Using Intraday Statistics for the Estimation of the Return Variance

Transcription

1 Joural o Face ad Ivestmet Aalyss, vol., o., 03, -3 ISSN: (prt verso, -0996(ole Scepress Ltd, 03 Us Itraday Statstcs or the Estmato o the Retur Varace Erhard Reschehoer Abstract Ths paper proposes ew estmators or the daly retur varace whch are based o commo traday statstcs (ope, hh, low, ad clos prces. These estmators utlze ormato cotaed products o absolute values o ucorrelated traday statstcs. A emprcal study o e compoets o the Dow Joes Idustral Averae rom to shows that they outperorm exst estmators over all stocks ad tme perods. JEL classcato umbers: 3, G Keywords: Rae-based estmato, Browa moto, Folded ormal dstrbuto Itroducto Lopez [7] showed that the daly squared retur s a ubased but extremely mprecse estmator or daly volatlty. Estmators based o traday statstcs are, o course, much more precse. Parkso [8] proposed a rae-based estmator ad Garma ad Klass [6] optmzed ths estmator by troduc wehts ad tak jot eects to accout. Ulke estmators obtaed rom squared hh-requecy traday returs (realzed volatlty; see, e.., [], these rae-based estmators are relatvely robust to varous types o traday patters (or a more detaled dscusso, see [9]. Ths paper proposes to mprove the optmzed estmator o Garma ad Klass [6] by clud products o absolute values o ucorrelated traday statstcs addto to the already cluded squares o traday statstcs ad products o correlated traday statstcs. The ew estmators are troduced the thrd subsecto o the ext secto. The rst Departmet o Statstcs ad Operatos Research, Uversty o Vea, Uverstätsstr. 5, A-00 Vea, Austra, Artcle Io: Receved : March 9, 03. Revsed : Aprl, 03. Publshed ole : May 5, 03

2 Erhard Reschehoer two subsectos dscuss several exst estmators. The estmators cosdered Subsecto. are ully speced whereas those cosdered Subsecto. deped o a ukow parameter. The latter subsecto also vestates the cosequeces o msspeccato o the ukow parameter. Secto 3 presets the results o a emprcal study whch compares the perormace o the deret estmators. Secto cocludes. Methods. Fully Speced Estmators or the Retur Varace Feller [5] derved the asymptotc desty ucto: ( r t 8 o the rae: k ( k k kr t R( t max ( W( s m ( W( s 0st 0st o a Weer process W(s, s0, the terval [0,t], where deotes the ormal desty ucto wth zero mea ad ut varace. It ollows that: k k k ( ( k (t E( R ( t k or k{,3,,5,...} ad k k k ( k (t E ( R ( t or k= [8], where: ( u j j u s the Rema zeta ucto ad: ( u j ( k j u (or tables o values o see []. I partcular, we have: ( 0, ( lo(, Thus, ( 6, ( E( R( t t, E ( R ( t lo( t,

3 Us Itraday Statstcs or the Estmato o the Retur Varace , 3 E( R ( t t ad E( R ( t 9 (3 t. Because lo stock prces are commoly modeled as a Browa moto wth ukow volatlty (ad drt, the momets E ( R, E ( R lo(, E ( R ( t, E ( R 9 (3, 3 o the rae R o the Browa moto B(s=W(s, s0, o the terval [0,] ca be used or the estmato o the varace o the retur over a ut tme perod as well as or the evaluato o the propertes o the respectve estmators. I the case o daly prces, s elble compared to ad ca thereore be dsrearded. Suppose that the ope, hh, low, ad clos prces o N trad days are avalable. Because volatlty chaes over tme, oly the last <N trad days are used or the estmato o the preset volatlty. Let (O,H,L,, =,...,, be the correspod sample o the lo traday statstcs. The most obvous mprovemet over the stadard estmator ˆ0 ( s obtaed smply by dvd each trad day a perod whe the market s closed ad a perod whe t s ope. Deot the correspod retur varaces by ad, respectvely, ad assum that, leads strahtorwardly to the estmator ˆ ˆ ˆ ( O ( O. ( A urther mprovemet s obtaed by utlz addtoal ormato whch s avalable or the secod perod. Assume or the momet that the tme terval [0,] represets oly the tme o the trad sesso, ot the whole hours trad day. The the applcato o the ormula or the secod ocetral momet o the rae o a Browa moto yelds estmators or ad, respectvely, whch are based o the traday raes R =H L, =,...,,.e., lo( R (3 ˆP [8] ad ˆ ˆ ˆ. ( R P A less obvous but possbly more ecet verso o ( was proposed by Garma ad Klass [6]. Ther estmator wll be dscussed more detal the ext subsecto. As poted out by Ya ad Zha [], the o drt assumpto may ot be approprate certa stuatos, e.., stro bull markets. A estmator or whch s depedet o s ve by RS H ( H L ( L ˆ, (5 where H H O, L L O, ad O [0, ]. However, the

4 Erhard Reschehoer composte estmator ˆ crs ˆ ˆ RS (6 aa depeds o, albet to a lesser deree tha the other estmators. Ya ad Zha [] thereore proposed the urther developmet ˆ YZ ( O ( ( ˆ O k k RS, (7 Where: O deotes the sample mea o O O, =,...,, whch s truly depedet o.. Not surprsly, ths depedece comes wth a prce. The estmator (7 caot be based o a sle day,.e., must be reater tha oe. Ya ad Zha [] recommeded to set the costat k to k order to mmze the varace. I the case o =0 (two weeks, they expected that the varace o the result estmator s typcally more tha seve tmes smaller tha that o the bechmark estmator (. Volatlty estmators based o traday statstcs are ote used or the evaluato o volatlty models such as ARH [] ad GARH [3] models. These models der the way they descrbe the depedece o today's varace o the volatlty the past. I s lare, the varace estmates produced by the estmators dscussed above mht closely resemble the codtoal varace estmates produced by some o the volatlty models. Oly the case =, a ar comparso o volatlty models s thereore possble. Estmators such as (7 are thereore ot uversally usable ad wll thereore ot be cosdered the rest o ths paper.. Estmators Deped o a Ukow Parameter The estmators o the prevous subsecto were derved uder dealz assumptos, e.., the ormalty ad seral ucorrelatedess o returs. More ecet estmators ca be derved uder addtoal assumptos. The key assumpto ths subsecto s that the parameter / s kow. Uder ths assumpto, Garma ad Klass [6] showed that eve the smple estmator ( O ( O ( ˆ (8 s already "two tmes better" tha the bechmark estmator. Ideed, the expected value o each term s ve by E( O E( ( ( O E ad the varace by ( ( O Var Var O ( ( Var Var(.

5 Us Itraday Statstcs or the Estmato o the Retur Varace 5 To urther mprove ececy, Garma ad Klass [6] proposed ther composte estmator: R a a cgk O lo( ˆ, (9 ad ther "best" estmator a GK O ˆ + a L H L H R ( (0 The rst s closely related to estmator ( ad the secod takes also to accout the jot eects betwee the deret traday statstcs. The varaces o the estmators (9 ad (0 are mmzed whe a=0.7 ad a=0., respectvely, ad these cases they are more tha sx tmes ad more tha eht tmes, respectvely, "better" tha the bechmark estmator (. The applcato o the estmators dscussed ths subsecto requres the speccato o the ukow parameter. learly, t caot smply be obtaed rom the physcal tme terval dur whch markets are closed. It must rather be estmated rom hstorcal data sets. Ya ad Zha [] dd just that ad oud values betwee 0.8 ad The value chose a real applcatos mht thereore der scatly rom the true value. To llustrate the possble cosequeces o such a msspeccato, the smple estmator (8 s used. I a deret value s used stead o, the bas o ths estmator s or each term ve by ( ( ( ( O O E, ts varace by ( ( ( ( ( ( ( O Var O Var, ad ts mea squared error (MSE by ( ( (. Fure.e shows that =0.5 s a sae choce. Ideed, the MSE s ths case always smaller tha that o the bechmark estmator (, because ( ( 0<<. Fures.a-c show the MSE as a ucto o or =0.3, =0.5, ad =0.7, respectvely. Not surprsly, the MSE s small wheever s close to, but t creases quckly as moves away rom the true value. Note that the choce = s optmal oly =0.5. Uder the plausble assumpto that 0.5, a choce o <0.5 would also be sae. The solutos o ( ( ( = are just the roots o the polyomal P(= +(8 3 (8 +5 +(8 +-3 =0.

6 Erhard Reschehoer I the worst case,.e., =0.5, there are our real roots, two o whch are the terval (0,, amely = ad =-= Fures.d- show the MSE as a ucto o or =, =0.5, ad =, respectvely. The MSE s or = always smaller tha that o the bechmark estmator <0.5, equalty holds =0.5. Fure shows or each caledar year rom 96 to 0 ad or each o e stocks the estmate o obtaed by dvd the sample varace o O O by the sample varace o. Smlar estmates were obtaed whe o-cetral secod momets were used or whe the sum o the secod momets o O O ad O was used as dvsor. For each stock, the estmates were well below 0.5 or most o the tme, hece choos = seems reasoable. I practce, t wll hardly make ay derece whether s used or smply 0.5 or 0.3. (a (d (b (e (c ( Fure : MSE o the estmator (8 as a ucto o or =0.3 (a, =0.5 (b, =0.7 (c ad as a ucto o or = (d, =0.5 (e, = (, respectvely

7 Us Itraday Statstcs or the Estmato o the Retur Varace Fure : Estmates o the parameter obtaed by dvd the sample varace o O by the sample varace o or the caledar years rom 96 to 0. Data: AA (red, BA (pk, AT (orae, DD (old, DIS (brow, GE (ree, HPQ (blue, IBM (ray, KO (black.3 Estmators us Addtoal Iormato I the case o kow, the estmators (9 ad (0 seem to be the methods o choce. The ma advatae o (9 s ts smplcty, whereas (0 uses more o the avalable ormato by tak also the jot eects betwee H, L, ad to accout. However, there are other product terms whch are possbly more mportat. Because o the approxmate ucorrelatedess o the actors, t may be justed to ore products such as O or O R. But ths s certaly ot true or products such as O or O R. The costructo o mproved estmators whch utlze the latter quattes s strahtorward. Uder the usual dealz assumptos, the dettes: E O E ( (

8 8 Erhard Reschehoer ad E O ER ( ( ca mmedately be derved just by ot that the expected value o the rae o a Browa moto s ve by / ad the expected value o the olded ormal dstrbuto by O ( /. Ths motvates the troducto o the smple estmators ˆ O ( ad OR ( ˆ O R, ( as well as o composte estmators such as: ˆ or co a ˆ GK ( a ˆ O ˆ cor a ˆ GK ( a ˆ OR (3. ( It s ot clear whether try to choose a order to mmze the varaces o (3 ad ( s a worthwhle exercse. Not oly s ukow, whch s aravated by the act that ths parameter s ether costat across all stocks or across all tme perods (see Fure, but the other multplcatve costats are also ucerta, because they have bee derved uder the urealstc assumpto o ormalty. Fally, the restrcto that the sum o the "wehts" equals oe s also questoable. I vew o the may ucertates, ths restrcto caot relably esure ubasedess. Moreover, accept a small bas s usually a eectve way to reduce the MSE. Bascally, there are two alteratves. The rst s to use equal wehts (.e., a=0.5 ad a "reasoable" value or the parameter (e.., =. The secod s to use hstorcal data to estmate all wehts occurr a estmator such as H ˆ ao br c O R. (5 3 Emprcal Results To assess the perormace o the volatlty estmators dscussed Secto, the daly ope, hh, low ad clos prces o those compoets o the Dow Joes Idustral Averae (DJI were dowloaded rom Yahoo!Face whch have the loest hstory. The selected stocks are Alcoa (AA, Boe (BA, aterpllar (AT, Du Pot (DD, Walt Dsey (DIS, Geeral Electrc (GE, Hewlett-Packard (HPQ, IBM (IBM, ad oca-ola (KO. Ther prces are avalable sce Jauary, 96. The sample perod eds o March 3, 03. Fure 3 shows the (lo absolute returs (a, the squared returs (b, ad the ourth powers o the returs (c. Obvously, ay statstcal erece based o the ourth powers would be dubous at best. The outcome would deped oly o a very small umber o extreme returs. The MSE o the compet estmators was thereore estmated a ostadard way. Statstcs o squared returs were averaed beore they were squared aa. More precsely, the varace V o a estmator (, whch uses oly ormato rom the th trad day, was estmated by V ˆ( N, where:

9 Us Itraday Statstcs or the Estmato o the Retur Varace 9 m ˆ( ( ( V m N m> ad V ˆ( 0. Note that (, ( are..d. N( B, V, the: ( ( ~ N(0,V ad E ( ( V The bas B was estmated by B ˆ( N, where: Bˆ ( m m ( ( N (ote that the mss value 0 was replaced by O. I eeral, a roll wdow s more approprate or bas estmato tha a expad wdow. However, sce all compet estmators use oly the latest ormato, the bas s maly caused by the use o wro wehts or the deret traday statstcs. There s o classcal bas-varace trade-o. Fure 3: Plots o (lo absolute returs (a, squared returs (b, ad ourth powers o returs (c. Data: AA (red, BA (pk, AT (orae, DD (old, DIS (brow, GE (ree, HPQ (blue, IBM (ray, KO (black

10 0 Erhard Reschehoer To detect possble chaes over tme, the creas sums Vˆ ( m Bˆ ( m, m=,...,n, were plotted aast tme. Fure shows that the stadard estmator ( ad the smplest estmator based o traday statstcs ( are cosstetly outperormed by the rae based estmator ( ad the drt-robust estmator (6. Further mprovemets ca be obtaed by us the more sophstcated estmators deped o the ukow parameter. Fure 5 shows that the choce o s ot crtcal. Ay value below 0.5 but ot too close to zero yelds a very compettve verso o the estmator (9. To avod ay suspco o data m, = wll be used the ollow because ths choce s based o theoretcal arumets oly. Fure 6 shows that ths case (0 s deed a mprovemet over (9. The perormace s creased jot eects betwee the deret traday statstcs are take to accout. However, us addto also O R leads to a urther mprovemet. The ew estmators ( (wth a=0.5 ad = ad (5 have the smallest MSE. For (5, a=0 was used. The rst term was excluded because t s ar less relable tha the other two. The wehts b=0.33 ad c=36 were oud rom hstorcal data. However, the daer o a data-m bas s relatvely small because the same wehts were used or all stocks ad all tme perods. Fure : omparso o deret retur varace estmators wth respect to the cumulatve MSE (volet: stadard estmator (, lhtree: smplest estmator based o traday statstcs (, red: rae based estmator (, old: drt-robust estmator (6. Data: (a AA, (b BA, (c AT, (d DD, (e DIS, ( GE, ( HPQ, (h IBM, ( KO

11 Us Itraday Statstcs or the Estmato o the Retur Varace Fure 5: omparso o the smple rae based retur varace estmator ( (red wth deret versos o the optmzed estmator (9 (=0.: darkree, =0., 0.3, 0.: ree, =0.5, 0.6., 0.7: lhtblue wth respect to the cumulatve MSE. Data: (a AA, (b BA, (c AT, (d DD, (e DIS, ( GE, ( HPQ, (h IBM, ( KO

12 Erhard Reschehoer Fure 6: omparso o deret retur varace estmators wth respect to the cumulatve MSE (red: rae based estmator (, cya: optmzed estmator (9, blue: mproved optmzed estmator (0, orae: composte estmator (, black: pramatc estmator (5. Data: (a AA, (b BA, (c AT, (d DD, (e DIS, ( GE, ( HPQ, (h IBM, ( KO

13 Us Itraday Statstcs or the Estmato o the Retur Varace 3 ocluso The emprcal study preseted Secto 3 showed that the estmators or the daly retur varaces proposed Secto outperorm the exst estmators cosstetly over all stocks ad all tme perods. The compet estmators were compared wth respect to the MSE whch s the usual way to optmze the trade-o betwee bas ad varace. The problem the case o acal applcatos s that the ourth powers o stock returs are extremely volatle. The varaces o the varace estmators were thereore ot obtaed drectly rom ourth powers but rather rom squared averaes o squares. The stablty o the results shows that ths approach s approprate. Thus, t seems that the ew estmators are deed the methods o choce or the estmato o the daly retur varaces based o ope, hh, low, ad clos prces. Reereces [] M. Abramowtz ad I. A. Steu, Hadbook o mathematcal uctos wth ormulas, raphs, ad mathematcal tables, Natoal Bureau o Stadards, 96. [] T. G. Aderse ad T. Bollerslev, Aswer the skeptcs: yes, stadard volatlty models do provde accurate orecasts, Iteratoal Ecoomc Revew, 39, (998, [3] T. Bollerslev, Geeralzed autoreressve codtoal heteroskedastcty, Joural o Ecoometrcs, 3, (986, [] R. F. Ele, Autoreressve codtoal heteroscedastcty wth estmates o the varace o Uted Kdom lato, Ecoometrca, 50, (98, [5] W. Feller, The asymptotc dstrbuto o the rae o sums o depedet radom varables, The Aals o Mathematcal Statstcs,, (95, 7-3. [6] M. B. Garma ad M. J. Klass, O the estmato o securty prce volatltes rom hstorcal data, The Joural o Busess", 53, (980, [7] J. A. Lopez, Evaluat the predctve accuracy o volatlty models, Joural o Forecast, 0, (00, [8] M. Parkso, The extreme value method or estmat the varace o the rate o retur, The Joural o Busess 53, (980, [9] S. H. Poo, A practcal ude to orecast acal market volatlty, Wley, 005. [0] L.. G. Roers ad S. E. Satchell, Estmat varace rom hh, low ad clos prces, Aals o Appled Probablty,, (99, 50. [] L.. G. Roers, S. E. Satchell ad Y. Yoo, Estmat the volatlty o stock prces: A comparso o methods that use hh ad low prces, Appled Facal Ecoomcs,, (99, 7. [] D. Ya ad Q. Zha, Drt-depedet volatlty estmato based o hh, low, ope, ad close prces, The Joural o Busess, 73, (000, 77-9.