  chrisaycock Feb 28 '13 at 12:10

It took me a while to figure-out what you were asking. If I understand you correctly, you have minute-level OHLC market data and you want to compute the returns from this time series.

Returns are normally computed close-to-close. Whether you have one-minute bins or you have daily aggregates, you should ignore the opens and just compute returns from one close to the next.

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