The estimation of daily volatility using high frequency data in the South African equity market

Financial market volatility is central to the theory and practice of asset pricing, option pricing, asset allocation, portfolio selection, portfolio rebalancing and hedging strategies as well as various risk management applications. Most textbooks assume volatility to be constant; however in practice this is a very dangerous assumption to make and has lead to a research program regarding the distributional and dynamic properties of financial markets. Given that financial markets display high speeds of adjustment, studies based upon daily observations may fail to capture information contained in intraday or high frequency market movements and until relatively recently the use of daily or equally spaced data was considered the highest meaningful sampling frequency for financial market data. Recently, the volatility modelling literature took a significant step forward. Andersen et al. (2001) proposed a new approach called 'realized' volatility that exploits the information in high frequency returns. Basically, the approach is to estimate daily volatility by taking the square root of the sum of the squared intraday returns which are sampled at very short intervals. We discuss several theoretical measures for volatility of which quadratic variation (QV), integrated variance (IV) and conditional variance (CV) are the most popular. Realized variance is a consistent estimator for QV and can approximate IV and CV under various conditions. GARCH models are only concerned with estimating CV. Furthermore we will discuss the NIG-GARCH model proposed by Venter and de Jongh (2002, 2004) and show how these models may be fitted to daily data to provide estimates and forecasts for daily volatility. We will also investigate three realized volatility methods which use intraday data to estimate dialy volatility namely; Jm, d m and d m . Here SSR is the sum of the squared intraday returns, ABDE is the realized variance estimator proposed by Andersen, Bollerslev, Diebold and Ebens (2001) and VARHAC is the Vector Autoregressive Heteroscedastic and Autocorrelation Consistent estimator introduced by Den Haan and Levin (1 996). Our study will be conducted along the lines of the study by Andersen et al. (2001), utilising South African equity data. However, as far as the scaled returns are concerned we will investigate alternative volatility estimators for daily return volatilities. In particular, the VARHAC estimator (see e.g., Bollen and lnder (2002)) will be compared with the ABDE estimator (see e.g., Andersen et al. (2001)) as well as with daily volatility estimators based on the NIG-GARCH model proposed by Venter and de Jongh (2002, 2004). To the best of our knowledge a similar study has not yet been performed in the South African equity market context. Consistent with the Andersen et al. (2001) study we found the unconditional distributions of the realized variances to be positively skewed (right skewed), while the realized logarithmic standard deviations are approximately Gaussian, as are the distributions of the returns scaled by realized standard deviations or realized volatilities. This conclusion holds for all the realized variance estimators (ABDE, SSR and VARHAC). The distribution of the returns scaled by the GARCH and NIG-GARCH volatility estimates are also close to normality. However, the distributions of the daily variance estimates and the log of the volatility (standard deviations) estimates based on the GARCH and NIG-GARCH models are clearly not normally distributed. We therefore conclude that our findings in the South African market context do agree with the claims made by Andersen et al. (2001) for the realized volatility estimation techniques and that the realized volatility techniques are a much better match to the criteria as set out by Andersen et al. (2001) than the GARCH and NIG-GARCH volatility estimates. Volatiliteit in die finansiele mark is sentraal tot die teorie en die praktyk van bateprysing, opsieprysing, bate-allokasie, portefeulje seleksie, portefeulje herbalansering en verskansingstrategiee sowel as verskeie riskobeheer toepassings. Meeste handleidings veronderstel dat volatiliteit konstant is; maar in die praktyk is dit nie 'n geldige aanname nie en he! dit ondermeer gelei tot 'n nuwe navorsingsveld wat die verdeling en dinamiese eienskappe van finansiele markte ondersoek. Gegewe dat die finansiele mark vinnig aanpas by veranderende omstandighede, sal studies wat gebaseer word op daaglikse waarnemings misleidend wees, aangesien hoe frekwensie mark bewegings nie weerspieel word nie. Daaglikse data (einde van die dag sluitingspryse) was tot onlangs die mees betekenisvolle steekproef frekwensie data vir die finansiele mark. Betekenisvolle vooruitgang in die veld van volatiliteit modellering is onlangs gemaak. Navorsing deur Andersen et al. (2001) beveel 'n nuwe benadering aan; naamlik "gerealiseerde volatiliteit" wat al die inligting in hoe frekwensie opbrengste (soos gebaseer op intradag prysdata) gebruik om daaglikse volatiliteit te beraam. Die gerealiseerde volatiliteitsbenadering skat daaglikse volatiliteit deur die vierkanstwortel van die som van die gekwadreerde intra-daaglikse obrengste oor kort intervalle te bereken. In hierdie studie word getoets of die bevindinge van Andersen et al. (2001) ook van toepassing gemaak kan word op Suid-Afrikaanse aandeel data. Andersen et al. (2001) bestudeer die SSR en ABDE beramers, wat beide op intradag data gebaseer is. SSR word gedefineer as die som van die gekwadreerde intradaaglikse obrengste en is 'n beramer van daaglikse variansie, ABDE is die gerealiseerde variansie beramer soos bekend gestel deur Andersen, Bollerslev, Diebold and Ebens (2001). Anders as Andersen et al. (2001) sal ons ook ander beramers vir daaglikse variansie bestudeer, naamlik die VARHAC beramer van Bollen en lnder (2002). wat op intra-dag data gebaseer is, en die GARCH en NIG-GARCH model van Venter en de Jongh (2002, 2004), wat op daaglikse data gebaseer is VARHAC is die Vector Autoregressive Heteroscedastic en Autocorrelation Consistent beramer soos bekend gestel deur Den Haan and Levin (1996). Tot die beste van ons wete is 'n soortgelyke studie nog nie op die Suid-Afrikaanse effektebeurs aangepak nie. Verskeie teoretiese maatstawwe vir daaglikse variansie word in die tesis bespreek, soos kwadratiese variansie (QV), ge'integreerde variansie (IV) en voorwaardelike variansie (CV). Gerealiseerde variansie (SSR) is 'n konsekwente beramer vir QV en kan IV en CV benader onder sekere omstandighede. Daarenteen word GARCH modelle gebruik om voorwaardelike variansie (CV) te beraam. In lyn met Andersen et al. (2001) se bevindinge het ons die volgende gevind. Die onvoorwaardelike verdelings van gerealiseerde volatiliteit soos beraam deur die vierkantswortels van SSR, ABDE en VARHAC is positief skeef. Die gerealiseerde logaritmiese standaardafwykings en die verdeling van die opbrengste wat gestandaardiseer is deur die gerealiseerde standaard afwykings is rnin of meer norrnaal verdeel. Hierdie gevolgtrekking geld vir al die gerealiseerde variansieberarners (ABDE, SSR en VARHAC). Ook het ons bevind dat die verdeling van die opbrengste gestandaardiseer deur die GARCH en NIG-GARCH variansieberamers naby normal verdeel is. Die verdelings van die daaglikse variansie is nie normaal verdeel vir enige beramer nie, rnaar die verdeling van die logaritme van die volatiliteite (standaard afwykings) soos beraarn deur die gerealiseerde variansie berarners is we1 normal verdeel, terwyl dit nie die geval is vir die GARCH en NIG-GARCH beramers nie. Ons rnaak dus die gevolgtrekking dat ons resultate en bevindings soos gebaseer op Suid-Afrikaanse effektebeurs data, met Andersen et al. (2001) se afleidings saarnstem.