PERIODICITIES OF FOREIGN EXCHANGE MARKETS AND THE DIRECTIONAL CHANGE POWER LAW

• Published date: 01 July 2013
• Author: Nick Constantinou,Wing Lon Ng,Iacopo Giampaoli
• DOI: http://doi.org/10.1002/isaf.1343
• Date: 01 July 2013

PERIODICITIES OF FOREIGN EXCHANGE MARKETS AND THE

DIRECTIONAL CHANGE POWER LAW

IACOPO GIAMPAOLI,

a

WING LON NG

a

*AND NICK CONSTANTINOU

b

a

Centre for Computational Finance and Economic Agents (CCFEA), University of Essex, Colchester,UK

b

Essex Business School, University of Essex, Colchester, UK

SUMMARY

This paper utilizes advanced methods from Fourier analysis in order to describe periodicities in financial ultrahigh

frequency foreign exchange data. The Lomb–Scargle Fourier transform is used to take into account the irregularity

in spacing in the time domain. It provides a natural framework for the power spectra of different inhomogeneous

time-series processes to be easily and quickly estimated. Furthermore, an event-based approach in intrinsic time

based on a power-law relationship is employed using different event thresholds to filter the foreign exchange

tick-data. The calculated spectral density demonstrates that the price process in intrinsic time contains different

periodic components for directional changes, especially in the medium–long term, implying the existence of

stylized facts of ultrahigh frequency data in the frequency domain. Copyright © 2013 John Wiley & Sons, Ltd.

Keywords: ultrahigh frequency transaction data; foreign exchange; irregularly spaced data; Lomb–Scargle Fourier

transforms; spectral density

1. INTRODUCTION

The concept of physical time plays a key role in classical economics through the discounting of future

cash flows. In particular, in exponential discounting, classical economics yields a time-independent

discount rate; agents have consistent time preferences (e.g. Takahashiet al., 2008). However, alternative

concepts of time have also provided additional insights. Recently, Takahashi et al. (2008) and Han and

Takahashi (2012) investigated the effect of psychological time (logarithmic in physical time) leading

to hyperbolic discounting. The model was then employed to explain empirical anomalies in classical

economics, such as time reversal preference (agents prefer small immediate rewards over larger future

rewards) and the ‘sign-effect’(in which delayed losses are more steeply discounted than gains). This

model of psychological time was also recently extended by Takahashi (2011) to obtain insights into

Prelec’s probability weighting functions (Prelec, 1998) as applied to standard prospect theory of

decisions under risk. In this paper we consider another alternative to physical time in analysing the

financial market, namely event-based intrinsic time.

In financial markets, trading activity typically tends to vary depending on the time of day, shrinking

during lunch-time and over weekends, and increasing prior to major scheduled news announcements;

for example, see Chordia et al. (2001). These observations are usually measured in physical time.

Another approach that has recently been put forward and which has led to a rich array of scaling

behaviour in the foreign exchange (FX) market is a time scale that is defined via events; that is,

the so-called intrinsic time –see Derman (2002) and Glattfelder et al. (2011).

* Correspondence to: Wing Lon Ng, Centre for Computational Finance and Economic Agents (CCFEA), University of Essex,

Colchester, UK. E-mail: wlng@essex.ac.uk

Copyright © 2013 John Wiley & Sons, Ltd.

INTELLIGENT SYSTEMS IN ACCOUNTING, FINANCE AND MANAGEMENT

Intell. Sys. Acc. Fin. Mgmt. 20, 189–206 (2013)

Published online in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/isaf.1343

Scaling laws, which are ubiquitous in finance, have a significant history and have been used to

discriminate between competing models that describe the price process –for a review, see Mandelbrot

(2001). The directional-change scaling law that is the focus of this paper has been found to be

particularly useful. This scaling law has recently been utilized to develop FX trading algorithms

(Dupuis and Olsen, 2012) and in risk management. Its use in risk management, via the construction

of the ‘scale of market quakes’(Bisig et al., 2012), is interesting in that the approach is analogous to

the Richter scale for earthquakes, which is a logarithmic scale based on the empirical observation that

the probability of an earthquake of a given energy follows a scaling law.

In particular, the presence of different patterns of trading activity in financial markets makes the flow

of physical time discontinuous, as the number of transactions tends to increase or decrease at different

time intervals. This empirical observation, which led to the definition of intrinsic ‘trading time’, coined

by Mandelbrot and Taylor(1967), suggests that the concept of physical time may not be the fundamental

time scale and research should additionally focus on an event-based time scale. In this paper, the analysis

of price movements focuses on an intrinsic time scale defined by ‘directional-change’events: price

movements exceeding a given threshold which are independent of the notion of physical time. Recent

literature has demonstrated a rich landscape of scaling laws in intrinsic time for ‘ultrahigh frequency’

(UHF) FX data (see Bouchaud (2001, 2002) for an early survey), including a scaling law for the

directional-change events considered in this paper.

Following the intrinsic time paradigm by Glattfelder et al. (2011), we filter our UHF data by

analysing those points where there has been a directional change (the event), consisting of price

movements away from a local maximum (or minimum), which exceed a given threshold –also see

Hautsch (2004: 36) for a more formal description. The newly created time series naturally includes

fewer observations, but it still contains significant information about the price evolution via its

associated scaling law and about the temporal structures through its directional-change duration, and

is irregular-spaced in time. Indeed, time series in intrinsic time are fundamentally unevenly spaced in

time and there is no justification in artificially making them equally spaced –see also Dacorogna

et al. (2001) and Giampaoli et al. (2009).

Aldridge (2010) argued that trading opportunities are indeed present at various data frequencies

(i.e. observation intervals), hence suggesting that agents with different time horizons and risk preferences

would be able to trade profitably in the market. One of the key features common to all typ es of high-

frequency data is the persistence of underlying trading opportunities; these range from fractions-of-

a-second price m ovements, arising f rom market-making trades, t o several-minutes-long strat egies

based on momentum forecasted by microstructure theories, to several-hours-long market moves deriving

from cyclical events and deviations from statistical relationships (Aldridge, 2010). The development of

high-frequency trading strategies beginswith the identificationof recurrent (i.e. periodic)profitable trading

opportunitiespresent in the data, and their profitability depends upon the chosentrading frequency. As the

data frequency increases, the possible range of each price movement shrinks, but the number of ranges

increases, thus potentially increasing trades profitability.

1

The main goal of this paper is to detect potential periodic patterns of UHF FX market data. For

example, it is well known that intraday data have a consistent diurnal patter n of trading activities over

the course of a trading day due to institutional characteristics of organized financial markets, such as

opening and closing hours or intraday auctions. In fact, FX markets have stronger seasonal patterns

1

In fact, Aldridge (2010) calculates the maximum gain as the sum of price ranges at each frequency, whereas the maximum

potential gain (loss) at every frequency is determined by the (negative) sum of all per-periodranges at that frequency.

190 I. GIAMPAOLI ET AL.

Copyright © 2013 John Wiley & Sons, Ltd. Intell. Sys. Acc. Fin. Mgmt. 20, 189–206 (2013)

DOI: 10.1002/isaf