The ideal frictionless market of Merton (1969) does not adequately simulate the more complex real market. First of all, price dynamics obviously depend on an agent’s actions in the market; moreover, there is no single characteristic of an asset’s market value (price). Since the 1990s, electronic trading through limit order books (LOB) has been gaining popularity, providing the market with a set of orders with different volumes and prices during any trading period. Inability to close a deal at an estimated price led to the necessity of including transaction costs in portfolio management models and price impact modeling. For the past two decades, research in this field has provided complex models that allow for time varying forms of LOBs, temporary and permanent price impact, resilience etc.
The most sophisticated and yet also fundamental way of estimating transaction costs is estimating the whole structure of LOB. Usually the market is represented as a complex Poisson process where each event is interpreted as the arrival/liquida – tion/cancellation of orders at specific depth levels. Large (2007) considers the arrival of ten kinds of market events (market bid/ask order limit bid/ask order, cancellation of bid/ask order, etc.) according to a multivariate Hawkes process with intensity depending on the past trajectory. Intensity in Large’s model does not depend on order depth (distance from best quote).
Cont and Larrard (2012) introduced a complex Poisson model with time and depthvarying intensity and obtained theoretical results on the subject. Unfortunately, due to the extreme complexity of the general approach, it is extremely difficult to calibrate the parameters. Thus, some simplifying assumptions, based on empirical observations of a particular market, are necessary. On the other hand, the Poisson model must be flexible enough to reflect dynamics of real events, otherwise forecast errors will make the result useless for practice.

D – – Initial cumulative volume


Cumulative volume in 30 sec






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 spacetime 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 powerlaw 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.







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.


Resilience has been recently included in impact models and is usually described in exponential form with a priori given intensity: Suppose that Kt0 is immediate response after a trade at time t0, then
t
f
— p(u)du
Temporary Impactt = Kt0e t0 .
Almgren and Chriss (1999) considered instantaneous replenishment: pu = 1; Obizhaeva and Wang (2012), Gatheral et al. (2011) and others assumed exponential resilience with constant intensity: pu = const. General law of deterministic resilience rate has been presented in recent papers of Gatheral (2010), Gatheral et al. (2012), Alfonsi et al. (2009), and Fruth et al. (2011).
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