Page 1

334 Finance a úv?r – Czech Journal of Economics and Finance, 59, 2009, no. 4

JEL Classification: C1, C5, G1

Keywords: intraday data, realized variance, return and volatility distributions, heterogeneous autoregressive

model

Distribution and Dynamics of Central-European

Exchange Rates: Evidence from Intraday Data*

Vít BUBÁK – Institute of Economic Studies, Charles University in Prague and CES, Université

Paris I. Panthéon-Sorbonne (vit.bubak@malix.univ-paris1.fr)

Filip ŽIKEŠ – Imperial College London, Business School (fzikes@imperial.ac.uk)

Abstract

This paper investigates the behavior of the EUR/CZK, EUR/HUF and EUR/PLN spot ex-

change rates in the period 2002–2008, using 5-minute intraday data. We find that daily

returns on the corresponding exchange rates scaled by model-free estimates of daily re-

alized volatility are approximately normally distributed and independent over time. On

the other hand, daily realized variances exhibit substantial positive skewness and very

persistent, long-memory type of dynamics. We estimate a simple three-equation model

for daily returns, realized variance and the time-varying volatility of realized variance.

The model captures all salient features of the data very well and can be successfully em-

ployed for constructing point, as well as density forecasts for future volatility. We also

discuss some issues associated with measuring volatility from the noisy high-frequency

data and employ a simple correction that accounts for the distortions present in our data-

set.

1. Introduction

The recent economic downturn has put an end to a period of relative stability

that the Czech koruna, Hungarian forint and Polish z?oty enjoyed over the last years.

The considerable increase in the volatility of these currencies raises a question about

the ability of the Czech Republic, Hungary and Poland to fulfill the exchange rate

stability criteria stipulated in the Maastricht Treaty. Indeed, these criteria require that

for at least two years prior to the entry into the Eurozone, the applicant country's cur-

rency remain within a normal fluctuation band around the central parity, effectively

setting limits to the currency’s volatility during the pre-accession period (Antal and

Holub, 2007). There is no doubt that while the choice of the appropriate monetary

and exchange rate policies will be crucial to ensure that the currency meets the con-

vergence criteria, the design and implementation of such policies would not be pos-

sible without a thorough understanding of the statistical properties of the currencies

in question. A practical framework for accurate modeling and forecasting of the ex-

change rate volatility in particular could ultimately help in making the relevant poli-

cies more efficient.

A good knowledge of the Central European (CE) exchange rates dynamics is

equally relevant for asset pricing and risk management. Understanding the condition-

al probability distribution of the exchange rate returns and their volatility is critical

for accurate estimation of various models used in pricing and hedging derivative se-

curities written on the exchange rate. On a more general level, frequent and poten-

* The authors thank Jozef Baruník for helpful comments on an earlier version of this paper. Bubak grate-

fully acknowledges financial support from the Czech Ministry of Education, grant code MSM0021620841.

Page 2

Finance a úv?r – Czech Journal of Economics and Finance, 59, 2009, no. 4 335

tially large unexpected exchange rate movements adversely affect the performance of

export-oriented businesses. Papaioannou (2006) discusses specific types of exchange

rates risk that these companies face at times of increased currency volatility, include-

ing transaction costs associated with hedging against unfavorable exchange rate

movements and economic costs arising from increased uncertainty about future rela-

tive competitiveness. As the CE currencies continue to suffer from relatively high

volatility triggered by the global economic crisis, containing these and related risks

demands effective risk management decisions that are impossible without a sound

knowledge of the underlying exchange rate behavior.

The CE currencies have been subject to a wide range of studies. The most re-

cent focus on understanding the effectiveness of foreign exchange interventions con-

ducted by Central Banks (see Geršl, 2004; Geršl, 2006; Geršl and Holub, 2006; Égert

and Komárek, 2006; Égert, 2007), the sustainability of the real exchange rates (Bulí?

and Šmídková, 2005), or the equilibrium real exchange rate determination (Melecký

and Komárek, 2008), among others.

In contrast, only a limited number of studies have attempted to model the dy-

namics of the spot exchange rates for the CE currencies. Ko?enda and Valachy (2006)

provide a detailed analysis of the exchange rate volatility in the Visegrád countries,

with a particular focus on the period in which these countries abandoned tight FX

regimes for more flexible ones. Using daily nominal exchange rate data, the authors

employ an augmented version of a threshold GARCH-in-Mean (T-GARCH) model

to study the effects of path dependency, asymmetric shocks, and movements in in-

terest rates on exchange rate volatility during the regime transition. The study shows

that the introduction of the more flexible regime lead to a general increase in ex-

change rate volatility, with the level of volatility persistence becoming roughly the same

across the exchange rates analyzed. The authors also find a significant and negative

effect of asymmetric shocks on the volatility of Polish z?oty and Hungarian forint under

the floating regime.

In a related paper, Fidrmuc and Horváth (2008) analyze the exchange rate dy-

namics in the selected EU members including the Czech Republic, Hungary and Po-

land, using daily data from 1999 to 2006. The authors apply both a GARCH model

and an extended version of the TARCH model to assess the exchange rate volatility

in connection with the estimated target exchange rate and the credibility of exchange

rate management. Among other findings, the study shows that the daily exchange rate

volatility exhibits strong persistence as well as systematic asymmetric effects, with

the latter being especially pronounced during the periods of exchange rate appre-

ciation.

Horváth (2005) investigates the medium-term determinants of the bilateral ex-

change rate volatility of Central and Eastern European countries (CEEc) based on

the optimal currency area criteria. As part of the analysis, the author also compares

the actual and predicted exchange rate variability between the Euro area countries

and the CEEc. Although limited to the use of quarterly data and a relatively short

sample period from 1999 to 2004, the study shows that the actual exchange rate vari-

ability is larger in the CEEc compared to what it had been in the Euro area before its

creation. In addition, the author finds the predicted exchange rate variability to be

close to the Eurozone levels, with the difference between the latter and the actual

Page 3

336 Finance a úv?r – Czech Journal of Economics and Finance, 59, 2009, no. 4

variability caused by the Euro area countries participating in the ERM during the sam-

ple period.

Finally, Frömmel (2007) provides an interesting investigation of the changes

between volatility regimes in five Central and Eastern European countries, including

the Czech Republic, Poland and Hungary. Frömmel employs a Markov-Switching

GARCH model to study whether the changes between the volatility regimes are consist-

ent with changes in the official exchange rate arrangements. Among other findings,

the author concludes that an increase in the flexibility of the exchange rate regime

leads to an increase in exchange rate volatility.

The goal of this paper is to examine the conditional distribution of the Czech

koruna, Hungarian forint and Polish z?oty exchange rates vis-à-vis the Euro in the pe-

riod 2002–2008. Employing a 5-minute intraday data, we examine the distributional

properties and time-series dynamics of both daily exchange rate returns, as well as

daily realized variance. Unlike the existing empirical literature that employs almost

exclusively a GARCH framework to study the dynamics of the exchange rate, our work

relies on model-free nonparametric measures of ex-post volatility based on the use of

intraday data. This approach, pioneered by Andersen and Bollerslev (1998), has at-

tracted substantial attention in the recent financial econometric literature; see e.g.

McAleer and Medeiros (2008) for a recent review. It offers a number of advantages.

First, no parametric assumptions are needed to ensure that the realized vari-

ance and related measures are consistent for the true, unobserved volatility, apart

from some mild regularity conditions. This is in stark contrast to the GARCH frame-

work, where all results concerning the behavior of volatility hinge on a particular

specification of the GARCH variance equation.

Second, realized variance captures the total variation in the price or exchange

rate over a given period of time, unlike a GARCH-type model that focuses on condi-

tional volatility of the price at time t, given the information set available at time t – 1.

In other words, realized variance combines both the volatility expectations as well as

the innovations to volatility. This carries important implications for studying the con-

ditional distributions of one-period returns as pointed out by Andersen, Bollerslev

and Dobrev (2007): while the one-period financial returns standardized by condition-

al volatility typically appear to be leptokurtic, standardizing by realized volatility

produces approximately Gaussian innovations. This in turn lends empirical support to

a large class of continuous-time stochastic volatility models widely employed in the as-

set pricing literature.

Finally, since the realized variance and alternative related measures render vol-

atility essentially observable up to a measurement error that vanishes as the sampling

frequency increases, simple time-series models can be used to model and accurately

forecast future volatility (see Andersen, Bollerslev, Diebold and Labys, 2003; Ander-

sen, Bollerslev and Dobrev, 2007, among others). This includes not only point fore-

casts, that is, the expected future volatility, but the entire predictive density for future

volatility, allowing for construction of confidence intervals around the point forecast

or, similarly, estimation of the probability that future volatility remains within a cer-

tain fluctuation band. The ability to provide the predictive density for future volatility

also facilitates the measurement and management of risk associated with trading re-

alized volatility, which has become very popular in recent years (e.g. Bondarenko,

Page 4

Finance a úv?r – Czech Journal of Economics and Finance, 59, 2009, no. 4 337

2007). In this paper, we only focus on a simple model for returns and variance since

our primary interest lies in studying the dynamics and conditional distributions of

the EUR/CZK, EUR/HUF and EUR/PLN spot exchange rates.

Our empirical results confirm some stylized facts about the behavior of re-

turns and volatility of foreign exchange rates. We find that daily returns on the ex-

change rates are approximately normally distributed and independent over time, when

properly scaled by model-free estimates of daily realized variance. Daily realized

variance, on the other hand, exhibits substantial positive skewness as well as a very

persistent, long-memory type of dynamics. We propose a relatively simple model for

daily returns, realized variance and the time-varying volatility of realized variance,

finding that it very well captures all salient features of the data. In addition, the model

is shown to perform remarkably well out-of-sample, delivering accurate volatility fore-

casts. It may therefore serve well as an auxiliary model for estimating various con-

tinuous-time stochastic volatility models used for pricing derivative securities written

on the exchange rate (Bollerslev, Kretschmer, Pigorsch and Tauchen, 2009).

The rest of the paper is organized as follows. In Section 2 we describe our

theoretical framework and discuss some distributional predictions that it generates

for the EUR/CZK, EUR/HUF and EUR/PLN returns. In Section 3, we follow with

a definition of the realized variance as a model-free measure of variation in asset

prices and some of the issues associated with measuring volatility from noisy high-

-frequency data. In Section 4 we describe the data and in Section 5 we report the em-

pirical results. In particular, we present the results of the tests of normality and in-

dependence of returns standardized by realized volatility, the estimation of a joint

model for daily returns, realized variance and the volatility of realized variance, and

the results of an out-of-sample volatility forecasting exercise. Section 6 concludes

the paper with some suggestions for future work.

2. Theoretical Framework

Following a vast body of recent literature in financial econometrics, we adopt

a relatively simple, yet very general continuous-time framework. Working in con-

tinuous time has a number of technical advantages, but more importantly it provides

a direct link to the asset pricing literature, which establishes a number of important

results concerning the restrictions on admissible models governing asset prices in

an arbitrage-free environment (Back, 1991). A detailed overview of this and related

issues is beyond the scope of this paper and we refer the interested reader to an ex-

cellent discussion in Andersen, Bollerslev, Diebold and Labys (2003).

We assume that the logarithmic spot exchange rate, st, follows a stochastic

volatility model given by

t

tu

su

???

?

00

dd

t

uu

W

?

?

(1)

where ?t and ?t denote the drift and volatility processes, respectively, and Wt is

a standard Brownian motion. Both ?t and ?t are allowed to be general stochastic pro-

cesses and we do not impose any parametric assumption regarding their respective

laws of motion. Also, no restrictions are placed on the dependence between volatility

(?t) and the Brownian motion (Wt) driving the exchange rate innovations.

Page 5

338 Finance a úv?r – Czech Journal of Economics and Finance, 59, 2009, no. 4

A few remarks regarding the model in equation (1) are in order. First, the sam-

ple paths of the exchange rate are continuous, hence ruling out the presence of jumps.

We choose to make this assumption to keep our framework simple for the sake of

exposition, but nothing prevents us from including a jump process to the drift and

diffusion components in equation (1). Indeed, the measures of volatility that we em-

ploy later in the paper can capture both parts of the variation, i.e. the diffusion part

and jump part, if present, and hence there is no loss of generality in this sense by

doing otherwise.

Second, the model nests a wide variety of arbitrage-free stochastic volatility

models employed in the asset pricing literature. The well-known Black-Scholes model,

where both the drift and volatility are constant, is a prominent example. For more

general and empirically relevant specification see Chernov, Gallant, Ghysels and

Tauchen (2003) and the references therein.

Finally, the model delivers testable distributional predictions: the one-period

returns defined as rt = st – st–1 are conditionally on the sample path of drift and vol-

atility, normally distributed. Formally:

?

t

?

?

?

2

u

1

11

|{,}~ d ,ud

tt

t

tu u tu

t

rNu

? ???

?

?

?

(2)

Since the drift is typically negligible at daily and weekly frequencies, espe-

cially in the case of foreign exchange rates, the key quantity that we are interested in

is the so-called integrated variance,

2

u

1

d

t

t

t

IVu

?

?

??

(3)

which, as equation (2) shows, is the natural measure of variation in the one-period

returns. The conditional normality of rt further implies that in the absence of depen-

dence between the volatility process and the Brownian motion driving the exchange

rate (Wt), the one-period standardized returns follow the standard normal distribution,

t

tu

t

?

?

??

1

1/2

2

u

1

d

~(0,1)

d

iid

t

t

ru

N

u

?

?

?

??

(4)

Similar predictions can be derived when the volatility process correlates with

the Brownian motion. The normality of properly standardized returns has found an over-

whelming empirical support across different assets classes; see e.g. Andersen, Bol-

lerslev and Dobrev (2007), Andersen, Bollerslev, Frederiksen and Nielsen (2009) and

Žikeš (2008) for recent evidence from equity index futures, individual stocks, and for-

eign exchange rates, respectively. It is worth reiterating that this distributional assump-

tion can be tested without making any parametric assumptions about the volatility

process since the integrated volatility appearing in the denominator of the standardized

returns can be consistently estimated by nonparametric methods, which we describe in

the next section.

3. Measuring Daily Variance

Suppose we obtain a sample of size T(M+1) corresponding to T days, each hav-

ing M + 1 intraday observations of the logarithmic spot exchange rate. We denote by