Modelling Volatility Using High, Low, Open and Closing Prices: Evidence from Four S&P Indices

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1 Inernaional Research Journal of Finance and Economics ISSN Issue 28 (2009) EuroJournals Publishing, Inc hp:// Modelling Volailiy Using High, Low, Open and Closing Prices: Evidence from Four S&P Indices Chrisos Floros Senior Lecurer in Banking and Finance, Deparmen of Economics Universiy of Porsmouh Porsmouh Business School, Porsmouh, PO1 3DE, UK Tel: +44 (0) Absrac This paper uses several models (Alizadeh, Brand and Diebold, 1999; Parkinson, 1980; Garman and Klass, 1980; Rogers and Sachell, 1991) for he calculaion of volailiy based on high, low, open and closing prices. We use recen daily daa from four S&P indices, namely S&P 100, S&P 400, S&P 500 and S&P Small Cap 600. The resuls show ha a simple measure of volailiy (defined as he firs logarihmic difference beween he high and low prices) overesimaes he oher hree measures. Keywords: Volailiy, S&P indices, High price, Low price, Open Price, Closing Price, US. JEL Classificaion Codes: C14, C15, G13, G Inroducion Financial heories are ofen based on assumpions concerning he srucure of price daa (sock reurns, exchange and ineres raes), see Andreou e al (2001). For example, Efficien marke hypohesis (EMH) assumes ha speculaive prices can be modelled as random walks (Fama, 1970) while CAPM and Black-Scholes opion pricing model assume ha reurns are Normally disribued and follow he probabilisic assumpions of uncorrelaedness and saionariy. Empirical evidence of he ime series of daily sock reurns include (1) lepokurosis (fa ails relaive o he normal disribuion), (2) skewness, and (3) volailiy clusering (large reurns are expeced o follow large reurns, and small reurns o follow small reurns). According o Pagan (1996), volailiy is relaed o uncerainy and shows how much asse prices are moving around 1. There has been a significan emphasis on ime series models o explain he empirical observaion of volailiy clusering. Empirical models include he Generalised Auorgressive Condiional Heeroskedasiciy (GARCH) models developed by Engle (1982) and Bollerslev (1986), and sochasic volailiy (SV) models (see Hwang and Sachell, 2000). 1 Financial ime series exhibi periods where he volailiy is consisenly low ha alernae wih periods of consisenly high volailiy. This variaion of volailiy can be linked o he arrivals of informaion (see Mandelbro and Taylor, 1967) and rading volume (see Karpoff, 1987).

2 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) 199 A number of research papers model sock index reurns o es he performance of GARCH and SV models (Heyen and Ka, 1994; Andersen, 1994; Kim e al., 1998; Andersen, 1996; Chan and Lien, 2003) and repor ha making he choice beween GARCH and SV models is no easy 2. For his aricle, we consider four volailiy measures, oher han GARCH and SV, for US sock indices following he works of Alizadeh, Brand and Diebold (1999), Gallan, Hsu and Tauchen (1999), Parkinson (1980), Garman and Klass (1980), and Rogers, Sacell and Yoon (1994). We model daily volailiy using opening, closing, high and low prices from four S&P indices, namely S&P 100, S&P 400, S&P 500 and S&P Small Cap 600. The purpose of his paper is o re-evaluae he performance of several volailiy measures using recen daily range daa from he US. Our aim is wofold: (i) we examine he performance of volailiy esimaes when using US daa from S&P sock marke, and (ii) we es if volailiy esimaors based on opening, high, low and closing prices are efficien measures for S&P indices. The paper is organised as follows: Secion 2 provides he mehodology and daa informaion. Secion 3 presens he main empirical resuls, while Secion 4 concludes he paper and summarises our findings. 2. Mehodology and Daa Descripion Numerous recen sudies have been direced a modelling he sock marke volailiy using ime series modelling (Canarella and Pollard, 2007; Floros, 2007; Floros and Vougas, 2006). However, hey only use closing prices, and herefore, heir examinaions fail o consider a full range of prices (high, low, open as well as closing prices) in each rading day. To furher es he efficiency of volailiy measures in our daa, we model he non-consan volailiy parameer using four models based on he opening, closing, high and low prices 3. Volailiy measures (Chan and Lien, 2003) Le O, C, H, L denoe he opening, closing, high and low prices a day, respecively. A simple measure of volailiy is defined as he firs logarihmic difference beween he high and low prices (Alizadeh, Brand and Diebold, 1999; Gallan, Hsu and Tauchen, 1999): V = ln( H ) ln( L ) (1) S, Parkinson (1980) proposes a volailiy measure assuming an underlying geomeric Brownian moion wih no drif for he prices: [ H / L ] 2 V = = (2) 2 P, 0.361R ln( ) According o Chan and Lien (2003), V P, could be as much as 8.5 imes more efficien han logsquared reurns. A furher volailiy measure is based on opening and closing prices. Garman and Klass (1980) sugges he following measure: 1 V [ ] 2 [ ][ ] 2 GK, = ln( H ) ln( L ) 2ln 2 1 ln( C ) ln( O ) (3) 2 According o Chan and Lien (2003), boh measures are unbiased when he sample daa are coninuously observed wih V GK, being more efficien han V P,. 2 Heyen and Ka (1994) show ha GARCH models ouperform SV models in modelling exchange raes, while Kim e al. (1998) repor ha SV models are superior o GARCH models. Furher, Hwang and Sachell (2000) argue ha GARCH models are more suiable for describing volailiy. 3 In he curren lieraure here are oher popular specificaions, he GARCH models and he SV models; in his aricle, we only adop he volailiy framework based on he opening, closing, high and low prices. Our aim is o see which volailiy measure (oher han GARCH and SV) dominaes he oher.

3 200 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) When he drif erm is no zero, neiher he Parkinson nor he Garman-Klass measures are efficien (Chan and Lien, 2003). Hence, an alernaive measure wih independen drif is required. Rogers and Sachell (1991) and Rogers, Sachell and Yoon (1994) propose a volailiy measure which is subjec o a downward bias problem: V [ ln( H ) ln( O )][ ln( H ) ln( C )] + [ ln( L ) ln( O )][ ln( L ) ln( C )] = (4) RS, The daa employed in his sudy comprise 2010 daily observaions on he S&P sock indices: S&P 100, S&P 400, S&P 500 and S&P Small Cap 600. The daa covers he period 3 January December Closing, Open, High and Low prices for sock indices were obained from Daasream Inernaional and Bloomberg. Table 1 gives he descripive saisics for daily sock prices. We presen hree saisics which are calculaed using he observaions in he full sample: Skewness, Kurosis, and Jarque-Bera. Almos all US series have negaive skewness implying ha he disribuion has a long lef ail (only S&P 500 index shows a negaive skewness). The values for kurosis are less han hree in all cases. In oher words, he disribuions are no peaked relaive o normal. Moreover, he Jarque-Bera es rejecs normaliy a he 5% level for all disribuions. So, he sample has all financial characerisics: volailiy clusering and playkurosis. Furhermore, he resuls from he ADF uni roo ess (no repored here) indicae ha all series are I(1), and herefore, quaniaive models can be used o measure daily volailiy.

4 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) 201 Table 1: Descripive Saisics (Prices) H S&P 100 Close High Low Open Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions I S&P 500 Close High Low Open Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions J S&P 600 Close High Low Open Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions K S&P 400 Close High Low Open Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions Noes: Skewness is a measure of asymmery of he disribuion of he series around is mean. Kurosis measures he peakedness or flaness of he disribuion of he series. Jarque-Bera is a es saisic for esing wheher he series is normally disribued. 3. Empirical Resuls According o Cheung e al. (2009), daily highs and lows of sock indices do no diverge over ime. The same applies for opening and closing prices for our sudy 4. The resuls from equaions (1)-(4) are presened in Table 2. In all cases Vs overesimaes Vgk, Vp and Vrs, and i ranges from 1.5% (S&P 100) o 1.36% (S&P 500). Furhermore, S&P 100 has he highes Vgk, S&P Small Cap 600 shows a high Vp, S&P 100 has a high Vrs and S&P Small Cap The coinegraion framework for daily high-low and open-close prices show srong evidence of long-run relaionships (he resuls are available upon reques).

5 202 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) has a high Vs. Hence, boh S&P 100 and S&P Small Cap 600 show an increase in volailiy measures when specific models are used. Table 2: Volailiy Esimaes A S&P 100 Vgk Vp Vrs Vs Mean 9.05E E Median 4.55E E E Maximum Minimum 2.23E E Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions B S&P 500 Vgk Vp Vrs Vs Mean 8.09E E E Median 4.32E E E Maximum Minimum 1.59E E Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions C S&P 600 Vgk Vp Vrs Vs Mean 8.35E E Median 5.21E E E Maximum Minimum 2.93E E Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions D S&P 400 Vgk Vp Vrs Vs Mean 8.44E E Median 4.78E E E Maximum Minimum 2.47E E Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions Noes: Skewness is a measure of asymmery of he disribuion of he series around is mean. Kurosis measures he peakedness or flaness of he disribuion of he series. Jarque-Bera is a es saisic for esing wheher he series is normally disribued.

6 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) 203 Figure 1: S&P S&P Vgk Vp Vrs Vs S&P Figure 2: S&P S&P Vgk Vp Vrs Vs.09 S&P

7 204 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) Figure 3: S&P Small Cap S&P Small Cap Vgk Vp Vrs Vs S&P Small Cap Figure 4: S&P S&P 400 MidCap Vgk Vp Vrs Vs.10 S&P 400 MidCap

8 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) Summary and Conclusions Volailiy in financial markes has araced growing aenion by praciioners, policy makers and researchers as i is a measuremen of risk. The resuls repored in his paper show esimaes of volailiy in he US. We model volailiy using four models based on open, closing, high and low daily prices. We consider daily daa from four US sock indices (S&P 100, S&P 400, S&P 500 and S&P Small Cap 600) o es which measure dominaes each oher. Firs, we find srong evidence ha daily prices can be characerised by volailiy models. In paricular, we repor ha he prices have all financial characerisics: volailiy clusering, playkurosis and nonsaionariy. Finally, we use four models o calculae daily volailiy. The resuls show ha Vs, a simple measure of volailiy defined as he firs logarihmic difference beween he high and low prices, overesimaes Vgk, Vp and Vrs. These findings are srongly recommended o risk managers and modellers dealing wih he US financial indices. Fuure research should examine he performance of sochasic volailiy mehods o describe boh volailiy and marke risk of major sock indices. References [1] Alizadeh, S., Brand, M. W., and Diebold, F. X., Range-based esimaor of sochasic volailiy models. Working paper, Universiy of Pennsylvania. [2] Andersen, T., Sochasic auoregressive volailiy: a framework for volailiy modeling. Mahemaical Finance 4, [3] Andersen, T., Reurn volailiy and rading volume: an informaion flow inerpreaion of sochasic volailiy. Journal of Finance 51, [4] Andreou E., Piis, N., and Spanos, A., Modelling sock reurns: The empirical lieraure. Journal of Economic Surveys 15(2), [5] Bollerslev, T., Generalised auoregressive condiional heeroscedasiciy. Journal of Economerics 31, [6] Canarella, G. and Pollard, S. K., A swiching ARCH (SWARCH) model of sock marke volailiy: some evidence from Lain America. Inernaional Review of Economics 54(4), [7] Chan, L. and Lien, D., Using high, low, open, and closing prices o esimae he effecs of cash selemen on fuures prices. Inernaional Review of Financial Analysis 12, [8] Engle, R. F., Auoregressive condiional heeroscedasiciy wih esimaes of he variance of UK inflaion. Economerica 50, [9] Fama, E., Efficien capial markes: A review of heory and empirical work. Journal of Finance 25, [10] Floros, C., The use of GARCH models for he calculaion of Minimum Capial Risk Requiremens: Inernaional Evidence. Inernaional Journal of Managerial Finance 3(4), [11] Floros, C. and Vougas, D. V., Index fuures rading, informaion and sock marke volailiy: The case of Greece. Derivaives Use, Trading and Regulaion 12, [12] Gallan, A. R., Hsu, C. T., and Tauchen, G., Using daily range daa o calibrae volailiy diffusion and exrac he forward inegraed variance. Working paper, Universiy of Norh Carolina, Chapel Hill. [13] Garman, M. B., and Klass, M. J., On he esimaion of securiy price volailiies from hisorical daa. Journal of Business 53, [14] Heynen, R. C., and Ka, H. M., Volailiy predicion: A comparison of he sochasic volailiy, GARCH (1,1), and EGARCH (1,1) models. The Journal of Derivaives 4, [15] Hwang, S., and Sachell, S., Marke risk and he concep of fundamenal volailiy: Measuring volailiy across asse and derivaive markes and esing for he impacs of derivaives markes on financial markes. Journal of Banking and Finance 24,

9 206 Inernaional Research Journal of Finance and Economics - Issue 28 (2009) [16] Karpoff, J. M., The Relaion Beween Price Changes and Trading Volume. Journal of Financial and Quaniaive Analysis 22(1), [17] Kim, S., Shephard, N. and Cchib, S., Sochasic volailiy: Likelihood inference and comparison wih ARCH models. Review of Economic Sudies 65. [18] Mandelbro, B. and Taylor, H., On he Disribuion of Sock Price Differences. Operaions Research 15, [19] Pagan, A., The economerics of financial markes. Journal of Empirical Finance 3, [20] Parkinson, M., The exreme value mehod for esimaing he variance of he rae of reurn. Journal of Business 53, [21] Rogers, L. C. G., and Sachell, S. E., Esimaing variance from high, low, and closing prices. Annals of Applied Probabiliy 1, [22] Rogers, L. C. G., Sachell, S. E., and Yoon, Y., Esimaing he volailiy of sock prices: a comparison of mehods ha use high and low prices. Applied Financial Economics 4,