Volatility Forecasting with High Frequency Data

Transcription

1 Voliliy Forecsing wi Hig Frequency D oungjun Jng Deprmen of Economics Snford Universiy Snford, CA 945 jyjglory@gmil.com Advisor: Professor Peer Hnsen y 7, 7 ABSRAC e dily voliliy is ypiclly unobserved bu cn be esimed using ig frequen ick-by-ick d. In is pper, we sudy e problem of forecsing e unobserved voliliy using ps vlues of mesured voliliy. Specificlly, we use dily esimes of voliliy bsed on ig frequency d, clled relized vrince, nd consruc e opiml liner forecs of fuure voliliy. Uilizing single exponenil smooing, we develop formule yield e opiml coefficiens for our forecs. We compre e precision of our forecs wi ose of wo populr forecsing models, e HAR regression model nd e ocl evel model, in erms of men squred errors. In empiricl nlysis of e seven DJIA socks, our model performs beer n wo compeing models in mos of e cses. Keywords: Relized Vrince, Hig-Frequency D, Insrumenl Vribles, ocl evel odel, HAR model, liner forecsing

2 . Inroducion In common erms, voliliy refers o e flucuions observed in some penomenon. In economic erms, i refers o e vribiliy of e rndom componen of ime series. Specificlly, in finncil economics, voliliy cn be defined s e sndrd deviion of e rndom Wiener-driven componen in coninuous-ime diffusion model (Andersen e l. 5.) esuring nd forecsing voliliy s wide rnge of pplicions. Poin forecsing, inervl forecsing, probbiliy forecsing, nd densiy forecsing re ll good exmples of sisicl pplicions. For finncil pplicions, risk mngemen, sse llocion wi ime-vrying covrinces, nd opion vluion wi dynmic voliliy re represenive ones (Andersen e l. 5.) In eir seminl pper Answering e Skepics: es, Sndrd Voliliy odels do Provide Accure Forecss, Andersen nd Bollerslev (998) sowed voliliy models produce good forecss for e len voliliy fcor wic is imporn for mny finncil pplicions. CAP (Cpil Asse Pricing odel) ssumes invesors re only concerned wi e re of reurn nd risk, wic is represened s sndrd deviion of reurns. However, e sndrd deviion of e reurn mig be oo simple o represen e risk fcing invesors. Reurn voliliy is primry inpu o opion pricing nd porfolio llocion problems. oreover, good forecss of voliliy re essenil o implemen nd evlue sse pricing, rding, nd edging eories (Andersen nd Bollerslev, 998.) Recenly, compliced meods suc s Vlue--Risk (VR) nd Expeced Sorfll (ES) uilize voliliy forecsing ecniques ve been used o mesure risks more ccurely.

3 Forecsing dily voliliy is cenrl for finncil risk mngemens. e Relized Vrince (RV), wic is sum of squred inrdy reurns, is e simples mesure of dily voliliy bsed on ig-frequen d. A drwbck of is esimor is i is sensiive o mrke microsrucure noise, suc s rounding errors in e inrdy prices. So e empiricl esimes of dily voliliy used in is pper re e kernel-bsed esimes of Brndorff-ielsen, Hnsen, unde, nd Seprd (7). ese re closely reled o e RV, bu robus o mrke microsrucure noise. For simpliciy, we will simply refer o ese esimes s e RV. Among severl meods, e HAR (Heerogeneous Auoregressive model of e Relized Voliliy) model nd e (ocl evel) model re widely used o forecs voliliy, since ey sow nice predicive performnce empiriclly nd cn be esily compued. In is pper, we will model e len voliliy wi n Auoregressive (AR) model. For simpliciy, we will minly focus on n AR() model, owever n AR() model wi uni roo will lso be used. e problem we seek o solve is o deermine fuure voliliy expressed s liner combinion of previous dys RV, including ody s RV. Secion summrizes e reled reserc. In Secion, we deil our noion nd sudy severl models. In Secion 4, seven equiies of Dow Jones Indusril Averge (DJIA) d re exrced from rde nd Quoe (AQ) dbse, smpled from o 4, will be used o deermine opiml coefficiens for models. e precision of forecs will be mesured by e vlue of e men squred error (SE) is clculed by compring e dily forecss in 5 wi e observed voliliies, RVs. In Secion 5, we conclude wi suggesions for fuure reserc nd discussion of our resuls.

4 . ierure Review Since e join disribuionl crcerisics of sse reurns re essenil o evlue e risk-reurn rdeoff, ey re primry fcors o price finncil insrumens. oreover, ey re imporn o mnge risks s ey re reled o e condiionl porfolio reurn disribuions wic decide e likeliood of exreme sifs in porfolio vlue (Andersen e l. ). e second momen srucure of e condiionl disribuions is e mjor elemen of ime-vrying crcerisics of e disribuion. us, mny resercers ve invesiged forecsing nd modeling reurn voliliy in dep (Andersen e l. ). Wi e developmen of compuer nd nework sysems, d cn be colleced for sorer ime orizons. is improvemen in e vilbiliy of d cnged e focus from qurerly or monly modeling o weekly or dily modeling. Furer, inrdy or ick-byick d nlyses ve lso become possible. Voliliy forecsing is especilly effecive on ig-frequency bsis, suc s ourly or dily (Crisoffersen nd Diebold, ). However, e sndrd voliliy models used o forecs e dily level cnno ccommode inrdy informion in deil, nd lso models developed for inrdy level cnno explin dily d sufficienly well (Andersen e l. ). is discrepncy is cused by e presence of mrke microsrucure noise in ig-frequency finncil d. rke microsrucure noise mkes sndrd esimors suc s RV unrelible, since i induces uocorrelion in e inrdy reurns wic mkes esimors bised (Hnsen nd unde, 6). In response o is problem, Brndorff- ielsen e l. (7) develop e clss of relized kernel esimors of qudric vriion. As lredy menioned in e inroducion, in is pper, we will use ese relized kernel-

5 bsed esimes nd cll ese esimes s RV for simpliciy. Corsi presens simple AR-ype model of e relized voliliy considers voliliies relized over differen ime orizons (4). He erms is model Heerogeneous Auoregressive model of e Relized Voliliy (HAR). e HAR model is bsed on e HARCH model of uller e l. (997) wic ws inspired by Heerogeneous rke Hypoesis nd e symmeric propgion of voliliy beween vrious ime orizons. Using yers (from December 89 o July ) of ick-by-ick logrimic middle prices of USD/CHF F res, Corsi finds is simple HAR model successfully reproduces some of e min empiricl crcerisics of e finncil d: f ils, long memory in e voliliy, nd disribuionl properies of relized voliliy (4). Beginning wi e HAR model, we will use e nex secion o discuss eoreicl resuls of severl models, including e model nd n AR() model.

6 . eoreicl Frmework Firs of ll, we disinguis cul voliliy from observed voliliy. Acul voliliy is len vrible, wile observed voliliy is noisy esimor of cul voliliy. For exmple, e dily inegred vrince is cul voliliy nd e relized vrince, RV, is observed voliliy (Hnsen nd unde, 6). Suppose prices re given by dp σ dw were is sndrd Brownin W process, p is e logrim of insnneous price, nd σ is e ime-vrying voliliy. Inegred vrince ssocied wi dy is defined by e inegrl of e insnneous voliliy over e ime inervl, σ s ds. e sndrd definiion of e RV ssocied wi dy, is j RV r jδ, were d Δ nd r jδ inrdy reurns smpled ime inervl Δ (Corsi, 4). Brndorff-ielsen e l. (7) define ( jδ ( j ) δ )( ( j ) δ ( j ) γ ( δ ) ) δ,,,,,,,, for ny process. In is pper, we define e relized kernel H ( δ K( δ ) w γ ) H nd cll is s RV for simpliciy. w re non-rndom weigs ere. See Brndorff-ielsen e l. (7) for deils. We use noion o denoe log (RV relized vrince observed voliliy) ime, nd o denoe log (Inegred vrince cul voliliy) ime. e difference beween cul voliliy nd observed voliliy cn be represened s mesuremen error, were { } re iid wie noise re no correled wi nd

7 . emiclly,. We wn o find opiml liner forecs of given,. We define e opimliy o minimize ( ) Since ˆ E, E ˆ were ˆ f (,, ) w i [ ] E ( ) ( ˆ ) E( ) { ( ˆ E ) } ( ˆ ) i i. E σ E ( ˆ ) nd σ is consn, minimizing E ( ˆ is sme s minimizing. ) E ( ˆ ) As E ( ˆ ) is esier o compue, we will focus on minimizing is erm from now on. I. HAR odel Following Corsi (4), we define weekly nd monly RV ime 5 ( w) Weekly RV ime RV ( 5 ) log( RV i ) 5 5 i s ( m) only RV ime RV ( ) log( RV i ) oe one week s five rding dys nd one mon s rding dys on verge. (w) (m) Corsi (4) proposes e HAR model esimes from, RV, RV, by i regressing on, 5 5 i i, i, nd. i In sor, 5 5 ˆ ( d ) (w) (m) w w i w i i i (c) w () (c) ( w is consn erm.)

8 II. ocl evel odel ocl evel () model srs wi nd, were { } nd { } re iid wie noise nd independen of ec oer. Since s, s vr σ O O σ I ( A) We wn o find e opiml liner combinion of opiml w ( w, w ) o minimize vr ˆ ( ω) ι (,, ) ( ), ˆ ω w, w, ( ) * grnge muliplier, we derive ( ),, w ( w, w ) ; our gol is o coose w' Aw suc w ' ι were. Using e meod of w A ι ι' A ι from e firs order condiions. oe * w (,,,) wen, s sould be e cse in e bsence of mesuremen errors. σ σ A Define nd A σ σ * opiml weigs given by ( ) eoreiclly, for model, w A ι ι A ι. O O I. us, we find e corr ( Δ, Δ ) (Hrvey, 99) ( ). We cn esime ˆ by compuing corr Δ, Δ ) from ~4 d. For (

9 insnce, G (Generl oors) sock s correlion -.47 nd.96. Afer clculing opiml weigs, we esime ˆ w ˆ ( ) e number of previous dys we use o esime. For consisency wi e HAR model, is used in is pper. eoreiclly, s, { } w forms geomeric sequence. In priculr, were *, w ) ( p p p ( ) 4 (Hnsen, 7) III. Sionry AR() ore generlly, for (were < ) we ge A σ A O O. 4 O erefore, we cn compue opiml weigs if we know, or esime, vlues of nd. For model,. Afer clculing opiml weigs, we esime. w ˆ. AR() wi uni roos For AR() cse, we cn find opiml weigs similrly. oe is procedure does no require uni roo ssumpion., for AR(). By iering, we ge

10 b b b Vr b b b Vr c c c c c c c c c O σ d e e d e d O O O O σ were ), (, i i i k i k k i i k k i b e b d c for. is is specil cse of AR(p). Generl cses re nlyzed in Hnsen nd unde (6). V. Esimion wi Insrumenl Vrible Z We use simple AR() model; nd μ ( < ) nd use s n insrumenl vrible (Since pplying gives beer esimes, we use i insed of.) en Ordinry es Squres esimor is Z ( )( ) ( ) ˆ S nd Insrumenl Vrible esimor is () 5 ( )( ) ( )( ) ˆ ( ) 6 ˆS ) vr( ), cov( ( ) ( ) vr, cov σ σ σ σ σ σ ) ( ( S S ) ˆ lim nd ( ) ˆ lim. en we ge S. ( ) 7 Single exponenil smooing is very populr sceme o produce smooed ime

11 series ssigns exponenilly decresing weigs on ps observions. e be n observion nd be smooed vlue for ime. Sring from s, we recursively s compue s from e bsic equion of exponenil smooing, s ( β ) s β for (e expnded equion is ( ) ( ) i observions re given relively more weig in forecsing n e older observions re. i s β β β ). oe recen β is clled e smooing consn; is e smooing consn in our model. Uilizing single exponenil smooing, we subsiue μ wi ( )δ. Insed of i δ ( s in e bsic equion), we se δ in our model. Inuiively, δ is n verge of ll ps vlues, insed of single ps observion or single smooed vlue. We ve μ ( ) δ ( δ ) δ ) (. Since we ve esimes of nd, we cn cieve opiml weigs * w ( w ), w. ( ) δ en, ˆ w ( δ ) w w δ. erefore, ˆ w ( 8) w δ oe w, by e definiion of e exponenil smooing. Iniilly, we se δ. Here, δ represens n verge of ll previous vlues vilble. e, i is plusible o define δ s n verge of e recen vlues. For exmple, we cn ke n verge of ps mon (δ were ), ps qurer ( δ 5 ), or ps yer (δ 5 ). We will exmine ll suggesed

12 δ o find wic δ provides bes esimes, i.e., esimes give minimum men squred error (SE). S Insed of esiming from, we cn esime nd by e predicion bsed crierion (Hnsen nd unde, 6). We define Q(,) ( ˆ ), were ˆ w nd e ls dy of e vilble d, December, 4 in is cse. en Q is smple SE funcion; we esime ( ˆ,, ˆ ) Q rg min Q(,, ) ( 9). We lso es is new se of esimors ( Q, ˆ, ); keep, bu use Q, ˆ insed of. For comprison, we lso pply is meod o e model nd esime ˆ Q, insed of from (), ˆ corr ( Δ, Δ ). VI. Esimion of AR() For is secion, we ssume. As is uni roo condiion will elimine e consn in e model, we cn use Q(,) ( ˆ ) were ˆ w s we did in e previous secion (Wiou e uni roo condiion, e formul for Ŷ becomes ˆ w g( ˆ) μ.) We re especilly ineresed in e vlues of ; if is close o zero, i is likely pplying n AR() model by dding n ddiionl erm from n AR() model does no improve forecsing significnly.

13 VII. Possible Exension: uliple Insrumenl Vribles We sr from n AR() model, nd μ. We used Z in pr V s n insrumenl vrible. In is secion, we consider muliple insrumenl vribles,,, k. We cn esime by using e wo sge, les squres (SS) esimor. e Z,,,, ). ( k μ μ ( ) () For e firs-sge regression of SS, we regress on Z. We ge Π Z v, were Π ( π π, π ).,, k By OS regression, ˆ Π Z Z ( Z ). We denoe ˆ s e prediced vlue from is regression. Afer replcing wi in ˆ, we run e secondsge regression of SS. Finlly, we ge n esime of. ˆ SS ( ˆ ˆ )( ) ( ˆ ˆ ) ( ) By uilizing more insrumenl vribles, we cn improve e precision of e resuling esimor. For is pper, we do no ke dvnge of is pproc o esimion.

14 4. Empiricl Anlysis We nlyze empiricl d of sock reurns for e 7 equiies of DJIA. Since ick size ws reduced from /6 of dollr o cen on Jnury 9,, we coose e smple period from Februry, o December, 5. e d were exrced from rde nd Quoe (AQ) dbse nd dys wi less n 5 ours of rding were removed from e smple. o es models, we esime coefficiens for (), (), (8) using d from Februry, o December, 4. Afer esiming coefficiens, we forecs yer 5 wi ree models nd compue SE wi e cul yer 5 d. oe δ (8) is e only vrible cn be cnged during e forecsing procedures. I. Esiming coefficiens We use Alco (AA) cse for n exmple. For HAR model, coefficiens of () re given by e OS regression. ˆ.4 Since i. 7 5 i i. 97, from (). i corr ( Δ, Δ ).485 from (), ˆ Afer clculing wi our formule, we ge ˆ.6 from (9), ˆ.65 from (5), Q, ˆ.96 from (6),. 6 from (7), 9.86 from (9). e big difference ˆ Q, S beween ˆ S nd ˆ indices e use of e insrumenl vrible is essenil; is muc more likely o be correled wi e noise ime n is. From compued nd, we compue opiml weigs for e model nd e model.

15 Grp summrizes is resul. Grp. Opiml weigs of AA (Alco) for ree models Opiml Weigs for AA coefficiens.45 HAR lgged dys e grp sows for ll models, coefficiens decy exponenilly s e number of lgged periods increses. Generlly speking, recen d couns muc more. For e model, eoreiclly clculed weigs using (4) coincide wi e derived opiml weigs. For e model, e rio ( w i w i /.59) is consisen by single exponenil smooing. sly, sum of weigs is for e model, wile < for e model due o e δ erm.

16 II. SE comprisons For convenience, le () is e model ˆ w w δ wi δ (ll ps d). Similrly, e (m) model uses δ were (ps mon), e (q) model uses δ δ 5 5 (ps yer). (ps qurer), nd (y) uses en Squred Error (SE) of AA re sown ble. Beer resuls (smller SEs) re underlined. ble. en Squred Errors of AA (Alco) HAR () (m) (q) (y) ˆ or Q, ˆ or Q, ˆ From ble, we observe using ˆ or ˆ provides beer (smller) SE for mos of e cses. Also, (m) nd (q) sows beer resuls n (), nd (y) provides e bes resuls mong ll s. ese resuls old for ll seven socks excep G. Q, Q I is likely ere is cerin opiml leng of e period o clcule δ gives e bes esimes. SE for seven socks for ll forecsing models re sown ble. As before, underlined numbers re e smlles SEs.

17 ble. en Squred Errors of seven socks AA BA CA DIS GE G IB HAR Q, ˆ ˆ () Q, ˆ (m) Q, ˆ (q) Q, ˆ (y) Q, ˆ G is n unusul cse. SE of G is more n wice of oer socks SE. Excep G, (y) provides lower or similr SE compred o e or HAR model.

18 III. Forecss Grp plos cul yer 5 vlues of of Alco (AA) sock nd forecss from e model. Grp. Oer socks sow similr perns. Acul d nd model forecss for 5, AA (Alco) Acul d nd model forecss Acul D HAR (y) ime

19 Grp sows e rnge of is lrger for e G cse. Grp. Acul d nd model forecss for 5, G (Generl oors) Acul d nd model forecss Acul D HAR (y) ime

20 Grp sows e model s iger weigs on recen d n oer models ve; is fc explins wy e model s e bigges flucuions mong ree models in grp nd. On e oer nd, e model s e smlles flucuions due o e exisence of e δ erm. e δ erm, e verge of ps vlues (for cerin period; one yer is used in is cse), cs like buffer nd reduces vriions.., ˆ ˆ, ˆ Q, Q, ble., ˆ ˆ, vlues nd rios ˆ Q, Q, AA BA CA DIS GE G IB Averge Q, ˆ Rio ( Q, ˆ / ) ˆ Q, ˆ Rio ( ˆ / ) Q, ˆ Overll, Q, ˆ is bou 4.59 imes of nd ˆ is bou.6 imes of ˆ. Q, For e model, lmbds from e predicion bsed crierion, ˆ Q,, re similr o lmbds from e OS regression,. e, for e model, ˆ re more n 4 imes of ˆ Q, in generl. However, in bo models, rios re firly close o verges. is resul suggess ere re cerin relionsips beween lmbds derived from e OS regression nd

21 lmbds obined from e predicion bsed crierion. In is cse, ˆ (or ) could Q, Q, ˆ be esimed from ˆ (or ), if we need em since ey provide beer forecss. V. Vlues of from n AR() wi uni roos ble 4. Vlues of nd from AR() under e uni roo condiion AA BA CA DIS GE G IB ( ) (, for ll seven socks. e erm s coefficien very close o, ) zero. In oer words, e erm ppers o conribue lile for e predicion power. erefore, using n AR() model provides lmos eqully effecive forecsing resuls nd requires less compuions, compred o n AR() model under e uni roo condiion.

22 5. Summry nd Discussion As we ve seen in ble : en Squred Errors of seven socks, e (y) model provides beer resuls n oer models (including e HAR model nd e model) do, excep for e G cse. Also, from e sme ble, we observe (y) model works beer n (). is indices using only recen d o compue δ (e verge of ps d), insed of ll vilble ps d, gives beer esimes. One possible explnion could be e voliliy clusering in e empiricl d. However, (y) model lso works beer n (m) nd (q) even oug ese wo models use muc limied recen d (only ps mon or ps qurer, wic re sorer periods compred o e ps yer is used in e (y) model.) is comprison beween (y) nd oer models implies coosing proper leng of e ps period o ge δ resuls in beer esimes. Quesions including w re e proper lengs nd w fcors deermine em sould be nswered by fuure reserc. Anoer key poin of ble is using ˆ Q, provides beer esimes n using. Also, for e model, using ˆ works beer n using ˆ. ble sows Q, Q, ˆ is bou 4.59 imes of nd ˆ is bou.6 imes of ˆ. ese rios Q, re firly consisen mong socks; rio ˆ / vries from.57 o 5.47 nd Rio Q, ˆ Q, / ˆ vries from. o.5. I is possible ere re cerin eoreicl relionsips beween Q, ˆ nd or ˆ nd ˆ define rios. Q, Grps plo cul d nd esimes of AA nd G sow G sock s more volile RV. For G, e rnge of is - o 4, pproximely; for AA nd oer socks, e rnge of is - o, pproximely. Also, G s e lrges SE mong

23 seven socks; e SE of G is more n wice of ose of oer socks. esing more socks or esing longer periods could elp us o confirm e G cse is n unusul cse. ble 4 sows ( ) (,) for ll socks. e fc coefficiens of e, ddiionl erms,, re lmos zero suggess dding e second erm does no improve forecss significnly. Using e insrumenl vrible Z, we find our simple AR() model cpures lmos ll e dynmics is imporn for forecsing. However, sudying non-uni roo AR() model cse could be n ineresing fuure opic o pursue. Admiedly, since is pper focuses on finding beer forecsing model for e specific cse, using ~4 d o forecs yer 5 for seven socks, ere is no gurnee e (y) model performs beer n e HAR model nd e model in oer periods or for oer socks. esing e proposed (y) model on oer d ses could be elpful o suppor e effeciveness of e model. e mos imporn cievemen of is pper is i s uilized simple pproc o esime fuure voliliies.

24 Reference Andersen, orben G. nd im Bollerslev, 998, Answering e Skepics: es, Sndrd Voliliy odels do Provide Accure Forecss, Inernionl Economic Review, Vol. 9, o. 4, Symposium on Forecsing nd Empiricl eods in croeconomics nd Finnce, pp Andersen, orben G., im Bollerslev, Frncis. Diebold nd Pul bys, P.,. "odeling nd Forecsing Relized Voliliy," Economeric, (7): pp Andersen, orben G., im Bollerslev, Peer F. Crisoffersen, nd Frncis. Diebold, 5, Voliliy Forecsing, PIER Working pper 5- Brndorff-ielsen, O.E., Hnsen, P. R., unde, A., nd Seprd,., 7, Designing relized kernels o mesure e ex-pos vriion of equiy prices in e presence of noise. Unpublised mnuscrip. Crisoffersen, Peer F. nd Frncis. Diebold,, How Relevn is Voliliy Forecsing for Finncil Risk ngemen? e Review of Economics nd Sisics, Vol. 8, o., pp. -. Corsi, Fulvio, 4. A Simple ong emory odel of Relized Voliliy, Working pper, Insiue of Finnce, Universiy of ugno. Hnsen, Peer R. nd Asger unde, 6. Relized Vrince nd rke icrosrucure oise, Journl of Business & Economic Sisics, Vol.4, o.94: pp. 7~8. Hnsen, Peer R. nd Asger unde, 6, Voliliy Persisence nd Opiml Forecsing, working pper. Hnsen, Peer R. Economics 75: ime Series ecure noes, Snford Universiy, 7. Hrvey, A.C., 99, ime series odels, edn, Hrveser Wesef, Hemel Hempsed. uller, U., Dcorogn,., Dv, R., Pice, O., Olsen, R. nd von Weizscker J., 997. Voliliies of differen ime resoluions nlyzing e dynmics of mrke componens, Journl of Empiricl Finnce, 4: pp. -9