The Equity Index Skew and Asymmetric Normal Mixture GARCH

Transcription

1 Th Unvry of Rng THE BUSINESS SCHOOL FOR FINANCIAL MARETS Th Equy In Sw n Aymmrc Norml Mur GARCH ISMA Cnr Dcuon Pr n Fnnc 4-4 Fr Vron: Ocobr 4 Crol Alnr ISMA Cnr Unvry of Rng Em Lzr ISMA Cnr Unvry of Rng Coyrgh 4 Alnr n Lzr. All rgh rrv. ISMA Cnr Th Bun School Unvry of Rng Whngh PO Bo 4 Rng RG6 6BA U Tl: F: Wb: Drcor: Profor John Bor Chr n Fnnc Th ISMA Cnr uor by h Inrnonl Scur Mr Aocon

2 Abrc Th wn n hycl rbuon of quy n rurn n h ml volly w n h r nurl mur r ubjc of nv cmc rrch. Much non now bng focu on mol h r bl o cur m-vryng cononl wn n uro. For h ron norml mur GARCH mol hv bcom vry oulr n fnncl conomrc. W nrouc furhr ymmr no h cl of mol by mofyng h GARCH vrnc roc o w vrnc roc wh lvrg ffc. Th ymmrc norml mur GARCH mol cn ffrn bwn wo ffrn ourc of ymmry: h rn ymmry u o h ffrn mn n h cononl norml mur rbuon n h ynmc ymmry u o h w GARCH roc. Emrcl rul on fv mjor quy nc fr mloy mny cl crr o rmn whhr ymmrc norml mur GARCH mol cn mrov on ymmrc norml n Sun - GARCH cfcon. Norml - n norml mur ymmrc GARCH mol wr lo u o mul ml volly ml for h S&P n n w fn h much h mo rlc w r mul by GARCH mol wh mur of wo GJR vrnc comonn. JEL Clfcon Co: C3 G3. ywor: GARCH roc norml mur quy w mr crh w rnc lvrg ffc Auhor Dl: Profor Crol Alnr Chr of R Mngmn n Drcor of Rrch ISMA Cnr Bun School Th Unvry of Rng PO Bo 4 Rng RG6 6BA Un ngom Eml: c.lnr@mcnr.rg.c.u Tl: ISMA Cnr F: Em Lzr PhD Sun ISMA Cnr Bun School Th Unvry of Rng PO Bo 4 Rng RG6 6BA Un ngom Eml:.lzr@mcnr.rg.c.u Tl: ISMA Cnr F: W woul l o hn Rz Dmrr n h rcn of h FMA 4 Confrnc for hr hlful commn n uggon.

3 ISMA Cnr Dcuon Pr n Fnnc 4-4 I Inroucon Ou-of-h-mony u oon on n quy n r n rcv form of nurnc for rr h fr gnrl mr cln. Wh n oon mr mr n rlvly hor uly h mr rc of h oon r ofn fr hghr hn h Blc-Schol 973 mol rc b on h -h-mony volly. Conqunly h ml volly of h oon commonly foun o b hghr hn h ml volly of -h-mony cll n u oon n ou-of-h-mony cll. Th w or mr n quy n ml volly h bn vry ronounc nc h globl oc mr crh n 987. Th quy n ml volly w oc wh ngvly w ml r nurl rurn ny. B 997 fn h wn cn ccoun for roun 4% of SP5 oon rc n ugg h h r nurl n rhr hn h hycl n of quy nc h hv chng nc h crh. If h hycl n o n rmn l w n lourc hn h r nurl hn obl lnon n ncr n h r rmum of h vrg rr nc h crh of 987. Mo oon rcng mol mly h boh wn n louro n h r nurl ny houl mnh ubnlly mury ncr.g. onov n Mn n ohr. Alo h wn n louro of hycl rurn n houl f h cnrl lm horm l mnh h ro of rurn ncr. Nvrhl Crr n Wu 3 fn ronounc r nurl w n h SP5 n h r no oon wh ovr on yr o ry. Bh l 3 focu on h rlonh bwn h hghr momn of h r nurl ny n h hghr momn of h hycl ny of n rurn. Thy how h ngv wn n h r nurl mur cn r vn whn h hycl ny ymmrc rov hr c uro n h hycl ny n h rr hv ov r rmum. Lvrg ml h h n w houl b l ronounc hn nvul quy w h ffc of vrfcon bu Bh l fn h n h OEX n cn hn h h n w mor ronounc hn ll nvul quy w. On ylz fc h h mrg from h rrch h l n h US oc mr nc h crh of 987 h n w r boh oo ronounc n oo rn o ccor wh h nr m-r nly of h cononl n of n rurn n cn only b ln by h r rmum. S B 99 Rubnn 994 Jcwrh n Rubnn 996 Drmn n ml 997 Tomn n mny ohr rov nv mrcl vnc of h ongong hnomnon. Coyrgh Alnr n Lzr. All Rgh Rrv.

4 ISMA Cnr Dcuon Pr n Fnnc 4-4 Th lrg lrur on r nurl w mch by n vn lrgr lrur on cononl momn n h hycl mur. Th r lmo cluvly moll n h gnrlz uorgrv cononl hroccy frmwor nrouc by Engl 98 n Bollrlv 986. Th obrv non-norml n boh cononl n uncononl rurn hghr hn cn b rc by norml GARCH mol. Hnc Bollrlv 987 nrouc h Sun - GARCH mol n Frnnz n Sl 998 n h o h w -rbuon. Th -GARCH mol hv no m-vron n h cononl hghr momn o h -GARCH ml quy w my only chng wh mury f hr m vryng r rmum. Howvr mny u.g. Chroffrn Hon n Jcob 4; B 99 n ohr mhz h mornc of m-vrbly n h hycl cononl wn. Mor rcn rrch by H l 4 n Alnr n Lzr 4 on GARCH mol wh mur of norml GARCH vrnc roc h hown h h mol rov br f o hycl cononl n hn mny ohr y of GARCH mol. Th norml comonn n h mur rbuon cn b nrr ymbolng ffrn mr crcumnc or grou of ffrn nvmn bhvour. Snc h norml mur GARCH mol o hv mvryng cononl hghr momn hy coul b cbl of rlcng volly ml vn whou r rmum. Hvng h Alnr n Lzr 4b how h h connuou lm of norml mur GARCH mol ochc volly mol; no n h ronl n of vrnc ffuon bu mol wh Byn uncrny ovr h obl vrnc comonn. Snc h uncrny cnno b hg r rmum mu ly n h r nurl ny n h m vron n h r rmum nhnc boh h w n h rm rucur n ml voll. Alnr n Lzr 4 how h f h mol h mor hn wo vrnc comonn vr b n rmr m r lly o rul n conqunly h m cononl wn n c uro cn b unccounbly unbl ovr m. A l for mollng mjor chng r m-r hy fn h h mur of wo GARCH comonn mol ourform boh ymmrc n ymmrc -GARCH mol n norml mur GARCH mol wh mor hn wo comonn. Howvr whl h mur of wo norml vrnc comonn cfcon my f chng r wll h mol r no uffcnly flbl o ly o quy nc. Th bcu hr only on ourc of wn n h hycl rurn n.. h rng from h ffrn mn n h comonn of h norml mur cononl ny. Th mol gnor h lvrg ffc n qu h h bn ocumn by Blc 976 n mny ohr. 3 Unl lcly moll for nnc n Hnn 994 n Hrvy n Squ A fll n h quy rc m h frm mor hghly lvrg n volly ncr bcu h frm fuur bcom mor uncrn. Howvr commnur r n rc wll no hv h ymmrc ffc on volly. Coyrgh Alnr n Lzr. All Rgh Rrv.

5 ISMA Cnr Dcuon Pr n Fnnc 4-4 Whn h cr m-vryng volly mol of quy rurn hr v lrur on ymmrc GARCH mol Engl n Ng 993; Glon l 993; Nlon 99 n mny ohr. Th mol gn cur only on ourc of wn.. h lvrg ffc. Clrly onl rucur rqur o cur h mrcl obrvon of Bh l 3 n ohr bou h nur of wn n h r nurl quy n w. Svrl r.g. Br n Wu ; Wu uy h corrlon bwn rurn n volly. A wll mhzng h mornc of m-vrbly n h r rmum h uhor how h h lvrg ffc rhr hn volly fbc h rmn h ngv corrlon bwn rurn n volly. Hnc n ymmrc GARCH vrnc roc houl b mor owrful hn h GARCH-n-Mn roc for lnng w ffc n h hycl mur. In rcn uy Chroffrn n Jcob 4b comr om GARCH mol for oon vluon. Thy lo conclu h ml ymmrc GARCH or n ny GARCH mol h cur h lvrg ffc rform b. Sll h mol hy conr whch no nclu h norml mur GARCH wr no bl o cur hr h full n of h wn or c uro n h. In h r w fr n h norml mur GARCH mol o nrouc n ffrn wo nc ourc of wn on ynmc n h ohr rn. W hll ly h mol o horcl on fv mjor nrnonl oc mr nc howng h rov h b f of h ffn ffrn GARCH mol conr hr ymmrc n wlv ymmrc GARCH mol. By rcovrng rr blf bou h llhoo of oc mr crh n h r n rurn h my b rnc urng crh mlmnng n ymmrc norml mur GARCH llow on rw nw ngh bou h hycl ny of oc mr rurn. W u h n ohr GARCH mol o nly h rmnn of h n w n h oc mr. Evn whou r rmum h volly w ml by h ymmrc norml mur GARCH mol hb ronounc w h r hough mnhng for long- oon. By conr non of h ohr mol cn ln h obrv chrcrc of r nurl w n oc n mr whou r rmum. Th r rucur follow: Scon II fn h gnrl ymmrc norml mur GARCH mol n nvg h ror uch cononl n uncononl momn of om cfc vrn. Scon III crb h quy n for fv mjor quy mr Frnc Grmny U Jn n US n h mon mhoology. Scon IV ror h mon rul for ymmrc n ymmrc norml mur GARCH mol wh wo vrnc comonn n for vrl lrnv mol nclung ymmrc n w -GARCH wh boh ymmrc n ymmrc vrnc roc. W ly vrl mol lcon crr o nfy h b mol. Scon V mn h rmr m from h norml mur GARCH mol n m nfrnc on h llhoo of n bhvour urng uul mr crcumnc n quy Coyrgh Alnr n Lzr. All Rgh Rrv. 3

6 ISMA Cnr Dcuon Pr n Fnnc 4-4 mr crh. Thn con VI l h mol o mul h ml quy n w n h SP5 n n Scon VII ummrz n conclu. II Th Aymmrc Norml Mur GARCH Mol Th cfcon of h mol h quon. For mlcy h cononl mn quon wrn y. I conn no lnory vrbl h cn b m rly. Th rror rm um o hv cononl norml mur ny wh zro mn whch robbly wgh vrg of norml ny funcon wh ffrn mn n vrnc. W wr: I ~ NM... µ... µ... µ Th h cononl ny of h rror rm whr f f rrn norml ny funcon wh ffrn conn mn µ n ffrn m- vryng vrnc for. Th cononl vrnc bhvour crb by vrnc comonn n h chrcrz ccorng o on nrron ffrn mr crcumnc. Th vrnc cn follow ny GARCH roc bu for h uro of h r w um hr r hr obl. In on o h ymmrc GARCH nvly u n Alnr n Lzr 4 w conr wo y of ymmrc roc: NM-GARCH: NM-AGARCH for NM-GJRGARCH: for for ; whr f < n ohrw In ll c h ovrll cononl vrnc µ For > h nc of con hr n fourh momn r ur by mong l rngn conon hn n h ngl comonn mol. For nnc Alnr n Lzr 4 how h w no longr rqur <. In H l 4 hv omm foun > on Coyrgh Alnr n Lzr. All Rgh Rrv. 4

7 ISMA Cnr Dcuon Pr n Fnnc 4-4 Coyrgh Alnr n Lzr. All Rgh Rrv. 5 h con n hghr vrnc comonn. Th wy w only rqur h followng of conon for h non-ngvy of vrnc n h fnn of hr momn. 4 For w mu hv: < < < < < n n h NM-GARCH mol : µ > m > n n > n m Alo n h NM-AGARCH mol w rqur h: µ > m > n n > n m or n h NM-GJRGARCH mol : µ > m 5 >. n n 5 > n m. Thr r wo nc ourc of ymmry n h mol: Prn Aymmry: Th r n ll hr mol whn h cononl rurn ny mur of norml ny comonn hvng ffrn mn. An how h vn h uncononl ny wll hv non-zro wn n h h ncr wh h ffrnon of h comonn mn. For nnc whn hr ngv wn n h ovrll cononl rurn ny whn h comonn wh h hghr vrnc h ngv mn n ov wn n h ovrll cononl rurn ny whn h comonn wh h hghr vrnc h ov mn. Dynmc Aymmry: Th u o h rmr n h comonn vrnc roc of mol n whch cur hor-rm ymmr u o h lvrg ffc. If ov h cononl vrnc n h comonn hghr followng ngv unc rurn m hn followng ov unc rurn. In quy mr whr b nw normlly corron o ngv unc rurn w c ov. On h ohr hn ngv lvrg coffcn my b m from commoy rurn. 5 Tn oghr h ourc of wn n h hycl cononl rurn ny offr uffcnly rch rucur for curng h bhvour of quy n w. Th uncononl momn of h wo ymmrc norml mur GARCH mol n r n An 4 Thr no rghforwr rmr conrn for nc of fourh momn. W mly rqur hn h < E 4 <. 5 No h h h comonn of h cononl vrnc n on h ron of h unc rurn no roun mn µ n h nvul ny bu roun h ovrll mn. Th ron roun lwy grr. Hnc h hr ffc nuc mor wn n ch comonn cononl rurn ny hn n h ovrll cononl rurn ny.

8 ISMA Cnr Dcuon Pr n Fnnc 4-4. No h h uncononl wn n c uro r non-zro o nhr h wn nor h c uro wll convrg o zro: h cnrl lm horm o no ly. Of rculr nr h fc h h uncononl wn n ch comonn zro 6 o h uncononl wn n h ovrll n rurn ny m rmrly from h rn ymmry.. from h ffrn mn n h comonn of h norml mur cononl ny. 6 Snc w hv cononl normly for ch comonn h cononl wn for ch comonn zro lng o zro uncononl wn. Coyrgh Alnr n Lzr. All Rgh Rrv. 6

9 ISMA Cnr Dcuon Pr n Fnnc 4-4 III D n Prmr Emon Our rul wll b b on h ly clong rc of fv quy mr nc: CAC4 DAX3 FTSE N5 n S&P 5 from Jnury 99 o My 3. Th n rc r hown n Fgur 5 n h followng ummr h gnrl chrcrc of h ly rurn: 7 CAC DAX FTSE N S&P Volly 3.4% 4.9% 8.7% 4.36% 7.45% Swn ***.56.47***.6 Ec uro.739*** 3.964***.657***.47*** 3.378*** Th wn ngv c for h N whr vry gnfcnly ov. Th wn lo hghly gnfcn n ngv n h DAX. Mor c uro vn mor o n h DAX n h S&P n h FTSE n S&P nc hv bn l voll hn h ohr urng h ro. For ch n w m h cononl vrnc rmr rly on h rul from AR cononl mn quon. All nc h gnfcn ov uocorrlon h ly frquncy n ung nformon crr w nf u o 4 lg. 8 Thn mmzng h llhoo or quvlnly mmzng T L ln [ ] gv h oml rmr vlu gvn h. On mjor roblm n ny y of omon h rch for ror rng vlu o nur h h omon roc l o h globl omum n of locl on. To ovrcom h roblm ugg by Doorn n nl gr rch rform. Howvr h ffculy of omon ncr wh h numbr of rmr hu wh h numbr of comonn n h mur. Th ung formul h h followng form whr g h grn vcor H h Hn mr n rrn h -lngh: 7 For n nnul rurn X h fr four momn of rbuon r h mn µ E X vrnc E [X - µ ] wn E [X - µ 3 ] / 3 n c uro E [X - µ 4 ] / 4 3. Th nr rror.. of h ml m of h rmr r follow:.. ml mn / T.. ml vrnc /T.. of h ml wn 6 / T.. of h ml c uro 4 / T whr T rrn h ol numbr of obrvon. In h bl *** rrn rul gnfcnly ffrn from zro h.% lvl. 8 W f h followng AR mol: CAC: r r-.47 r-.494 r-3.95 r-4 DAX: r r-.38 r- FTSE: r.36.8 r-.56 r-.493 r-3 NIEI: r r-.485 r- SP: r.6.33 r- Coyrgh Alnr n Lzr. All Rgh Rrv. 7

10 ISMA Cnr Dcuon Pr n Fnnc 4-4 m m m [ H ] g To comu h Hn mr n h grn vcor w cn u hr nlyc or numrcl fr n con orr rvv of L wh rc o An. 9 m IV Emrcl Rul W f hr ymmrc n wlv ymmrc GARCH mol o h quy n : Symmrc norml GARCH Norml AGARCH 3 Norml GJR 4 Symmrc GARCH wh Sun rbu rror 5 AGARCH wh Sun rbu rror 6 GJR wh Sun rbu rror 7 GARCH wh Sw Sun rbu rror 8 AGARCH wh Sw Sun rbu rror 9 GJR wh Sw Sun rbu rror Symmrc NM-GARCH wh zro mn n h mur NM-AGARCH wh zro mn n h mur NM-GJR wh zro mn n h mur 3 NM-GARCH 4 NM-AGARCH 5 NM-GJR Th mon r ror n Tbl 5. Th ur fgur n ch cll ror h rmr m n h lowr fgur h -ro. No h n h bl h fr row ror h gr of from for h -GARCH mol 4 9 n h hgh wgh of h wo comonn n h NM- GARCH mol. Smlrly h hr row ror h wn rmr for mol 7 9 n h mn of h fr norml ny for h norml mur GARCH mol. Mol Slcon Th four mol lcon crr u r Th mmum llhoo: To ccoun for rmony h A Informon Crron AIC n Schwrz Byn Informon Crron BIC wr lo mn hough no ror n h Tbl. 9 Th rul wr gnr ung C n O vron 3.3 Doorn n h G@rch cg vron 3. Lurn S. n Pr J.-P. In Tbl 5 h rmr r m by MLE. Numbr n rnh rrn -ro wh * n ** gnfyng gnfcnc h 5% n % lvl rcvly. Coyrgh Alnr n Lzr. All Rgh Rrv. 8

11 ISMA Cnr Dcuon Pr n Fnnc 4-4 b Th Nwy 985 momn cfcon : Followng Nwy 985 w for normly n h nrz rul chcng h fr four momn n for zro uocorrlon n h owr ung Wl. Thr r ol of conon nclung jon on ll conon. T c for h momn hv rbuon n for h cumulv hv rbuon. Th Tbl ror h numbr of ou of h r rjc %. c Th uncononl ny : Th ny on h hogrm f bwn h mol mul n h orgnl. Th on of h mo ffcul for GARCH mol o for h uncononl rbuonl f. Th mol rurn r mul n hr hogrm m ung nonrmrc rnl roch. Svrl lrnv r vlbl for h rnl our chon funcon bng h of Enchnov 969. Thn h mol lcon crron b on h mof olmogorov-smrnov S c olmogoroff 933 Smrnov 939 My 95 n hm. Th Auocorrlon Funcon ACF : By conr o h cur h ynmc ror of h mol qur rurn nmly h mrcl uocorrlon of h qur rurn. An 3 h horcl uocorrlon funcon of h ffrn mol n w ly h Mn Squr Error MSE crron o h f wh h mrcl uocorrlon. Th rul of h cfcon r hown n h l four row of Tbl 5. Th r cu n urn: Llhoo: All r fvour h NM-GJRGARCH mol wh non-zro mn n h comonn c h N for whch h NM-AGARCH rfrbl. Th no ju bcu hr r mor rmr n h mol h AIC uor h llhoo rul. No h h BIC rfr h -GARCH mol 4 6 for om r. b Momn cfcon : Th how h h mo bc mol.. 3 o no cur h hghr momn. Bu byon h obrvon h momn o no nguh wll bwn h mol. W fn h hr mo of h mol ll n h c of h FTSE n or mo hv vrl rjcon n h DAX. Ovrll w cn y h h -GARCH mol 4 9 rouc mrgnlly br rul on h momn. c Uncononl ny: Th how clr rfrnc for h NM-GJRGARCH wh ll hr ourc of ymmry.. wh ffrn comonn mn c for h CAC whch rfr zro mn. W mul rurn b on h m rmr n o nur h h mul ny no ffc by mll ml z w u 5 rlcon. Alo o vo ny nflunc of h rng vlu ch mulon h h n m bu w only u h l mul rurn. Coyrgh Alnr n Lzr. All Rgh Rrv. 9

12 ISMA Cnr Dcuon Pr n Fnnc 4-4 ACF : Th lo fvour h ymmrc NM mol. Fr no h whl on or mor of h -GARCH mol my o ronbl job o cur h ynmc ror of h qur rul h mol rform bly for l on n. Th b -GARCH mol ccorng o h crron mol 5 AGARCH wh Sun rbu rror. Nvrhl h ll h MSE of 3.54 n h DAX.539 n h N n.8548 n h S&P 5. By conr ll h NM mol 5 rform vry wll ccorng o h crr. In ummry Th wo mo morn c n nc clr urory of h NM-GARCH mol ovr -GARCH mol n h ml ymmrc n ymmrc GARCH mol lhough lghly ffrn cfcon o br for ffrn r. Th rn ourc of ymmry r o b morn n ll nc c h CAC. Th h non-zro mn mol 4 n 5 r gnrlly rfrr. Th ynmc ymmr.. ho u o lvrg r lo vry morn h boh y of ymmrc comonn AGARCH n GJRGARCH grly mrov h f. Th NM-GARCH mol rform wll ccorng o crr n b wll. Th norml GARCH mol h wor f by ll crr. Th Sun -GARCH mol 4 9 f wll ccorng o wh h BIC crr n lb long wh ohr mol hy lo o wll on h momn. Howvr unl n AGARCH or GJR cfcon u h mol yl rculouly hgh uncononl volly. Inrngly h mol 7 9 on rform much br hn h mol 4 6. Thu whn h vrnc roc hr AGARCH or GJRGARCH h onl ymmry ffor by ung h w -rbuon n lc of h nr Sun -rbuon r o b unncry. V Norml Mur: GARCH wh Norml n Crh Comonn W now focu on h rmr m for h ymmrc norml mur GARCH mol 4 n 5. From h rul n Alnr n Lzr 4b w now h h connuou lm of h norml mur GARCH mol ochc volly norml mur ffuon roc wh norml mur rnon n n norml mur mrgnl ny. Th mng lw corron n bhvourl rm o h uncrn vw bou volly h r hl n homognou oulon of rr. Snc n ll of h m norml mur GARCH mol for quy nc w hv hgh volly comonn wh low robbly n low robbly comonn wh hgh robbly clr h h norml mur GARCH mol r curng Coyrgh Alnr n Lzr. All Rgh Rrv.

13 ISMA Cnr Dcuon Pr n Fnnc 4-4 uul mr crcumnc comonn n crh comonn n quy mr. Th robbl oc wh h comonn rrn h rr blf.. h robbl h gn o norml mr n o h occurrnc of crh urng h forc horzon. On lmon of h norml mur GARCH frmwor h h robbly um conn.. rr hv no lrnng ooruny o h hr ror blf r no u. If rr r o lrn hn h robbl woul n o b m-vryng for nnc n h cl of Mrov Swchng GARCH Mol nrouc by Hmlon n Suml 994 n C 994. Nvrhl norml mur GARCH mol lry gv conrbly mor ngh o h bhvour of hycl rurn n for quy nc hn h rvouly bn rcovr from hycl. In rculr h fr comonn.. uul mr comonn h h followng fur: A hgh robbly: wgh of romly.96 CAC DAX.9 FTSE.95 N n.93 S&P. An nnul rurn of.4% CAC.3% DAX.6% FTSE.4% N n.6% S&P. No h h Jn mr h con hvng ngv c rurn unr uul mr crcumnc. A lowr uncononl vrnc wh volly of romly % CAC DAX 6% FTSE 3% N n 5% S&P. Uul GARCH rmr vlu. Th lvrg ffc gnfcn h h c gn n no oo rong. Th con comonn.. h crh comonn h h followng fur: A low robbly: wgh of bwn.4 n.8 n h mur A ngv mn rurn c n h N o hr h con comonn cully n uwr jum comonn. I vr conrbly much mor hn h mn of h uul mr comonn: from 3% for h FTSE o 8% for h CAC n h S&P. A hgh uncononl volly of roun 45 5% n h CAC DAX n N bu much lowr roun 3% for h FTSE n S&P. A gnfcnly hghr conn lowr rnc rmr n lrgr rcon rmr hn n h fr comonn. A vry ronounc lvrg ffc urng crh mr n h S&P n CAC. No h h no rn from h ml wn c m on n rurn. Whn h mn r non-zro n mol 4 n 5. Coyrgh Alnr n Lzr. All Rgh Rrv.

14 ISMA Cnr Dcuon Pr n Fnnc 4-4 An morn vng of h norml mur GARCH mol h bly o mol m vron n h cononl hghr momn of rurn.. m vryng cononl wn n uro. In orr o u h m w n fr o comr h uncononl hghr momn of rurn m by h norml mur GARCH mol wh h uncononl hghr momn of rurn m ung ml ml c: For ll nc c h N h ngv wn r from hvng low vrnc comonn wh ov mn n ngv mn on h hgh vrnc comonn. Th ov wn n h N cur by h ngv mn on h low vrnc comonn n corronng ov mn on h hgh vrnc comonn. For h DAX FTSE n N h mol uncononl wn commnur wh h ml c n Scon III. Howvr for h S&P h norml mur GARCH mol rvl mor ngv wn hn rc mon of uncononl wn on rurn. Th hn bcu h lvrg coffcn m n h con comonn conlly lrg n h S&P. 3 Whn ymmrc GARCH comonn r u n mol n 3 gnrlly h mol c uro much oo hgh. Howvr h nroucon of ymmrc comonn brng h c uro own o rlc lvl vry clo o ho obrv n h ml momn. W uc h h onl ymmr nrouc by AGARCH or GJRGARCH comonn n h norml mur GARCH mol r vry morn for h ccur mollng of hghr momn of rurn. Th NM-GARCH wh ymmrc comonn h bn bl o nguh vrl bhvourl chrcrc of h fv mjor worl quy mr for m r nly of h rurn lon. Th mn fur of h crh ro r follow:. Crh ro n h US mr crry robbly of roun 6%. Th mr chrcrz by vry lrg ownwr jum on vrg roun 8% n nnul rm. Th vrg volly urng crh roun 3%. Alo h m h lvrg ffc vry ronounc.. Crh n h U mr occur wh robbly of roun 8% bu h r ccomn by mllr jum of romly %. Th vrg volly crh m mlr o h on n h US bu h lvrg ffc l gnfcn. 3. In h Frnch n Grmn mr crh hv n occurrnc r of 4% n h jum urng uch ro vrg 8%. Bu h mr mor voll urng mr crh 3 Th coffcn r lo lrg n h CAC. Howvr w h mol whou lvrg h rov br f o h CAC. Coyrgh Alnr n Lzr. All Rgh Rrv.

15 ISMA Cnr Dcuon Pr n Fnnc 4-4 hn n h ohr counr wh volly of roun 45%. Th lvrg ffc lo gnfcn bu lghly l hn n h US. 4. Th Jn mr bhv vry ffrnly from h ohr four mr h mr h bn n cln for mny yr. Thr r no crh ro n fc h unuul h on n whch hr n uwr jum n h n! Th o no hn oo ofn wh robbly of only 5% n hn h vrg volly mor hn 5% h c rurn ov n hr mor ronounc ngv w. VI Equy In Iml Volly Sw A ylz fc h h mrg from rcn lrur rvw n h nroucon h n w r boh oo ronounc n oo rn o ccor wh h nr m-r nly of h cononl n of n rurn. Bu h ymmrc norml mur GARCH roc no nr m-r mol. I h m-vryng cononl wn n uro o h volly w wll no b h m for ll mur. I lo bl o nguh bwn wo ourc of ymmry n hycl rurn rbuon ynmc lvrg ffc n mor rn ymmry n h w. Furhrmor rcovr rr blf bou h llhoo of mr crh n h rurn n volly bhvour urng rnqul n crh ro. A nurl quon o hrfor how h ror r rflc n h quy w ml by h mol. In h con w comr h ml volly w gnr by ymmrc norml mur GARCH mol wh ho ml by ohr GARCH mol n h hycl mur. In h con w how h h ymmrc norml mur GARCH mol gnr rlc w vn n h bnc of r rmum. W ubqunly ffrn bwn wo ymmrc ffc: rnc n h w h cur by h ffrnc n mn of h vrnc comonn n lvrg ffc h cur by h ymmry n ch vrnc comonn. W hll u h rmr m of h S&P n rurn gvn n Tbl 5 for: h norml - n ymmrc -GARCH mol; h GJR rmrzon for h nr - n ymmrc - GARCH mol; n h zro-mn NM-GARCH wh ymmrc n w b on h GJR mol comonn. 4 W now mul volly w ung ch of h mol n comr hr chrcrc. Snc our rul r b on ly rurn w lo mul ly rurn. Srng wh S n ung r.3 w mul h ynmc of h n vlu : 4 Smul w for h ohr quy nc r vlbl from h uhor on rqu. Coyrgh Alnr n Lzr. All Rgh Rrv. 3

16 ISMA Cnr Dcuon Pr n Fnnc 4-4 S S r / A Euron cll oon rc of r n mury T comu : rt E m S c T Rng h rocur 5 m n comung hr vrg gv our m of h oon rc. Thn lyng h nvr Blc-Schol formul gv h mul ml volly T. Fgur rn h ml b on h nn mol mnon bov. For comron w hv r o u h m vrcl cl from 5% volly for ch ml. 5 Snc h rcnly bn hown h h connuou lm of h norml GARCH mol h m vryng rmnc vrnc roc hr houl b no r rmum n h mol. 6 Noc h h norml GARCH w lmo comlly fl hr bng nohng n h mol o cur ymmry or rm rucur c mn rvron n h rmnc vrnc roc. Th GJR w n fgur f r mor rlc wh ubnlly hghr volly for ITM cll hn OTM cll. Alo h onl ymmry ffor by ung h w -rbuon n lc of h nr Sun -rbuon r o b unncry hr vry ll ffrnc bwn fgur n f. Inrngly w rch cly h m concluon b on our cl. Bu gn hr no uncrny n h mol n hnc w fn vry ll rm rucur n h w. A c h NM-GJR mol rouc much h mo rlc w: no only h w ronounc n l lnr hn n h ngl comonn GJR mol hr much mor vron of volly ovr m. Comrng fgur g wh h hr mll ncr n h w rnc whn non-zro mn r m n fgur h. Bu mll. On h ohr hn comrng fgur wh h b of h ngl comonn GJR rmrzon rgubly fgur f h onl comonn n fgur llow for rchr rucur n h w wh nocbl rm rucur cly w h ho. VII Summry n Concluon Th r h nrouc nw y of ymmry no wh rgubly h mo uccful n rcbl GARCH mol n common u.. h norml mur GARCH mol. Evn whou 5 Howvr for h wo -GARCH w fgur b n c h w mobl. Rcll from Tbl 5 h h long rm volly m from h mol wr mrobbly hgh 4.% for h -GARCH n 3.43% for h ymmrc - GARCH. No urr hn h hr volly w r comlly unrlc. 6 S Alnr n Lzr 4b. Coyrgh Alnr n Lzr. All Rgh Rrv. 4

17 ISMA Cnr Dcuon Pr n Fnnc 4-4 ymmry n h vrnc comonn h ymmrc norml GARCH lry h m-vryng cononl wn n uro. So ncry o furhr ourc of ymmry ung for nnc GJR n AGARCH comonn vrnc roc Th nwr o h quon unoubly y. Boh h cl crr n h mulon of h n w jufy h on of h con y of ymmry. Th GJR or AGARCH comonn cur lvrg ffc whl h ffrn comonn mn cur mor rn w ffc. I only wh ffrn comonn mn h h uncononl wn non-zro n h c. Bu our rul how h h ynmc ymmry of h GJR or AGARCH comonn r o b much h mo morn ffc n quy mr. Our bry of cl how h h on of ynmc ymmry vry hghly gnfcn n rmclly mrov h f of h norml mur mol. Howvr h w rnc h cur by h u of wo comonn wh ffrn mn only mrgnlly gnfcn. To ummrz our mrcl fnng w hv monr h clr urory of norml mur GARCH mol ovr ny ngl comonn GARCH mol nclung h GJR rmrzon n h w n nr Sun -GARCH mol. Our rul r lo uor by vry owrful bhvourl nrron for h horcl mol whr rr blf bou h llhoo of crh n h rurn n volly n rnqul m n urng h crh ro cn b rcovr from h hycl. Ovr h ro conr Jnury 99 o My 3 h rcv llhoo of crh w l n h Jn Frnch n Grmn mr bou 4%; n com h US wh h crh llhoo of bou 6% n fnlly h U whr rr oc robbly of 8% o h crh cnro. W hv foun h h crh mr n h US much rongr hn n h U n h h n jum own furhr. Nvrhl h U n US mr hv h low crh voll of 3% comr wh 45-5% n h Jn Frnch n Grmn mr. Thr vry ronounc lvrg ffc urng crh mr n h US n Frnc n h nc rr blv h h mr coul crh lb wh low robbly h lvrg ffc omn h long rm rnc fur n h quy w. Our fuur rrch wll fr n h lmng rul of Alnr n Lzr 4b o h c of ymmrc GARCH vrnc comonn n hu rv h funconl form of h m vryng r rmum. Comron of h r nurl w gnr by h NM-GJR mol woul b of rculr nr. Thn h cr n connuou vron of h NM-GJR mol coul b clbr o hycl n oon mulnouly followng h wor of Chrnov n Ghyl n ohr. Thrfr rm wh rcbl lognorml mur rnon n mrgnl rc n h mol houl rov uful ool for rcng n hgng h nn oon. Coyrgh Alnr n Lzr. All Rgh Rrv. 5

18 ISMA Cnr Dcuon Pr n Fnnc 4-4 Rfrnc Alnr C. n Lzr E. 4: Norml mur GARCH: Alcon o chng r mollng ISMA Cnr Dcuon Pr 4-6 forhcomng n h Journl of Al Economrc Alnr C. n E. Lzr 4b Th connuou lm of norml mur GARCH roc ISMA Cnr Dcuon Pr n Fnnc DP4- vlbl from B D.S. 99: Th crh of '87: W c Th vnc from oon mr Journl of Fnnc B D.S. 997: Th wn rmum: oon rcng unr ymmrc roc Avnc n Fuur n Oon Rrch Vol B D.S. : Po-'87 Crh fr n h S&P 5 fuur oon mr Journl of Economrc Br G. n Wu. G. : Aymmrc volly n r n quy mr Th Rvw of Fnncl Su Vol. 3 No. -4 Blc F. 976: Su of oc rc volly chng Procng of h 976 Mng of h Amrcn Scl Aocon 77-8 Blc F. n Schol M. 973: Th rcng of oon n coror lbl Journl of Polcl Economy Brgo D. n F. Mrcuro : Dlc n mur ffuon for nlyclly-rcbl ml mol n: Gmn H. Mn D.B. Pl S.R. Vor A.C.F. Eor Mhmcl Fnnc Bchlr Congr Srngr Brln Chrnov M. n E. Ghyl : A Suy owr unf roch o h jon mon of objcv n r nurl mur for h uro of oon vluon Journl of Fnncl Economc Vol C J. 994: A Mrov mol of wchng-rgm ARCH Journl of Bun & Economc Sc Vol Crr P. n Wu L. 3: Th fn momn log bl roc n oon rcng Journl of Fnnc Chroffrn P. Hon S n Jcob. 4: Oon vluon wh cononl wn forhcomng n Th Journl of Economrc Chroffrn P. n Jcob. 4: Whch GARCH mol for oon vluon Mngmn Scnc Vol Bh G. N. n Mn D. 3: 'Soc rurn chrcrc w lw n h ffrnl rcng for nvul quy oon Th Rvw of Fnncl Su 6-43 Drmn E. n ml M. 997: Th rn of chng n ml n voll Qunv Srg Rrch No Golmn Sch Engl R. F. n Ng V.. 993: Murng n ng h mc of nw on volly Journl of Fnnc Frnnz C. n Sl M. 998: On Byn mollng of f l n wn Journl of h Amrcn Scl Aocon Glon L. R Jgnnhn R. n Runl D. E. 993: On h rlonh bwn c vlu n h volly of c rurn on oc Journl of Fnnc H M. Mn S. n Polll M.S. 4: M norml cononl hrocy Journl of Fnncl Economrc No. -5 Hmlon J.D. n Suml R. 994: Auorgrv cononl hroccy n chng n rgm Journl of Economrc Vol Jcwrh J.C. n Rubnn M. 996: Rcovrng robbl from oon rc Journl of Fnnc Coyrgh Alnr n Lzr. All Rgh Rrv. 6

19 ISMA Cnr Dcuon Pr n Fnnc 4-4 onov M. n Mn D. : Oon rcng ung vrnc gmm mrov chn Rvw of Drvv Rrch Nlon D. B. 99: Cononl hroccy n rurn: nw roch Economrc Nwy W.. 985: Gnrlz mho of momn cfcon ng Journl of Economrc Rubnn M. 994: Iml bnoml r Journl of Fnnc Tomn R.G. : Iml volly urfc: uncovrng rgulr for oon on fnncl fuur Th Euron Journl of Fnnc Wu G. : Th rmnn of ymmrc volly Th Rvw of Fnncl Su Vol. 4 No Coyrgh Alnr n Lzr. All Rgh Rrv. 7

20 ISMA Cnr Dcuon Pr n Fnnc 4-4 Coyrgh Alnr n Lzr. All Rgh Rrv. 8 An : Momn of h Aymmrc Norml Mur GARCH Mol W u h followng noon: E E n E y for. Tng con of n gv for NM-AGARCH E E µ y Tng con of n nong.5 by gv for NM-GJRGARCH E E µ y umng romly h n r nnn. Tng con of 3 gv: y µ Th hr momn : y E E h 3 3 µ 3 µ n h wn cn b r 3/ h. Th c uro n boh mol : z E E whr g B f B ' ' E z Th fourh momn u h followng noon:

21 ISMA Cnr Dcuon Pr n Fnnc 4-4 Coyrgh Alnr n Lzr. All Rgh Rrv. 9 ' y 4 µ µ 6 M M M M B whr n. n 5 n j j j n j M M M M M g whr j j j j j n n n. w c c w M f Furhrmor: n y n y h w j j j j j j q y r c wh q µ n [ ] 4 y y h r n y y r n. Th full rvon of h rul vlbl from h uhor on rqu.

22 ISMA Cnr Dcuon Pr n Fnnc 4-4 Coyrgh Alnr n Lzr. All Rgh Rrv. An : Numrcl Drvv of h Aymmrc Norml Mur GARCH Mol Th only ffrnc from h NM-GARCH mol numrcl rvv Alnr n Lzr 4 h fr n con orr rvv of wh rc o n h r follow: z σ β σ ' ' w σ β σ T A A w For NM-AGARCH: ' z. Th rng vlu for h ron for r: ' whr T T ' A Th rng vlu for h comuon r ' For NM-GJRGARCH: ' z. Th rng vlu for h ron for r: '. 5 whr T T ' A n h rng vlu for h comuon r '

23 ISMA Cnr Dcuon Pr n Fnnc 4-4 Coyrgh Alnr n Lzr. All Rgh Rrv. An 3: Th Auocorrlon Funcon of h Squr Error n h Aymmrc Norml Mur GARCH Mol Th uocorrlon of h qur rror cn b r : 7 4 z c ] E[ ] E[ Vr Cov Corr [ ] [ ] b E E c µ µ For NM-AGARCH: > h b c b b c b For NM-GJRGARCH: 5 b c. b Th rng vlu r: z c n z ' z c b g f B. 7 Snc h vrnc of h NM-GARCH mol cn b r GARCH vrnc ccorng o Bollrlv 986 h uocorrlon cn lo b wrn n AR roc.

24 ISMA Cnr Dcuon Pr n Fnnc 4-4 Tbl. Emon rul for h CAC 4 In Mol f. for µs..for E-3 3.E-4 9.9E-4 6.8E-4.E-4 7.5E-4 6.8E-4.E-4 7.4E-4 4.3E-4 -.E-5 5.6E-4 4.E-4 -.6E-5 5.4E µ Uncononl.34%.83%.83%.75%.7%.7%.75%.6%.4% 3.%.36%.% 3.9% 3.%.3% Uncononl.4%.4%.97%.65%.7%.3% Uncononl 4.5% 43.73% 4.3% 4.% 46.56% 4.3% Uncononl Uncononl Logllhoo Momn % Dny ACF Coyrgh Alnr n Lzr. All Rgh Rrv.

25 ISMA Cnr Dcuon Pr n Fnnc 4-4 Tbl. Emon rul for h DAX 3 In Mol f. for µs..for E-3 4.9E-4.E-3.9E-4 5.5E-5 4.E-4.9E-4 7.E-5 4.E-4 8.E-5-6.8E-5.6E-4 7.8E-5-7.E-5.5E µ Uncononl 4.3% 3.% 3.9% 37.% 9.9% 9.75% 37.6% 3.% 9.54% 4.87% 4.% 3.6% 4.8% 4.% 3.54% Uncononl 3.7%.69%.% 3.6%.64%.9% Uncononl 49.7% 49.5% 47.% 48.5% 47.97% 45.89% Uncononl Uncononl NA NA NA NA NA Logllhoo Momn % Dny ACF Coyrgh Alnr n Lzr. All Rgh Rrv. 3

26 ISMA Cnr Dcuon Pr n Fnnc 4-4 Tbl 3. Emon rul for h FTSE In Mol f. for µs..for E-4-4.4E-5.9E-4.6E-4 -.5E-4.8E-4.6E-4 -.4E-4.8E-4.E-4-3.9E-4.E-4 6.E-5-3.8E-4.9E µ Uncononl 8.53% 6.97% 7.4% 8.9% 7.5% 7.73% 8.9% 7.4% 7.68% 7.95% 7.% 7.9% 7.99% 7.7% 7.7% Uncononl 6.5% 5.73% 6.% 6.9% 5.78% 6.9% Uncononl 6.3% 6.9% 7.39% 4.48% 7.% 7.5% Uncononl Uncononl Logllhoo Momn % Dny ACF Coyrgh Alnr n Lzr. All Rgh Rrv. 4

27 ISMA Cnr Dcuon Pr n Fnnc 4-4 Tbl 4. Emon rul for h NIEI 5 In Mol f. for µs..for E-3 -.6E-4.5E-3.E-3-6.6E-4 8.9E-4.E-3-6.7E-4 9.E-4 6.8E-4-8.E-4 6.E-4 7.4E-4-7.7E-4 6.3E µ Uncononl 4.99% 4.8% 5.96% 5.85% 4.66% 7.5% 5.87% 4.66% 7.5% 5.6% 4.3% 5.9% 5.9% 4.46% 6.3% Uncononl.44%.8% 3.98%.75%.39% 4.% Uncononl 47.55% 5.38% 5.63% 49.36% 5.7% 5.7% Uncononl Uncononl NA Logllhoo Momn % 3 Dny ACF Coyrgh Alnr n Lzr. All Rgh Rrv. 5

28 ISMA Cnr Dcuon Pr n Fnnc 4-4 Tbl 5. Emon rul for h S&P 5 In Mol f. for µs..for E-4 -.9E-4.E-4 6.8E-5 -.3E-4.7E-4 7.6E-5 -.3E-4.8E-4.E-5 -.8E-4.3E-4.6E-5 -.8E-4.3E µ Uncononl 9.5% 6.63% 6.% 4.% 7.93% 7.77% 3.43% 7.7% 7.47% 7.56% 6.5% 6.4% 7.64% 6.3% 6.3% Uncononl 5.% 4.7% 5.3% 5.6% 4.64% 4.77% Uncononl 9.78% 3.49% 3.77% 3.5% 9.6% 9.38% Uncononl Uncononl NA NA NA Logllhoo Momn % 3 Dny ACF Coyrgh Alnr n Lzr. All Rgh Rrv. 6

29 ISMA Cnr Dcuon Pr n Fnnc 4-4 Fg.. Smul Equy In Sw Norml GARCH Volly 5% % 5% % 5% % Monyn.6..8 m m5 m8 m3 m6m9 Mury b -GARCH Volly 45% 4% 35% 3% 5% % 5% % 5% % Monyn.6..8 m m5 m8 m3 m6m9 Mury c Aymmrc -GARCH Volly 35% 3% 5% % 5% % 5% % Monyn.6..8 m m5 m8 m3 m6m9 Mury Coyrgh Alnr n Lzr. All Rgh Rrv. 7

30 ISMA Cnr Dcuon Pr n Fnnc 4-4 GJR Volly 5% % 5% % 5% % Monyn.6..8 m3 m6m9 m m5 m8 Mury GJR- Volly 5% % 5% % 5% % Monyn.6..8 m3 m6m9 m m5 m8 Mury f GJR-ymmrc Volly 5% % 5% % 5% % Monyn.6..8 m m5 m8 m3 m6m9 Mury Coyrgh Alnr n Lzr. All Rgh Rrv. 8

31 ISMA Cnr Dcuon Pr n Fnnc 4-4 g NM- GARCH zro mn Volly 5% % 5% % 5% % m8 m5 m Monyn.6..8 m3 m6m9 Mury h NM-GARCH Volly 5% % 5% % 5% % m m5 m Monyn.6..8 m3 m6m9 Mury NM-GJR Volly 5% % 5% % 5% % Monyn.6..8 m m5 m8 m3 m6m9 Mury Coyrgh Alnr n Lzr. All Rgh Rrv. 9