Linear Models for the Impact of Order Flow on Prices I. Propagators: Transient vs. History Dependent Impact

Abstract

Market impact is a key concept in the study of financial markets and several models have been proposed in the literature so far. The Transient Impact Model (TIM) posits that the price at high frequency time scales is a linear combination of the signs of the past executed market orders, weighted by a so-called propagator function. An alternative descriptionthe History Dependent Impact Model (HDIM) -assumes that the deviation between the realised order sign and its expected level impacts the price linearly and permanently. The two models, however, should be extended since prices are a priori influenced not only by the past order flow, but also by the past realisation of returns themselves. In this paper, we propose a two-event framework, where price-changing and non price-changing events are considered separately. Two-event propagator models provide a remarkable improvement of the description of the market impact, especially for large tick stocks, where the events of price changes are very rare and very informative. Specifically the extended approach captures the excess anti-correlation between past returns and subsequent order flow which is missing in one-event models. Our results document the superior performances of the HDIMs even though only in minor relative terms compared to TIMs. This is somewhat surprising, because HDIMs are well grounded theoretically, while TIMs are, strictly speaking, inconsistent.

Nonlinear price impact from linear models

“…HDIM2 best reproduces the detailed shape of D( ), both for small ticks and for large ticks. This is interesting since previous work, using an approximate calibration of HDIMs based on a two-point factorisation of three-point correlations, reported disappointing results for HDIM2 in the case of small ticks [8][9][10]. The same effect is seen in Fig.…”

“…As shown in appendix C, bias-free calibration is not always guaranteed, but it can be verified by measuring the cross-correlation between the model input and its prediction-error. Taken together, our methodological improvements also solved the long-standing question why HDIM models were previously found to often underperform the theoretically less well-founded and sometimes inconsistent TIM models [8][9][10]).…”

“…This shows that non-price-changing orders, which can be caused by taking less liquidity than available and by stimulated refill, are indeed the main cause of the vanishing aggregate-sign impact for biased order flows-simply because long sequences of predominantly buy (or sell) market orders are also long sequences of non-price changing events. It also shows that the nonlinear impact curves are reproduced the HDIM2, which is generally considered a linear model [10], because the crucial nonlinearity lies in the distinction of whether an order will change the price at all. In this sense, it is a linear model with preclassified inputs.…”

“…As stated above, the original aim of three of the considered models was to remove the strong correlations injected by the order flow from the models' output returns (see e.g. [8][9][10]). The underlying insight is that this would be impossible if the impact of a trade were permanent and time-invariant.…”

Back Matter

“…For an overview on propagators, see [Taranto et al, 2016]. To start, take this model for the mid-price dynamics:…”