Forecasting Volatility:Evidence from the German Stock Market

Transcription

1 Economics Deparmen Economics Working Papers The Universiy of Auckland Year 2001 Forecasing Volailiy:Evidence from he German Sock Marke Hagen Bluhm Jun Yu Universiy of Auckland, This paper is posed a ResearchSpace@Auckland. hp://researchspace.auckland.ac.nz/ecwp/219

2 Forecasing Volailiy: Evidence from he German Sock Marke * Hagen H.W. Bluhm a,junyu b February 2001 Absrac In his paper we compare wo basic approaches o forecas volailiy in he German sock marke. The firs approach uses various univariae ime series echniques while he second approach makes use of volailiy implied in opion prices. The ime series models include he hisorical mean model, he exponenially weighed moving average (EWMA) model, four ARCH-ype models and a sochasic volailiy (SV) model. Based on he uilizaion of volailiy forecass in opion pricing and Value-a-Risk (VaR), various forecas horizons and forecas error measuremens are used o assess he abiliy of volailiy forecass. We show ha he model rankings are sensiive o he error measuremens as well as he forecas horizons. The resul indicaes ha i is difficul o sae which mehod is he clear winner. However, when opion pricing is he primary ineres, he SV model and implied volailiy should be used. On he oher hand, when VaR is he objecive, he ARCH-ype models are useful. Furhermore, a rading sraegy suggess ha he ime series models are no beer han he implied volailiy in predicing volailiy. JEL classificaion:g12,g15 Keywords: Forecasing Volailiy; ARCH Model; SV Model; Implied Volailiy; VaR; Germany * We would like o hank Deusche Boerse AG, he German sock exchange, for providing he daa, and he Universiy of Auckland Research Commiee for financial suppor. a Bankgesellschaf Berlin AG; Hagen.Bluhm@ib.bankgesellschaf.de b Deparmen of Economics, The Universiy of Auckland, Privae Bag 92019, Auckland, New Zealand; j.yu@auckland.ac.nz.

3 1. Inroducion Using daily daa from he German sock marke, his paper compares wo basic approaches o forecas volailiy. The firs approach uses various univariae ime series echniques while he second approach makes use of volailiy implied in opion prices. The ime series models include four ARCH-ype models and a sochasic volailiy (SV) model. We focus on he forecas horizons of 1, 10, and 180 rading days and 45 calendar days. The evaluaion crieria used are mean squared predicion error (MSPE), bounded violaions, and he LINEX loss funcion. A rading sraegy is also used o examine he usefulness of he ime series models in predicing volailiy. Forecasing financial marke volailiy has received exensive aenion in he lieraure by academicians and praciioners in recen ime; see Poon and Granger (2000) (hereafer PG) for an excellen review of he lieraure. Alhough volailiy forecasing is a nooriously difficul ask according o Brailsford and Faff (1996) (hereafer BF), i is generally agreed ha volailiy is predicable and hence he marke for volailiy is no as efficien as ha for reurns. Broadly speaking here are wo ways o forecas volailiy. The firs mehod uses he hisorical reurn informaion only while he second makes use of volailiy implied in opion prices. The exising empirical evidence is conflicing in hree ways. Firs, wihin he firs mehod, he performance of he models depends on he daa, forecasing horizon, sampling frequency, and evaluaion crieria (cf BF). Second, wihin he second mehod, because of he volailiy smile, a ypical feaure of implied volailiy, i is no enirely clear how o exrac volailiy from opion prices (cf PG). Third, when comparing ime series forecass wih opion forecass, people have found conflicing evidence; see, for example, Jorion (1995) and Canina and Figlewski (1993) for evidence for and agains he opion forecass respecively. This paper complemens he lieraure in hree ways. Firs, we use daa from a counry which receives lile aenion in he lieraure and ye is imporan in he inernaional framework. Second, we compare SV forecass wih opion forecass. The SV model provides more realisic and flexible modeling of financial ime series han he ARCH-ype models, since i essenially involves wo noise processes. The beer in-sample fi of he SV model over he ARCH-ype models has been documened in he lieraure (see, for example, Danielsson (1994), Geweke (1994), and Kim e al. (1998)), however, he SV model receives much less aenion in he volailiy forecasing lieraure. To our presen knowledge, here is only one paper published. Using New Zealand daa, Yu (1999) finds ha he SV model performs beer han all he oher univariae ime series models, including he ARCH-ype models. Alhough recen effors have compared ARCH forecass and opion forecass, nohing has been done o compare SV forecass and opion forecass. Third, forecas horizons and error measuremens have been arbirarily chosen in he lieraure. In his paper, he forecas horizons and error measuremens are seleced based on he uilizaion of volailiy forecass in he financial indusry. In paricular, we use opion pricing and Value-a-Risk (VaR) as he pracical guidance o choose forecas horizons and error measuremens.

4 The paper is organized as follows. Secion 2 reviews he feaures of he German sock marke. The daa and descripive saisics are in Secion 3. Secion 4 oulines he mehods used in his paper for volailiy forecass. Secion 5 describes he forecas horizons and he error measuremens. Secion 6 discusses he empirical resuls and Secion 7 concludes. 2. The German Sock Marke The imporance of he German economy is refleced in is sock marke, which ranks fourh in erms of marke capializaion and hird in erms of urnover. In paricular, he fas-growing German opion and fuure exchange, which is he second larges in he world, demonsraes ha he German sock marke has drawn a lo of inernaional aenion; see Figure 1 for he comparison of he marke capializaion and urnover a six major inernaional markes in he world Marke Capializaion of domesic Equiies Turnover in domesic Equiies Trillion EURO USA Japan Unied Kingdom Germany France Ialy Figure 1: Imporance of Inernaional Sock Markes The German sock marke consiss of eigh regional sock markes, where he Frankfur exchange wih 78% of he combined urnover is he mos imporan. All hese regional markes are based on open-oucry rading. Socks a he regional sock markes are raded in wo differen ways. The opening, midday and closing prices of each sock are calculaed using an aucion sysem. Trading in beween akes place in he convenional (coninuous) way. All orders in one sock ha arrive before he opening aucion a 8:30am are gahered. The opening price is he one a which he highes number of socks is raded. The same happens a he 1pm aucion and a he closing aucion a 5pm. Since here is a minimum order size for coninuous rading, he aucion sysem ensures ha all orders, especially small orders, are execued. Mos imporanly, he closing aucion sysem ensures ha for mos socks he closing price will be based on rade a 5pm. In addiion o he regional open-oucry markes, Germany has an elecronic rading sysem called XETRA. Since is inroducion in 1991 XETRA seadily increased is urnover share in he bigges 2

5 socks. A he end of 1998, for example, XETRA represened 68% of he urnover in he DAX equiies raded in he whole counry. Trading in XETRA akes place from 8:30am unil 5:15pm, wih an opening aucion a 8:30am and a closing aucion a 5:15pm The Deusche Akien index (DAX) represens he 30 larges domesic socks raded in Germany. A he end of 1998, socks included in he DAX index accouned for 76% of he oal marke capializaion in Germany and 80% of he equiy urnover in Frankfur, making he DAX a highly represenaive index for he German sock marke. To more easily signal volailiy o invesors, he German sock exchange inroduced a volailiy index based on implied volailiies of DAX opions in December I is called VDAX. The VDAX index is based on linear inerpolaion of he volailiies of he wo sub-indices ha are neares o a remaining lifeime of 45 calendar days. More deails abou he VDAX index are provided in Secion DATA 3.1 DAX Reurns One daa series we have is he daily DAX index from January 1, 1988 o June 30, 1999 and is based on daily closing aucion prices a he Frankfur sock exchange. We use logarihmic reurns calculaed from he DAX series, resuling in 2,876 daily reurn observaions. Inspecing he DAX reurn series as depiced in Figure 2 reveals ha, as expeced, volailiy is no consan over ime and moreover ends o cluser. Periods of high volailiy can be disinguished from low volailiy periods. Apar from he clusering of large negaive reurns in Augus 1998, here are hree ousanding reurns ha are large in absolue value. The firs is he 13.7% fall on Ocober 16, 1989 in he wake of he burs merger bubble in he Unied Saes. The second is he - 9.9% reurn on Augus 19, 1991, he day of he coup agains Gorbachev in he Sovie Union, an even ha severely affeced he German marke. The main reason for he fall of 8.4% on Ocober 28, 1997 was he Asian crisis. The mean daily reurn of he DAX series is %. The sandard deviaion of he daily reurns is , which is equivalen o an annualized volailiy of 20%. The series also exhibis a negaive skewness of and an excess kurosis of 9.8, indicaing ha he reurns are no normally disribued. The Jarque-Bera saisic of 11,824 also shows ha we have o rejec normaliy wih a p-value of one. These findings are consisen wih oher financial ime series. Figure 3 plos a hisogram of he daa and a normal densiy whose mean and variance mach sample esimaes. I shows ha numerous reurns are above four sandard deviaions, which is highly unlikely in he normal disribuion. This is evidence of lepokurosis and commonly found in financial ime series. 3

6 Jan-88 Jan-89 Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Figure 2: Reurn Series for FRA-DAX. The ovals indicae a low volailiy period and a high volailiy period Figure 3: FRA-DAX Normalized Reurn Disribuion The Ljung-Box Q-saisics for he firs 50 lags is for reurns and for squared reurns. Therefore, we canno rejec he hypohesis ha here is no serial correlaion in he level of reurns bu we have o rejec he same hypohesis in squared reurns. Using he augmened Dickey-Fuller (ADF) uni roo es we can clearly rejec, as expeced, he hypohesis of a uni roo in he reurn process. The ADF -saisic is which rejecs he uni roo hypohesis wih a confidence level of more han 99%. Furhermore we also have o rejec he hypohesis of a uni roo in he squared reurn process, which is an approximaion of he volailiy process, where he ADF- es saisic is equal o

7 Based on he various univariae ime series models which will be reviewed in Secion 4, he DAX reurn series is used o forecas volailiy. 3.2 VDAX Figure 4 plos VDAX for he period from January 1992 o July Two special evens sand ou in his figure. Firs, he sharp increase in Ocober 1997 in he wake of he Asian crisis. Second, he jump in implied volailiy afer he Russian bond defaul, where he VDAX increased from 29.73% on Augus 21, 1998 o a high of 56.31% on Ocober 2, The VDAX index will also be used o forecas volailiy. 60% VDAX 50% Annual Volailiy 40% 30% 20% 10% 0% Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Figure 4: DAX Volailiy Index (VDAX) 3.3 Acual Volailiy To assess he performance of various mehods, we need o compare forecased volailiies wih acual volailiies. Unforunaely, he acual volailiy is no direcly observed and hence i has o be esimaed. A common approach in he lieraure is o use he absolue or squared daily reurn o esimae he daily volailiy. In he more recen lieraure, daily volailiy has been esimaed from high frequency daa. When using high frequency daa, such as ick-by-ick daa, paricular aenion has o be paid o he possible negaive auocorrelaion caused by he bid-ask bounce. If prices are recorded from ransacions, he price of each ransacion migh bounce beween he bid and ask price causing negaive auocorrelaion. Neglec of he negaive serial auocorrelaion will lead o an upwardbiased esimaion of daily volailiy. 5

8 I has been found ha improvemens can be made in esimaing acual volailiy based on high frequency daa. For example, using 5-minue inra-daily daa, Andersen and Bollerslev (1998) find ha he GARCH model provides accurae volailiy forecass. Blair, Poon and Taylor (2000) suppor his by reporing an 11.5% o 41.4% increase in one-day-ahead predicion 2 R when daily volailiy has been esimaed using high-frequency daa. Since we only have a daa se a he daily frequency, we calculae he volailiy in a cerain period simply as he square roo of he sum of squared daily reurns in ha period; ha is, N T 2 T = r = 1 σ (3.1) where r is he daily reurn on day and NT is he number of rading days in ha period Volailiy Forecasing Techniques 4.1 Forecasing models using he hisorical reurn informaion only The forecasing models using he hisorical reurn informaion only include he hisorical mean model, he exponenially weighed moving average (EWMA) model, 2 he GARCH model (Bollerslev (1986)), he GJR-GRACH model (Glosen, e al (1993)), he EGARCH model (Nelson (1990)), he GARCH-M model (Engle e al (1987)), and he SV model (Taylor (1986)). The hisorical mean model, he EWMA model, he GARCH(p,q) model and he GJR-GRACH(p,q) model are defined in BF. We now define he oher models. In all cases we assume r is he reurn a period and ε 1 ~ N(0,1). I The EGARCH(p,q) model is defined by r = µ + σ ε The GARCH-M(p,q) model is defined by q q αi ε i + γ iε i + i= 1 i= 1 2 logσ = ω + β logσ. p j = 1 j 2 j r = µ + δσ + σ ε, 2 1 Obviously, volailiy is defined as sandard deviaion insead of variance in his paper. 2 This model is referred o in BF as he exponenial smoohing model alhough i is more convenional o refer i o in he forecasing lieraure and indusry as he EWMA model. 6

9 q p 2 = ω air i i= 1 j = 1 σ β σ. i 2 j The sysem parameers in all he ARCH-ype models are esimaed using he maximum likelihood mehod. The lag parameers p and q are seleced using he BIC crierion. To obain h-sep ahead forecas of he volailiy, we follow Baillie and Bollerslev (1992). The SV model is defined by r = µ + exp( / 2) ε, h h = ω + α 1 + η, h 2 σ η where η ~ N(0, ). Compared o he ARCH-ype models, he SV model provides a more flexible modeling of financial ime series, since i essenially involves wo noise processes, one for he observaions, and one for he laen volailiies. Unforunaely, he likelihood funcion for he SV model has no closed form expression and herefore maximum likelihood (ML) esimaion is applicable. Recenly several mehods have been proposed o esimae he SV model. Such mehods include Quasi ML (QML) proposed by Harvey e al (1994), simulaed ML by Danielsson (1994), GMM by Andersen and Sorensen (1996), Markov Chain Mone Carlo (MCMC) by Jacquier, Polson and Rossi (1994) and Meyer and Yu (2000), Efficien Mehod of Momens by Gallan and Tauchen (1994), Mone Carlo maximum likelihood by Sandmann and Koopman (1998), and he empirical characerisic funcion mehod by Knigh, Sachell and Yu (1998). Some of hese mehods, such as QML and MCMC, also produce forecass of volailiy as by-producs. MCMC provides he exac opimal predicors of volailiy, however, i is compuaionally inensive. Despie is inefficiency, he QML mehod is consisen and very easy o implemen numerically. Following Yu (1999), we use QML o esimae parameers in he SV model and obain h-day ahead volailiy forecass. There remain wo quesions before we can perform our model evaluaion. The firs is how we divide he sample. The 2,126 observaions from January 4, 1988 o June 28, 1996 are used o fi he models and he ou-of-sample period covers he remaining 750 observaions from July 1, 1996 o June 30, This division is arbirary, bu he ou-of-sample period covers periods of boh low volailiy and exremely high volailiy in 1998, making accurae volailiy forecass more difficul. The oher quesion is wha sample we should use for model fiing as addiional observaions beyond June 28, 1996 become available. In his paper we use he mehod of expanding forecas windows. Tha is, beginning wih he las day of he ou-of-sample period we compue a volailiy forecas for each horizon. For he second day of he ou-of-sample period he volailiy forecass are 7

10 obained in he same way, bu he informaion se includes he realized reurn of he firs ou-ofsample day. 4.2 Deriving Implied Volailiies While fuure volailiy can be forecased using hisorical reurn informaion, i can also be derived from opion prices. For example, he Black-Scholes formula for pricing a European call opion on a non-dividend paying sock needs five inpu variables: he mauriy of he opion, he srike price, he curren price of he underlying, he risk-free ineres rae associaed o he mauriy of he opion and he volailiy of he underlying sock over he lifeime of he opion. Once he opion is raded, he only unknown parameer is he volailiy. Theoreically, herefore, he Black-Scholes formula can be used o derive he implied volailiy which should express he marke volailiy forecas of he underlying asse. Unforunaely, in pracice, he implied volailiy depends on he opion pricing model used and is sricly associaed wih he mauriy and srike of he opion. Therefore, we canno derive a unique implied volailiy for he underlying securiy. A ypical feaure of implied volailiy is he so-called volailiy smile, which resuls from he observaion ha ou-of-he-money and in-he-money opions have a higher implied volailiy han a-he-money opions. Figure 5 shows an implied volailiy marix for DAX opions on June 21, The price of he underlying was 5,414 (a-he-money poin). Volailiy smile can be easily idenified. 70% 60% 50% 40% 30% 20% 10% 0% Srike May-00 Aug-99 Mauriy Figure 5: Implied Volailiy Marix for DAX Call Opions on June 21, 1999 There are several oher problems involved in deriving implied volailiies from opion prices. Firs, mos opion price series are no synchronous wih prices of he underlying asses. Using daily 8

11 closing prices for an opion and is underlying migh have he effec ha he las price of an opion is based on an earlier price of he more liquid underlying and no on he closing price. Second, relaive bid-ask spreads are large for opions, especially for ou-of-he-money and in-he-money opions, making i difficul o derive he correc opion price. Third, he opion forecass work less well if he opion marke is less liquid. In he lieraure mos of he papers using he opion forecass focus on he U.S. marke where he mos liquid daa are available. 3 To our presen knowledge, here are only hree papers using daa from ouside he U.S. marke. In paricular, Edey and Ellio (1992) use he Ausralian daa while Vaselellis and Meade (1996) and Gemmill (1986) use he Briish daa. In his paper, we use VDAX, a German daa se, o forecas fuure volailiy. The VDAX is based on linear inerpolaion of he volailiies of he wo sub-indices ha are neares o a remaining lifeime of 45 calendar days. 4 The mauriy of 45 days of he synheic underlying opion remains consan. I means ha he VDAX is lifeime-independen and herefore does no expire, eliminaing he effecs of srong flucuaions of volailiy, which ypically occur close o expiry. The sub-indices correspond o he mauriies of he currenly raded DAX opions and are calculaed using four opions (wo calls and wo pus) whose srike is closes o he curren price of he underlying. 5 Using he implied volailiy of hese four opions, based on he Black-Scholes model, he volailiy value of he sub-index is calculaed as follows: where V i Pu Call Pu Call (Xh F i) * (v + v ) + (Fi X ) * (vh + vh ) Vi =, 2(X X ) h is he implied volailiy sub index corresponding o mauriy i; F i he forward or fuures price corresponding o mauriy i; X he srike price of an individual opion wih h referring o a srike price of he opion above he curren underlying price and o a srike price of he opion below he curren underlying price; v he implied volailiy of an individual opion. I should be sressed ha VDAX akes only a-he-money opions ino accoun, neglecing he ou-of-he-money and in-he-money opions ha are no liquid. By consrucion VDAX can overcome some of he aforemenioned problems in deriving implied volailiy from opion prices. 6 Firs, i records every en seconds all necessary inpu daa and calculaes new values for sub-indices only when he relevan opion prices have changed. Since he underlying is raded very frequenly, he opion price always corresponds o he underlying price. Second, only opions whose bid-ask spread is no more han 15% of he bid price are aken ino accoun and hence can reduce 3 See Blair, Poon and Taylor (2000) and he references herein. 4 Using he same idea as obaining volailiy forecass for horizon of 45 calendar days from VDAX, we can obain volailiy forecass for oher horizons. 5 Insead of he DAX index iself as he underlying, he forward price of he DAX index is used, which can be easily obained from he DAX fuure prices. 6 How o forecas fuure volailiy from opions wih differen srikes has recenly received a grea deal of aenion in he lieraure; see, for example, PG for a review. 9

12 measuremen errors. Third, VDAX is based on he second larges opion and fuure exchange marke in he world and hence is reasonably liquid. 5. Forecas Horizons and Error Measuremens 5.1 Forecas horizons Alhough a variey of forecas horizons have been used in he lieraure, hey are arbirarily chosen wihou resoring o pracical guidance. I is known ha volailiy forecass have been widely used in financial insiuions for various purposes, of which opion pricing and VaR are of paricular ineres from a pracical viewpoin. In his paper we use he forecas horizons by aking ino accoun he pracical requiremens of volailiy forecass in opion pricing and VaR. Unil recen years volailiy forecass have been almos exclusively needed as an inpu variable in he Black-Scholes opion pricing model. This ypically requires forecas horizons beween a monh and a year. Furhermore, he mos liquid opions raded a opion exchanges are hose wih a shor mauriy, usually one o hree monhs. To compare he ime series forecass wih he opion forecass we choose a forecas horizon of 45 calendar days, which is exacly he same as VDAX. Since many exchanges also rade long-erm opions wih mauriies up o 2 years, we wan o es he abiliy of he models o forecas volailiy over longer horizons. In addiion o he 45 calendar days, we also use a forecas horizon of 180 rading days (abou nine calendar monhs). Wih he growing usage of VaR as a risk managemen ool, he need for shor-erm variance and covariance forecass becomes greaer. Mos invesmen banks are ineresed in forecasing he risk of heir porfolios unil he closing of he nex rading day. In addiion, he Basle Capial Accord in 1998 (Basle Commiee On Banking Supervision (1999)) requires banks o compue VaR over a horizon of en rading days. A relaed applicaion for volailiy forecass is margining a fuure and opion exchanges. To ensure ha marke paricipans are able o fulfill her financial obligaion resuling from fuure and opion conracs, he exchange requires hem o deposi a margin. This ofen-called iniial margin should cover he anicipaed price risk of heir posiions. The German fuure and opion exchange (EUREX) currenly calculaes iniial margins based on 30 and 250 days of hisorical volailiies. These volailiies are used o forecas he 99% confidence inerval for omorrow s price of he underlying. Resuling from his price inerval and he posiion of he marke paricipans, he EUREX calculaes he iniial margin. Therefore, for he purpose of VaR and riskbased margining we use forecas horizons of one and en rading days. 5.2 Error Measuremens In he lieraure a variey of saisics have been used o evaluae and compare forecas errors These include roo mean square error (RMSE), mean absolue error (MAE), mean absolue 10

13 percenage error (MAPE), mean mixed error (MME), he Theil-U saisic, and he LINEX loss funcion. RMSE, MAE and MAPE are convenionally used error saisics. MME is proposed by BF while Yu (1999) advocaes he use of he Theil-U and LINEX loss funcion. As wih he choice of forecas horizons, however, he evaluaion saisics are also arbirarily chosen. Deciding which error measuremen should be applied o he volailiy forecass, we believe, wo quesions have o be answered. Firs, is he absolue or relaive deviaion imporan? Second, does i maer wheher he volailiy was over-prediced or under-prediced? Where opion pricing is concerned, he firs quesion is equivalen o wheher relaive or absolue rading profis maer. This is because he price of an a-he-money opion is a linear funcion of he volailiy according o he Black-Scholes model. Since in he financial world relaive profis are more imporan i can be assumed ha an opion rader is ineresed in he relaive deviaion beween forecased and realized volailiy. This is why we use MAPE which is defined by, MAPE 1 = T T ˆ σ τ + h σ σ τ = 1 τ + h τ + h, (5.1) where T is he number of ou-of-sample observaions minus he number of days of he forecas horizon; σ he acual volailiy a he period ; σˆ he forecased volailiy a he period. Since he Basle Commiee's Marke Risk Amendmen o he Capial Accord in 1998 specifies ha banks have o calculae he price risk of heir financial aciviies and se aside sufficien capial o cover his marke risk, VaR is oday a sandard ool used o comply wih Basle Accord requiremens. VaR uses a volailiy forecas and a confidence level based on normally disribued reurns o yield a poenial loss. According o he Basle Capial Accord, banks have o se aside reserve capial as big as hree imes he poenial loss ha will no be exceeded wih 99% probabiliy. Depending on he accuracy of he volailiy forecas, however, here can be more or less han 1% of he reurns ouside he boundary implied by he poenial loss. The regulaion inroduced by he Capial Accord emphasizes he need for accurae volailiy predicion. Over-predicion of fuure volailiy over a long period requires more cosly capial, while under-predicion leads o more boundary violaions han implied by he confidence level. If VaR is used only as an inernal risk managemen ool, he boundary violaions would be he concern and hey should a leas be as low as wha he confidence level implies. If VaR is used in line wih he Basle Capial Accord, he price risk of he porfolio has o be covered by equiy, and hence overand under-predicion maers. I can be assumed ha an under-predicion maers more han overpredicion. This is because he model will no be acceped by he regulaory body if he boundary violaions are higher han implied. The siuaion is cerainly worse han providing a lile bi more capial when he price risk was overesimaed. 11

14 Where VaR is concerned, herefore, we will apply wo error measuremens, he number of boundary violaions ha occur during he forecas horizon and he LINEX loss funcion. Deviaing from he LINEX loss funcion used in Yu (1999) we will use a LINEX loss funcion defined by T 1 L ( a) = ˆ σ + exp( ( ˆ τ στ α στ στ )) 1, (5.2) T τ = 1 where α is a given parameer which capures he degree of asymmery. The LINEX loss funcion in Yu (1999) would yield for α = 0 a forecas error equal o zero even hough σˆ τ differs from σ τ. In (5.2), however, we sill have a non-rivial symmeric loss funcion when α = 0. Figure 5 plos he LINEX loss funcion wih differen values of α. 7 There are no rules as o how o choose he opimal α, unforunaely. In general i should depend on how much one dislikes/likes under-predicion relaive o over-predicion. In his paper we choose α = 30, which implies he LINEX value from under-predicion is abou 25% higher han ha from over-predicion. Also we chooseα = 0. Combinaions of he forecas horizons and he error measuremens produce four es seings, which are summarized in Table LINEX value Asymmery Parameer Forecas Error (forecasrealized) Figure 5: LINEX Loss Funcion 6. Empirical Resuls 6.1 In-sample Fi 12

15 To frame our discussion, we briefly presen some resuls from in-sample fi based on he firs of our expanding samples. For he EWMA model, he value of damper coefficien is chosen o produce he bes fi by minimizing he sum of squared in-sample forecas errors. This yields he opimal value of As o he choice of p and q in he ARCH-ype models, he BIC crierion selecs GARCH(1,3), GJR-GARCH(1,3), EGARCH(2,1) and GARCH-M(1,3) respecively. Moreover, wihin he ARCH family, he EGARCH model has he lowes BIC value and hence he bes in-sample fi. The superior in-sample fi of he EGARCH model is also suppored by he news impac curve es inroduced by Engle and Ng (1993). In erms of he esimaed persisence in he variance equaion, he GJR-GARCH model and he EGARCH model are he lowes, followed by he GARCH and he GARCH-M models. More deailed in-sample resuls are available on reques. Tes Uiliy Forecas Horizon 1 Opion pricing 45 calendar days Error Measuremen MAPE 2 Opion pricing 180 rading days MAPE 3 VaR 1 rading day Boundary violaions/ LINEX loss funcion 4 VaR 10 rading days Table 1: Overview of he Tes Seings Boundary violaions/ LINEX loss funcion 6.2 Ou-of-sample Forecass In Table 2 we repor he acual value of forecas error measuremens and ranking for all he ime series models and VDAX under Tes 1 and Tes 2, and hose for he ime series models only under Tes 3 and Tes 4. There are several resuls emerging from Table 2. Firsly, no single mehod is clearly superior. For example, Tes 1 indicaes ha he SV model provides he mos accurae forecass followed by VDAX as second while under Tes 2 VDAX ranks firs. Tes 3 favors he GJR-GARCH model and he EGARCH model, bu Tes 4 favors EWMA, he GARCH-M model, and he SV model. The model rankings depend on he forecas horizon as well as he forecas error measuremen. This observaion is consisen wih he exising lieraure when only he various ime series models compee; see, for example, BF. Secondly, as expeced, implied volailiy appears o be a good predicor of fuure volailiy. I ranks firs under Tes 1 and second under Tes 2 and clearly ouperforms he ARCH-ype forecass under boh ess. This observaion is consisen wih he findings of Jorion (1995). When comparing 7 Unforunaely he LINEX loss funcion given in (5.2), unlike Yu s LINEX loss funcion, has no closed form expression for he LINEX opimal forecas. 13

16 SV forecass and implied volailiy forecass, however, neiher is dominan and wha we can conclude is here is a rade-off beween hem. Thirdly, when VaR is he concern, i appears ha various ARCH-ype models are useful. For example, he GJR-GARCH ranks firs for 1-day-ahead forecas under boundary violaions and he LINEX loss L(-30); he EGARCH model ranks firs for 1-day-ahead forecas under L(0); GARCH- M model ranks firs for 10-day-ahead forecas under L(-30). Alhough no ARCH-ype model ranks firs for 10-day-ahead forecas under boundary violaions or for 10-day-ahead forecas under L(0), he GARCH model and he EGARCH model rank a close second in boh cases respecively. Tes 1 Tes 2 Tes 3 Tes 4 Horizon 45cds 180ds 1 d 10 ds Saisics MAPE MAPE BV(99%) L(-30) L(0) BV(99%) L(-30) L(0) Hisorical Mean 29.45% (7) 27.35% (4) 3.87% (7) (6) (7) 5.13% (5) (7) (7) EWMA 29.07% (4) 30.87% (5) 2.80% (2) (4) (6) 2.83% (1) (4) (6) GARCH 29.58% (8) 22.68% (2) 3.07% (3) (5) (5) 2.97% (2) (2) (4) GJR- GARCH 29.39% (5) 34.38% (7) 2.67% (1) (1) (3) 5.13% (5) (6) (5) EGARCH 28.92% (3) 49.79% (8) 3.47% (6) (2) (1) 5.13% (5) (5) (2) GARCH % 22.79% 3.07% % M (5) (3) (3) (3) (4) (3) (1) (3) SV 23.09% 33.09% 3.20% % (1) (6) (5) (7) (2) (4) (3) (1) VDAX 25.16% 21.24% (2) (1) Table 2: Resuls of ou-of-sample forecass. MAPE is defined by (5.1); BV(99%) is he percenage boundary violaions for a confidence level of 99%; L(-30) and L(0) are he LINEX values for α=-30 and α=0 scaled by The number in brackes is he ranking under each es. Fourhly, he performance of he ARCH-ype models wih less persisence ges worse when he forecas horizon ges longer. For example, by he boundary violaions and L(-30), he GJR-GARCH model, a favorie model in BF, ranks firs under Tes 3, and ranks dismally fifh and sixh under Tes 4. Also, is MSPE increases from 29.39% o 34.38% when he forecas horizon increases from 45 calendar days o 180 rading days. The reason for his is he rapid decline in he forecased squared reurns due o he lowes persisence found in he in-sample esimaion of he GJR-GARCH 14

17 model. Since he volailiy is deermined by he sum of he squared reurns, low persisence ends o generae lower volailiy forecass over longer horizons. BF also finds ha he GJR-GARCH model under-predics volailiy more ofen han oher models. However, hey offer no reason for his behavior. The same observaion and explanaion apply o he EGARCH model. Moreover, he low persisence also aribues o he wider range of he MSPE values across he models for he horizon of 45 calendar days relaive o hose for he horizon of 180 rading days. Fifhly, he forecas performance of he simple average model is raher poor, especially for shor horizons. In paricular, under boh Tes 3 and Tes 4 i has he highes number of boundary violaions and he LINEX loss values. The oher simple model, he EWMA model, however, performs reasonably well. I has he lowes boundary violaions in Tes 4 and he second lowes in Tes 3. The finding suggess ha he EWMA model could be a suiable model for forecasing volailiy in he VaR framework. Given ha he EWMA esimaor is exremely easy o calculae, his finding helps o jusify why EWMA is ofen used in financial insiuions. Dimson and Marsh (1990) also repor evidence o suppor EWMA. Sixhly and finally, judged by he relaive number of boundary violaions under Tes 3 and Tes 4, all ime series models end o under-predic volailiy. However, given ha he consrucion of boundaries relies on he assumpion of normally disribued reurns, a large value of boundary violaions can be he resul of he lepokuric reurn disribuion raher han he under-predicion of volailiy. 6.3 Trading Sraegy We need o emphasize ha he ess in Secion 6.2 are based on average errors. One can use a es saisic o compare he differences beween wo error disribuions; see, for example, Wes and Cho (1996). Alhough such a es allows one o make a saisical inference abou he model performance, in finance a more appealing approach is o use economic reasoning for he comparison. In his paper we use a rading sraegy o es he usefulness of ime series models. This rading sraegy is based on buying or selling call opions on he DAX index. The opion pricing formula derived by Black and Scholes (1973) is a posiive funcion of he expeced volailiy of he underlying sock price. If we know he fuure volailiy, we can consruc a riskless porfolio by buying an opion and selling he underlying sock or index. However, since he opion s expeced or implied volailiy is only a forecas, his sraegy is no riskless and can yield a profi or loss depending on he rue volailiy. If he rue volailiy is less han he implied volailiy, buying he undervalued opion and selling he underlying sock reurns a profi. Similarly, selling an apparenly overvalued opion and buying he underlying sock will reurn a profi. If he volailiy forecass generaed by ime series models are superior o implied volailiies, one can use his rading sraegy o generae profis. 8 8 A deailed descripion of his rading sraegy can be found in Figlewski (1989). 15

18 Since he opions used for his rading sraegy have mauriies beween 7 and 67 calendar days, he opimal ime-series model is seleced based on he average of he MAPE values for Tes 1 (45 days mauriy) and Tes 4 (abou 14 days mauriy). As Table 3 shows, his yields he EGARCH(2,1) model as he bes model. 10cds 45cds Mean Hisorical Mean 449% 29.5% 239% EWMA 466% 29.1% 247% GARCH 471% 29.6% 250% GJR-GARCH 410% 29.4% 220% EGARCH 374% 28.9% 201% GARCH-M 467% 29.4% 248% SV 425% 23.1% 224% Table 3: Average MAPE of Tes 1 and Tes 4 Using selemen (closing) prices for DAX opions from July 2, 1999 o Ocober 15, 1999, we calculae for each day an implied volailiy based on an a-he-money call opion wih he shores mauriy available (minimum 7 days) and an implied volailiy based on a medium erm opion wih 37 and 67 days mauriy. These wo implied volailiies are hen compared wih he forecass generaed by he EGARCH model for horizons corresponding o he mauriies of he opions. If he EGARCH model s forecas is 20% higher han he implied volailiy, he call opion is bough; if he EGARCH model s forecas is 20% lower, he call opion is sold. The reurn is he discouned coss (revenue) of buying (selling) he underlying index divided by he opion premium. Since rading he index iself is exremely cosly and given ha he high indivisibiliy is acually impossible, we will use as underlying he fuure on he DAX index. The ransacion coss ha are aken ino accoun here are he ones ha arise from rading he fuure, which are also he main coss of a dela-hedge sraegy 9. The fuure s ransacion coss consis of a bid-ask-spread of ± 0.5 index poins of he selemen price and a fee of 1-EURO per raded fuure ha he EUREX charges. These ransacion coss are ypical for insiuional invesors, and only a fracion of he coss ha reail invesors would pay performing volailiy arbirage. The low persisence of he EGARCH model is once again obvious. For he opion wih he shorer mauriy (denoed as opion 1), on 28 ou of 71 rading days he EGARCH model s forecas is 20% lower han he implied volailiy and on one day 20% higher. For he opion wih he longer mauriy (denoed as opion 2), on 45 ou of 51 rading days 10 he EGARCH model s volailiy 9 Since his sraegy resuls in a spread of shor posiion in a call opion and long posiion in he underlying, he iniial margin is small. Also, he coss for he iniial opion rade are negligible. 10 The difference in rading days beween opion 1 and 2 arises because we have no marke daa in opion 2 afer Sepember 11 which maures in November.. 16

19 forecas is 20% lower han he implied volailiy and never higher han he implied volailiy. This resulsin29radesforopion1and45radesforopion2. As a resul we would expec ha if our model predics ha he realized volailiy is lower han implied in he opion price, ha rade would yield a profi. A loss would arise when he realized volailiy urns ou o be higher han he implied volailiy. 80% 60% 40% 20% -400% -300% -200% -100% 0% 100% 200% 300% 0% -20% Figure 6: Scaer plo of a rade s reurn and he deviaion beween implied and realized volailiy. The x-axis depics he reurn in percenage erms and he y-axis he percenage difference beween he implied volailiy and he realized volailiy. Since all he rades depiced in his figure are sales of a call opion ha he EGARCH model regards as overpriced, he more posiive he difference beween he implied and he realized volailiy, he higher he expeced reurn and vice versa. Figure 6 shows ha our daa suppors his expecaion since mos poins are in he firs or hird quadran, while he deviaions are caused by ransacion coss and marke imperfecions. I also shows ha no all rades generae a posiive reurn. In fac he average reurn is negaive (-9.1%), alhough i is very small and parly caused by he ransacion coss. More ineresingly, for 29 rades of opion 1 he average reurn is posiive (0.89%), while 45 rades of opion 2 generae an average reurn of -18.9%. This resul is due o he low persisence, causing he EGARCH model o underpredic volailiy more for opion 2 han for opion 1. The conclusion from his rading sraegy is ha ime series models are no beer a predicing volailiy han he implied volailiy, alhough admiedly his rading sraegy is hardly represenaive. 7. Conclusions This paper has compared wo basic approaches o forecas volailiy in he German sock marke. The firs approach uses various univariae ime series echniques while he second approach makes 17

20 use of volailiy implied in opion prices. The ime series models include he hisorical mean model, he EWMA model, four ARCH-ype models and a SV model. Based on he uilizaion of volailiy forecass in opion pricing, we choose he forecas horizons of 45 calendar days and 180 rading days and he error measuremen of MSPE. Based on he uilizaion of volailiy forecass in VaR, we choose he forecas horizons of one and en rading days and he error measuremens of bounded violaions and he LINEX loss funcion. The resuls sugges ha he rankings are sensiive o error measuremens as well as forecas horizons. Consisen wih he evidence found in BF, he findings indicae ha no single mehod is he clear winner. However, when opion pricing is he primary ineres, he SV model and implied volailiy should be used. On he oher hand, when VaR is he concern, he ARCH-ype models are useful. Furhermore, a rading sraegy suggess ha ime series models are no beer han implied volailiies a predicing volailiy, consisen wih he findings of Blair, Poon, and Taylor (2000) and Jorion (1995). References Andersen, T.G., and T. Bollerslev, 1998, Answering he skepics: yes, sandard volailiy models do provide accurae forecass, Inernaional Economic Review, 39, 4, Andersen, T.G., and B. Sorensen, "GMM esimaion of a sochasic volailiy model: A Mone Carlo sudy", Journal of Business and Economic Saisics, 14, 1994, pp Baillie, R., and T. Bollerslev, 1992, Predicion in dynamic models wih ime-dependen condiional variances, Journal of Economerics, 52, Basle Commiee On Banking Supervision, 1999, "Performance of models-based capial charges for marke risk 1 July-31 December 1998", Basle Black, F. and M. Scholes, 1973, Poliical Economy, 81, The Pricing of Opions and Corporae Liabiliies, Journal of Blair, B., S-H Poon, and S.J. Taylor, 2000, Forecasing S&P 100 Volailiy: The Incremenal Informaion Conen of Implied Volailiies and High Frequency Index Reurns, Journal of Banking and Finance, Forhcoming Bollerslev, T., 1986, Generalized auoregressive condiional heeroskedasiciy, Journal of Economerics 31, Brailsford, T.J. and R.W. Faff, 1996, "An evaluaion of volailiy forecasing echniques", Journal of Banking and Finance 20,

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