# Order Book Dynamics

I have been following this paper: Price dynamics in a Markovian limit order market, by Rama Cont and Adrien De Larrard

The model is especially pertinent as I only have access to L1 data. The model is clear and intuitive and I have implemented the analytical model. Also it (and the related paper A stochastic model for order book dynamics Rama Cont, Sasha Stoikov, Rishi Talreja) have solutions to key questions like:

1. The probability that next move will be an up move.
2. The time till the next move.
3. Probability of executing an order before the market moves.

Which are precisely the questions I need answers to.

However when I calculate the empirical conditional probabilities of a price movement based on the current book shape (as shown on p12 of the paper), I find that the empirical surface is almost flat (this is based on a months worth of data), suggesting that the current book shape has very little influence on the next tick move?

To calculate the empirical conditional probabilities I simply count up how often each bid/ask size combination occurs and in addition what proportion of these initial bid/ask combinations lead to an up move. (I assume this is correct it should be very simple).

It seems counter-intuitive to me that the book shape shouldn't hold any information regarding the future book movement?

Are there other models that might be better suited to my situation (but that can still allow me to answer the questions shown above?

The Queue Reactive Model (by Huang, L and Rosenbaum) is an improvement of what Cont and de Larrard (CL) did.

This model is capturing

• the inflows and outflows in each queue given the current state of the orderbook (it is one of your remark)
• but more importantly, once one queue depletes, the discovered quantity is not taken at random (like in the CL model)

I guess it is what you want. The only point is that to implement the full queue reactive you need more than level I. But you can yourself adjust the model to level one. Or you can have a look at the model of this paper: Limit Order Strategic Placement with Adverse Selection Risk and the Role of Latency it is very close to a level I version of the queue reactive.

• The first link should probably point here: arxiv.org/abs/1312.0563.pdf Jan 24, 2018 at 15:22
• Is there a github repo associated with a good order book dynamics? Dec 16, 2018 at 3:17

Actually, I argue in a paper that this lack of information is very important. The paper is at

Harris, D.E.(2017) The Distribution of Returns. Journal of Mathematical Finance, 7, 769-804

Let us assume that a security is traded in a double-auction with many buyers and sellers. Since the winner's curse isn't operative in equilibrium in a double auction the only rational behavior is to bid your expectation. The limit book, when combined with private unplaced orders when the market is away from an individual's expectation, should be normally distributed according to the central limit theorem since it is a distribution of expectations.

This means the order book is nothing except random noise. Returns are the ratio of sell price over buy price. So it is the ratio of normally distributed noise divided by normally distributed noise. If the equilibrium prices are considered the center of location and as (0,0) in the error space, you will end up with a Cauchy distribution as your distribution of returns, by well-known theorem.

This is nice since the distribution of disaggregated returns, when shell companies, bankruptcies, and mergers are removed is a very slightly skewed truncated Cauchy distribution. The skew is created by the intertemporal budget constraint. A return can only exist given a trade and there is a probability that a price will exceed the budget of the potential buyer resulting in not trade at all.

Since there is a 100% chance someone would accept 100 shares of IBM at zero dollars per share and a zero percent chance at infinity, the sigmoid probability function skews the truncated Cauchy distribution.

The current book is intrinsically noise.