Forecasting High‐Frequency Risk Measures

• Published date: 01 April 2016
• Author: Christophe Hurlin,Sessi Tokpavi,Denisa Banulescu,Gilbert Colletaz
• DOI: http://doi.org/10.1002/for.2374
• Date: 01 April 2016

Journal of Forecasting,J. Forecast. 35, 224–249 (2016)

Published online 25 November 2015 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2374

Forecasting High-Frequency Risk Measures

DENISA BANULESCU,1GILBERT COLLETAZ1CHRISTOPHE HURLIN1AND

SESSI TOKPAVI2

1

Université d’Orléans, DEG, Rue deBlois, BP 26739, 45067, Orléans, Cedex 2, France

2

EconomiX, Université Paris Ouest NanterreLa Défense, 200 avenue de la République, 92000,

Nanterre Cedex, France

ABSTRACT

This article proposes intraday high-frequency risk (HFR) measures for market risk in the case of irregularly spaced

high-frequency data. In this context, we distinguish three concepts of value-at-risk (VaR): the total VaR,the marginal

(or per-time-unit) VaR and the instantaneous VaR. Since the market risk is obviously related to the duration between

two consecutive trades, these measures are completed with a duration risk measure, i.e. the time-at-risk (TaR). We

propose a forecasting procedure for VaR and TaR for each trade or other market microstructure event. Subsequently,

we perform a backtesting procedure specifically designed to assess the validity of the VaR and TaR forecasts

on irregularly spaced data. The performance of the HFR measure is illustrated in an empirical application for two

stocks (Bank of America and Microsoft) and an exchange-traded fund based on Standard & Poor’s 500 index.

We showthat the intraday HFR forecasts capture accurately the volatility and duration dynamics for these three assets.

Copyright © 2015 John Wiley & Sons, Ltd.

KEY WORDS high-frequency risk measure; value at risk; time at risk; backtesting

INTRODUCTION

This paper proposes intraday high-frequency risk (HFR) measures for market risk in the case of irregularly spaced

high-frequency data. We distinguish three concepts of value-at-risk (VaR): the total VaR, the marginal (or per-time-

unit) VaR and the instantaneous VaR. The total VaR corresponds to the maximum expected loss for the time horizon

of the next trade, whereas the marginal VaR refers to the time horizon of the next time unit, which is generally defined

in seconds. The third concept corresponds to the maximum expected loss for the time interval between tiand tiC

for an infinitesimal time increase ,wheretidenotes the time of the last trade.

When considering tick-by-tick data, the market risk is obviously related to the duration between two consecutive

trades (Diamond and Verrecchia, 1987; Easley and O’Hara, 1992). As a consequence, our market risk measures are

completed with a duration risk measure. Here, we consider the time-at-risk (TaR), initially introduced by Ghysels

et al. (2004), which is defined as the minimum duration prior to the next trade with a given confidence level.

To our knowledge, only two papers derive intraday market risk models using tick-by-tick data. Giot (2005) relies

on two approaches.1First, he proposes to resample the data along a pre-specified time grid, which yields tempo-

rally equidistant observations (10 or 30 minutes). Then, standard time series models (RiskMetrics- or generalized

autoregressive conditional heteroskedasticity (GARCH)-type models) are used to forecast the conditional volatility

and the VaR for equidistantly time-spaced returns. But this approach neglects the irregular timing of trades. Second,

Giot derives the returns volatility (and thus the VaR) from the conditional intensity process associated with the price

duration, defined as the time necessary for the price of an asset to change by a given amount. Following Engle and

Russell 1998, he considers an autoregressive conditional duration (ACD)-type model to describe the dynamics of

duration and compute the irregularly spaced VaR for price events. Then, this VaR is rescaled at fixed-time intervals

for backtesting purposes. More recently, Dionne et al. (2009) proposed an intraday VaR (IVaR), which is based on an

ultra-high-frequency (UHF)-GARCH-type model (Engle, 2000) and a Monte Carlo simulation approach, to infer VaR

at any fixed-time horizon. However, in both studies the market risk measure is always rescaled at fixed time intervals

because of the way it is constructed or for backtesting purposes. Alternatively,we propose to evaluate the market risk

at each trade, while taking into account the irregular timing of transactions.

One advantage of our measures is that they can be extended to any type of market microstructure event by con-

sidering a subset of trades with specific characteristics or marks. We can define a VaR and TaR for the transactions

associated, for instance, with significant price changes (i.e. price events) or with a minimum volume (i.e. volume

events). This feature differentiates our measure from the intraday VaR proposed by (Giot, 2005), which is only valid

for price events.

Correspondence to: Christophe Hurlin, University of Orléans, LEO, UMR CNRS 7322, France E-mail: christophe.hurlin@univ-orleans.fr

1We do not mention here the papers that propose intraday VaRs for regularly spaced returns (Coroneo and Veredas,2012 etc.). Note that Kozhan

and Tham (2012) have recently proposed a measure for the execution risk in high-frequency arbitrage.

Copyright © 2015 John Wiley & Sons, Ltd

Forecasting High-Frequency Risk Measures 225

The VaR and TaR are defined as quantiles of two conditional distributions: the distribution of the intraday returns

and the distribution of durations. Since these variables are linked, many approaches can be used to model their joint

dynamics (Engle, 2000; Engle and Russell, 1997; , 1998; Ghysels and Jasiak, 1998; Gerhard and Hautsch, 2002;

Grammig and Wellner, 2002; Meddahi et al., 2006). In this context, the definition of the conditioning information set

and the exogeneity assumptions made for both processes are important. When forecasting the volatility (and thus the

VaR) for the next trade, which is indexed by iC1, two solutions can be adopted. The first one assumes that the past

prices and the other marks are known until the ith trade (Ghysels and Jasiak, 1998). The second solution assumes

that, in addition to this information, the duration between the ith and the ith+1 trades is also known (Engle, 2000;

Meddahi et al., 2006). This difference is important for risk management; the market risk is generally evaluated after

each transaction for the time horizon of the future transaction and not 1 millisecond before its realization. Besides, the

duration risk, as measured by the TaR, is not relevant if we assume that the duration before the next trade is known.

Therefore, we consider only the information available at the current trade to compute our HFR forecasts, which is

one of the primary differences of our approach from the IVaR of Dionne et al. (2009).

We present a simple forecasting algorithm for the HFR measure based on an EACD model for the conditional

duration process and a time-varying GARCH model (Ghysels and Jasiak, 1998) for the conditional volatility. The

two models are estimated by quasi-maximum likelihood (QML). To compute then the intraday TaR and VaR, we pro-

pose a semi-parametric approach similar to that considered by Manganelli and Engle (2001) in the day-to-day VaR

perspective. No specific assumptions (except those required by the QML estimation method) are made regarding the

conditional distributions of durations and returns. This forecasting algorithm is evaluated with a backtesting proce-

dure specifically designed to assess the validity of the VaR and TaR forecasts obtained for each trade. In contrast

to the previous studies, we do not rescale the VaR forecasts to fixed time intervals to apply the typical backtest-

ing procedures. The forecasts are evaluated at each transaction up to the time horizon of the next transaction, as

is generally done in the context of HFR management. We use three backtests compatible with irregularly spaced

data: the LR test of Christoffersen (1998) based on a Markov chain model; the duration-based test of Christoffersen

and Pelletier (2004); and the generalized method of moments (GMM) duration-based test proposed by Candelon

et al. (2011).

In an empirical application, we consider three financial assets: Bank of America and Microsoft stocks and an

exchange-traded fund (ETF) that tracks the S&P) 500 index. The use of an ETF is justified by the increasing

importance of these assets in the fund management industry.2For each asset, we compute a sequence of one-step-

ahead forecasts for 1% VaR and 1% TaR based on the trade or price events recorded in September 2010. In all

cases, the VaR and TaR forecasts accurately capture the volatility and duration dynamics. The VaR and TaR viola-

tions, which are defined as circumstances in which the ex post tick-by-tick returns (durations) are smaller (higher)

than the ex ante VaR (TaR) forecasts, satisfy the unconditional coverage and independence assumptions (Christof-

fersen, 1998). The frequency of violations is always statistically not different from the level of risk, i.e. 1%in

our case. More importantly, these violations are not clustered, which indicates that TaR and VaR forecasts adjust

rapidly to the changes observed in past durations and returns. Finally, we show that these backtesting results are

valid throughout the day and the week, even though the EACD-GARCH model is not re-estimated. In addition,

our findings are also robust to the choice of split point used to separate the in-sample and out-of-sample periods

(Hansen and Timmermann, 2012).

The remainder of the paper is organized as follows. In the next section, we define the HFR measures and introduce

the method used to model the dynamics of returns and durations. The third section proposes the forecasting algo-

rithm and the intraday periodicity adjustment. The fourth section presents the empirical application and backtesting

procedure, whereas the fifth section discusses marginal VaR and instantaneous VaR. The last section concludes and

suggests further research.

HIGH-FREQUENCY RISK MEASURES

The tick-by-tick data for a given stock are described by two variables: the time of the transactions and a vector of

marked point processes. The latter variable contains, for example, the volume, the bid–ask spreads and the price of

the contract observed at the time of the transaction. Consider a trade that occurred at time tiat a log-price piand

denote by ´ithe corresponding vector of marks other than the price. Thus the duration between two consecutive

trades is defined as xiDtiti1and the corresponding continuously compounded return is riDpipi1.The

information set available at time ti1is denoted by Fi1and includes all past durations and marked point processes:

Fi1D®xj;p

j;´

j;j i1¯.

2At the end of August 2011, 2982 ETFs worldwide were managing USD 1348 billion, which represents 5.6% of the assets under the management

of the fund management industry. Additionally, the total ETF turnover that occurs on-exchange via the electronic order book was 8.5% of the

equity turnover (Fuhr D, 2011).

Copyright © 2015 John Wiley & Sons, Ltd J. Forecast. 35, 224–249 (2016)