# "Noise traders only" order book model by Bak et al.

Last time we have discussed what order book is, and now we will present a simple model for the order book [1], which was inspired by reaction-diffusion model from physics.

Note that the full model considered in [1] is more complex than we discuss in this post. Here we only reproduce the results of the model with "noise traders" only (as discussed in Section IV B of the article).

## The "noise traders" only model

The model itself is rather simple. It is assumed that there are \( N \) agents (here \( N \) must be even). Half of the agents will always submit bid orders, while the other half will always submit ask orders.

Initially all agents submit their orders. Log-prices for the bid orders are uniformly distributed between \( -(\Delta P -1) /2 \) and \( 0 \). While prices for the ask orders are uniformly distributed between \( 0 \) and \( (\Delta P-1)/2 \). Note that, unlike in this interpretation, in the original formulation of the model linear prices were used.

After initialization, during each time tick, randomly selected agent revises his order price. He moves the order one unit toward the spread, with probability \( \frac{1+D}{2} \), or one unit away from the spread, with probability \( \frac{1-D}{2} \). If ask order meets bid order (or vice versa), both orders are anihilated and new current price is set. Afterwards the agents resubmit their orders - bid order price is picked randomly between \( - (\Delta P-1) /2 \) and \( P_t \), ask order price is picked randomly between \( P_t \) and \( (\Delta P -1) /2 \).

In real markets such behavior is possible, but not reasonable as some stock exchanges may apply charges. It would be too costly to adjust the order's price often. Hence the model appears to be not plausible. But the model has its redeemable feature - it reminds of certain physical system.

Imagine a long tube. You inject two different types of particles from the sides of the tube. Inside the tube particles move randomly (diffuse) until they collide with particles of other type and anihilate (react). The same model describes a physical and financial system.

## Interactive applets

To undestand the model better study the interactive applets below. The first applet shows us the structure of the order book. A similar applet was published in the previous text, but it used another model (which will be discussed in the near future). As in this model order book often has alot of orders, we have chosen lines instead of points to show the amount of order per price level (though the price levels are still discrete as in the previous example).

The second applet has a somewhat more traditional look. By using the applet below you can see the log-price and absolute return time series, as well as the main statistical features of absolute return (PDF and spectral density). Note that this model does not have diserable statistical features, we show them simply out of tradition and our own curiosity.

**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.

## References

- P. Bak, M. Paczuski, M. Shubik. Price variations in a stock market with many agents. Physica A 246: 430-453 (1997). doi: 10.1016/S0378-4371(00)00067-4.