3
$\begingroup$

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 $T\rightarrow\infty$ ?

$\endgroup$

1 Answer 1

7
$\begingroup$

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 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.

$\endgroup$
2
  • 1
    $\begingroup$ very nice and coherent answer! $\endgroup$
    – Lars
    Jun 5 at 17:58
  • 1
    $\begingroup$ It is a great help and mow I get it clearly. $\endgroup$
    – user62408
    Jun 5 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.