# Realized volatility vs close to close return and open to close return

I am calculating daily volatility in 3 ways:

• Realized variance=> sum of square of 5 minute returns for each trading day(from 09:30 to 16:00)
• Close to close return=>(ln(close price at day i)-ln(close price at day i-1))^2
• Open to close return=>(ln(close price at day i)-ln(open price at day i))^2

There are almost 2000 days at the data.

Mean of square root of above calculations are so different from each other:

• volatility related realizations(5 min) is 0.0153
• volatility related to close to close price is 0.0125
• volatility related to open to close price is 0.0105

Whay are day so different from each other? Is it natural? Or do I have a mistake?

I will be very glad for any help.

• The first one is realized variance, not realized volatility. The third one doesn't include volatility from opening gaps compared to the second. Part of the variance in the price of something is due only to gaps. Commented Jun 7, 2020 at 12:40
• @javierazcoiti, why do you say first one isn't realized volatility? Isn't it an estimate of it? I realize the diffrence between 2nd one and 3rd one. But I guess, 1st one shpuld close to 3rd one since, both include the trading time interval. I am very confused about the situation. Is one of them biased? Thansk. Commented Jun 7, 2020 at 13:01
• The realized volatility is the squared root of the realized variance. Commented Jun 7, 2020 at 13:03
• @javiaerazcoiti, I editid the content. Thanks. Commented Jun 7, 2020 at 13:12
• Somewhat related: Molnar "Properties of range-based volatility estimators". Or maybe not so relevant... Commented Jun 7, 2020 at 18:24

...I am calculating daily volatility in 3 ways...

No, you are not computing the same quantity in 3 different ways.

You are computing volatility over different frequencies. The log daily returns and log-5 minute returns need not have the same volatility. Similarly, you could also compute, say, quarterly returns or 10 second returns and get different volatility series.

Take a hypothetical scenario where the open price and close price are the same but there is intra-day price movement due to trading. Then that date contributes zero to daily volatility. On the other hand, the realized volatility computed from a high-frequency intra-day return series would be non-zero.

These are volatility measures from the perspectives of investors with different holding periods and trading horizons. They are not the same. Indeed, traders who hold their positions for 5 minutes and for 5 days face different types of risk.

The open-to-close and close-to-close measures should also not be the same. They are not even the same returns. The close-to-close return accounts for off-hour, and then next day, trading.

• @Micheal thanks for the explanation. You are right. They are different calculatons. But I was thinking that at the long run, they shoud close to each other. All are different ways of estimating unobserved daily volatility. Just the time interval included at the calculations may create biases. But for the 1st and 3rd ones, the time intervals are same. Commented Jun 7, 2020 at 13:04
• "...the long run, they should close..."---that depends on different aspects of the instrument, e.g. nature of the instrument, liquidity, type of traders involved, etc. Your data also may not cover "long run", whatever that means in this specific case. Commented Jun 7, 2020 at 13:12
• Also, you're implying that in the long run, traders who hold their positions for 5 minutes and for 3 months ("long run") face the same risk. That is doubtful. Commented Jun 7, 2020 at 13:19
• @Micheal, I guess 2000 days enough for long run definition. But, I am not sure, I may be wrong. Ok. I wonder that, let I am forecasting one head day volatility everyday using previous open to close prices using a model. To compare the performance of the model; I compare the model forecasts and the realized obsered volatility for the day. Is an adaquate way to compare the model?. Because, I think realized volatility should be the best estimate for the unobserved volatility of the day. Commented Jun 7, 2020 at 13:20
• "Is an adequate way to compare the model?" That is the general procedure to judge predictive models, not just for vol. Compute the MSFE --- (forecast - realized)^2. Commented Jun 7, 2020 at 13:25