In the context of high frequency trading, I'm exploring the application of the mean absolute deviation estimate for high-frequency volatility calculation. What would be the optimal choice for this calculation:
If you have any other suggestions, I am open to explore the alternatives.
Thank you.
You probably ask the question because you noticed that is you take the traded price to compute a naive volatility estimate on the all the trades, this (apparent) volatility is very high. It is due to the bid-ask bounce, and can be seen via a very simple model.
If the price discovery follow a Markov chain like $P_{t} = P_{t-\delta t} + \sqrt{\delta t}\sigma\,\xi_{t}$ but is time to time traded on the bid side and time to time on the ask side. Let's model this "randomness" (bid or ask) via another random variable $\epsilon_t$. Then our process looks like $$P_t = P_{t-\delta t}+\sqrt{\delta t}\sigma \xi_{t} + \epsilon_t.$$ Using the naive estimate for volatility from time $t=0$ to time $t=T=K\sqrt{\delta t}$ (i.e. a full day) reads $$\hat\sigma^2 = \mathbb{E} \sum_{t=\delta t}^T (P_t - P_{t-1})^2= \sigma^2 +\mathbb{E}\sum_{t} (\epsilon_t - \epsilon_{t-\delta t})^2.$$ Taking the assumption that the randomness of the bid-ask bounce is i.i.d you get $$\hat\sigma^2 = \sigma^2+ 2 (K-1) \mathbb{V}(\epsilon)$$ that explodes when you have more and more trade in one day.
If you believe in the model, you can do better following Zhang, Lan, Per A. Mykland, and Yacine Aït-Sahalia. "A tale of two time scales: Determining integrated volatility with noisy high-frequency data." Journal of the American Statistical Association 100, no. 472 (2005): 1394-1411.
But is would require and independence of the series of $\epsilon$ and of the time of observation. If you want to do better, you should follow Hayashi, Takaki, and Nakahiro Yoshida. "Nonsynchronous covariation process and limit theorems." Stochastic processes and their applications 121, no. 10 (2011): 2416-2454.
Last but not least, the influence of the tick size is subtle, if you work on large tick assets (i.e. the spread being on average less than 1.7 ticks), you should use Robert, Christian Y., and Mathieu Rosenbaum. "A new approach for the dynamics of ultra-high-frequency data: The model with uncertainty zones." Journal of Financial Econometrics 9, no. 2 (2011): 344-366.
For a review of all that, you can have a look at L and Sophie Laruelle. Market microstructure in practice. World Scientific, 2nd Edition 2018.
