The Mike-Farmer Model Without the Cancellation Process (MFWC)

It is an interesting question about what there would be on the market if there were no cancellations. Would trading or the market be stable or not? We realize the MF model without cancellations (we call it MFWC).

3 Model Upgrading

The most important thing that we try to improve in the MF model is the distribution of order price. We cut distribution into two parts: one with a positive tail and one with a negative tail. We find that both tails of distribution fit a good by power-law distribution with a tail exponent = —2.15 for positive values and a tail exponent = —2.493 for negative values (we inversed the negative tails and after that the estimate coefficients). Power-law poorly describes the center of distribution, when orders are put at the best prices. We fit ±10 ticks from the best prices (x = 0)


Fig. 7 Fitting of tails of empirical distribution using the power-law


Fig. 8 Fitting of the empirical price distribution using two power-law distributions with Student’s t in the center

using Student’s t-distribution. On the Russian market traders see only the first ten prices for buy and sell, so this part of the orders should have another distribution (for example t-distribution) (Figs. 7 and 8).

Another additional improvement related to the order cancellation process is trying to take into account another metric of liquidity, for example RTCI:


. 1 Pi – p I • n

KICI = —^




i: order position in the order book, i = 1… k, k: total number of limit orders in the book, pc. price of order i,

nc. volume of order i, щ <0 for buy side orders, p: current market price.

Подпись: Table 3 Parameters of the upgrade model on the Russian market (AFLT, January 2012) Parameters Description Value & positive Exponent of the positive tail -2.15 & negative Exponent of the negative tail -2.493 A•10-5 Parameter for equation of the orders cancellation 2.87 D3 Parameter for the equation of the order cancellations -583.1 ft positive • 10 Scale factor for the positive tail 40 ft negative • 10 Scale factor for the negative tail 0.9

This metric allows the measurement of the sparseness of the order book. The order book may contain a large number of orders, but all the orders are far away from each other (in this case book it would be rarefied). For more details of this metric, see Arbuzov and Frolova (2012).

We calculated the probability of cancellation conditional on RTCI and found that it could be approximated by a linear function as in case of order book imbalance. In Fig. 9, we can see a reasonably expected result, that when orders in the order book are located far from each other, traders have no reasons to cancel their orders (Table 3).

We calculated an RTCI metric at each step of our simulation. The total condi­tional probability was calculated as:

Подпись: D1 exp Уі ) (n imb + B)


P (Ci yt, nimb, ntot, RTCI) = A(1

X (1 – D2exp~ntot) (RTCI + Dз)

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