International journal of advanced production and industrial engineering (A Blind Peer Reviewed Journal)

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1 IJAPIE , Vol 1(1), Inernaional journal of advanced producion and indusrial engineering (A Blind Peer Reviewed Journal) orecasing Volailiy Using GARCH: A Case Sudy Nand Kumar 1, Rishabh Verma 2, Punee Gupa 3 1 (Deparmen of Applied Science & Humaniie,Delhi Technological Universiy, India) 2,3 (Deparmen of Mahemaics,Delhi Technological Universiy, India) 1 nand.dce@gmail.com Absrac : orecasing volailiy is fundamenal o he risk managemen process in order o price derivaives, devise hedging sraegies and esimae he financial risk of a firm s porfolio of posiions. In recen years, Auoregressiv Condiional Heeroscedasicy (ARCH) ype models have become popular as a means of capuring observed characerisics of financial reurns like hick ails and volailiy clusering. These models use ime series daa on reurns o model condiional variance Our sudy shows ha GARCH volailiy capures he mos informaion of fuure volailiy The implied volailiy calculaed from he sudy subsumes only 46% of realized volailiy whereas GARCH Volailiy subsumes 70% of realized volailiy, herefore, Garch volailiy is a beer measure of volailiy in opion pricing. Keywords (11Bold): Geomeric Brownian moion, Black- Scholes model, Implied Volailiy, ARCH, GARCH Volailiy. I. INTRODUCTION Since he inroducion of he Black-Scholes model (1973), researchers have sudied he empirical performance of he model. The Black-Scholes model explains ha he price of heavily raded asses follow a geomeric Brownian moion ha looks like a smile or smirk wih consan drif and volailiy. When applied o a sock opion, he model incorporaes he consan price variaion of he sock, he ime value of money, he opion's srike price and he ime o he opion's expiry. According o he geomeric Brownian moion model, he reurns on a cerain sock in successive, equal periods of ime are independen and normally disribued. Thus hey form a Markov process. The assumpions on which his model is based mee he financial marke laws and rules imposed by he Marke Efficiency Hypohesis. These rules and laws suppose ha only presen informaion abou a sock is sufficien o deermine he fuure price of ha sock. So heoreically he geomeric Brownian moion seems o be a good way o model fuure sock. The nex major sep of his paper is o deermine esimaes of volailiy. In recen years, Auoregressive Condiional Heeroscedasicy (ARCH) ype models have become popular as a means of capuring observed characerisics of financial reurns like hick ails and volailiy clusering. These models use ime series daa on reurns o model condiional variance. An alernaive way o esimae fuure volailiy is o use opions prices, which reflec he marke s expecaion of volailiy. Day and Lewis (1993) compare he relaive informaion conen and predicive power of implied volailiy and ARCH-ype forecass for crude oil fuures. A similar sudy by Xu and Taylor (1996) examines he informaional efficiency of he PHLX currency opions marke in predicing volailiy. Duffie and Gray (1995) compare he forecasing accuracy of ARCH ype models, Markov swiching models, and implied volailiies for crude oil, heaing oil and naural gas markes. Our sudy aemp o es he hypohesis ha GARCH volailiies subsume informaion conain in reurns and provides he bes monh-ahead volailiy forecass for SBI equiy for which we have aken closing price daa of SBI equiy from 23rd Ocober 2007 o 30 h Ocober However for finding implied volailiy, daa is considered from 26 h July, 2013 o 30 h Ocober, 2013 because he opions in Indian Capial Markes have been inroduced only recenly and sufficien ime-series opions daa is no available. Also we have divided our daa in In-Sample and Ou-of-Sample daa and performed regression analysis on boh. The Ou-of- Sample daa we have considered for OLS regression is from 8 h Ocober 2013 o 30 h Ocober II. DATA AND METHODOLOGY Geomeric Brownian Moion- To simulae he sock price using Geomeric Brownian Moion we performed Mone Carlo Simulaion on he SBI sock price from 2 nd Augus 2013 o 8 h Ocober We ake he closing price of he sock on 2 nd Augus 2013(Rs ) as ime 0. Referring he above menioned price as S 0 we find one of he oucomes of day 1 by using he log normal propery. ln S ~ [ln S o + (μ-σ 2 /2)T, σ 2 T ] IJAPIE ISSN: Vol. 1 Issue

2 Nand Kumar e al., Table 1: Calculaed Parameers for Mone Carlo Simulaion μ % σ 2.804% Price 1680 annual volailiy % The average (μ )and sandard deviaion(σ) of log reurn is calculaed.thus he consan annualized volailiy comes ou o be % shown in he above able. Then simulaion is done by Mone Carlo Simulaion using μ and σ as parameers. Various movemens of sock prices observed are shown in figure 1.a,b,c. (a) volailiy such as Implied volailiy, Hisorical volailiy, EWMA volailiy and GARCH volailiy in our sudy. Implied Volailiy- Implied volailiy of an opion conrac is ha value of he volailiy of he underlying insrumen which when feeded as an inpu in an opion pricing model (such as Black-Scholes) will reurn a heoreical value equal o he c, curren marke price of he opion. Thus and p, was calculaed for each day from 26 h July,2013 o 8 h Ocober,2013 (for In-Sample comparison) by using he sock price, srike price, rae of ineres (4 percen),opion marke c p,, price, ime of expiraion. Here and denoe he implied volailiy of call and pu opions respecively a ime. And weighed average of boh he volailiies is considered as follows: = ( c, + p, ) / 2 Such a weighed average is simple o implemen and i avoids he noise, creaed by single implied volailiy, ha migh impac an observaion. Hisorical volailiy- Meron (1980) has shown ha he accuracy of an esimae of volailiy using pas volailiy increases wih he sampling frequency wihin a given overall observaion period. We hus choose o use daily daa and preferred o omi he usual esimaor of he mean o avoid excessive noise, and use he following formula: h, 252 i R 2 i where R i denoes he log-reurn on day i and is calculaed as R i = ln(s i /S i-1), where Si is he index level on he same day i. Exponenially-weighed average volailiy (b) Assuming ha volailiy varies wih ime, he EW version compensaes o some exen for one of he shorcomings of simple hisorical volailiy by giving greaer weigh o he recen observaion. or each closing price observaion a ime, we also measure he volailiy using an exponenially-weighed average of pas daily volailiy, including he day of he observaion, wih a decay facor of We use he formula: (c) igure 1: a,b,c Various movemens of sock prices observed Relaxing he assumpion of consan volailiy in Black Scholes Model, we have considered various sochasic f, 252*0.06* 0.94 i0 where 0.94 is he decay facor, 0.06 he sum of he weighs, and where R denoes he log-reurn on day i. We calculaed all volailiies from 23 rd Ocober, 2007 o 8 h Ocober, 2013 on all rading days n R 2 i IJAPIE ISSN: Vol. 1 Issue

3 Nand Kumar e al., GARCH ig ure 2 Reurn series from 23 rd Ocober 2007 o 8 h Ocober 2013 Reurns series are preferred over prices in analysis of financial ime series because hey have aracive saisical properies like saionariy. Bu here he acual disribuion of reurn series has faer ails compared o fied normal disribuion.thick ails can be modeled by assuming a condiional normal disribuion for reurns; where condiional normaliy implies ha reurns are normally disribued on each day, bu ha parameers of he disribuion change from day o day. Reurns are hus no idenically disribued wih mean 0 and variance σ a each poin in ime. Insead, i is fair o say ha σ changes wih ime. The persisence of volailiy in opion marke is an indicaion of auocorrelaion in variances. Tesing for Saionariy Ljung box es is used for esing he saionariy characer of reurn series and reurn square series. The null hypohesis for his es is ha he firs m auocorrelaions(ρ 1, ρ 2,..., ρ m) are joinly zero. H 0 : ρ 1 = ρ 2 =... = ρ m = 0 Using malab he p value for reurn series is Thus our null hypohesis is acceped i.e., (h=0). Bu in case of reurn square series, p value is approximaely zero.thus our null hypohesis is rejeced which concludes ha reurn square series is no saionary. Tesing for ARCH effec Using Malab he null hypohesis is rejeced (h = 1, p = 0) in favour of he ARCH(2) alernaive. The saisic for he es is 44.53, much larger han he criical value from he disribuion wih wo degrees of freedom, The es concludes ha here is significan volailiy clusering in he residual series. Also from correlogram diagram as shown below we can say ha here is no significan auocorrelaion in reurn series (ig 3). Bu here is significan auocorrelaion in reurn square series (ig 4) suggesing volailiy dependen on pas volailiies. igure 3. Correlogram showing auocorrelaion of reurn series a various lags igure 4.Correlogram showing auocorrelaion of reurn square series a various lags Time varying volailiy is modeled saisically by esimaing a condiional variance equaion in addiion o he reurns generaing process.in pracice, he GARCH (1, 1) model comprising only hree parameers in he condiional variance equaion is sufficien o capure he volailiy clusering in he daa. The condiional variance equaion of GARCH (1,1) model is: 10% 8% 6% 4% 2% 0% -2% -4% -6% -8% -10% 25% 20% 15% 10% 5% 0% -5% -10% AC AC a0 a1 r 1 1 AC Reurn series is considered from 23 rd Ocober, 2007 o 8 h Ocober, 2013 (for In-Sample comparison). We have calculaed he parameers (α o, α 1, β 1) using maximum log likelihood mehod in excel. UL LL AC UL LL IJAPIE ISSN: Vol. 1 Issue

4 Dae 3/12/2008 8/7/2008 1/6/2009 6/2/ /20/2009 3/11/2010 8/3/ /24/2010 5/20/ /13/2011 3/6/2012 7/24/ /11/2012 4/30/2013 9/17/2013 Nand Kumar e al., Table 2: Long Run Volailiy Likelihood ω α β α o 0.01 Long run volailiy σ 2.69% E-06 Annualized σ 42.78% Long run annual volailiy came ou o be percen. Using GARCH(1,1) model and esimaed parameers, we calculaed volailiies from 23 rd Ocober 2007 o 8 h Ocober Nex we predic volailiies by differen models o perform he OLS regression agains realized volailiies. Realized Volailiy or each sock price observaion a ime, we measure he Realized volailiy by he sample sandard deviaion of he daily index reurns over he remaining life R of he opion. Again, we deliberaely omi he esimaor of he mean, which would have been oo noise-sensiive. We use he following formula: f, 252 R i 1 2 i where R i denoes he log-reurn on day i. The sudy aemps o repor he resuls of OLS regression of he realized volailiy on In-Sample values and forecas values given by he various models and esimaors. Also in order o compare bias and efficiency of various esimaors and models for esimaion, we have calculaed he following errors such as Bias = Ε (σ rs σ ), Mean square error MSE) = E [(σ rs - σ ) 2 ], Relaive bias = E [(σ rs - σ )/ σ ], Mean absolue error (MAE) = E [Abs (σ rs - σ )] RESULTS AND DISCUSSIONS Sock Prices of SBI equiy were simulaed by Geomeric Brownian moion. The behavior of simulaed prices was found o be similar o he pah followed by closing prices of he given equiy as shown earlier. Differen volailiies were calculaed for all rading days from 23 rd Ocober 2007 o 8 h Ocober 2013 and each volailiy was ploed agains ime as below: 100.0% 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% Volailiies Realized Volailiy Hisorical volailiy EWMA Garch Volailiy igure 5. showing he relaionship beween realized volailiy, hisorical, implied and GARCH volailiies. Errors were calculaed for each ype of volailiy as shown in able 1a(appendix). Hisorical Volailiy was rejeced as i has maximum error. The Bias, Mean square error and Mean absolue error for EWMA was greaer as compared o hese errors found in implied volailiy. The comparison beween GARCH and Implied Volailiy canno be conclusive based on hese errors. Therefore o ge a clearer picure, we regress all he four volailiies wih he realized volailiy. Performing regression on In-Sample daa beween Hisorical and Realized Volailiy (Table 2b), he R-square comes ou o be and a negligible p-value. In he case of EWMA (Table 2c), he R-square comes ou o be and he p- value is again negligible. In he case of GARCH (Table 2d), he R-square is and p-value is negligible. or he regression beween Implied and Realized Volailiy (Table 2a), R-square comes ou o be and a negligible p-value. The sudy sugges ha in case of regression on In-Sample daa no volailiies is explaining he realized volailiiy o he saisfacory level. Similarly performing regression on Ou-of-Sample daa beween Hisorical and Realized Volailiy (Table 2f), he R- square comes ou o be and a negligible p-value. In he case of EWMA (Table 2g), he R-square comes ou o be and he p-value is again negligible. In he case of Implied (Table 2e), he R-square is and a negligible p-valu. or he regression beween GARCH and Realized Volailiies (Table 2h), R-square comes ou o be and a negligible p-value. The sudy sugges ha GARCH Volailiy ouperforms all he oher hree volailiies as i bes explains he realized volailiy in he case of Ou-of-Sample daa. IJAPIE ISSN: Vol. 1 Issue

5 Nand Kumar e al., V. CONCLUSIONS In his paper, firsly we have validaed he performance of geomeric Brownian moion on SBI sock price by seeing he pah followed by simulaed prices and closing prices. Then we have compared differen volailiies o find ou he correc measure of volailiy which can be used as an inpu in Black Scholes formula. The resuls sugges ha errors for GARCH volailiy are found o be less as compared o oher volailiies. The OLS regression of forward looking implied volailiy is less han 50% which implies ha implied volailiy subsumes less par of fuure volailiy whereas he OLS regression for forecased GARCH volailiies agains realized volailiy, is more han 70% which sugges ha GARCH volailiy subsumes maximum par of fuure volailiy. Thus GARCH forecased volailiy comes ou o be an effecive measure of volailiy and may be used by raders and hedgers in indian derivaives marke. REERENCES [1] Abdelmoula Dmouj (2006,Nov). Sock price modeling: Theory and Pracice. aculy of Sciences, Amserdam:BMI Paper. [2] Ahoniemi (2005). Modeling and orecasing implied volailiy an economeric analysis of he VIX index, Helsinki Cenre of Economic Research. [3] Ajay Pandey (2005,June). Volailiy Models and heir Performance in Indian Capial Markes, Vikalpa (Vol. 30), 11 [4] Akgiray, V (1989). Condiional Heeroskedasiciy in Time Series of Sock Reurns: Evidence and orecass, Journal of Buisness, 62(1) [5] Bollerslev. (1986),Generalized auoregressive condiional heeroskedasiciy, Journal of Economerics 31 (1986). Norh-Holland [6] Carol Alexander (1998), Risk Managemen and Analysis: Measuring and Modelling inancial Risk, Wileys [7] Chrisensen, B.J & Prabhala, N.R (1978). The relaion beween implied and realized volailiy, Journal of inancial Economics [8] Chrisensen,B.J.& Hansen, C.S (2002). Sock marke volailiy and he informaion conen of sock index opions, Journal of inancial Economics. [9] Day, T.E & Lewis, C.M(1992). Sock marke volailiy and he informaion conen of sock index opions, Journal of inancial Economics. [10] Engle, R and Paon, AJ(2001). Wha Good is a Volailiy Model?.Quaniaive inance,1(2). [11] ama, E(1965). The Behaviour of Sock-Marke Prices. Journal of Business,38(1) [12] Hamao,Y,Masulis, RW and Ng, VK(1990). Correlaions in Price Changes and Volailiy across Inernaional Sock Markes, Review of inancial Sudies, 3(2). [13] Huang Kun (2011), Modeling Volailiy of S&P 500 Index Daily Reurns: A comparison beween model based forecass and implied volailiy. [14] Jaykumar Dagha( ).Derivaives: orwards,uures & Opions, The Bes Way o Miigae Volailiy. Alliance Buisness School [15] John C. Hull, Opions, uures and Oher Derivaives. 7 h Ed. Pearson Educaion :Prenice Hall Publishers(2008), 4-5 [16] Namia Sharma (1998). orecasing Oil Price Volailiy, aculy of he Virginia Polyechnic Insiue and Sae Universiy, 2-3 [17] Peer Gross (2006). Parameer Esimaion for Black- Scholes Equaion, URA [18] S.C. Gupa and V.K. Kapoor (1970). undamenals of Mahemaical Saisics. 11 h ed. Sulan Chand & Sons [19] Schwer, G W (1989). Why Does Sock Marke Volailiy Change over Time?. Journal of inance,44(5) [20] Sheldon M. Ross (2010), Inroducion o Probabiliy and Saisics for Engineers and Scieniss. 4 h ed. Academic Press an Imprin of Elsevier [21] Vladimir M. Ionesco (2011). The performance of implied volailiy in forecasing fuure volailiy: an analysis of hree major equiy indices from 2004 o MIT Sloan School of Managemen, 6-8 IJAPIE ISSN: Vol. 1 Issue

6 Nand Kumar e al., Table 1a: Bias, Mean Square Error, Relaive Bias, Mean Absolue error and OLS regression on In-Sample daa and OLS regression on Ou- of- Sample daa Appendix: Volailiy Bias Mean Square Error Relaive Bias Mean Absolue error Regression square(r 2 ) (In-Sample) Regression Square(R 2 ) (Ou-of-Sample ) Hisorical Volailiy EWMA Implied Volailiy GARCH Volailiy Table 2a: OLS regression beween Implied and Realized Volailiy (In-Sample daa) Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 56 df SS MS Regression E-05 Residual Toal IJAPIE ISSN: Vol. 1 Issue

7 Nand Kumar e al., s Sandard Inercep X Variable Error Sa P-value Table 2b: OLS regression beween Hisorical and Realized Volailiy (In-Sample daa) Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 1512 Df SS MS Regression E-11 Residual Toal E- 4.78E- s Sandard Error Sa P-value Inercep E X Variable E Table 2c: OLS regression beween EWMA and Realized Volailiy (In-Sample daa) Regression Saisics Muliple R R Square IJAPIE ISSN: Vol. 1 Issue

8 Adjused R Square Sandard Error Observaions 1512 Nand Kumar e al., df SS MS Regression E-61 Residual Toal s Sandard Error Sa P-value Inercep X Variable E Table 2d: OLS regression beween GARCH and Realized Volailiy (In-Sample) Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 1498 Df SS MS Regression E Residual IJAPIE ISSN: Vol. 1 Issue

9 Toal Nand Kumar e al., s Sandard Error Sa P-value Inercep X Variable E Table 2e: OLS regression beween Implied and Realized Volailiy(Ou- of- Sample daa ) Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 12 df SS MS Regression Residual Toal s Sandard Error Sa P-value Inercep X Variable Table 2f: OLS regression beween Hisorical and Realized Volailiy(Ou-of-Sample daa) IJAPIE ISSN: Vol. 1 Issue

10 Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 12 Nand Kumar e al., df SS MS Regression Residual Toal s Sandard Error Sa P-value Inercep X Variable Table 2g: OLS regression beween EWMA and Realized Volailiy(Ou-of-Sample daa) Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 12 df SS MS IJAPIE ISSN: Vol. 1 Issue

11 Nand Kumar e al., Regression Residual Toal s Sandard Error Sa P-value Inercep E X Variable Table 2h: OLS regression beween GARCH and Realized Volailiy(Ou-of-Sample daa) Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 13 df SS MS Regression Residual Toal s Sandard Error Sa P-value Inercep X Variable IJAPIE ISSN: Vol. 1 Issue