Contemporary Price Impact Modeling

x 10

D - - Initial cumulative volume

Cumulative volume in 30 sec

Forecast for 30 seconds

90% confidence bounds

Order Depth

Fig. 1 LOB forecast in terms of cumulative volume as a function of depth for MICEX RTKM shares, 18 January 2006

Consider a simple LOB model with only two types of events: arrival and cancellation of limit order at one side of the book. Intensities are stationary and independent but depend on depth. Volume of each order is a random variable with a priori given parametric distribution with unknown parameters depending on depth. Thus, LOB is modelled via compound homogeneous space-time Poisson process. We calibrated the following model to real MICEX data, assuming from empirical observations that

1. event volume distribution is a mixture of discrete and lognormal;

2. intensities as functions of depth are power-law functions.

We estimate parameters в of the model from order flow history using maximum likelihood and Bayesian methods. Then, using LOB structure Lt0 as initial state of the system we model Lt0+T Lt0, в and take Lt0+T = E (Lt0+T Lt0, в) as a forecast. Results of forecasting structure for 30 s horizon and 90 % confidence bounds are presented in Fig. 1. We see that even for small horizon confidence interval is too wide for any practical use of such forecast. This is partly explained by presence of discrete part in volume mixture distribution, which is usually difficult to estimate from training sample. Atoms of volume distribution stand for volume values preferred by participants (100, 1,000, 5,000 lot etc.), orders with preferred volumes can amount up to 50 % of total number of orders.

Due to technical difficulties and intention to integrate an LOB model into portfolio optimization, a simple a priori form of the book is usually considered.

Time

Fig. 2 Price impact aspects

Accent in modeling is made on the price impact function itself. Three main aspects are considered in such an approach:

• Immediate response of best price after a trade, which affects future costs until book replenishes.

• Resilience of LOB, i. e. ability to replenish after a trade; together with immediate response, this is often called temporary price impact. Infinite resiliency means that LOB replenishes instantaneously.

• Permanent price impact, or the effect of replenishment to a level other than pre­trade value; this effect describes the incorporation of information from the trade, which affects market expectations about ‘fundamental price’ of the asset (Fig. 2).

Permanent price impact is not considered in many classical models of optimal portfolio selection. For a particular case—optimal liquidation—many works assume the simplest dependence, where impact is a linear function of trade volume (i. e., Kyle 1985). Linear approximation can be considered appropriate in most practical cases because of difficulty in calibration of a more complex function in the presence of many agents.

Immediate response function is usually considered linear in volume, which is equivalent to the assumption of the flat structure of LOB (Obizhaeva and Wang 2012), or the assumption that trade volumes a priori are less than current market depth. Andreev et al. (2011) consider a polynomial form of immediate response function with stochastic coefficients. Fruth (2011) presents the most general law of immediate response in the form of a diffusion process under several mild conditions.