Bid-Ask Bounce and High-Frequency Volatility Estimation
Summary
The document explains why volatility estimated from every transaction price can be overstated at high frequency. It models observed prices as a latent price that evolves with market uncertainty plus a separate component reflecting whether a trade occurs at the bid or ask. When the bid-ask component changes between observations, transaction returns include that bounce as well as the underlying price movement.
Under an independent, identically distributed noise assumption, the added contribution to realized variance grows with the number of observations, so sampling more trades can make the naive estimate increasingly inflated. The response points to multi-timescale methods for integrated volatility, methods for nonsynchronous observations, and uncertainty-zone models that account for tick-size effects in large-tick assets. These approaches rely on assumptions about noise and observation times, and the document does not directly compare executed, mid, and weighted-mid prices or establish one universally optimal input. It recommends treating market microstructure and tick size as important features of the estimator choice.
Key ideas
- Transaction prices can alternate between bid and ask around the latent price, creating bid-ask bounce.
- The bounce adds noise to high-frequency price changes and can inflate naive realized volatility.
- Under the stated independent-noise model, the estimated variance rises with the number of observations.
- Multi-timescale estimators address some forms of high-frequency observation noise.
- Tick size and observation timing can affect which volatility method is appropriate.
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Full text
# Optimal Price Metric for High-Frequency Volatility: Executed Price, Mid Price, or Weighted Mid Price?
# Optimal Price Metric for High-Frequency Volatility: Executed Price, Mid Price, or Weighted Mid Price?
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:
- the executed price associated with each executed message
- the mid price associated with each tick
- the weighted mid price associated with each tick
If you have any other suggestions, I am open to explore the alternatives.
Thank you.
## Answer by lehalle (score 1, accepted)
https://quant.stackexchange.com/a/79435
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.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.