Value-at-Risk Estimation of Long Memory Data

Transcription

1 Value-a-sk Esmao of Log Memory Daa Y-Pg Chag, Mg-Ch Hug, Ye-Hao L Dearme of Busess Mahemacs, Soochow Uversy, Tae(00), Tawa Absrac The volaly of facal me seres lays a mora role may alcaos, esecally he feld of rsk maageme. More recely, may sudes sugges ha he log memory heomeo do exs he codoal volaly of facal daa. The ew class of fracoally egraed geeralzed auoregressve codoal heeroscedascy (FIGACH) model roosed by Balle, Bollerslev, ad Mkkelse (996) ca allow for hs log memory roery he codoal varace. I hs aer, we use he FIGACH model o comue Value-a-sk () for varous seculave reurs. We comare he erformace of several models hrough Moe Carlo smulao sudes ad fd ha he roery of log memory cao be modeled aroraely by radoal GACH or Iegraed GACH (IGACH) models. Keywords: FIGACH, Log memory, Moe Carlo Smulao, Value-a-sk () Emal addresses: ychag@bmah.scu.edu.w (Y. Chag), hugg@bmah.scu.edu.w (M. Hug), PDF creaed wh dffacory ral verso

2 . Iroduco Sce Bak of Ieraoal Seleme (BIS) recommeded he Value-a-sk () as a way o uafy marke rsk, has recely become a mora ool o marke rsk maageme ad facal suo regulaory urose. I order o esmae, s euvale o comug a uale of a secfc dsrbuo, ad may auhors have looked for a arorae class of dsrbuos o descrbe he characerscs of asse reurs. I emrcal sudes, s well kow ha he asse reurs have he dsrbuos wh hgh kuross ad heavy al. May models have bee roosed he modelg of facal rsk rece years. The mac of models o rsk maageme has bee rofoud. However, a creasg amou of evdece has hghlghed he here lmaos of rsk modelg (Daelsso 00), ad has bee wdely agreed ha models should be used cauously. summarzes he wors loss over a holdg erod wh a cera level of cofdece,.e. descrbes he lef -uale of a asse or orfolo reur dsrbuo for a seleced coverage robably. I a more formal sascal ermology, o esmae s o locae he lef -uale of a dsrbuo. Tycally small values lke 0.05, 0.0, or are used for o rerese a hgh level of cofdece (.e. - s ear o ). For relaed heorecal ad emrcal sudes, we refer o Morga (996), Duffe ad Pa (997), Joro (000), Peza ad Basal (00) ec. Deoe X as he asse rce a me ad as he reur durg [-, ]. Uder a couous comoudg assumo, he reur s defed as X l = l( X ) l( X ) X = whle he a me wh cofdece level s formally defed as P < =. (.) ( + ) I hs aer, we comare he erformace of hree GACH ye models hrough PDF creaed wh dffacory ral verso

3 Moe Carlo smulao sudy ad dscuss he sesvy of rsk esmao whe he asse reur s wh heavy-aled dsrbuo. The remag of hs aer s srucured as follows. Seco roduces he FIGACH model whch ess he GACH ad IGACH models as secal cases. Fve dffere FIGACH relaed comuao models were roosed Seco 3. Seco 4 roduces he daa geerag rocesses ad he smulao resuls for comarg he erformace of varous esmao models. Fally, Seco 5 cocludes hs aer.. FIGACH Model Le { a } deoe a dscree-me real-valued sochasc rocess, ad F he formao se of all formao hrough me. Followg Bollerslev (986), he rocess { a } s he sad o follow a GACH model f here exs a rereseao such ha a = σ ε, E( ε F ) = 0, E( ε F ) =, ad he codoal varace σ s measurable wh resec o F ad s arameerzed as a dsrbued lag of as suared ovaos, = ω + βσ + γ a ω + β ( L) σ γ ( L) a, = = σ + β + ( L) = βl + β L + Λ β L, γ + ( L) = γ L + γ L + Λ γ L, ad L deoes he lag or backshf oeraor;.e. Lσ = σ ad L a = a. The above euao ca be rearraged o [ γ ( L) β( L)] a = ω + [ β ( L)] ν, ν a σ ca be vewed as ovaos ad hus E( ν ) = 0. I s commoly F PDF creaed wh dffacory ral verso

4 foud may emrcal sudes ha he sum of he esmaed arameers a GACH(,) model ( γ ˆ ˆ + β ) s very close o oe (see, for examle, Taylor 986, Dg, Grager, ad Egle 993). Movaed by hs emrcal heomeo, he Iegraed GACH, or IGACH(,), s he roduced by Egle ad Bollerslev (986) o model he log-ru volaly erssece. The olyomal he IGACH(,) model, whch resrcs γ ( L) + β ( L) =, has oe u roo. Thus oe ca facorze hs olyomal as γ ( L) β ( L) φ( L)( L) ad he IGACH(,) model may be wre as [ φ ( L)( L)] a = ω + [ β ( L)] ν, I a IGACH model a al shock o he codoal varace remas sesve for all fuure redcos of all horzos. Alhough he erssece of he esmaed codoal varace aeared may emrcal alcaos (Bollerslev, Chou, ad Kroer 99), geeral uo sugges ha mos cases he volaly s mea-reverg. To recocle hs coflcg evdece, Balle, Bollerslev, ad Mkkelse (996) suggesed he Fracoally Iegraed GACH (or FIGACH) model as [ d φ ( L)( L) ] a = ω + [ β ( L)] ν, d s a fracoal umber for allowg fracoal orders of egrao ad he roos of φ ( L) = 0 le ousde he u crcle. We ca smly rewre he FIGACH(,d,) model as follows σ = ω + β φ. (.) d ( L) σ + ( β( L) ( L)( L) ) a The aral dffereal oeraor d ( L) has he Taylor s exaso as 3 PDF creaed wh dffacory ral verso

5 d j= 0 ( L) = π L, j j π 0 = ad d π j = for j > 0 j =. By smle oeraos, we have d φ ( L)( L) = ( = φ L )( = 0 π L ) =+ = φπ ) L ( π φ π Λ = =0 λ L, λ = λ = π φ π Λ φ π, >. Hece he codoal varace 0, σ he FIGACH model ca be exressed as = βa = = σ = ω+ βσ λa. (.) Ths FIGACH(,d,) model ess he GACH model for d=0 ad he IGACH model for d=. 3. FIGACH Value-a-sk Whle a covarace-saoary GACH model s codoal volaly decay a a geomerc rae ca oly caure he shor-ru emoral deedeces, he forecass of all horzos a IGACH model rema sesve o he al (revous) codos. Thus we use FIGACH model he esmao of ad comare he resul wh hose from 4 PDF creaed wh dffacory ral verso

6 GACH ad IGACH models. We roose hree FIGACH comuao models as follows. 3. FIGACH-Normal Uder he assumo of ormally dsrbued log-reurs,, =,,, s wre as = µ + a, (3.) a = σ ε follows a FIGACH(,d,) model, ε ~..d. N(0,) ad he codoal varace σ sasfy euao (.). We call hs as a FIGACH-ormal model. Followg he defo of (.) = P( + µ < F ) = P( σ µ < F ) + + σ + Hece he forecas of = µ + σ z, + bewee me ad + s z sasfy Φ(z ) = ad Φ ( ) deoes he cumulave dsrbuo fuco of a sadard ormal dsrbuo. The mos commo aroach for esmag he ukow arameers µ ad + σ reles o he maxmzao of a codoal lkelhood fuco. For oao smlfcaos, we se = +,.e., =,,, rerese for he h eares observed log-reurs. Le θ = µ, d, β, Λ, β, φ, Λ, φ ). Uder he assumo of ormally ( dsrbued oe-se-ahead redco errors, he codoal log lkelhood fuco euals ( µ ) log L( θ ;, Λ, ) = { log(π ) + [log( σ + ) + ]}, (3.) σ = + σ sasfy euao (.). Le µˆ, dˆ, βˆ, Λ, βˆ, φˆ,, φˆ be he codoal + Λ maxmum lkelhood esmaor (CMLE) of µ, φ,.e. o maxmze he, d, β, Λ, β, φ, Λ euao (3.). By luggg he esmaes o euao (.), 5 PDF creaed wh dffacory ral verso

7 + = ωˆ + βσ ˆ ˆ βˆ aˆ λˆ aˆ = = = σ ˆ, (3.3) he esmaor of uder he assumo of ormal oe-se-ahead redco error ε a FIGACH model s ( FIGACH N ) ˆ + = µ ˆ + σˆ z. 3. FIGACH- model I s well kow ha he me seres asse reurs usually have he dsrbuos wh heavy als, hgh kuross, ad he heomeo of volaly cluserg. Assumg ha he log-reurs, =,,, are as model (3.) bu wh ε ~..d. ( ν ) / ν T ν, he degrees of freedom ν > ad he codoal varace σ sasfy euao (.). We call hs a FIGACH- model. Smlar o he FIGACH-ormal model, he forecas of bewee me ad + s = µ + σ + ( ν ) ν, / (ν ) s he h ( ) uale of he sude s dsrbuo. Uder he assumo of sude dsrbued oe-se-ahead redco errors, he codoal log lkelhood fuco euals ν + ν log L( θ ;, Λ, ) = [logγ( ) log Γ( ) log( π ( ν ))] ( µ ) [log( σ + ) + ( ν + ) log( + )]. (3.4) σ ( ν ) = + By luggg CMLE o ˆ + σ euao (3.3), he esmaor of uder he assumo of -dsrbued oe-se-ahead redco error ε a FIGACH model s ( FIGACH ) ˆ = ˆ + σˆ + ( νˆ ) / νˆ µ ( νˆ). 6 PDF creaed wh dffacory ral verso

8 3.3 FIGACH-HS model Ay assumo of a secfc famly of dsrbuos bears he rsk of dsrbuo mssecfcao from whch severe sysemac esmao bas may hae. Whou assumg ay reur dsrbuo, here we roduce hree oaramerc mehods o esmae he dsrbuo uale () ad s esmao rsk. Assumg ha he log-reurs,=,,, are as model (3.) bu wh ε ~..d. couous radom varable wh dsrbuo fuco F ad E( ε ) 0, Var( ε ) =. Whe he dsrbuo of oe-se-ahead = redco error ε, F s kow, followg he rocess FIGACH-ormal model, he becomes ( FIGACH F ) ˆ + = µ ˆ + σˆ. s he h -uale of F ad ˆ + σ s as euao (3.3). Whe F s ukow, alog wh, µ, ad σ, all eed o be esmaed. For he esmao of µ ad σ, alhough he assumo of codoal ormally dsrbued sadardzed ovaos s volaed may emrcal sudes, followg Wess (986) ad Bollerslev ad Wooldrdge (99), he ormal uas codoal maxmum lkelhood esmaor (QCMLE) resulg from euao (3.) s asymocally vald ad cosse. Followg hese argume, here we use QCMLE µˆ, dˆ, βˆ, Λ, βˆ, φˆ,, φˆ ad euao (3.3) o esmae µ ad Λ σ +. For he esmao of ˆ µ ε ˆ =, =, Λ, σˆ +, defe ad deoe εˆ ( ) εˆ ˆ () ε ( ) Λ as her order sascs. We roose hree oaramerc mehods for he esmaos of h -uale of F,, as follows. 7 PDF creaed wh dffacory ral verso

9 () Frs we use emrcal uale of εˆ ˆ, εˆ, Λ, ε as he esmaor of,.e. ( HS ) ˆ = ( w) ε ( m) + wε ( m+ ), m = [ + 0.5], [ ] sads for he Gaussa fuco, ad w = m The he emrcal esmaor of, ˆ ( FIGACH E) s he defed as ( FIGACH E) ˆ + = µ ˆ + σˆ ˆ. () Harrell ad Davs (98) roosed aoher o-kerel uale esmaed by he lear combao of order sascs. The HD mehod s more effce for geeral dsrbuos ad here s o eed for he deermao of he kerel fuco ad he wdow wdh. Defe he emrcal dsrbuo fuco F ˆ ( a ) as I ( ) s a dcaor fuco,.e. Fˆ ( ε ) = I ( ε ( ) ε ), = I( ) = 0 f ε f ε > ε, F ˆ ( a ) s he a ubased esmaor of F (a). Followg Harrel ad Davs (98), he execed value of ε (k ) s k k E( ε ( k ) ) = F ε ε ε dε B k k ( ) ( ), (, + ) 0 Β ( s, ) s a Bea fuco ad he HD esmaor of -uale s defed as ˆ ( HD) = B{( + ),( + )( )} = = B Fˆ ( ε ) ε {( + ),( + )( )} B 0 ( + ) ( ) ( ε ) ( + )( ) dε {( + ), ( + )( )} ε ( ), ˆ F s he verse fuco of Fˆ, B x ( a, b) s a comlee Bea fuco. The he HD esmaor of, ˆ ( FIGACH HD) s he defed as 8 PDF creaed wh dffacory ral verso

10 ( FIGACH HD) ˆ ˆ + σˆ + = µ ˆ. HD (3) Huag (00) combed he HD uale (Harrell ad Davs, 98) wh he ueual weghs emrcal fuco (Huag ad Brll, 995) ad roosed aoher oaramerc esmaor. Huag ad Brll (995) defed he ueual wegh emrcal fuco F ˆ ( a) = w, w =, I(, a) ( ( ) ) w, s w, Huag (00) defed -uale, =,, = ( ), =,3, Κ. ( ) ( HDlc) ˆ as ˆ ( HDlc) = B{( + ), ( + )( )} = = k, ( ), 0 Fˆ w ( y) y ( + ) ( y) ( + )( ) dy F ˆ w s he ueual wegh emrcal fuco of Fˆ w ad k, = B j= {( + ), ( + )( )} B,,, = w j,, =, Κ, ; 0, = 0. {( + ), ( + )( )}, The he HDlc esmaor of, V ˆa ( HDlc), s defed as ( FIGACH HDlc) ˆ ˆ + σˆ + = µ ˆ. HDlc 4. SIMULATION COMPAISON We erformed a smulao sudy o comare he erformaces of esmao for 9 PDF creaed wh dffacory ral verso

11 each of he models descrbed he revous seco. Models: FIGACH, IGACH, GACH Error dsrbuo: N(0,), T(5), T(0), ST(0.5,6), ST(-.5,6), ST(.5,9), ad ST(-.5,9). To dscuss he robusess of varous models, we have d=0, 0.5, 0.75 Follow BBM- we se samle sze 3000 I he arameer esmao, we follow Bollerslev (986) ad use he BHHH mehod whch was roosed by Berg, Hall, Hall, ad Hausma (974). Follow BBM(996) we se he re-samle values of he ovaos (for he codoal mea) ε ad he codoal varace σ eual o he ucodoal samle varace he rocess of recursve arameer esmao. We use MATLAB 6..0 for all he comuaos hs aer. The codoal varace euao (.), σ = ω + βσ βa = = = λa, whch clude fe lags mus be rucaed for he cosderao of comug me, however, rucag a oo low a lag may cause he ably caurg he log memory feaures. Followg he recommedao of BBM (996) we se he rucao lag a Daa Geerag Process Assumg ha he log-reurs,, =,,, ca be wre as euao (3.),.e. = µ + a, a = σ ε follows a FIGACH(,d,) model ad ε, =,,, are deedely ad decally dsrbued wh E( ε ) = 0, E ε ) =. We se µ = 0 F ( F ad four dsrbuoal forms for he codoal (oe-se-ahead) redco ovaos ( ε ) were cosdered he rue DGP. The frs dsrbuo we examed was he sadard ormal dsrbuo,.e. ε ~..d. N(0,). Ths model s he oe mos commoly used for rsk maageme uroses. The secod dsrbuo we examed was he sadardzed 0 PDF creaed wh dffacory ral verso

12 -dsrbuo,.e. ca be wre as euao (3.) bu wh ε ~.. d. ( d ) d T ( d), T (d) s a radom varable wh sadardzed -dsrbuo ad d s he degree of freedom. Here d=5 ad d=0 are used. The hrd model we examed was he Double Exoeal (DE) dsrbuo whch ca be wre as euao (3.) bu wh a ~... d DE( ). Lde (00) foud ha he double exoeal dsrbuo s suable descrbg he behavor of asse reur daa. Fally, he fourh dsrbuoal form we examed was he skewed- dsrbuo (deoed as ST( ν, ξ )) roosed Feadez ad Sell (988) ad Lamber ad Laure (00). Iroduce.d.f. here. Four coeffce ses of skewess ad kuross cosdered here are (.5, 9), (-.5, 9), (0.5, 6), ad (-0.5, 6). Ther corresodg ( ν, ξ ) are (6.56,.7995), (6.56, ), (6.46,.647), ad (6.46, ), resecvely. The rasformao formula bewee (S,K) ad ( ν, ξ ) roduced here. 4.. Evaluao Crera Two crera were used evaluag he forecasg erformace of varous GACH-based models hs smulao sudy. Frs he mea absolue error (MAE) s defed as N MAE= ˆ N = ad he secod crera used here s he mea absolue relave error (MAE) defed as N MAE= ( ˆ ) /. N = The s he rue uale ad he ˆ s he redced. Smaller value for boh crera rereses a beer model Smulao esul For each of he four dsrbuos descrbed he revous daa geerag rocess, 0000 reeaed samles of 50 observaos were radomly geeraed. For each samle, he PDF creaed wh dffacory ral verso

13 esmao was calculaed. Due o he hghly smlar aer, we oly reor he resuls for he cases =0.0 (whch s he BIS regulaory reured) ad he resulg MAE here. We om he resuls for all he cases abou =0.05 ad he resulg MAE. They are avalable hrough emal from he auhors. For all he geeraed samles, esmaors ad cofdece ervals were calculaed by usg he FIGACH, GACH, ad IGACH models. Based o he reored MAE from our smulao resuls Table o Table 3, we fd ha whe he daa geerag rocess s FIGACH models are more robus ha GACH models ad yeld more sable resuls, esecally he case whe he fracoal umber d s large. Table. MAE whe d=0.75. model N(0,) T 5 T 0 DE ST(.5,9) ST(-.5,9) ST(0.5,6) ST(-0.5,6) FIGACH-N GACH-N IGACH-N FIGACH-T GACH-T IGACH-T FIGACH-E GACH-E IGACH-E FIGACH-HD GACH-HD IGACH-HD FIGACH-HD lc GACH-HD lc IGACH-HD lc PDF creaed wh dffacory ral verso

14 Table. MAE whe d=0.5. model N(0,) T 5 T 0 DE ST(.5,9) ST(-.5,9) ST(0.5,6) ST(-0.5,6) FIGACH-N GACH-N IGACH-N FIGACH-T GACH-T IGACH-T FIGACH-E GACH-E IGACH-E FIGACH-HD GACH-HD IGACH-HD FIGACH-HD lc GACH-HD lc IGACH-HD lc PDF creaed wh dffacory ral verso

15 Table 3. MAE whe d=0. model N(0,) T 5 T 0 DE ST(.5,9) ST(-.5,9) ST(0.5,6) ST(-0.5,6) FIGACH-N GACH-N IGACH-N FIGACH-T GACH-T IGACH-T FIGACH-E GACH-E IGACH-E FIGACH-HD GACH-HD IGACH-HD FIGACH-HDlc GACH-HDlc IGACH-HDlc Cocluso ad dscusso summarzes he wors loss over a holdg erod wh a cera level of cofdece ad has become a sadard ool used by may facal suos o measure marke rsk. Due o s here lmaos of rsk modelg, however, has bee wdely agreed ha models should be used cauously. I hs aer we fd ha he roery of log memory cao be modeled aroraely by radoal GACH or Iegraed GACH (IGACH) models. 4 PDF creaed wh dffacory ral verso

16 Lmao of FIGACH σ <0. Bred, Crao, ad De Lma (998) o well-defed. eferece. Balle,.T., Bollerslev, T. ad Mkkelse, H.O., 996. Fracoally egraed geeralzed auoregressve codoal heeroskedascy. \ex{joural of Ecoomercs}, 74, Balle,.T., Cece, A.A. ad Ha, Y.W., 000. Hgh freuecy Deusche Mark-US dollar reurs: FIGACH rereseaos ad oleares. Mulaoal Face Joural, 4, Ber, E.K., Hall, B.H., Hall,.E., ad Hausma, J.A., 974. Esmao ad ferece olear srucural models. Aals of Ecoomc ad Socal Measureme, 3/4, Bollerslev, T., 986. Geeralzed auoregressve codoal heeroskedascy. Joural of Ecoomercs, 3, Bollerslev, T., 987. A codoal heeroskedasc me seres model for seculave rces ad raes of reur. evew of Ecoomcs ad Sascs, 69, Bollerslev, T., Chou,.Y. ad Kroer, K.F., 99. ACH modelg face: a revew of he heory ad emrcal evdece. Joural of Ecoomercs, 5, Bollerslev, T. ad Mkkelse, H.O., 999. Log-erm euy acao secures ad sock marke volaly dyamcs. Joural of Ecoomercs, 9, Bollerslev, T. ad Mkkelse, H.O., 996. Modelg ad rcg log-memory sock marke volaly. Joural of Ecoomercs, 73, Bollerslev, T. ad Wooldrdge, J.M., 99. Quas-maxmum lkelhood esmao ad ferece dyamcmodels wh me-varyg covaraces. Ecoomerc evews,, PDF creaed wh dffacory ral verso

17 0. Bred, F.J., Crao, N. ad de Lma, P., 998. The deeco ad esmao of log memory sochasc volaly. Joural of Ecoomercs, 83, Chug, C.-F., 999. Esmag he fracoally ergraed GACH model. Naoal Tawa Uversy, workg aer.. Dg, Z. ad Grager, C.W.J., 996. Modelg volaly erssece of seculave reurs: A ew aroach. Joural of Ecoomercs, 73, Dg, Z., Grager, C.W.J. ad Egle,.F., 993. A log memory roery of sock marke reurs ad a ew model. Joural of Emrcal Face,, Egle,.F. ad Bollerslev, T., 986. Modelg he erssece of codoal varace. Ecoomerc evews, 5, Feradez, C. ad Seel, M., 998. O Baysa modelg of fa als ad skewess. Joural of he Amerca Sascal Assocao, 93, Harrel, F.E., ad Davs, C.E., (98), A ew dsrbuo - free uale esmaor, Bomerka, 69, Huag, M.L., ad Brll, P.H., (995), A ew weghed desy esmao mehod, Comuaoal Sascs & Daa Aalyss, 37, Huag, M.L., (00), O a dsrbuo-free uale esmaor, Comuaoal Sascs & Daa Aalyss, 37, Hull, J., ad Whe, A., (998), Value a rsk whe daly chages marke varables are o ormally dsrbued, The Joural of Dervaves, 5, Joro, P., (000), Value a sk - The New Bechmark for Corollg Marke sk, McGraw-Hll, New York.. Lamber, P. ad Laure, S., 00. Modelg facal me seres usg GACH-ye models ad a skewed sude desy. Mmeo, Uversy de Lege.. Lde, M., (00), A model for sock reur dsrbuo, Ieraoal Joural of Face ad Ecoomcs, 6, PDF creaed wh dffacory ral verso

18 3. Nelso, D.B., (99), Codoal heeroskedascy asse reurs: a ew aroach, Ecoomerca, 59, ay, B.K. ad Tsay,.S., 000. Log-rage deedece daly sock volales. Joural of Busess ad Ecoomc Sascs, 8, PDF creaed wh dffacory ral verso