When Realized Variance Estimates Quadratic Variation
Summary
The document describes realized variance as the sum of squared intraday log returns and explains when it estimates a day’s quadratic variation. Under a diffusion price process and in the absence of market microstructure noise, increasingly frequent observations make realized variance converge in probability to quadratic variation, which in this setting equals integrated variance. The discussion also notes conditions under which the estimator is unbiased and says drift and risk-premium effects are often small in practice.
The main limitation is microstructure noise: as sampling becomes more frequent, noise can contaminate realized variance and cause its bias to grow rather than disappear. Sparse intraday sampling is offered as one practical way to reduce that problem. The result therefore depends on the price-process assumptions and data quality; the answer does not prescribe an optimal sampling interval or compare alternative noise-robust estimators.
Key ideas
- Realized variance is computed by summing squared intraday returns.
- Without microstructure noise, increasingly frequent observations yield a consistent estimate of quadratic variation under the stated diffusion setup.
- In that setup, quadratic variation equals integrated variance over the day.
- Microstructure noise can make very frequent sampling bias realized variance, so sparse sampling may help.
Tags
Full text
# Daily realized volatility and true daily volatility
# 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 $T\rightarrow\infty$ ?
## Answer by Pleb (score 11, accepted)
https://quant.stackexchange.com/a/71151
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:
- $RV_t$ is also an unbiased estimator when $\mu = \beta = 0$.
- 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).
- Under the diffusion setting, when $\mu = \beta = 0$, and assuming absence of noise, $RV_t$ is a consistent estimator of Integrated variance/volatility $IV_t$: $$ \lim_{n \rightarrow \infty}RV_t \overset{\mathbb{P}}{\longrightarrow} QV_t = \int_{t-1}^t \sigma^2_s \: ds. $$
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.