# Efficient market maker order book model

We continue our series of posts on order book models by considering extremely simple order book model. In this model most agents submit only market orders, while there is also a single market maker, which provides liquidity for them.

## The model

The model is initialized with a full order book. Here, full order book means that for every possible quote there is a single limit order at that quote. All quotes below current price are filled with bid market orders, while all quotes above current price are filled with ask market orders.

Market orders arrive at rates \( \lambda^{-} \) and \( \lambda^{+} \) for buy and sell orders respectively. As soon as market order arrives it consumes best limit order of the opposite side. As soon as the limit order is consumed market maker places another limit order. The new limit order replaces the old one, but it maybe placed on either side of the order book. The new limit order is placed on the same side of the order book as the market order with probability \( p \), likewise a limit order is placed on the opposite side of the order book with probability \( 1 - p \).

Note that order book model formulated in this way is essentially an almost perfect replica of the Walrasian market maker.

## Interactive applets

To understand the model and its dynamics better you should study the interactive applets below.

The first applet shows us how the structure of the order book evolves as the time goes on. Note that the profiles of the both sides remain the same in this model and only the price moves. The shapes of the profiles do not change as order book is always filled with one order per quote. What changes is only the boundary between the sides of the order book.

The second applet is in certain sense traditional Physics of Risk applet for all financial market models. Though there are some specific differences related to the model in question.

First of all instead traditional price, or log-price, series plot on the bottom left you see relative price series plot. We have selected relative price instead of absolute to price to avoid introducing boundary condition at 0 (as there cannot be negative absolute prices). Simply add some large enough number to the relative price to obtain absolute price.

Also instead of return in traditional sense (difference of log-price at two different points in time) we have "price change" and use \( | \Delta p | \) notation. This model lacks some important ingredients to reproduce power-law "price change" PDF. Actually "price change" in this model is normally distributed.

Also as the order flows are uncorrelated the spectrum of price changes is flat. The price in this case undergoes simple Brownian motion.

**Acknowledgment.** This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics'' (09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students through Practical Research Activities'' measure.