Realized Variance for Simple Returns

Summary

The document asks whether realized variance can be calculated from simple returns instead of log returns. It defines realized variance as the sum of squared intraday returns and describes a stochastic volatility model in which each return is a volatility term multiplied by a standard normal shock. Under that setup, the expected sum of squared returns corresponds to expected realized variance, while the variance of the summed returns provides a related population quantity.

The central issue is how to interpret or estimate realized variance when returns are raw percentage changes, which do not add across time in the same way as log returns. The document poses this question but includes no answer, derivation, empirical evidence, or estimator for simple returns. It therefore frames a useful distinction between return conventions without resolving how compounding affects the desired variance measure or its estimate.

Key ideas

• Realized variance is presented as the sum of squared intraday returns.
• The stated stochastic volatility model links expected realized variance to the sum of return variances.
• Simple returns compound multiplicatively, which complicates comparison with sums of log returns.
• The document raises the estimation question but does not provide a method or conclusion.

Tags

# How can I compute a realized variance for raw instead of log returns?


# How can I compute a realized variance for raw instead of log returns?












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.