The Daniels Model (2003)

The first example of this model was presented in Daniels et al. (2003). After that there were a few papers published with an analysis of this model (Farmer et al. 2005, 2006). The main assumption of this model was that orders are come onto the market randomly. There are market orders that are executed immediately and limit orders, which are placed at a fixed price level and executed only when there is a counter-party on the market which wants to trade at this price. All orders have an intensity of incoming, an intensity of canceling, a volume and a price (see Fig. 1). All these parameters can be measured using empirical data, and this is the main advantage of this model. We try to create a model as in the original papers.

For the estimation of parameter a we calculated the difference between the best prices and incoming orders (relative price). The model suggested that most orders come near the best prices (see Fig. 2a). According to the model, we estimated only 58 % of the distribution of the relative price for effective limit order placement.

We calculated Qpper = 12 tick size (the 60 percentile of distribution of the relative price for effective limit order) and Qtower = — 11 tick size (the two percentile of distribution of the relative price for effective limit order) and the total number of effective limit orders 1,655,646 in this interval, so a = 3,427 orders/per day • per price or a = 0.108 orders/per second • per price. The total number of effective orders was 869 market orders and 3,390 limit orders with imme­diate execution, so |r = 0.0064 orders /per second (for more details of parameter estimation see Appendix A.1 in Farmer et al. 2005). There is one important remark that we estimated parameter 1 in terms of time (not in terms of events as in the original work). We found that the average time of an order’s life is equal to 3.5 s before cancellation.

 Volume

 Sell limit orders

 Sell market orders

 V V -3

 Price

 Buy market orders Q о « n

 Orders cancellation

 Fig. 1 Scheme of the Daniels model Fig. 2 Histogram of (a) entering order price differences from the best price and (b) power-law tails of order price Table 1 Parameters of the Daniels model on the Russian market (AFLT, January 2012) All the parameters of this model you can see in Table 1. We understand that the 1 month of our sample cannot be as representative as the length of 1 year and some seasonal effects should be taken into account, but as a comparison of models it would be good enough, so we are using these parameters in the comparison of models. We would like to thank Oksana A. Kostousova for discussions on this model.