# Independence of latent price and market microstructure noise

When examining how Market Microstructure works and affects price formation, there is talked about the:

"Assumed independence of of latent price and microstructure noise"

• From "On the Correlation Structure of Microstrucure Noise: A Financial Economic Approach", Diebold and Strasser, 2013.

I am struggling to understand what this means exactly?

For example, when examining a lot of financial trades and quotes, and looking for a underlying latent price - what does it mean to assume independence of the noise? And what do you assume the noise is then specifically? "Wrong" trades? Computer-errors?

The microstructure noise is, roughly speaking, the small-scale noise introduced into the market as a result of the way the market is designed.

For example, suppose that there is an asset and the "real" price for this, is 126.6. That is, if we magically knew everyone's honest, perfect maximum buy and minimum sell prices and could match them off and arrive at the equilibrium price, then it would be 126.6.

However, imagine that the market only quotes in whole numbers. That means you can either buy at 127 or sell at 126, but it is physically impossible to trade at 126.6. Intuitively we expect to see a sequence of trades switching between 126 and 127. This is called "bid ask bounce" and is perhaps the simplest example of microstructure.

Even with such a trivial example, we can start to do interesting things. For example, since the price is closer to 127 than 126, we might expect to see more 127s than 126s. We could use a simple logit function or something to select the probability the next tick is on the near or far side. Also we can "what-if" the market has a different tick size? Clearly we expect in our model the bid ask bounce to reduce.

In our example the latent price is a constant, but that is not very realistic. For a better model one would usually assume some dynamic model for this hidden price process. Perhaps Brownian motion, for example. Now you can see how with our model from above, we will see trade prices bouncing either side of our hidden price. As the price gets closer to a whole number, more trades are done on that, until it crosses and then we start seeing trades on the next tick along.

As you can imagine, you can continue to play this game, making the latent price process more complex, and adding more complex microstructure models to generate the trades around that price. And all along we have assumed independence of latent price and microstructure noise. That is to say, our microstructure process does not impact the latent price, and the microstructure process stays the same, regardless of what the price is doing.

This is a useful assumption to make, since it separates the problem out - without it you probably wouldn't be able to solve much and the usefulness may be limited. That said, it is easy enough to think up possible models that would break this:

e.g.

• Modelling effect of limit orders in the order book when price breaks new ground
• Market maker inventory changes during long price runs in one direction.
• Psychological anchors set by previously traded prices impacting latent price.

I guess you're asking three inter-relatated though conceptually separate questions:

• What is the latent or efficient price?

• What is market microstructure noise (as opposed to non-market-microstructure noise)?

• What does it mean to assume that the efficient price process and the microstructure noise process are independent?

I won't attempt to provide a full examination of the issues raised by these questions: entire articles could (and have been) written about them. I'll only attempt a summary review.

• The efficient price process is, by assumption, unobserved. Nevertheless, under the maintained assumptions of frictionless markets, perfect and symmetric information, and rationality of all market participants, one can apply some basic theorems to infer some statistical properties of the efficient price process. Most importantly, the efficient price process must generally be a semimartingale with respect to some filtration. In a continuous-time setting, it is also straightforward to show that the continuous (i.e., non-jump) component of the efficient price process must be local Brownian motion with local drift and variance components.

If the observed price process isn't a semimartingale, it's either because there's an additional component (usually, but not always, called market microstructure noise) or because the maintained assumptions (rationality, perfect and symmetric information, etc) aren't valid to begin with. Let's pursue the first reason.

• To make market microstructure noise more than just a catch-all (and tautological) component, it's important to relate any observed or known deviations from the semimartingale property to actual properties of the ways in which trading is conducted. I'll mention three such phenomena (there are several more, of course):

• Trading is known to produce not just one price, but two prices: the bid and the ask price. The bid-ask spread is in general non-zero even in a competitive market, for reasons well-explored in the literature of the past 25 years. (In econometric practice, one frequently calculates the average of bid and ask prices and tries to model this average as some kind of martingale but, strictly speaking, that's not always valid.)

• Because of inventory and liquidity effects, there's often predictability of price movements at the very highest sampling frequencies. E.g., if prices are observed every second, it may be possible to predict price movements out to a few seconds. This predictability is at the heart of what algorithmic trading rules are all about.

• There may also be some mean-reversion of prices and volatilities at the very highest sampling frequencies due to the so-called "bid ask bounce": Empirically, it's been found that if a given trade happens at the bid, the very next trade is more likely to happen at the ask than at the bid, and vice versa. Because there's a spread between bid and ask prices, there's a bit of extra volatility that arises at the very highest sampling frequencies. This usually shows up in so-called volatility signature plots, which plot the intraday sampling freqency $\nu$ against the associated realized volatility $\sigma_\nu$: At the very highest sampling frequencies, $\sigma_\nu$ is frequently an increasing function of $\nu$.

Note that these "noise components" are generated as a consequence of the fact that trading takes place in real places among real people. As such, these noise components are not akin to those generated by computer programming errors. I.e., flash crashes in certain stocks are not an example of market microstructure noise.

• To assume, then, that the latent or efficient price process and the market microstructure noise process are mutually independent really means that they're generated by two underlying processes which don't interact with each other. E.g., if new information enters the market, it should affect the value of the semimartingale but not meaningfully alter the noise process. (The interaction between the two processes is usually assumed to be additive, but I suppose one could generate a statistical theory that works with multiplicative rather than with additive noise.)

Naturally, it is not possible to verify directly that the independence assumption is correct, as one of the two underlying processes isn't observed. However, if it were the case that some observed departures from the semimartingale property are such that they cannot be explained usefully/credibly as being related to various market imperfections (such as the bid-ask bounce), one would have reason to doubt the empirical usefulness of the entire approach to modelling observed prices.