A SIMULATION ANALYSIS OF HERDING AND UNIFRACTAL SCALING BEHAVIOUR

• Date: 01 January 2014
• DOI: http://doi.org/10.1002/isaf.1346
• Published date: 01 January 2014
• Author: Wing Lon Ng,Steve Phelps

A SIMULATION ANALYSIS OF HERDING AND UNIFRACTAL

SCALING BEHAVIOUR

STEVE PHELPS*AND WING LON NG

Centre for Computational Finance and EconomicAgents (CCFEA), University of Essex, Wivenhoe Park, Colchester, UK

SUMMARY

We model the financial market using a class of agent-based models in which agents’expectations are driven by

heuristic forecasting rules (in contrast to the rational expectations models used in traditional theories of financial

markets). We show that, within this framework, we can reproduce unifractal scaling with respect to three

well-known power laws relating (i) moments of the absolute price change to the time-scale over which they are

measured, (ii) magnitude of returns with respect to their probability and (iii) the autocorrelation of absolute returns

with respect to lag. In contrast to previous studies, we systematically analyse all three power laws simultaneously

using the same underlying model by making observations at different time-scales and higher moments. We show

that the first two scaling laws are remarkably robust to the time-scale over which observations are made,

irrespective of the model configuration. However,in contrast to previous studies, we show that herding may explain

why long memory is observed at all frequencies. Copyright © 2013 John Wiley & Sons, Ltd.

Keywords: scaling; agent-based modelling; adaptive expectations

1. INTRODUCTION

The recent financial crisis highlights the difficulties inherent in models such as the capital asset pricing

model which are based on rational expectations and efficient markets assumptions. Whilst it would be an

exaggeration to state that belief in the efficient markets hypothesis actually caused the crisis (Ball, 2009;

Brown, 2011), it is nevertheless now acknowledged that widely adopted theoretical models which

assume that returns are Gaussian, as per geometric Brownian motion, are not consistent with the data

from real-world financial exchanges (Lo and MacKinlay, 2001).

This had led to a resurgent interest in alternatives to models based on rational expectations models

and the efficient markets hypothesis; Lo (2005) proposed the ‘adaptive markets hypothesis’as an

alternative paradigm. The adaptive markets hypothesis posits that incremental learning processes may

be able to explain phenomena that cannot be explained if we assume that agents instantaneously adopt

a rational solutio n, and is inspired by mode ls such as the ‘El Farol Bar’problem (Arthur, 1994) in which

it is not clear that a rational expectations solution is coherent.

Agent-based models address these issues by modelling the system in a bottom-up fashion; in a typical

agent-based model we simulate the behaviour of the participants in the market –the agents –and equip

them with simple adaptive behaviours.Within the context of economic and financial models, agent-based

modelling can be used to simulate markets with a sufficient level of detail to capture realistic trading be-

haviourand the nuances of the detailed microstructural operationof the market: for example, the operation

of the underlying auction mechanism used to match orders in the exchange (Iori and Chiarella, 2002).

* Correspondence to: Steve Phelps, Centre for Computational Finance and Economic Agents (CCFEA), University of Essex,

Wivenhoe Park, Colchester, UK. E-mail: sphelps@essex.ac.uk

Copyright © 2013 John Wiley & Sons, Ltd.

INTELLIGENT SYSTEMS IN ACCOUNTING, FINANCE AND MANAGEMENT

Intell. Sys. Acc. Fin. Mgmt. 21, 39–58 (2014)

Published online 7 October 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/isaf.1346

The move to electronic trading in today’s markets has provided researchers with a vast quantity of

data which can be used to study the behaviour of real-world systems comprised of heterogeneous

autonomous agents interacting with each other. A recent area of research within the multi-agent

systems community (Cassell and Wellman, 2012; Palit et al., 2012; Rayner et al., 2013) attempts to take

a reverse-engineering approach in which agent-based models of markets are built to replicate specific

statistical properties that are universally observed in real-world data sets across different markets and

periods of time –that is, the stylized facts of financial asset returns (Cont, 2001).

In this paper we use an agent-based model to explore whether adaptation could be responsible for

some of the scaling laws observed in empirical financial time-series data. The main advantage of

scaling laws is their universality and scale invariance, allowing for both flexibility and consistency in

modelling. Scaling laws help to detect whether observed phenomena are to a certain degree similar

or even the same at different scales. The notable feature of this self-similarity concept is that the

characteristics and their implications would apply to both short-term and longer term behaviour of price

dynamics. Glattfelder et al. (2011), for instance, illustrated on high-frequency foreign exchange data

how one can exploit the plethora of intra-day data to construct robust trading models and then simply

scale models to address both short-term shocks and long-term fluctuations in market movements,

benefiting from the scale invariance property of the scaling law (also see Ng (2012)).

The existence of these scaling laws presents a mystery. Consider the fact that expected absolute price

changes are a power law of the time over which they are measured. This is highly surprising since the

underlying economic causes of changes to the price are very different depending on the time horizon

over which observations are made: at the highest frequency price changes are driven by the operation

of the limit-order book and low-latency algorithmic trading, whereas over the longer time periods the

interplay of the supply and demand of large institutional investors is the dominant factor. Thus, it is

remarkable that financial time-series data should exhibit such striking self-similarity at time-scales

varying from seconds to months, particularly since at higher frequencies we know that prices do not

follow geometric Brownian motion.

Although there has been great deal of literature documenting the existence of these scaling laws in

empirical data, to date that has been limited success in building m odels which explain how these

scaling phenomena arise. Existing agent-based models have had some success in showing that adaptive

expectations can give rise to some of these scaling laws. For example, LeBaron and Yamamoto (2007)

introduced an agent-based model in which power-law decay in the autocorrelation of volatility and

order-signs can only be replicated when agents learn from each others’forecasts via imitation, leading

to herding in expectations. When their model is analysed under a treatment in which this herding does

not occur, the corresponding long-memory properties disappear. However, this is not conclusive evi-

dence that these scaling properties arise from learning or herding behaviour; Tóth et al. (2011) exam-

ined long-memory in order-signs by comparing two different models, and showed that long-memory in

order-signs could be better explained as arising from the splitting of large institutional orders into

smaller orders in order to mitigate market impact –that is, it is likely caused by an exogenous influence

on the market rather than arising endogenously.

Nevertheless, there is not necessarily a single unified cause of all forms of scaling, and there remains

the possibility that endogenous factors could account for other scaling laws. Although scaling laws

relating to autocorrelation and size distributions have received a great deal of attention from the

agent-based modelling community, there are relatively few models which attempt to explain other

forms of scaling, such as the power law relating absolute price changes to the time period over which

they are measured. Indeed, there has been little attention paid to the fact that many scaling laws

continue to hold irrespective of the time period over which the quantities of interest are measured. In

40 S. PHELPS AND W. L. NG

Copyright © 2013 John Wiley & Sons, Ltd. Intell. Sys. Acc. Fin. Mgmt., 21,39–58 (2014)

DOI: 10.1002/isaf