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Whenever I read about calculating realized variance, people are using log returns. However, I was asking myself whether it is possible to calculate realized variance also for simple, raw returns.

Realized variance is defined as

$RV^{(n)} = \sum_{j = 1}^{n} r_{j,n}^{2}$

and $r_{j,n}$ are for example 5 min log intraday returns and RV is the realized variance of that given day.

Assuming that the returns follow a stochastic volatility process

$r_{j,n} = \sigma_{j,n}u_{j,n}$ with $u_{j,n} \sim N(0,1), j = 1,\ldots,n$

then we have

$\mathbb{V}[\sum_{j = 1}^{n}r_{j,n}] = \mathbb{E}[\sum_{j = 1}^{n}\sigma^{2}_{j,n}] = \mathbb{E}[RV^{(n)}]$.

How does this work with raw returns, which we cannot easily sum up? How do we get the RV estimate of them?

Thanks!

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