WHAT GOOD IS A VOLATILITY MODEL? *

Transcription

1 WHAT GOOD IS A VOLATILITY MODEL? * Rober F. Engle and Andrew J. Paon Deparmen of Finance, NYU Sern School of Business, and Deparmen of Economics, Universiy of California, San Diego, 9500 Gilman Drive, La Jolla, CA , USA 9 January, 001. ABSTRACT A volailiy model mus be able o forecas volailiy; his is he cenral requiremen in almos all financial applicaions. In his paper we ouline some sylised facs abou volailiy ha should be incorporaed in a model; pronounced persisence and meanreversion, asymmery such ha he sign of an innovaion also affecs volailiy and he possibiliy of exogenous or pre-deermined variables influencing volailiy. We use daa on he Dow Jones Indusrial index o illusrae hese sylised facs, and he abiliy of GARCH-ype models o capure hese feaures. We conclude wih some challenges for fuure research in his area. Keywords: volailiy modelling, ARCH, GARCH, volailiy forecasing. JEL Classificaion Code : C * Please send commens or quesions o rengle@sern.nyu.edu.

2 1. INTRODUCTION A volailiy model should be able o forecas volailiy. Virually all he financial uses of volailiy models enail forecasing aspecs of fuure reurns. Typically a volailiy model is used o forecas he absolue magniude of reurns, bu i may also be used o predic quaniles or in fac, he enire densiy. Such forecass are used in risk managemen, derivaive pricing and hedging, marke making, marke iming, porfolio selecion, and many oher financial aciviies. In each, i is he predicabiliy of volailiy ha is required. A risk manager mus know oday he likelihood ha his porfolio will decline in he fuure. An opion rader will wan o know he volailiy ha can be expeced over he fuure life of he conrac. To hedge his conrac he will also wan o know how volaile is his forecas volailiy. A porfolio manager may wan o sell a sock or a porfolio before i becomes oo volaile. A marke maker may wan o se he bid ask spread wider when he fuure is believed o be more volaile. There is now an enormous body of research on volailiy models. This has been surveyed in several aricles and coninues o be a fruiful line of research for boh praciioners and academics. As new approaches are proposed and esed, i is helpful o formulae he properies ha hese models should saisfy. A he same ime, i is useful o discuss properies ha sandard volailiy models do no appear o saisfy. We will concern ourselves in his paper only wih he volailiy of univariae series. Many of he same issues will arise in mulivariae models. We will focus on he volailiy of asse reurns and consequenly will pay very lile aenion o expeced reurns.

3 Firs we will esablish noaion. Le P be he asse price a ime and r = ln(p) ln(p-1) be he coninuously compounded reurn on he asse over he period -1 o. We define he condiional mean and condiional variance as: (1) m = E [r ] 1 () h = E 1[r m ] where E-1[u] is he expecaion of some variable u given he informaion se a ime -1 which is ofen denoed [u I ]. Wihou loss of generaliy his implies ha R is E 1 generaed according o he following process: (3) R = m + h ε, where E-1[ε] = 0 and V-1[ε] = 1 In his paper we are ofen concerned wih he condiional variance of he process and he disribuion of reurns. Clearly he disribuion of ε is cenral in his definiion. Someimes a model will assume: (3) {ε} ~ i.i.d. F( ) where F is he cdf of ε. We can also define he uncondiional momens of he process. The mean and variance are naurally defined as (4) µ = E [r µ ], σ = E[r ] and he uncondiional disribuion is defined (5) ( µ )/ σ ~ G r where G is he cdf of he normalized reurns. 3

4 A model specified as in equaions (1), () and (3) will imply properies of (4) and (5) alhough ofen wih considerable compuaion. A complee specificaion of (4) and (5) however, does no imply condiional disribuions since he degree of dependence is no formulaed. Consequenly, his does no deliver forecasing relaions. Various models for reurns and volailiies have been proposed and employed. Some such as he GARCH ype of models are formulaed in erms of he condiional momens. Ohers such as sochasic volailiy models are formulaed in erms of laen variables which make i easy o evaluae uncondiional momens and disribuions bu relaively difficul o evaluae condiional momens. Sill ohers, such as muli-fracals or sochasic srucural break models, are formulaed in erms of he uncondiional disribuions. These models ofen require reformulaion o give forecasing relaions. Higher momens of he process ofen figure prominenly in volailiy models. The uncondiional skewness and kurosis are defined as usual by (6) ξ E 3 ( r µ ) E( r µ ) =, ζ = 3 σ σ 4 4 The condiional skewness and kurosis are similarly defined (7) s = E 3 ( r m ) E ( r m ) 1, 3/ h 1 k = 1 1 h 4 Furhermore, we can define he proporional change in condiional variance as h h (8) variance reurn= h

5 Some of he variance reurn is predicable and some is an innovaion. The volailiy of he variance is herefore he sandard deviaion of his innovaion. This definiion is direcly analogous o price volailiy. (9) VoV = V( variance reurn) A model will also generae a erm srucure of volailiy. Defining h + k E[r+ k], he erm srucure of volailiy is he forecas sandard deviaion of reurns of various mauriies, all saring a dae. Thus for an asse wih mauriy a ime +k, his is defined as k k (10) ν + k V r + j E ( r + j) j= 1 j= 1 The erm srucure of volailiy summarizes all he forecasing properies of second momens. From such forecass, several specific feaures of volailiy processes are easily defined.. STYLIZED FACTS ABOUT ASSET PRICE VOLATILITY A number of sylised facs abou he volailiy of financial asse prices have emerged over he years, and been confirmed in numerous sudies. A good volailiy model, hen, mus be able o capure and reflec hese sylised facs. In his secion we documen some of he common feaures of asse price volailiy processes. 5

6 .1 Volailiy exhibis persisence The clusering of large moves and small moves (of eiher sign) in he price process was one of he firs documened feaures of he volailiy process of asse prices. Mandelbro (1963) and Fama (1965) boh repored evidence ha large changes in he price of an asse are ofen followed by oher large changes, and small changes are ofen followed by small changes. This behavior has been repored by numerous oher sudies, such as Baillie e al. (1996), Chou (1988) and Schwer (1989). The implicaion of such volailiy clusering is ha volailiy shocks oday will influence he expecaion of volailiy many periods in he fuure. Figure 3., which will be described in he following secion, displays he daily reurns on he Dow Jones Indusrials Index over a welve year period and shows evidence ha he volailiy of reurns varies over ime. To make a precise definiion of volailiy persisence, le he expeced value of he variance of reurns k periods in he fuure be defined as (11) h + k E (r+ k m + k). The forecas of fuure volailiy hen will depend upon informaion in oday s informaion se such as oday s reurns. Volailiy is said o be persisen if oday s reurn has a large effec on he forecas variance many periods in he fuure. Taking parial derivaives, he forward persisence is: h+ k (1) θ + k = r This is a dimensionless number as squared reurns and condiional variance are in he same unis. 6

7 For many volailiy models his declines geomerically bu may be imporan even a year in he fuure. A closely relaed measure is he cumulaive persisence, which is he impac of a reurn shock on he average variance of he asse reurn over he period from o +k. I is defined as (13) 1 ( ( h + h h )) k + k + k φ + k = = ( θ 1... k + k + θ + k + + θ r + 1 The response of long-erm opion prices o volailiy shocks suggess ha volailiy models should have significan cumulaive persisence a year in he fuure. ) A furher measure of he persisence in a volailiy model is he half-life of volailiy. This is defined as he ime aken for he volailiy o move halfway back owards is uncondiional mean following a deviaion from i. (14) 1 τ = k : h + k σ = h + 1 σ. Volailiy is mean revering Volailiy clusering implies ha volailiy comes and goes. Thus a period of high volailiy will evenually give way o more normal volailiy and similarly, a period of low volailiy will be followed by a rise. Mean reversion in volailiy is generally inerpreed as meaning ha here is a normal level of volailiy o which volailiy will evenually reurn. Very long run forecass of volailiy should all converge o his same normal level of volailiy, no maer when hey are made. While mos praciioners believe his is a characerisic of volailiy, hey migh differ on he normal level of volailiy and wheher i is consan over all ime and insiuional changes. 7

8 More precisely, mean reversion in volailiy implies ha curren informaion has no effec on he long run forecas. Hence (15) plim θ 0, for all. k + k = which is more commonly expressed as (16) plim h = σ <, for all. k + k even hough hey are no quie equivalen. I is possible o generalize he concep of mean reversion o cover processes wihou finie variance. Consider some oher saisic such as he inerquarile range or he 5% quanile and call i q. The same definiions in (1), (15) and (16) can be used o describe persisence and mean reversion. The cumulaive versions however ypically do no have he same simple form as (13), see for example he CAViaR model of Engle and Manganelli (1999). Opions prices are generally viewed as consisen wih mean reversion. Under simple assumpions on opion pricing, he implied volailiies of long mauriy opions are less volaile han hose of shor mauriy opions. They usually are closer o he long run average volailiy of he asse han shor mauriy opions.. Innovaions may have an asymmeric impac on volailiy Many proposed volailiy models impose he assumpion ha he condiional volailiy of he asse is affeced symmerically by posiive and negaive innovaions. The 8

9 GARCH(1,1) model, for example, allows he variance o be affeced only by he square of he lagged innovaion; compleely disregarding he sign of ha innovaion. For equiy reurns i is paricularly unlikely ha posiive and negaive shocks have he same impac on he volailiy. This asymmery is someimes ascribed o a leverage effec and someimes o a risk premium effec. In he former heory, as he price of a sock falls, is deb-o-equiy raio rises, increasing he volailiy of reurns o equiy holders. In he laer sory, news of increasing volailiy reduces he demand for a sock because of risk aversion. The consequen decline in sock value is followed by he increased volailiy as forecas by he news. Black (1976), Chrisie (198), Nelson (1991), Glosen e al. (1993) and Engle and Ng (1993) all find evidence of volailiy being negaively relaed o equiy reurns. In general, such evidence has no been found for exchange raes. For ineres raes a similar asymmery arises from he boundary of zero ineres raes. When raes fall, (prices increase) hey become less volaile in many models and in mos empirical esimaes, see Engle Ng and Rohschild, Chan e al. (199) and Brenner e al. (1996). In diffusion models wih sochasic volailiy, his phenomenon is associaed wih correlaion beween he shock o reurns and he shock o volailiy. The asymmeric srucure of volailiy generaes skewed disribuions of forecas prices and under simple derivaive pricing assumpions, his gives opion implied volailiy surfaces which have a skew. Tha is, he implied volailiies of ou-of-he-money pu 9

10 opions are higher han hose of a-he-money opions, which in urn are higher han he implieds of in-he-money pus..3 Exogenous variables may influence volailiy Mos of he volailiy characerisics oulined above have been univariae; relaing he volailiy of he series o only informaion conained in ha series hisory. Of course, noone believes ha financial asse prices evolve independenly of he marke around hem, and so we expec ha oher variables may conain relevan informaion for he volailiy of a series. Such evidence has been found by, iner alia, Bollerslev and Melvin (1994), Engle and Mezrich (1996), Engle, Io and Lin (1990) and Engle, Ng and Rohschild (1990). In addiion o oher asses having an impac on he volailiy series, i is possible ha deerminisic evens also have an impac. Such hings as scheduled company announcemens, macroeconomic announcemens and even deerminisic ime-of-day effecs may all have an influence on he volailiy process. Andersen and Bollerslev (1998a), for example, find ha he volailiy of he deusche mark dollar exchange rae increases markedly around he ime of he announcemen of U.S. macroeconomic daa, such as he Employmen Repor, he Producer Price Index or he quarerly GDP. Glosen e al. (1993) find ha indicaor variables for Ocober and January assis in explaining some of he dynamics of he condiional volailiy of equiy reurns. 10

11 .4 Tail Probabiliies I is well esablished ha he uncondiional disribuion of asse reurns has heavy ails. Typical kurosis esimaes range from 4 o 50 indicaing very exreme non-normaliy. This is a feaure ha should be incorporaed in any volailiy model. The relaion beween he condiional densiy of reurns and he uncondiional densiy parially reveals he source of he heavy ails. If he condiional densiy is Gaussian, hen he uncondiional densiy will have excess kurosis due simply o he mixure of Gaussian densiies wih differen volailiies. However here is no reason o assume ha he condiional densiy iself is Gaussian, and many volailiy models assume ha he condiional densiy is iself fa ailed, generaing sill greaer kurosis in he uncondiional densiy. Depending on he dependence srucure of he volailiy process, he reurns may sill saisfy sandard exreme value heorems..5 Forecas Evaluaion Esablishing he effeciveness of a volailiy forecas is no sraighforward since volailiy iself is no observed. The mehod mos consisen wih he esimaed models is simply o ake each reurn divided by is one-sep ahead forecas sandard deviaion and hen apply any ype of es o see if he square of his variable is predicable. An alernaive ype of es is o examine he forecas accuracy of he model in predicing realized volailiy, fuure values of sample variances. For a one period problem, his amouns o regressing squared reurns on a consan and he condiional variance. The es is wheher he inercep is zero and he slope is one. Various forecass can be enered ino his equaion o deermine which is he bes. 11

12 (17) r = a + bh + u This approach is no recommended for several reasons. Because r is heeroskedasic, r will be much more heeroskedasic; hence his regression will be very inefficien and will have misleading sandard errors. Robus sandard errors should be used, however hese may no make an adequae adjusmen. Correcing for he heeroskedasiciy would involve dividing boh sides by h, leading simply o he original approach. A second drawback is ha r is a noisy esimae of he volailiy o be esimaed. Hence he maximum R ha can be achieved by his regression, if all is perfecly correc, is very low. To improve his, invesigaors may use volailiy measured over longer periods such as weekly or monhly realized volailiies. When non-overlapping periods are used, he sample becomes much smaller, and when overlapping daa are used, he sandard errors become far more problemaic. See for example Sock and Richardson (1989). Andersen and Bollerslev (1998b) proposed using a measure of realized volailiy based on observaions wihin he period. For forecasing daily volailiy, hey used 5 minue daa o consruc a daily volailiy. This improves he efficiency of his regression grealy. There is however a limi as high frequency daa has los of poenial pifalls due o bid ask bounce and irregular spacing of he price quoes. A hird drawback o his approach is ha i measures he level of variance errors raher han he more realisic proporional errors. This crierion will assess primarily he performance for high volailiies. A soluion migh be o ake logs of he realized volailiy and is forecas. For more discussion see Bollerslev e al. (1994). 1

13 3. AN EMPIRICAL EXAMPLE To illusrae he above poins, we now presen a concree example. We use daily close price daa on he Dow Jones Indusrials index, over he period 3 Augus, 1988 o Augus, 000, represening 3,131 observaions 1. The Dow Jones Indusrials index is comprised of 30 indusrial companies socks, and represen abou a fifh of he oal value of he U.S. sock marke. We ake he log-difference of he value of he index, so as o conver he daa ino coninuously compounded reurns. Figures 3.1 and 3. plo he price level and he reurns on he index over he sample period. 3.1 Summary of he daa Some summary saisics on he daa are presened in Table 3.1 below. As his able shows, he index had a small posiive average reurn of abou one-wenieh of a percen per day. The daily variance was 0.854, implying an average annualized volailiy of 14.4%. The skewness coefficien indicaes ha he reurns disribuion is subsanially negaively skewed; a common feaure of equiy reurns. Finally, he kurosis coefficien, which is a measure of he hickness of he ails of he disribuion, is very high. A Gaussian disribuion has kurosis of 3, implying ha he assumpion of Gaussianiy for he disribuion of reurns is dubious for his series. 1 These daa in ASCII forma are available from he second auhor s web sie a hp:// The Jarque-Bera es for normaliy of he reurns disribuion yields a saisic of , much greaer han any criical value a convenional confidence levels, hus rejecing he null hypohesis of normally disribued reurns. 13

14 TABLE 3.1: DOW JONES INDUSTRIAL INDEX RETURNS SUMMARY STATISTICS. Mean Variance Skewness Kurosis An analysis of he correlogram of he reurns, presened in Figure 3.3, indicaes only weak dependence in he mean of he series, and so for he remainder of he paper we will assume a consan condiional mean. The correlogram of he squared reurns, however, indicaes subsanial dependence in he volailiy of reurns. 3. A volailiy model A widely used class of models for he condiional volailiy is he auoregressive condiionally heeroskedasic class of models inroduced by Engle (198), and exended by Bollerslev (1986), Engle e al (1987), Nelson (1991), Glosen e al (1993), amongs many ohers. See Bollerslev e al. (199) or Bollerslev e al. (1994) for summaries of his family of models. A popular member of he ARCH class of models is he GARCH(p,q) model: (18) h = ω + αi ( R i µ ) + βjh p i= 1 q j= 1 j 14

15 This model can be esimaed via maximum likelihood once a disribuion for he innovaions, ε, has been specified. A commonly employed assumpion is ha he innovaions are Gaussian 3. Using he Schwarz Informaion Crierion we found ha he bes model in he GARCH(p,q) class for p [1,5] and q [1,] was a GARCH(1,1). The resuls for his model are presened below: TABLE 3.: RESULTS FROM THE GARCH(1,1) MODEL Coefficien Robus sandard error consan ω α β A es for wheher his volailiy model has adequaely capured all of he persisence in he variance of reurns is o look a he correlogram of he sandardized squared residuals. If he model is adequae, hen he sandardized squared residuals should be serially uncorrelaed. The Ljung-Box Q-saisic a he wenieh lag of he sandardized squared residuals was , indicaing ha he sandardized squared residuals are indeed serially uncorrelaed. 3 Bollerslev and Wooldridge (199) showed ha he maximum likelihood esimaes of he parameers of he GARCH model assuming Gaussian errors are consisen even if he rue disribuion of he innovaions is no Gaussian. The usual sandard errors of he esimaors are no consisen when he assumpion of Gaussianiy of he errors is violaed, so Bollerslev and Wooldridge supply a mehod for obaining consisen esimaes of hese. 15

16 3.3 Mean reversion and persisence in volailiy The resuls above indicae ha he volailiy of reurns is quie persisen, wih he sum of α and β being , implying a volailiy half-life of abou 73 days. Alhough he reurns volailiy appears o have quie long memory, i is sill mean revering: he sum of α and β is significanly less han one 4, implying ha alhough i akes a long ime, he volailiy process does reurn o is mean. The uncondiional mean of he GARCH(1,1) process is calculaed as he raio of ω o he difference beween 1 and he sum of α and β. For he Dow Jones over he sample period his urns ou o be 0.854, which implies ha he mean annualized volailiy over he sample was 14.67%; very close o he sample esimae of he uncondiional volailiy given in Table 3.1. A plo of he annualized condiional volailiy esimaes over he sample period is given in Figure 3.4. As described in secion.1, a measure of he persisence in volailiy is he parial derivaive of he overnigh reurn volailiy a ime +k wih respec o he squared reurn a ime, denoed θ+k,. A plo of θ+k, for k ranging from 1 o 100 is given in Figure 3.5. This plo shows ha he impac decays geomerically, and is essenially zero beyond one hundred days. The limi of his sequence is zero, confirming ha his volailiy process is mean revering. The equivalen measure for he volailiy of k-period reurns, denoed φ+k, in secion.1, also declines oward zero, hough a a slower rae, as equaion (13) suggess ha i should. 4 A one-sided -es ha he sum of alpha and bea is greaer han or equal o one yields a es saisic of.54, which is greaer (in absolue value) han he 5% criical value of

17 An alernaive way o observe he mean-revering behavior in h is in he srucure of long-erm forecass of volailiy. Figure 3.6 presens forecass a 3 Augus, 1995 and 3 Augus, 1997 of he annualized daily reurn volailiy ou o a year from each of hose daes. The firs of hese forecass was made a a dae wih unusually high volailiy, and so he forecass of volailiy decline gradually o he uncondiional variance level. The second of hese forecass was made during a ranquil period, and so he sequence of forecass is increasing oward he uncondiional volailiy level. One way o examine he volailiy of volailiy, VoV, is o plo he one period ahead volailiy and he k-periods ahead forecas volailiy. In Figure 3.7 we presen hese forecass for he one day, one quarer, one year, and wo year cumulaive forecass. I is immediaely apparen ha he movemens in he one day horizon are larger han he movemens in he year horizon. The inermediae horizons lie beween. This is an implicaion of he mean reversion in volailiy. The annualized esimaes of he volailiy of volailiy for hese forecass are given below. TABLE 3.3: VOLATILITY OF VOLATILITY FOR VARIOUS FORECAST HORIZONS FROM GARCH(1,1) One Day One Quarer One Year Two Years Sd. Dev An asymmeric volailiy model As menioned in he previous secion, he sign of he innovaion may influence he volailiy in addiion o is magniude. There are a number of ways of parameerising 17

18 his idea, one of which is he Threshold GARCH (or TARCH) model. This model was proposed by Glosen e al. (1993) and Zakoian (1994) and was moivaed by he EGARCH model of Nelson (1991). p (19) h = ω + αi ( R i µ ) + βjh j + δ kλk ( R k µ ) i= 1 q j= 1 where δ-k is an indicaor variable, aking he value one if he residual a ime -k was negaive, and zero elsewhere. r k= 1 The TARCH model implies ha a posiive innovaion a ime has an impac on he volailiy a ime +1 equal o α imes he residual squared, while a negaive innovaion has impac equal o (α+γ) imes he residual squared. The presence of he leverage effec would imply ha he coefficien γ is posiive; ha is, ha a negaive innovaion has a greaer impac han a posiive innovaion. We esimaed he TARCH(1,1,1) model, and presen he resuls in Table 3.3 below. These resuls indicae ha he sign of he innovaion has a significan influence on he volailiy of reurns. The coefficien on negaive residuals squared is large and significan, and implies ha a negaive innovaion a ime increases he volailiy a ime +1 by over four imes as much as a posiive innovaion of he same magniude. 18

19 Table 3.4: Resuls from he TARCH(1,1,1) model Coefficien Robus sandard error consan ω α γ β A model wih exogenous volailiy regressors I may also be of ineres o gauge he impac of exogenous variables on he volailiy process. This ype of model could offer a srucural or economic explanaion for volailiy. Such a model may be wrien as: p (0) h = ω + αi ( R i µ ) + βjh j + ϕ. X 1 i= 1 q j= 1 As an example, we used he lagged level of he 3 monh U.S. Treasury bill rae as an exogenous regressor in our model of Dow Jones Indusrials index reurns volailiy. The T-bill rae is correlaed wih he cos of borrowing o firms, and hus may carry informaion ha is relevan o he volailiy of he Dow Jones Indusrials index. TABLE 3.5: RESULTS FROM THE GARCH(1,1)-X MODEL Coefficien Robus sandard error Consan ω α β ϕ As he reader can see, he impac of he T-bill rae on he volailiy process of he Dow Jones Indusrials is small, bu quie significan. The posiive sign on his coefficien 19

20 indicaes ha high ineres raes are generally associaed wih higher levels of equiy reurn volailiy. This resul confirms ha of Glosen e al. (1993) who also find ha he Treasury bill rae is posiively relaed o equiy reurn volailiy. 3.5 Aggregaion of volailiy models Despie he success of GARCH models in capuring he salien feaures of condiional volailiy, i has some undesirable characerisics. Mos noably, he heoreical observaion ha if a GARCH model is correcly specified for one frequency of daa, hen i will be misspecified for daa wih differen ime scales, makes a researcher uneasy. Similarly, if asses follow a GARCH model, hen porfolios do no exacly do so. Below, we presen some evidence of his for our example daa se. We consider he esimaion of he simple GARCH(1,1) model on he daa, sampled a various frequencies. The resuls are presened in Table 3.6. TABLE 3.6: GARCH(1,1) PARAMETER ESTIMATES FOR DATA OF DIFFERING FREQUENCIES Daily daa -Day 3-Day 4-Day period period period Weekly daa Consan ω α β These resuls indicae ha he sampling frequency does indeed affec he resuls obained. As an example, he implied half-life of volailiy implied by each of he models (in days) is 73, 168, 183, 508 and 365. Clearly hese are subsanial differences alhough 0

21 he saisical and forecas significance of hese differences should be assessed. To some exen, he inerpreaion of hese models wih aggregae daa is slighly differen. Ideas such as he weak GARCH specificaion of Dros and Nijman (1993) may represen an alernaive soluion. However, he empirical esimaes on differen ime scales or porfolios are ypically reasonable, suggesing ha GARCH can be inerpreed as an approximaion or filer raher han a full saisical specificaion. Seps in his direcion are developed by Nelson and Foser (1994). 4. CONCLUSIONS AND CHALLENGES FOR FUTURE RESEARCH The goal of his paper has been o characerize a good volailiy model by is abiliy o forecas and capure he commonly held sylised facs abou condiional volailiy. The sylised facs include such hings as he persisence in volailiy, is mean-revering behaviour, he asymmeric impac of negaive versus posiive reurn innovaions and he possibiliy ha exogenous or pre-deermined variables may have a significan influence on volailiy. We used welve years of daily daa on he Dow Jones Indusrials index o illusrae hese sylised facs, and he abiliy of models from he GARCH family o capure hese characerisics. The condiional volailiy of he Dow Jones Indusrials index was found o be quie persisen, wih a volailiy half-life of abou 73 days, ye ess for nonsaionariy indicaed ha i is mean revering. A negaive lagged reurn innovaion was found o have an impac on condiional variance roughly four imes as large as a 1

22 posiive reurn innovaion, and he 3-monh U.S. Treasury bill rae was found o be posiively correlaed wih volailiy, implying ha higher ineres raes lead o higher equiy reurn volailiy. Finally, we found evidence consisen wih he heoreical resul ha he empirical resuls obained are dependen on he sampling frequency a drawback of he GARCH specificaion. Various aspecs of he volailiy process are imporan opics of research. The need for a model o forecas 100 or even 1000 seps ino he fuure has suggesed long memory or fracionally inegraed processes. In spie of subsanial research inpu, he value for hese forecas siuaions has no ye been esablished. Shifs in he volailiy process are someimes hough o be discree evens; only he Hamilon and Susmel (1994) model and is exension by Gray (1996) have been developed for his ask. Time varying higher condiional momens are clearly of ineres bu have proven difficul o esimae. Hansen (1994) and more recenly Harvey and Sidiqui (1999) have had some success.

23 5. BIBLIOGRAPHY Andersen, Torben G., and Bollerslev, Tim, 1998a, Deusche Mark Dollar Volailiy: Inraday Aciviy Paerns, Macroeconomic Announcemens, and Longer Run Dependencies, Journal of Finance, 53(1), Andersen, Torben G., and Bollerslev Tim, 1998b, Answering he Skepics: Yes, Sandard Volailiy Models Do Provide Accurae Forecass, Inernaional Economic Review, 39, Andersen, Torben G., Bollerslev, Tim, Diebold, Francis X. and Labys, Paul, 1999, The Disribuion of Exchange Rae Volailiy, Wharon Financial Insiuions Cener Working Paper and NBER Working Paper Baillie, Richard T., Bollerslev, Tim and Mikkelsen, Hans Ole, 1996, Fracionally Inegraed Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 74(1), Black, Fischer, 1976, Sudies of Sock Marke Volailiy Changes, Proceedings of he 1976 Meeings of he American Saisical Associaion, Business and Economic Saisics Secion, Bollerslev, Tim, 1986, Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 31, Bollerlsev, Tim, Chou, Ray Y. and Kroner, Kenneh F., 199, ARCH Modeling in Finance: A Review of he Theory and Empirical Evidence, Journal of Economerics, 5, Bollerslev, Tim, Engle, Rober F., and Nelson, Daneil B., 1994, ARCH Models, in he Handbook of Economerics, Vol IV, ed. R. F. Engle and D. McFadden, Amserdam: Norh Holland, Bollerslev, Tim and Melvin, Michael, 1994, Bid-Ask Spreads and he Volailiy in he Foreign Exchange Marke: An Empirical Analysis, Journal of Inernaional Economics, 36, Bollerslev, Tim and Wooldridge, Jeffrey M., 199, Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time-Varying Covariances, Economeric Reviews, 11(), Brenner, Robin J., Harjes, Richard H., and Kroner, Kenneh F., 1996, Anoher Look a Models of he Shor-Term Ineres Rae, Journal of Financial and Quaniaive Analysis, 31(1),

24 Chan, K.C., Karolyi, G. Andrew, Longsaff, Francis A., and Sanders, Anhony B., 199, An Empirical Comparison of Alernaive Models of he Shor-Term Ineres Rae, Journal of Finance, 47(3), Chou, Ray Yeuien, 1988, Volailiy Persisence and Sock Valuaions: Some Empirical Evidence using GARCH, Journal of Applied Economerics, 3, Chrisie, Andrew A., 198, The Sochasic Behavior of Common Sock Variances: Value, Leverage and Ineres Rae Effecs, Journal of Financial Economics, 10, Dros, Feike C. and Nijman, Theo E., 1993, Temporal Aggregaion of GARCH Processes, Economerica, 61(4), Engle, Rober F., 198, Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50(4), Engle, Rober F., Io, Takaoshi, and Lin, Wen-Ling, 1990, Meeor Showers or Hea Waves? Heeroscedasic Inra-Daily Volailiy in he Foreign Exchange Marke, Economerica, 58(3), Engle, Rober F., Lilien, David, and Robins, Russell, 1987, Esimaion of Time Varying Risk Premia in he Term Srucure: he ARCH-M Model, Economerica, 55, Engle, Rober F., and Manganelli, Simone, 1999, CAViaR: Condiional Auoregressive Value a Risk by Regression Quaniles, Universiy of California, San Diego, Deparmen of Economics Working Paper Engle, Rober F., and Mezrich, Joseph, 1996, GARCH for Groups, RISK, 9(8), Engle, Rober F., and Ng, Vicor, 1993, Measuring and Tesing he Impac of News on Volailiy, Journal of Finance, 48, Engle, Rober F., Ng, V. K. and Rohschild, M., 1990, Asse Pricing wih a Facor-ARCH Covariance Srucure, Journal of Economerics, 45(), Fama, Eugene F., 1965, The Behavior of Sock-Marke Prices, Journal of Business, 38(1), Glosen, Lawrence R., Jagannahan, Ravi and Runkle, David E., 1993, On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurns on Socks, Journal of Finance, 48(5), Gray, Sephen F., 1996, Modeling he Condiional Disribuion of Ineres Raes as a Regime-Swiching Process, Journal of Financial Economics, 4(1), 7-6. Hamilon, James D., and Susmel, Raul, 1994, Auoregressive Condiional Heeroskedasiciy and Changes in Regime, Journal of Economerics, 64(1),

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26 FIGURE 3.1: THE DOW JONES INDUSTRIAL INDEX, 3 AUG, 1988 TO AUG, Price /3/88 7/4/90 6/3/9 5/4/94 4/3/96 3/4/98 //00 Dae FIGURE 3.: RETURNS ON THE DOW JONES INDUSTRIAL INDEX. 6 4 Percen Reurn /3/88 6/3/9 4/3/96 //00 Dae 6

27 FIGURE 3.3: CORRELOGRAMS OF RETURNS AND SQUARED RETURNS Correlaion Correlaion Lag Lag FIGURE 3.4: ESTIMATED CONDITIONAL VOLATILITY USING A GARCH(1,1) MODEL Annualised Volailiy /3/88 7/4/90 6/3/9 5/4/94 Dae 4/3/96 3/4/98 //00 7

28 FIGURE 3.5: THETA AND PHI FOR K RANGING FROM 1 TO THETA PHI 8 Marginal Impac Lag FIGURE 3.6: FORECASTS OF DAILY RETURN VOLATILITY USING THE GARCH(1,1) MODEL Forecas volailiy Forecas horizon in days AUG3_95 AUG3_97 MEAN_VOL 8

29 FIGURE 3.7 VOLATILITIES AT DIFFERENT HORIZONS FROM GARCH(1,1) /3/88 6/3/9 4/3/96 //00 HORIZOND HORIZONQ HORIZONY HORIZONY 9