Stock Index Volatility: the case of IPSA

MPRA Munich Personal RePEc Archive Sock Index Volailiy: he case of IPSA Rodrigo Alfaro and Carmen Gloria Silva 31. March 010 Online a hps://mpra.ub.uni-muenchen.de/5906/ MPRA Paper No. 5906, posed 18. Ocober 010 1:51 UTC

STOCK INDEX VOLATILITY: THE CASE OF IPSA * RODRIGO A. ALFARO Cenral Bank of Chile CARMEN GLORIA SILVA Cenral Bank of Chile This paper inroduces alernaive measuremens ha use addiional informaion of prices during he day: opening, minimum, maximum, and closing prices. Using he binomial model as he disribuion of he sock price we prove ha hese alernaive measuremens are more efficien han he radiional ones ha rely only in closing price. Following Garman and Klass (1980) we compue he relaive efficiency of hese measuremens showing ha are 3 o imes more efficien han using closing prices. Using daily daa of he Chilean sock marke index we show ha a discree-ime approximaion of he sock price seems o be more accurae han he coninuous-ime model. Also, we prove ha here is a high correlaion beween inraday volailiy measuremens and implied ones obained from opions marke (VIX). For ha we propose he use of inraday informaion o esimae volailiy for he cases where he sock markes do no have an associaed opion marke. JEL: C, G11, G1 Keywords: Volailiy, Binomial Model, VIX, Bias and Efficiency. * We hank he valuable commens from Kevin Cowan, Pablo García, José Manuel Garrido, Felipe Jaque, Camilo Vio and an anonymous referee. An early version of his sudy was published in Spanish in The Lain American Journal of Economics (Cuadernos de Economía). Email: ralfaro@bcenral.cl, csilva@bcenral.cl 1

I. INTRODUCTION During he las years, financial markes have been affeced by volaile episodes which have increased he movemens on he asse prices. Knowing and undersanding he volailiy measures for key financial asses is he mos imporan ask for marke paricipans and for supervisors as well. For developed markes, he use of derivaives helps o esimae he volailiy of he underlying. Tha is he case of VIX which is a well-known measuremen of he sock index volailiy. Demeerfi e al. (1999) provides a formal proof for he VIX showing ha i is a general version of he implied volailiy, which is obained by assuming he Black-Scholes formula. In he case of counries wihou derivaive markes he sandard approach is o rely on saisical models in which a dynamic for he second momen of he reurn is added. Those are he cases of ARCH, GARCH, or EGARCH models which are esimaed by nonlinear mehods (Wilmo, 006). Those models rely on he asympoic of he esimaion mehods for ha a large sample is needed. Indeed, a rule of humb for esimaing a GARCH model, by maximum likelihood, is o use a leas 500 observaions. In his paper we sudy he use of inraday informaion for esimaing he volailiy of sock indexes. Following he works of Parkinson (1980), Garman and Klass (1980), and Rogers and Sachell (1991) we rely on four saisics of he sock price during he day: open, minimum, maximum, and close. This se of informaion is usually colleced by several rading companies such as Bloomberg or Reuers. Assuming ha he sock index can be characerized by he binomial disribuion we show ha inraday measuremens of volailiy are abou 3 o imes more efficien han he esimae obained by using only closing

prices. These findings provide accurae esimaes of he coninuous-ime resuls of Garman and Klass (1980), and Rogers and Sachell (1991). Also, we show ha he bias of inraday measures depends on he number of seps of he ree. This explains why he Parkinson s esimae is usually downward biases oward zero. We apply he inraday measuremens for he case of Chilean sock marke finding ha he empirical bias-correcion is lower han he coninuous-ime value. We inerpre his resul as he coninuous-ime model is a good approximaion for counries where housands of ransacions are made daily, meanwhile he binomial disribuion could be used for sock index in smaller financial markes. Finally, we apply he inraday measuremens o he S&P 500 index finding ha hose are highly correlaed wih he VIX. This evidence suppors he use of inraday informaion in equiy markes wihou derivaes. The paper is organized as follows: in he Secion II we discuss he inraday volailiy measuremens; in Secion III we provide an applicaion of hese measuremens o he Chilean Sock Index, IPSA; Secion IV compares he effecive and he implied volailiy measures for an inernaional sock index; and conclusions are given in Secion V, hen Appendix A covers he echnical deails. 3

II. INTRADAY MEASUREMENTS Tradiional volailiy measuremens use only closing prices and exclude imporan informaion which is generaed hrough he day. In his secion, opening, closing, minimum and maximum prices are incorporaed in he analysis. This informaion has been used previously by Parkinson (1980), Garman and Klass (1980) and Rogers and Sachell (1991). However, hese auhors have based heir sudies in he assumpion of a Brownian moion for he asse price. Such assumpion is valid for developed marke where here are a high number of daily ransacions. In he case of a less liquid marke, a binomial model developed by Cox, Ross and Rubinsein (1979) is proposed for he dynamic of he price. In his model, he asse price in he nex ime could be described by wo possible scenarios, which are deermined by he volailiy of he asse (σ ) : i) Up scenario: he asse price is increased by he facor u exp( σ 1 N ) = and ii) Down scenario: he asse price is decreased by d = 1 u, where N is he number of seps of he binomial ree. I is imporan o noe ha if N ends o infiniy hen he binomial model converges o he coninuous-ime Brownian moion. Thus, he number of seps of he binomial model represens he deph of he financial marke and he binomial model is he mos convenien model for he Chilean marke. The probabiliy of each scenario depends on he volailiy and he asse reurn and i can be represened by (Wilmo, 006):

p = 1 µ + σ 1 N As a simplificaion a zero reurn asse has been assumed ( µ = 0), hus he probabiliy of each scenario is equal o 0.5. Hereafer, observable inraday prices are assumed and he following definiions are applied: o is he logarihm of he opening price, which is assumed equal o 1 for simpliciy; c is he logarihm of he closing price; h is he logarihm of he maximum price; and l is he logarihm of he minimum price. Then, he following volailiy measures can be calculaed: σ ( c o) and σ ( h l) CC HL, where CC is he radiional volailiy measure based on closing prices and HL is he new esimaor which uses maximum and minimum prices ha are observed during he day, as i is suggesed by Parkinson (1980). As an example, Table 1 considers a binomial ree wih wo seps (N=). The resuls in he able correspond o he logarihm of he prices. Thus he closing price for he Up scenario is c = log(u) = σ 1 1 = σ which coincides wih he maximum price; and he minimum price is he logarihm of he normalized opening price. I can be seen ha only he Up-Up and Down-Down scenarios give informaion for he calculaion of CC whereas HL uses he available informaion in all scenarios for he volailiy esimaion. 5

Table 1: Resuls for a wo-seps model Scenarios Up Up c σ Up Down 0 Down Up 0 Down Down σ Source: Auhors calculaions. h σ σ 0 0 l 0 0 σ σ The expeced values for CC and HL may be calculaed as following: E E 1 1 ( σ ) = ( σ ) + ( σ ) CC + 1 1 σ 1 σ 1 ( σ ) = ( σ 0) + 0 + 0 + ( 0 + σ ) = 1.5σ HL = σ In a model wih wo seps, only CC is unbiased whereas HL overesimaes he rue variance in 0.5 imes. The bias is originaed for using discree daa of maximum and minimum prices in a wo-seps model. In pracice, he calculaion of he HL volailiy depends on he srong assumpion of a significan number of scenarios for he asse price. Parkinson (1980) demonsraes ha if he asse price follows a Brownian moion wihou = log() σ.773σ. This means rend and i can be observed coninuously hen ( ) E σ ha HL overesimaes he rue variance and he bias is condiional o he deph of he financial marke. HL 6

Alhough HL is biased, i is more efficien han CC in erms of Mean Squared Error (MSE). In paricular, for N= he formulas included in he appendix of his sudy show ha ECM ( σ ) = σ, and ( σ ) = 0.65σ CC ECM. HL An inuiive explanaion is ha he maximum and minimum prices conain more informaion abou he volailiy han he opening and closing prices, because he former obain heir values during he ransacion ime while he laer are only insan picures of he process. Defining a N as he bias facor of HL, hen for N = his facor is equal o a N = 1. 5. As a consequence, an unbiased esimaor (HL) can be obained from HL as follows: 1 σ HL σ HL = 0. 8σ HL E HL = an ( σ ) σ, and ECM ( σ ) = 0. σ HL 36 The new esimaor HL is unbiased as CC, bu i has lower variance. Anoher unbiased and efficien esimaor was proposed by Garman and Klass (1980) and is adapaion o he binomial model is such ha: σ GK 1 = a N ( h l) 1 ( c o) For consrucion, GK is unbiased and for he case of = ECM σ = 0.565σ. N, ( ) However, GK is more efficien han HL when N > 1 as i can be seen in he Table. GK 7

Previous resuls could be misleading if he asse price presens any rend during he day, his is µ 0. For insance, if he price increases hen h = c, and l = o, and HL and HL would be he resuls of spurious variances. As a way of conrolling for he rend of he asse, he opening and closing prices could be included in he variance formula as i is suggesed by Rogers and Sachell (1991) for a coninuous ime scenario: RS ( h o) ( h c) + ( l o) ( l c) σ =. This measuremen uses he maximum and minimum prices and incorporaes he opening and closing prices in order o subrac he inraday rend of he price. The basic saisics for RS assuming a binomial ree wih wo seps are he following: 1 σ 1 σ E ( σ ) = + = 0.5σ, and ECM ( σ ) = 0.65σ RS RS RS is also a biased esimaor and is bias is bigger han HL. Defining b N as he bias facor for RS, i is possible o calculae an unbiased esimaor of RS as follows: 1 RS = σ RS = σ RS E ( σ RS ) σ, and ECM ( σ RS ) = σ σ = 0.5 8

I is imporan o noe ha if he asse price can be characerized by a binomial ree wih wo seps, hen RS presens he same properies han CC. However RS does no presen rend. In heir original paper, Rogers and Sachell (1991) demonsrae ha RS is unbiased in a coninuous ime. In he model presened in his paper, he bias facor for RS converges o uniy when N is big enough. Up o now, he analysis has included he calculaion for he bias and he bias facor for he measuremens HL and RS for a wo-seps model and for asses wihou rend. An exension for a binomial model wih a greaer number of seps is exhibied in Table 1 /. In general, he advanages of he classical esimaor of volailiy CC are is simpliciy and is unbiasness. However, is main disadvanage is ha i ignores available informaion which could be imporan for he efficiency of he esimaor. In his way, Garman and Klass (1980) sae ha he CC would be a benchmark hrough which oher esimaors would be measured, and a Relaive Efficiency (RE) raio could be calculaed as follows: ( ) Var( σ CC ) RE σ A = Var( σ ) A Where Var( σ ) is he variance of CC and Var( σ ) corresponds o he variance of an CC unbiased esimaor ( σ ) for which he RE raio is being calculaed. A A 1 / I is imporan o noe ha CC and GK are unbiased for all number of seps. 9

Garman and Klass (1980) and Rogers and Sachell (1991) calculae RE for he esimaors of variance in a coninuous ime, whereas he heoreical values of RE in binomial rees wih seps lower han are presened in Table /. Table : Correcion facors and Relaive Efficiency Correcion facors Relaive efficiency o CC (*) N PK RS PK RS GK 1 1.000 0.000 1.50 0.50.78 1.00 1.78 3 1.500 0.333 3.8 1.33.37 1.59 0.06 3.09 1.77.3 5 1.700 0.50 3.8.06.73 6 1.760 0.90 3.7.8.83 7 1.86 0.518 3.39.7 3.0 8 1.868 0.5 3.0.63 3.1 9 1.91 0.56 3.9.77 3.31 10 1.96 0.583 3.5.89 3.0 11 1.980 0.599 3.58 3.00 3.53 1.005 0.613 3.59 3.09 3.61 13.033 0.66 3.65 3.18 3.7 1.053 0.638 3.67 3.6 3.79 15.075 0.68 3.71 3.33 3.89 16.09 0.657 3.73 3.0 3.95 17.111 0.666 3.76 3.5.0 18.15 0.67 3.78 3.51.09 19.11 0.68 3.81 3.57.16 0.15 0.689 3.8 3.61.1 1.168 0.695 3.85 3.66.8.179 0.701 3.86 3.70.33 3.191 0.707 3.89 3.7.39.01 0.713 3.90 3.79. Infinie.773 1.000 5.0 6.0 7.1 (*) Adjused by correcion facors. Sources: Garman and Klass (1980), Rogers and Sachell (1991) and auhors calculaion. / For seps equal o N, he number of possible scenarios is N. Due o compuaional limiaions, Table presens resuls only unil seps. 10

III. RESULTS FOR A CHILEAN STOCK INDEX In order o calculae he inraday volailiies for he IPSA 3 /, he correcion facors of he Chilean marke mus be known. Using daily daa for he prices of he IPSA from January, 1996 o January 18, 008, he hisorical values of volailiy are esimaed. The raio beween he non-correced volailiy measure of Parkinson and he daily sandard deviaion brings a median value of 1.5 (=.37). This means ha he correcion facor for he Chilean equiy marke is lower han he value suggesed by Parkinson (1980) for a coninuous-ime process (.773), bu i is greaer han he correcion facor for a seps of he binomial ree (.01). Wih his esimaed value of he correcion facor for he IPSA, he unbiased volailiy measure in annual erms is esimaed as follows /: ˆ σ HL, 50 H ˆ σ HL, = 50 = 1, 1.5.37 ( h ) 10 l L where H and L are he maximum and minimum prices for a specific day, respecively. Likewise, he median value for he raio beween he Rogers-Sachell volailiy and he daily sandard deviaion is 0.88 5 /, and he unbiased volailiy measure in annual erms for he IPSA in his case is calculaed as: 3 / IPSA is he Selecive Price Index of he Saniago Sock Exchange and i considers he 0h mos raded socks of he sock exchange. In inernaional comparison is widely used as he main sock index for Chile. / Recall ha HL incorporaes he daily rend of he price of he asse. Yang and Zhang (000) argue ha such rend should be small for inraday informaion as he invesors are no expecing significan movemens of he asse. 5 / The 95% confidence inerval for boh correcion facors implies he following esimaion ranges (1.50-1.58) and (0.8-0.9), respecively. 11

ˆ σ RS, ˆ σ RS, H H L L = 50 18 1 1 + 1 1, 0.88 C O C O where H, L, O and C are he maximum, minimum, opening and closing prices, respecively. Graph 1 shows ˆ σ HL,, ˆ σ RS,, and he sandard deviaion (CC) for a 1-days moving average. 50 0 Graph 1: Volailiy measuremens (percenage, monhly averages) CC HL RS 30 0 10 0 97 98 99 00 01 0 03 0 05 06 07 08 Sources: Bloomberg and auhors calculaion. From he graph, i is observed ha all he volailiy measuremens are posiively correlaed wih he urbulen episodes. In paricular, hrough his ime HL and CC (0.95) are he mos correlaed measuremens. This implies ha an inraday volailiy measure, as HL, is able o include he same informaion han a radiional one, as CC, bu HL would do i in a more efficien way. In oher words, we could have he same sandard error in HL han CC, bu using less days in he compuaion of he fis one. 1

IV. A COMPARISON OF EFFECTIVE AND IMPLIED VOLATILITY In order o validae he volailiy measuremens obained from he previous secion, hese measures are calculaed for equiy markes which presen available informaion of equiy opions. In paricular, a comparison of he effecive volailiy measuremens and he volailiy index calculaed wih opions (VIX) for he S&P 500 index is performed. Using daily daa for he 00-008 periods he CC, HL and RS for he S&P 500 are calculaed. As i is he case of a developed equiy marke, he sandard correcion facors suggesed by he lieraure are applied. Resuls show ha HL and RS exhibi lower sandard deviaion han CC (Table 3). This means ha he esimaed value of he volailiy is more efficien if he maximum and minimum prices are included. Moreover, RS presens a lower average han HL and CC because i does no consider he rend of he asse. Table 3: Volailiy of S&P 500 index (percenage) Mean Median Sandard deviaion Percenile 5% Percenile 95% CC 9.7 7.05 8.91 0.73 5.78 HL 9.80 8. 5.6 3.91 0. RS 8.6 7.50 5.78.08 18. VIX 15. 13.97.5 10.7 5.5 Sources: Bloomberg and auhors calculaion. On he oher hand, he VIX is greaer han he effecive volailiy of he S&P 500 Index (CC, HL or RS) while is sandard deviaion is lower. The firs issue arises because VIX conains a risk premium associaed wih he uncerainy of he derivaes, and he lower 13

sandard deviaion of he VIX emerges from he fac ha i is calculaed as he average volailiy for he nex 30 days. Finally, he effecive volailiy measuremens and he VIX are correlaed in 0.5 for daily daa and 0.9 for moving averages (Table ). This implies ha HL and RS capure properly he movemens of he VIX and hey could be good subsiues in he case of equiy markes wihou derivaes. Table : Correlaion beween VIX and volailiy measuremens for he S&P 500 (percenage) Daily measures Moving average (1) CC 0.5 0.89 HL 0.67 0.90 RS 0.5 0.86 (1) Exponenially weighed using a facor of 0.9. Sources: Bloomberg and auhors calculaion. V. CONCLUSIONS In his sudy we inroduced he use of inraday informaion o improve he efficiency in he esimaion of volailiy for sock prices. From he heoreical analysis, i is possible o conclude ha hose esimaes proposed are more efficien han relying only in closing prices. This is proved under he assumpion ha he sock price follows a binomial disribuion. The resul is close o he one presened in he lieraure by Parkinson (1980) or Rogers and Sachell (1991); however, we provide an exension ha shows ha he relaive efficiency of hose esimaes is 3 o imes bigger compared wih he use of close prices. Previous research provides higher level of efficiency because hey assume ha he sock prices follow a coninuous-ime Brownian moion. Moreover, he empirical analyses have showed ha financial series move away from heir hisorical averages in volaile episodes, and as a consequence, in hese periods only he mos recen informaion is relevan. 1

Based on he Chilean equiy marke, wo volailiy measuremens ha use inraday informaion have been proposed. The firs one is an adapaion of he volailiy index suggesed by Parkinson (1980) and he second one corresponds o an adapaion of he Rogers and Sachell (1991) volailiy esimaor. Boh are bias-correced for which we noe ha he correcion facors imply ha here is a finie number of seps ha accommodaes he binomial disribuion of he sock marke index. We confirm he use of hese inraday measuremens by showing he high level of correlaion beween hose applied o he S&P 500 index and he VIX, which is defined as an implied volailiy from opions over he same index. Wih ha evidence we conclude ha inraday volailiy measuremens are appropriae for equiy markes wihou opions markes. REFERENCES Alfaro, R., Silva, C.G., 008. Volailidad de Índices Accionarios: el caso del IPSA. Lain American Journal of Economics. Cuadernos de Economía, Vol. 5 (Noviembre): 17-33. Cox, J., Ross, S., Rubinsein, M., 1979. Opion pricing: a simplified approach. Journal of Financial Economics 7(3): 9-63. Demeerfi, K., Derman, E., Kamal, M., Zou, J., 1999. More Than You Ever Waned o Know Abou Volailiy Swaps. Goldman Sachs Quaniaive Sraegies Research Noes. Garman, M., Klass, M., 1980. On he Esimaion of Securiy Price Volailiies from Hisorical Daa. The Journal of Business 53(1): 67-78. 15

Parkinson, M., 1980. The Exreme Value Mehod for Esimaing he Variance of he Rae of Reurn. The Journal of Business 53(1): 61-65. Rogers, L., Sachell, S., 1991. Esimaing Variance from High, Low, and Closing Prices. The Annals of Applied Probabiliy 1(): 50-51. Wilmo, P., 006. Paul Wilmo on Quaniaive Finance, volume 3, second ediion. John Wiley & Sons, Ld. Yang, D., Zhang, Q., 000. Drif-Independen Volailiy Esimaion Based on High, Low, Open, and Close Prices. The Journal of Business 73(3): 77-91. APPENDIX A For calculaing he mean squared error i is necessary o know he fourh momen of he variances, whose compuaion is presened here: 1 1 1 1 σ E ( σ CC ) = ( σ ) + ( σ ) = σ, E ( σ ) ( σ ) HL = + =.15σ, 1 σ 1 σ E ( σ ) RS = = 0.15σ +, E 1 1 1,5 ( σ ) E( σ ) * 1 E ( h l) GK = 0.5.15σ + 0.375 σ + 0.375 σ 1,5 [ c ] + 1 E( σ ) = PK CC = 1.565σ 1 1 [( h l) c ] = ( σ ) ( σ ) + ( σ ) ( σ ) = σ E., 16