A Comparison of the Forecasting Ability of Immediate Price Impact Models

• Author: Huu Nhan Duong,Manh Cuong Pham,Paul Lajbcygier
• Published date: 01 December 2017
• DOI: http://doi.org/10.1002/for.2405
• Date: 01 December 2017

A Comparison of the Forecasting Ability of Immediate Price Impact

Models

MANH CUONG PHAM,

1

HUU NHAN DUONG

2

AND PAUL LAJBCYGIER

1,2

*

1

Department of Econometrics and Business Statistics, Monash Business School, Monash University,

Wellington Rd & Blackburn Road, Clayton, VIC 3800, Australia

2

Department of Banking and Finance, Monash Business School, Monash University, 900 Dandenong

Road, Caulfield East, VIC 3145, Australia

ABSTRACT

As a consequence of recent technological advances and the proliferation of algorithmic and high-frequency trading,

the cost of trading in financial markets has irrevocably changed. One important change, known as price impact, re-

lates to how trading affects prices. Price impact represents the largest cost associated with trading. Forecasting price

impact is very important as it can provide estimates of trading profits after costs and also suggest optimal execution

strategies. Although several models have recently been developed which may forecast the immediate price impact of

individual trades, limited work has been done to compare their relative performance. We provide a comprehensive

performance evaluation of these models and test for statistically significant outperformance amongst candidate

models using out-of-sample forecasts. We find that normalizing price impact by its average value significantly en-

hances the performance of traditional non-normalized models as the normalization factor captures some of the

dynamics of price impact. Copyright © 2016 John Wiley & Sons, Ltd.

key words price impact; trading costs; out-of-sample forecasting

INTRODUCTION

Prior studies in the market microstructure literature have shown that trading impacts asset prices in such a way that a

buy will bid prices up, while a sell will push prices down (see, for example, Glosten and Milgrom, 1985; Kyle, 1985).

This finding highlights that the largest part of total trading costs does not come from explicitly stated components such

as bid–ask spreads or commission fees, but from implicit costs such as price impact, and is usually unknown ex ante

(Keim and Madhavan, 1996, 1998; Kissell et al., 2004). The evolution of price impact has also changed substantially

due to the proliferation of algorithmic trading or high-frequency trading.

1

Thus it is important to understand and pre-

dict the behavior of price impact to determine the profitability of an investment or trading strategy.

2

The early work on price impact proposed different empirical models to gauge the impact caused by trades aggre-

gated over an interval of time (see, among others, Torre, 1997; Plerou et al., 2002). Recent studies focus on modelling

the impact that an individual trade places on prices, either immediately or after some fixed time following a trade ex-

ecution (see Bouchaud et al., 2009, for an excellent review).

Recently, attention has been focused on evaluating immediate price impact and three recent models have been

proffered: Lillo et al. (2003); Chen et al. (2005) and, most recently, Zhou (2012). For those smallest stocks which

are most sensitive to price impact, we calibrate each of these models and evaluate them using the most recent data

available. Our study is unique, as no other study compares all these relevant candidate models to determine which

provides the most accurate forecasts of price impact. Also, the pre-existing literature only evaluates models using

in-sample data. Even so, the in-sample empirical evidence supporting one model being superior to another is rather

weak (Chen et al., 2005; Zhou, 2012). To the best of our knowledge, no research has been conducted to compare

various price impact models based on out-of-sample forecasts. Such testing is important, as no model, whether the-

oretical or inductive, is immune to overfitting in-sample. Thus we aim to fill this gap in the literature by carefully

calibrating and comprehensively and thoroughly evaluating price impact models using out-of-sample forecasts of

price impact on recent data sets.

We find that among the immediate price impact models that have been proposed in the literature the model which

normalizes price impact by its prior average value, the Zhou (2012) model, outperforms all other models which

*Correspondence to: Paul Lajbcygier, Department of Banking and Finance & Department of Econometrics and Business Statistics, Monash

Business School, Monash University, Wellington Rd & Blackburn Road, Clayton, VIC 3800, Australia; Phone: +61 3 9905 9694; Fax: +61 3

9905 5475; E-mail: paul.lajbcygier@monash.edu

1

Algorithmictrading is trading that isautomated and managed using computeralgorithms (Hendershottet al., 2011). Algorithmictrading accounts for

73% of the tradingvolume in the USA (Hendershott et al., 2011)and 30–40% of total volumes traded on the AustralianSecurities Exchange (ASX,

2010). Hendershottet al. (2011) also show that algorithmic trading reducesprice impact on the New York Stock Exchange (NYSE).

2

For example, prior findings on whether or not active trading strategies and enhanced indexing outperform passive investments are mixed and

depend greatly on how trading costs are measured (see, among others, Knez and Ready, 1996; Korajczyk and Sadka, 2004; Arnott et al.,

2005; McQuarrie, 2008).

Journal of Forecasting,J. Forecast. 36, 898–918 (2017)

Published online 2 March 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2405

Copyright © 2016 John Wiley & Sons, Ltd.

employ unscaled price impact based on mean squared error (MSE) criterion. It also produces the second most accu-

rate forecast of immediate price impact amongst the four traditional models we consider when the forecast accuracy is

judged by mean absolute error (MAE) measure. This superior accuracy, obtained for both buyer-initiated and seller-

initiated trades, occurs for stocks in the two lowest stock deciles that stay in the S&P/ASX 200 index in 2012 and

2013 (where price impact is largest). We argue that the outperformance is due to normalizing price impact by its av-

erage value on prior trades on the same day. Owing to the co-movement between average price impact and price im-

pact itself, the dynamic behavior of price impact is partly captured by its average series, and hence is better revealed

when it is normalized, resulting in the superiority of the normalized model. We also show that when the average price

impact is added to the hitherto non-normalized traditional models both the in-sample fitting capability and the out-of-

sample predictive accuracy of these models are significantly improved.

Our findings highlight that the use of a price impact model with appropriate normalization (i.e. normalizing price

impact by its average value) is a good way of estimating and forecasting price impact. Our findings have significant

implications for traders and fund managers because the superior price impact forecasts can be employed to estimate

more accurate measures of price impact costs, which in turn enable more accurate evaluations of investment strategies

and may facilitate optimal execution.

The rest of the paper is organized as follows. The next section reviews related literature about immediate price im-

pact models. Then we provide the details of various immediate price impact models for single trades that have been

developed in the literature. We describe the data to be used in this study and the research methodology and the results

are discussed, followed by robustness analyses and finally we conclude.

LITERATURE REVIEW

One of the earliest price impact models was proposed in a theoretical work by Kyle (1985), in which market impact

is linearly related to the signed trading volume and permanent or persistent over time. Although simple, this linear

model was shown to be the only specification for the permanent price impact that prevents the existence of price

manipulations which consequently generate arbitrage profits (Huberman and Stanzl, 2004; Gatheral, 2010).

3

The

linear market impact specification is applied empirically in the work by Bertsimas and Lo (1998) and Dutta and

Madhavan (1995).

Later studies, however, find that the rate of increase in market impact costs diminishes as trading volumes get larger,

implying a concave rather than linear relation between price impact and transaction size.

4

Different functional forms

have been developed to capture this concavity nature of price impact,

5

ranging from power-law functions (e.g. Torre,

1997; Lillo et al., 2003; Farmer and Lillo, 2004; Almgren et al., 2005), logarithmic functions (e.g. Potters and

Bouchaud, 2003; Bouchaud et al., 2004) to a hyperbolic tangent (Plerou et al., 2002). When market impacts are

expressed as a power-law function of trading volume, the exponent is found to be around 0.5 for US stocks (Torre,

1997; Gabaix et al., 2006), approximately 0.3 for stocks listed on the London Stock Exchange (Farmer and Lillo,

2004; Farmer et al., 2005) and about 0.68 for Chinese stocks (Zhou, 2012). Different values of the exponent are driven

by the differences in the microstructure between different markets.

The study of market impact costs has been done extensively at aggregate transaction levels, where trades are

grouped over fixed time intervals. Using a 30-minute timescale, Torre (1997) finds that the price impact costs for

US stocks is proportional to the square root of trading volume, implying a concave market impact function. Further-

more, price impact is also estimated relative to stocks’daily volatility. The square root relation is confirmed by

Gabaix et al. (2003) when using 15-minute time intervals to analyze stocks on the NYSE and Paris Bourse ex-

changes. Other studies on different exchanges using similar timescales also come up with qualitatively similar

concave price impact functions (Kempf and Korn, 1999; Hopman, 2007). However, the nonlinear shape of market

impact becomes more linear when the time interval exploited gets larger (Plerou et al., 2002; Bouchaud et al., 2009).

More attention has been paid to the research of market impact at the individual transaction level over the last de-

cade. Using data from Trade and Quote database for the 1000 NYSE largest stocks from 1995 to 1998, Lillo et al.

(2003) find that immediate price impact can be characterized by a power-law function of normalized trading

volumes.

6

The exponent of the power-law relation varies from 0.1 to 0.5 across different stock groups in different

years and is typically higher for stocks with larger market capitalization. Furthermore, stock groups’liquidity is

positively associated to their average market capitalization via another power-law function whose power is

3

To rule out price manipulations, the permanent component of price impact is required to be a linear function of trading volume.

4

See, among others, Bouchaud et al. (2004, 2009); Chen et al. (2005); Keim and Madhavan (1996); Lillo et al. (2003); Lim and Coggins (2005);

Plerou et al. (2002).

5

Strictly speaking, the relation between price impact and trading volumes is concave for buyer-initiated transactions and convex for seller-initiated

trades because price impact is generally positive for buys but negative for sells. When taking the absolute value of price impact, the concavity

nature is obtained for both buys and sells.

6

Lillo et al. (2003) define normalized trading volume as the ratio of trading volumes in shares to their averages.

Forecasting Immediate Price Impact 899

Copyright © 2016 John Wiley & Sons, Ltd. J. Forecast. 36, 898–918 (2017)