# Daily realized volatility and true daily volatility

Can someone help if I am thinking correctly? If $$R(t,i)$$ is the i'th log-return for $$i = 1\ldots,M$$ of day $$t$$ for $$t = 1\ldots,T$$.

Can I assume that the daily realized volatility (denoted $$RV(t)$$) is a consistent estimator of the true daily volatility denoted $$QV(t)$$] in the sense that $$RV(t)\rightarrow QV(t)$$ when $$T\rightarrow\infty$$ ?

To keep it brief: the realized variance estimator, $$RV_t$$, is only a consistent estimator of Quadratic Variation (QV) under absence of microstructure noise.

Following the paper of Barndorff‐Nielsen, O. E., & Shephard, N. (2002) they show how the realized variance estimator, $$RV_t = \sum_{i=1}^n r_{i,t}^2,$$ is a consistent estimator of QV under absence of microstructure noise, when the number of intraday observations goes to infinity:

$$RV_t = \lim_{n \rightarrow \infty} \sum_{i=1}^n r_{i,t}^2 \overset{\mathbb{P}}{\longrightarrow} QV_t.$$

In their setup they model the log-price process following a diffusion on the form:

$$dp_t = (\mu + \beta \sigma^2_t) \: dt + \sigma_t dW_t,$$

where $$\mu$$ is the drift and $$\beta$$ is the risk-premium. Following from the diffusion setup of the log-price process, the quadratic variation $$QV_t$$ of the log-returns can be described as:

$$QV_t = \int_{t-1}^t \sigma^2_s \: ds,$$ which is equivalent to the integrated volatility/variance $$IV_t = \int_{t-1}^t \sigma^2_s \: ds$$ (only under a diffusion process, see paper).

#### I have highlighted some key findings from the above paper:

1. $$RV_t$$ is also an unbiased estimator when $$\mu = \beta = 0$$.

2. In practice, the effect of $$\mu$$ and $$\beta$$ on realized volatility/variance, is extremely small and is often safe to ignore in many cases (See section 5 of above paper).

3. Under the diffusion setting, when $$\mu = \beta = 0$$, and assuming absence of noise, $$RV_t$$ is a consistent estimator of Integrated variance/volatility $$IV_t$$: $$\lim_{n \rightarrow \infty}RV_t \overset{\mathbb{P}}{\longrightarrow} QV_t = \int_{t-1}^t \sigma^2_s \: ds.$$

4. Avoiding microstructure noise can be done by sparse-sampling intraday observations.

As a last note Zhang, L., Mykland, P. A., & Aït-Sahalia, Y. (2005) show that when microstructure noise is present, the bias of the $$RV_t$$ grows with $$n$$ and thus explodes when $$n \rightarrow \infty$$. Thus the realized volatility estimates not the true integrated volatility/variance, but rather a noise contaminated counterpart.

• very nice and coherent answer! Commented Jun 5, 2022 at 17:58
• It is a great help and mow I get it clearly. Commented Jun 5, 2022 at 20:53