Ad-hoc Analytic Option Pricing under Nonlinear GARCH with. NIG Lévy Innovations

Transcription

1 Ad-hoc Analyic Opion Picing nd Nonlina GARCH wih NIG Lévy Innovaions By Shaif Momd a Michal Dmpsy b M. Hmayn Kabi c* afiq Chodhy d a Dpamn of Mahmaics Univsiy of Dhaka Bangladsh shaif_mah@yahoo.com b School of Economics Financ and Making RMI Univsiy Mlbon Asalia michal.dmpsy@mi.d.a c School of Economics and Financ Massy Univsiy Palmson Noh Nw Zaland M.H.Kabi@massy.ac.n d School of Bsinss Univsiy of Sohampon UK .chodhy@soon.ac.k * Cosponding aho Absac his pap poposs an appoxima closd-fom opion-picing modl basd on a GARCH pocss wih NIG (Nomal Invs Gassian) Lévy innovaions. W dvlop h mahmaical famwok and dmonsa how o obain a closd-fom solion o h opion pic whn h n dynamics a chaacid by NIG innovaions ha fd volailiy fom a GARCH pocss wih NIG innovaions. Using a sampl of S&P 5 opions daa w caliba h poposd modl alongsid popla xising modls a diffn poins in im and fo diffn sampl sis. W pfom boh in-sampl and o-ofsampl fiing. Ovall w find ha o modl pfoms significanly b han xising modls boh insampl and o-of-sampl. Kywods: Lévy innovaions sochasic volailiy GARCH calibaion Nomal Invs Gassian

2 Ad-hoc Analyic Opion Picing nd Nonlina GARCH wih NIG Lévy Innovaions Absac his pap poposs an appoxima closd-fom opion-picing modl basd on a GARCH pocss wih NIG (Nomal Invs Gassian) Lévy innovaions. W dvlop h mahmaical famwok and dmonsa how o obain a closd-fom solion o h opion pic whn h n dynamics a chaacid by NIG innovaions ha fd volailiy fom a GARCH pocss wih NIG innovaions. Using a sampl of S&P 5 opions daa w caliba h poposd modl alongsid popla xising modls a diffn poins in im and fo diffn sampl sis. W pfom boh in-sampl and o-of-sampl fiing. Ovall w find ha o modl pfoms significanly b han xising modls boh in-sampl and o-of-sampl. Kywods: Lévy innovaions sochasic volailiy GARCH calibaion NIG. INRODUCION h wll-known dawbacks of h sminal Black-Schols opion-picing modl (97) a ha i fails o accon fo skwnss and havy ails in h ndlying ass and volailiy clsing ov im. A nmb of appoachs hav bn sggsd ha dal wih nmb of dawbacks. Noabl amongs hs a h jmp-diffsion modl (Mon 976; Psychoyios al. ) h sochasic volailiy modl (Hll and Whi 987; Hson 99; Bas ) h GARCH modl of Engl (98) and Bollslv (986) applid o opion picing (.g. Dan 995; Hson and Nandi ; Baon-Adsi al. 8) and h Lévy appoach (.g. Gman and Yo ; Gman ; Ca al. ; Dingc and Homann ).Whil h jmp-diffsion appoach consids disc jmps as wll as diffsion in sock pics h volailiy islf is modld as a scond sochasic pocss nd h sochasic volailiy appoach. h Lévy appoach consids h sock pics following a Lévy pocss. his appoach has bn fh xndd by Ca and W (4) and Hang and W (4) who s im-changd Lévy pocsss o ackl h mpiical fas of obsvd opion pics. Onhanalai (4) finds ha modls ha do no pic jmp isk nd

3 Lévy pocsss canno xplain h divgnc bwn h physical and isk-nal pobabiliy mass and ha h modl wih infini-aciviy jmps ah han Bownian incmns is mo lvan in ass picing. his pap addsss h infac of h GARCH and Lévy bodis of lia and poposs an alnaiv y complmnay appoach o hos of Ca and W (4) and Hang and W (4). O aim is o povid h fas y convincing amn of opion picing nd a spcial-cas Lévy pocss h Nomal Invs Gassian (NIG). W innd o achiv his placing h NIG innovaion wih h appoxima nomal innovaion. Moolman (8) Kim al. (8 ) and Onhanalai (4) incopoa GARCH wihin a Lévy pocss. Mo cnly Badsc al. (5) popos a non-gassian GARCH opion-picing modl. Howv hs sdis ly on h Mon Calo simlaion o pic opions and as sch do no bnfi fom an analyic appoach. Anoh noabl xampl in h sam spii is Mci (8) who combins GARCH wih mpd sabl (S) innovaions and povids an analyical solion compaabl o os. Howv his analysis is basd on a mo limid daas. Mo impoanly h GARCH- S volailiy dynamics sd by Mci (8) a applicabl only fo innovaions coming fom a posiiv Lévy pocss. W on h oh hand inodc a mahmaical famwok wih a nonlina volailiy dynamic appoximad by placing sandad NIG innovaions wih sandad nomal innovaions and dmonsa how ad-hoc analyic solions can b divd in h psnc of boh posiiv and ngaiv Lévy innovaions. Kim al. (8) popos KR disibion as on sch sbclass of h mpd sabl disibion and mpiically s and compa wih CGMY and h modifid mpd sabl (MS) disibion. Kim al. () inodc h apidly dcasing mpd sabl (RDS) GARCH modl wih an infinily divisibl disibd innovaion and compa h findings basd on classical mpd sabl (CS) GARCH modl.

4 W focs on h picing of a Eopan opion fo which w fis nd o know h isknal dnsiy a maiy. Howv a poblm wih h sandad GARCH s-p is ha w only know h isk-nal disibion on-sp ahad. o ovcom his difficly Hson and Nandi () popos a GARCH-lik modl wih nomal innovaions fo which hy comp h chaacisic fncion of h ndlying sing a csiv pocd and hn mploy h Foi invsion appoach of Hson (99) o pic opions. Unfonaly hi modl is no sffici n ly flxibl o xplain som wll-known opion biass paiclaly hos lad o sho-m maiy opions; Chisoffsn (6) conjcs ha his limiaion migh b d o h fac ha hi singl-piod innovaions a nomal. Vaios amps hav sbsqnly bn mad o lax h nomaliy assmpion in h conx of h long-hoion GARCH opion picing. Fo xampl Badsc and Klpg (8) plac nomaliy wih a smi-paamic appoach and hn s Mon Calo simlaion o pic h opions. his howv is im-consming o implmn and also infficin as i ss only sock pic infomaion no h infomaion conaind in opion pics. h psn pap sks o fill his gap by offing an appoxima closd-fom solion fo h picing of Eopan opions nd a GARCH famwok wh innovaions follow a NIG Lévy pocss. his is pacically sfl as i maks opion picing mch fas in al im. Sch a famwok may also b sfl fo GARCH opion picing nd oh Lévy pocsss. h maining sc of his pap is as follows: Scion divs h closd-fom GARCH opion-picing fomla wih NIG-Lévy innovaions. Scion discsss daa implmnaion isss calibaion and mpiical sls. Scion 4 conclds.

5 . CLOSED-FORM GARCH OPION PRICING WIH NIG-LÉVY INNOVAIONS wo impoan cn conibions in h GARCH-Lévy aa a Chisoffsn al. () and Chisoffsn al. (). h fom wok ss o h boad chaaciaions of Lévy dynamics b dos no apply hi modl o opions daa o off xplici divaions fo h Lévy innovaions ha hav bn xnsivly sd in h divaivs picing lia. h la sdy ackls affin GARCH dynamics wih Lévy innovaions and dmonsas ha closd-fom picing is no possibl fo innovaions basd on Lévy pocsss xhibiing boh posiiv and ngaiv jmps. W dvlop a mahmaical famwok o div appoxima closd-fom fomla simila o hos of Hson and Nandi (). Mo spcifically w plac hi condiional nomal innovaions wih innovaions divd fom Nomal Invs Gassian (NIG) pocss which is a fom of Lévy pocss. L s assm ha h sock pic follows h pocss S S X () wh X follows a im-vaying NIG Lévy pocss. h chaacisic fncion of (X) is ha of NIG ( ) in paicla h andom vaiabl (X) i is givn by { E isx xp ( is) }. () h momns of NIG() andom vaiabl X a shown in Appndix. 4

6 o pic an opion howv w also nd h condiional disibion of h ndlying ass a a mli-piod-ahad maiy '. W div sch a disibion following h csiv mhod dvlopd by Hson and Nandi ().hi csiv pocd is basd on h ida ha h condiional MGF can b xpssd as log( S ) [ I ] S xp A( ) B( ) [ ] E. () h goal is o solv fo A ( ) and B ( ) W pojc qaion () by on piod o obain log( S ) [ I ] S xp A( ) B( ) fo h NIG-Lévy innovaions chaaciing. [ ] E. (4) W assm ha h gnal fom of h condiional MGF holds fo im and s h iaiv popy of condiional xpcaions (which is h cnal fa of h Hson-Nandi csiv appoach) o obain h cosponding xpssion fo h condiional MGF a im : E log( S ) log( S ) [ I ] E E E S [ [ I ] I ] [ xp[ A( ) B( ) ] I ]. (5) X Using qaion () w hav S S. Plgging his ino qaion (5) w obain log( S ) X [ I ] E S xp A( ) B( ) E S [ [ ] I ] X E xp[ A( ) B( ) ] I [ ]. (6) Hnc no ma how many sps a h bwn and w can s qaions (4) and (5) csivly o div h condiional MGF a any maiy givn h infomaion availabl p o En passan w no ha o solion mhod can b xndd o oh Lévy innovaions sch as Vaianc Gamma (VG) and CGMY pocs s s. 5

7 . A compaison of () and (6) hn allows s o div h csiv laions fo h coffici ns ( ) and B ( ) A... NIG IME-CHANGED LÉVY INNOVAIONS W div h csiv cofficin laions fo h NIG-Lévy innovaions. Givn ha h sock pic follows h dynamics () h log n pocss follows in GARCH sings: X λ. (7) H ~ NIG ( ) I and h volailiy pocss follow h GARCH() spcificaio n (8) ( ) wh h abov scaling nss ni vaianc fo innovaions. Poposiion..: Fo h GARCH dynamics in (7) wih ~ NIG ( ) I wh is as in (8) h condiional skwnss and condiional kosis can b obaind as in (A7) and (A8) in Appndix A. [S Appndix A fo h poof.] 6

8 7 Poposiion..: Fo h GARCH dynamics in (7) wih ~ NIG I wh is as in (8) h qivaln maingal laionships among h paams a as givn by (B)- (B5) and (B8)-(B9). [S Appndix B fo h poof.] Now w s h abov wo qaions (7) and (8) o plac Xand in qaion (6) o obain h following GARCH NIG-Lévy dynamics: [ ] I I S B A E S E xp log λ (9) [ ] [ ] [ ] I I () xp ~ sinc xp xp B B B A S NIG B B B A S B B B A E S λ λ λ () Compaing qaions () and (9) w hn obain h following csiv laions:

9 8 [ ] λ B B B B A A () Sinc NIG ~ I ofn assms posiiv as wll as ngaiv vals w sic oslvs o nonlina dynamics of h Hson-Nandi yp. L s assm h following nonlina isk-nal dynamics 4 : sdnig NIG NIG γ µ γ µ µ γ () wh μ is h xpcd val of a NIG andom vaiabl and is givn by qaion (A) in Appndix A. Moov his modificaion of h volailiy dynamics -sablishs h hisoica l and isk-nal laion of and inodcs a nw hisoical and isk-nal laion fo h nw paam as dscibd in Appndix B. W can fh simplify h abov qaion o obain 4 his inodcion of non-linaiy dos affc h qivaln maingal laionships among h paams and is dmonsad in Appndix C. γ

10 sdnig γ. () 4 A poblm wih his chaaciaion of volailiy howv is ha whn is plggd ino qaion (6) i dos no yild xplici csiv laions fo A ( ) and B ( ) no closd-fom Eopan opion valaion is possibl wiho fh appoximaion. 5. Consqnly O poposd solion is o apply an appoximaion o h dynamics () ha pholds closd-fom valaion chniqs simila o hos of Hson and Nandi (). In paicla whn h dynamics () a chaacid fo NIG innovaions w popos an ad-hoc appoximaion ha psvs h chaaciaion of dynamics b placs h sandad NIG innovaions by sandad Nomal innovaions: sdnomal γ. (4) 4 his ad-hoc appoximaion is moivad by Hson-Nandi closd-fom picing basd on h following laion involving a sandad Nomal andom vaiabl E wh ~ N( ) [ xp{ a( b) }] xp ln( a). ab (5) a 5 A simila poblm was ncond by Onhanalai () in his sdy of GARCH-Lévy dynamics fo ass picing. H concldd ha fo Lévy innovaions capabl of xhibiing boh posiiv and ngaiv jmps h is no alnaiv o h Mon-Calo valaion of divaivs. Howv h poblm wih Mon Calo is ha i qis a long im o pic vn a singl opion. o amp sch picing w nd o consid a lag nmb of simlaions a las 5 ials a ndd a h xpns of hg compaional im ha nds qick calibaion pacically infasibl. 9

11 W apply h volailiy dynamics (4) (which a chaacid fo NIG innovaion b appoximad hogh h placmn of sandad NIG vaias by sandad Nomal vaias) in h gnal csiv laion (6): [ ] I I 4 log xp S B A E S E γ λ (6) W hn apply h laion (5) o (6) o simplify fh: [ ] [ ] [ ] I I 4 4 log log xp ~ sinc log xp xp S B B B B B A S NIG B B B B B A S E γ λ γ λ (7) A compaison of qaions (7) and () givs h following csiv laions:

12 B ( ) A( ) B( ) log[ B( ) ] A ( ) λ B( ) B ( ) γ B( ) 4 ( ) ( ). (8).. OPION PRICING AND CALIBRAION MEHODOLOGY W obain h opion pics hogh Foi Invsion as in Hson (99) and Hson and Nandi (). Fo h closd-fom (p o nmical ingaion) GARCH modl wih NIG innovaio ns w dno h modl pic byc cfgnig (wh h sbscip allds o closd-fom GARCH-NIG). his modl has svn paams o b simad 6 [ γ ]. Givn h paam consains w consid h calibaion as a consaind opimiaio n poblm ah han a simpl nonlina las-sqa on. h consains ais fom h GARCH sc as wll as h sal NIG paamiaion > and < 6 h on-day-ahad GARCH vaianc can also b ad as a paam. Howv aing on-day-ahad volailiy as a paam is only managabl fo a fw days mak daa bcas ach nw day cas a nw paam o b simad. So fo calibaions sing opion cods ov a long piod w nd o dicly fd h on-day-ahad volailiis in a dynamic fashion. Hson and Nandi () inp hs hogh a GARCH pocss ha focs h calibaion o ly havily on a long im sis of ass ns in addiion o h mak pic of opions. By conas w implmn h sam appoach sing GARCH-NIG closd-fom volailiy dynamics. O calibaion ofn povids a b fi han Hson and Nandi's modl dos. Anoh advanag of o appoach is ha i nds pas ass ns dndan.

13 i.. < >. hs h calibaion of h modl coms fom h solion o h following opimiaion poblm 7 [ ] [ ] b s c C n Minimi n i i cfgnig i mak γ γ A.. (9) wh A and b [ ]. W implmn diffn modls: () BS: Black-Schols (97) modl () VG: Vaianc Gamma of Madan al. (998) () NIG: Nomal Invs Gassian modl of Schon () (4) JD-DE: Exponnial dobl jmp modl of Ko () (5) CGMY: Ca al. s () modl (6) HS: Hson s (99) sochasic volailiy modl (7) HN(R): sicd vsion of Hson and Nandi s () GARCH modl (8) HN(U): nsicd vsion of Hson and Nandi s () GARCH modl (9) SC: Sco s (997) modl () B: Bas (996) modl () CH: 7 W implmn h consaind opimiaion sing h MALAB fncion fmincon.

14 Chisoffsn al. s (6) modl and () CFGNIG: o poposd closd-fom GARCH modl wih NIG innovaions. abl psns h chaacisics fncions of all modls.. DAA AND PILO SURVEY W s daily cods of opions win on h S&P5 indx add on h Chicago Boad Opions Exchang (CBOE). W iv h S&P5 indx p-and-call opion qos fom hompson R ick hisoy. h sampl piod ns fom Janay hogh o Dcmb 4. Fo calibaion and o-of-sampl assssmn w mploy daa p nil Jn 5 8. W s h daa on vy Wdnsday as i is h day of h wk las likly o b a holiday and i is lss likly o b impacd by day-of-h-wk ffcs (Onhanalai 4) 9. W hn clan o daa sing h sam ls applid by Hson and Nandi (): W do no consid any opion of a paicla monynss o maiy mo han onc in o sampl. his liminas a lag nmb of obsvaions. W xcld dp o-of-h-mony and dp in-h-mony opions bcas hs a ih infqnly add and/o hav low nogh pics fo h bid-ask spad o consi a majo poion of h pic. o b pcis only cods having an indx-o-sik aio bwn.9 and. a incldd in o sampl. W only incld opions ha hav bwn six and days o xpiaion. W limina vy long-m opions bcas hy a no acivly add and a pon o mispicing. 8 W discss diffn agggaion schms and cosponding in-sampl and o-of-sampl daa qimns in scion 5. 9 Dmas Flming and Whaly (998) discss h advanags of Wdnsday daa in dail.

15 Convsly w limina vy sho-m opions bcas hy hav sbsanial im dcay which maks i difficl o liably dmin h volailiy paam. Af applying h filing ls dscibd abov w hav 844 opions ov h im window. W fis cay o a pilo svy sing h opions codd on Wdnsday Jly 5. Sinc h disc im modls ak a considabl im o caliba Hson and Nandi () did no consid mo han six monhs cods a a im. In paicla hy ak a ya s s of cods bokn down ino wo six-monh piods considing h fis six monhs as an in-sampl piod and h scond six monhs as o-of-sampl. W follow h sam pocd and bak down annal cods in h sam way sing h fis six monhs cods fo in-sampl calibaion and h scond six monhs cods o assss o-of-sampl pfomanc. W also consid long piods of wo yas and h yas agggaion of opion cods. abl psns dscipiv saisics fo h opion qos basd on monynss and maiy. W dfin monynss as S/K wh S is h opion pic a maiy and K is h sik pic. 4. GOODNESS-OF-FI o assss in-sampl goodnss-of-fi w s h Roo Man Sqa Mas RMSE which consids qadaic dviaions bwn modl and mak pics: i modl mak Ci Ci n RMSE. () n o assss o-of-sampl goodnss-of-fi w s a naiv mas known as avag absol o (AAE) which consids lina dviaions bwn modl and mak pics: AAE N i modl pici mak N pic i. () 4

16 W also assss o-of-sampl goodnss-of-fi sing Man-Osid-Eo (MOE) which is sd o assss a modl s avag ndncy o ov-pic o ndpic: MOE N i [ ] ( modl pici aski ) Ι { modl pic ask } ( modl pici bidi ) Ι { modl pic bid } i i N i i. () Whn assssing o-of-sampl pfomanc w sic o pod sls o modls ha xplicily incopoa sochasic volailiy. W find ha modls ha incopoa jmps povid xclln fis ov vy sho piods (.g. on day o a copl of days obsvaions); b sch modls pfom mch wos fo long piods laiv o modls which consid xplici sochasic volailiy dynamics. Consqnly w sic o o-of-sampl pfomanc o svn diff n modls: Hson s (99) coninos-im sochasic volailiy modl sicd and nsicd vsions of Hson and Nandi s disc-im GARCH volailiy modl wih nomal innova io ns (Hson and Nandi ) Sco (997) Bas (996) Chisoffsn al. (6) and o CFGNIG modl. 5. CALIBRAION W po calibaions sing diffn coss-scions of opions codd on a wid im fam: fom Janay o Dcmb 4. W caliba modls nd h mpoal agggaio n schms. h fis schm cosponds o calibaion sing opions add on h fis six monhs of ach of h yas and 4 and hn w assss o-of-sampl pfomanc sing opions add on h maining six monhs of h lvan ya. abls 4A 5A and 6A po h in-sampl pfomanc and abls 4B 5B and 6B po h o-of-sampl pfomanc fo h and 4 calibaions spcivly. 5

17 O scond mpoal agggaion schm calibas modls sing infomaion conaind in opion conacs add ov wo-ya piods. Mo pcisly w caliba h modls sing opions add ding - and -4. In h fis cas w s h fis six monhs conacs of 4 o assss modls o-of-sampl pfomanc and in h scond cas w s conacs fo h fis six monhs of 5 fo o-of-sampl assssmn. abls 7A and 8A po h in-sampl sls and 7B and 8B po h o-of-sampl sls fo h - and - 4 calibaions spcivly. O hid agggaion schm consids h fll h-ya piod of -4. abl 9A pos h calibaion sls cosponding o his schm and abl 9B psns h cosponding o-of-sampl assssmn fo h piod fom Janay 5 o Jn RESUL ANALYSIS 6.. In-sampl calibaion sls abl shows h RMSE along wih paam simas of pilod -day calibaion fo all modls fo h opions add on Wdnsday Jly 5. Af claning h daa w hav 78 cods o consid on ha paicla day. Modls wih im-vaying volailiy nd GARCH wih nomal innovaions opfom oh sophisicad chaaciaions and modls wih imvaying volailiy nd GARCH wih NIG innovaions opfom modls wih GARCH volailiy wih nomal innovaions. 6

18 Compaing h paam simas in abls 4A 5A 6A 7A 8A and 9A w find ha wih h xcpion of h on-day calibaion in abl BS-implid volailiy is sabl ding diffn im piods and ov diffn sampl sis. In h VG modl h and υ paams a modaly sabl b h θ paam flcas fom sampl o sampl. Fo xampl h simas of θ basd on h fis half of vy ya s daa in abls 4A 5A and 6A a and -96 spcivly. h sima of h sam paam fo h piod Janay -Dcmb calibaion is.78 (abl 7A) which n o o b -.6 fo h piod Janay -Dcmb calibaion (abl 8A). In h NIG modl boh and paams flca acoss diffn sampls b h paam is mo sabl. In h JD-DE modl wih h xcpion of h on-day calibaion h paam is sabl acoss h diffn sampls; h λ and p paams bcom mo sabl as sampl si incass b h η and η flca ispciv of sampl si. h sima of p basd on h fis half of vy ya s daa angs fom.79 o.9998 whil fo calibaions basd on wo-ya and h-ya piods h simas main clos o.45. In h CGMY modl all fo paams flca significanly acoss h diffn sampls. In h HS HN(R) and HN(U) modls all h paams vay significanly fom on sampl o anoh. W find simila flcaions in paams acoss sampls fo boh h SC and B modls. In h CH modl wih h xcpion of h η paam all h oh paams flca acoss diffn sampls. Finally fo o poposd GARCH-NIG modl wih h xcpion of h paam all paams flca acoss h diffn sampls. In implmning sophisicad alnaivs o bnchmak BS modl w follow h xac pocd as dscibd and followd by h cosponding ahos. Whil calibaing Scos (997) Bas (996) and oh modls as iniial gss in opimiaion w sd h simas poposd by h cosponding ahos. Fo filing o h volailiy in Hson and Nandi () and Chisoffsn s GARCH-IG (6) modls w follow h simila mhodologis as poposd by h ahos. 7

19 In sho wih h xcpion of h BS-implid volailiy all h oh modls yild paams ha can vay ov im and acoss sampls. ning now o h RMSE w find ha ovall h BS modl povids h wos fi. h RMSE angs fom.9 o. fom.97 o.66 and fom.7 o.65 fo calibaions ov h sampl piod of h fis six monhs in (abl 4A) in (abl 5A) and in 4 (abl 6A) spcivly fo all modls. Wih h xcpion of h fis half of h calibaion o poposd GARCH-NIG (CFGNIF) povids h bs in-sampl fi RMSE is.74 fo h fis half of and.8 fo h fis half of. Fhmo as h sampl si incass h pfomanc impovs laiv o h pfomanc of oh modls h RMSEs fo h GARCH- NIG modl a.7 and.8 fo wo-ya in-sampl calibaions of - and -4 compad o h RMSEs in h angs of.7-. and fo h sam sampl calibaio n. Fo h h-ya calibaion in abl 9A h RMSE fo o modl is.6. In sm o poposd modl gnally opfoms xising modls basd on in-sampl RMSE fi. 6.. O-of-sampl Rsls abls 4B 5B 6B 7B 8B and 9B povid o-of-sampl fis fo opions wih h angs of imo-maiy: opions wih maiy < 4 days opions wih maiy ga han o qal o 4 days b lss han 7 days and opions wih maiy ga han o qal o 7 days b lss han days. W also po sls acoss diffn lvls of monynss: o-of-h-mony (.95 S/K<.99); a-h-mony (.99 S/K<.) and in-h-mony (. S/K<.5). h findings show ha o poposd ad-hoc analyic GARCH-NIG modl opfoms h oh modls basd on h o-of-sampl RMSE ciion in fiv o of six sampls. Only in h 8

20 sampl basd on h calibaion pfomd ding h fis half of is h GARCH-NIG modl ban and vn hn only in a singl insanc. Scond by h oh wo mass AAE and MOE h GARCH-NIG modl always opfoms h oh modls. hid h goodnss-of-fi sls achivd sing BS HS HN(R) and HN(U) SC B CH a of simila od of magni d amongs h diffn sampl sis. his wold imply fo xampl ha whn i coms o o-ofsampl fi a sochasic volailiy modl wih mlipl paams is no gaand o hav significan advanag ov h singl-paam BS modl. Foh as h calibaion piod lnghns h pfomanc of o poposd modl impovs laiv o h pfomanc of oh modls: his finding is noicabl in abls 8B and 9B. In sm o poposd ad-hoc analyic GARCH-NIG modl opfoms many xising modls by a vaiy of ciia acoss diffn sampl sis and im piods. his is d o h niq fa of o modl wh w a abl o incopoa volving condiional skwnss and kosis. W a hs abl o incopoa h obsvd mpiical chaacisics of h asymmy of h isk-nal disibion volailiy clsing and fa ails. 7. CONCLUSION his pap addsss h infac of h GARCH and Lévy bodis of lia and poposs an alnaiv appoach. O aim is o povid h fas y convincing amn of opion picing nd a spcial-cas Lévy pocss h Nomal Invs Gassian (NIG). h appoach povids a mahmaical famwok wih a nonlina volailiy dynamic appoximad by placing sandad NIG innovaions wih sandad nomal innovaions and dmonsas how ad-hoc analyic solions can b divd in h psnc of boh posiiv and ngaiv Lévy innovaions. Givn 9

21 ha h GARCH-NIG modl has mo flxibiliy o dscib h condiional volion of skwnss and h condiional volion of kosis w migh xpc i o accommoda coss-sik and coss-maiy fas b han wha oh modls do. Fo xampl skwnss and kosis in h Hson-Nandi () modl a dmind (and hnc consaind) by GARCH scal paams whas h GARCH-NIG modls skwnss and kosis in a mo flxibl imvaying fashion basd on non-nomal ah han nomal innovaions. Similaly h conino sim SVJ o SVJJ modls a limid by hi Makovian sc whas h GARCH-NIG modl is no. wo naal xnsions lnd hmslvs o fh invsigaion fo fh wok. h fis and mos obvios is o ndak compaabl analyss of oh Lévy pocsss h Vaianc- Gamma CGMY and so foh and hn o compa h pfomanc of hs diffn pocsss. A scond xnsion fis idnifid by Bas () is o fh invsiga and modl h xn o which coss-scional opion-picing pans cold b mad qaniaivly consisn wih h im-sis pans of h ndlying ass pic. Idally h isk-nal chaacisic fncio ns which a ofn sd o val coss-scional opion-picing pans cold b sd o val h im-sis popis of h ndlying ass as wll and so ns consisncy bwn h wo. Cnly howv his mains a challng. Pa of h poblm wold appa o b ha insananos opion-pic volion in sandad modls is no flly capd by ndlying ass pic movmns. Pa of h poblm migh also la o h fac ha h hoscdasiciy of GARCH conflics wih h saionay Makov im-sis popis of sandad modls. Sinc h GARCH-NIG is f of boh of hs limiaions w wold spcla ha h ga flxibiliy of h GARCH-NIG may hold h ky o achiving his objciv.

22 W s daily cods of opions win on h S&P5 indx add on h Chicago Boad Opions Exchang (CBOE). W iv h S&P5 indx p-and-call opion qos fom hompson R ick hisoy. h sampl piod ns fom Janay hogh o Dcmb 4. Fo calibaion and o-of-sampl assssmn w apply daa p nil Jn 5. Whn assssing o-of-sampl pfomanc w sic o pod sls o modls ha xplici ly incopoa sochasic volailiy. W find ha modls ha incopoa jmps povid xclln fis ov vy sho piods (.g. on day o a copl of days obsvaions); b sch modls pfom mch wos fo long piods laiv o modls which consid xplici sochasic volailiy dynamics. Consqnly w sic o o-of-sampl pfomanc assssmn o svn comping modls: Hson s (99) coninos-im sochasic volailiy modl sicd and nsicd vsions of Hson and Nandi s disc-im GARCH volailiy modl wih nomal innovaions (Hson and Nandi ) Sco s (997) modl Bas (996) modl Chisoffsn al. s (6) modl and o novl CFGNIG modl. O poposd ad-hoc analyic GARCH-NIG modl opfoms many xising modls by a vaiy of ciia acoss diffn sampl sis and im piods. his is d o h niq fa of o modl wh w a abl o incopoa volving condiional skwnss and kosis. W a hs abl o incopoa obsvd mpiical chaacisics of h asymmy of h isk-nal disibion volailiy clsing and fa ails.

23 Rfncs Badsc A. & Klpg R.J. (8). GARCH opion picing: A smipaamic appoach. Insanc: Mahmaics and Economics Badsc A. Ellio R.J. & Oga J.P. (5). Non-Gassian GARCH opion picing modls and hi diffsion limis. Eopan Jonal of Opaional Rsach Baon-Adsi G. Engl R. & Mancini L. (8). A GARCH opion picing modl wih fild hisoica l simlaion. Rviw of Financial Sdis -58 Bas D.S. (). Empiical opion picing: A Rospcion. Jonal of Economics Bas D.S. (996). Jmps and sochasic volailiy: Exchang a pocsss implici in Dsch Mak opions. Rviw of Financial Sdis Black F. & Schols M. (97). h picing of opions and copoa liabiliis. Jonal of Poliical Economy Bollslv. (986). Gnalid aogssiv condiional hoskdasiciy. Jonal of Economics 7-7. Ca P. Gman H. Madan D. B. & Yo M. (). h affin sc of ass ns: an mpiical invsigaion. Jonal of Bsinss Ca P. & W L. (4). im-changd Lévy pocsss and opion picing. Jonal of Financia l Economics 7-4. Chisoffsn P. Hson S.L. & Jacobs C. (6). Opion valaion wih condiional skwnss. Jonal of Economics Chisoffsn P. Doion C. Jacobs P. & Wang Y. (). Volailiy Componns Affin Rsicions and Nonnomal Innovaions. Jonal of Bsinss & Economic Saisics 8(4) Chisoffsn P. Elkamhi R. Fno B. & Jacobs K. (). Opion valaion wih condiional hoskdasiciy and nonnomaliy. Rviw of Financial Sdis (5) 9-8. Chisoffsn P. Jacobsc K. & Onhanalaia C. (). Dynamic jmp innsiis and isk pmims: Evidnc fom S&P5 ns and opions. Jonal of Financial Economics 6() Dingc K D. & Homann W. (). A gnal conol vaia mhod fo opion picing nd Lévy pocsss. Eopan Jonal of Opaional Rsach Dan J.C. (995). h GARCH opion picing modl. Mahmaical Financ 5 -. Dmas B. Flming J. & Whaly R. (998). Implid volailiy fncions: mpiical ss. Jonal of Financ Engl R. (98). Aogssiv condiional hoskdasiciy wih simas of h vaianc of UK inflaions. Economica Gman H. Madan D. & Yo M. (). im changs fo Lévy pocsss. Mahmaical Financ Gman H. (). P Jmp Lévy pocsss fo ass pic modlling. Jonal of Banking & Financ Gb H.U. & Shi E.S.W. (994). Opion picing by Essch ansfoms. ansacions of h Sociy of Acais Hson S. (99). A closd fom solion fo opions wih sochasic volailiy wih applicaions o bond and cncy opions. Rviw of Financial Sdis Hson S.L. & Nandi S. (). A closd fom GARCH opion valaion modl. Rviw of Financia l Sdis Hang J.Z. & W L. (4). Spcificaion analysis of opion picing modls basd on im-changd Lévy pocsss. Jonal of Financ

24 Hll J. & Whi A. (987). h picing of opions on asss wih sochasic volailiis. Jonal of Financ Kim K.S. Rachv S.. Bianchi M.L. & Faboi F.J. (8). Financial mak modls wih Lévy pocsss and im-vaying volailiy. Jonal of Banking and Financ Kim K.S. Rachv S.. Bianchi M.L. & Faboi F.J. (). mpd sabl and mpd infinily divisibl GARCH modls. Jonal of Banking and Financ Ko S. (). A jmp diffsion modl fo opion picing. Managmn Scinc Madan D. Caa P. & Chang E. (998). h Vaianc Gamma pocss and opion picing modl. Eopan Financ Rviw Mci L. (8). Opion picing in a GARCH modl wih mpd sabl innovaions. Financ Rsach Ls Mon R. (976). Opion picing whn ndlying sock ns a disconinos. Jonal of Financial Economics Moolman G.P. (8). Opion picing: A GARCH modl wih Lévy innovaions. Univsiy of Johannsbg. Dissaion. Onhanalai C. (). A nw class of ass picing modls wih Lévy pocsss: hoy and applicaions. Woking Pap Gogia Insi of chnology. Onhanalai C. (4). Lévy jmp isk: Evidnc fom opions and ns. Jonal of Financial Economics Psychoyios D. Dosis G. & Makllos R.N. (). A jmp diffsion modl fo VIX volailiy opions and fs. Rviw of Qaniaiv Financ and Acconing Schon W. (). Lévy Pocsss in Financ: Picing Financial Divaivs. John Wily & sons. Ld. Sco L.O. (997). Picing sock opions in a jmp-diffsion modl wih sochasic volailiy and ins as: Applicaion of Foi invsion mhods. Mahmaical Financ Shi.K. ong H. & Yang H. (4). On picing divaivs nd GARCH modls: A dynamic Gb- Shi appoach. Noh Amican Acaial Jonal 8 7-.

25 VG NIG JD-DE CGMY HS abl Jmp Modls and Risk-nal Chaacisic Fncions Modl Vaianc Gamma (Madan al. 998) Nomal Invs Gassian modl (Schon ) Exponnial dobl jmp modl (Ko JDDE ) ( s) Ca al. () SV modl (Hson 99) Chaacisic Fncions VG Φ ( s) xp i ln θγ γ s ln isθγ s γ X γ γ ( ) s NIG Φ ( s) xp i X ( is) ( p) pη η i b λ s η η Φ xp X pη ( p) η b s isλ η is is η Y Y Y Y i ( CΓ( Y )(( M ) M ( G ) G )) s CGMY Φ ( s) xp X Y Y Y Y CΓ( Y )(( M is) M ( G is) G ) { υ } f xp C D ix fo j j j j x x log S ; υ υ d ( ) kυ g j Cj i( ) ( bj iρξ d j)( ) ln ξ g j d ( b ) j iρξ d j bj iρξ d j Dj ; g ; j d j ( iρξ bj) i d j ξ g b j j iρξ d j ; ; b κ λ ρξ; b κ λ ( HN(R) Rsicd Hson and Nandi () GARCH { } f S xp A ; B ; wh A ; and B ; a givn by h csiv laions: A A B B ( θ ) B ( ; ) ( λ θ) θ B ( ; ) B ; ( ; ) ( ; ) ( ; ) ϖ ln( ( ; )) h isk nal vsion is obaind by plgging in λ and * placing θ wih θ θ λ 4

26 HN(U) SC B Unsicd Hson and Nandi () GARCH Jmp-diffsion wih Sochasic volailiy and Sochasic ins a modl (Sco 997) Jmp wih SV modl (Bas 996) { } f S xp A ; B ; wh A ; and B ; a givn by h csiv laions: A A B B ( θ ) B ( ; ) ( λ θ) θ B ( ; ) B ; ( ; ) ( ; ) ( ; ) ϖ ln( ( ; )) h isk nal vsion is obaind by plgging in λ and * placing θ wih θ θ λ ψ ( ) c R ln S ln J R ln S y y E E E { } ln J µ J / J µ J / J xp λ λ ( xp{ ( ) ξ η} ) E E R Exp ( ) R ( ) ( ρ ) Y ( ) η ( ) Y ( ) ρ ω κθ E E xp Y y wh ρ ω ( ) ( ) ( ) κ ρ Fo any complx s and s wih nonngaiv al pas sy j( ) syj( ) ( j( ) j) xp j j j Fo j wih j ( ) { } E y y a b y a ( /) κ j γ j κθ j j γ j ln γ j j γ j κ j γ j js γ j γ j γ j ( ) s κ js γ js( ) γ j γ j γ j κ j γ j js ( ) b wih γ κ s j j j j Φ ln ( S / S ) f j Φ SV E P j fo j µ j / Φ * * * jφ φ / xp Cj ; Φ Dj ; Φ V λ k k ( ; * * ) ln Cj Φ bλ k Φ ρ vφj γ j ( ρ ) vφj γ j v v γ j µ jφ Φ D ( ; ) j Φ γ j ; γ j ρ vφj v µ jφ Φ ρ vφj γ j γ j * * µ µ ρ v and γ j 5

27 CH IG-GARCH modl (Chisoffsn al. 6) { } xp ( ; ) ( ; ) f S A B A A wb ln a B 4 ( ) φ ( ) η ( ) 4 ( ) φ η η η ( ) ( ηφ ) B bb v a B cb h isk-nal vsion cosponds o Chisoffsn al. (6): ( ( η ) ) v η 6

28 abl S&P 5 Indx Opions daa -4 Panl A: Nmb of opion conacs 6 Maiy <4 4 Maiy<7 7 Maiy O-of-h-mony.9 S/K< A-h-mony.99 S/K<. 879 In-h-mony. S/K< All Panl B: Avag opion pics 6 Maiy <4 4 Maiy<7 7 Maiy O-of-h-mony.9 S/K< A-h-mony.99 S/K< In-h-mony. S/K< Panl C: Avag implid volailiy 6 Maiy <4 4 Maiy<7 7 Maiy O-of-h-mony.9 S/K< A-h-mony.99 S/K<.... In-h-mony. S/K< No: W s Eopan p-and-call opions on h S&P5 indx. h pics a akn fom nono ading volm qos on ach Wdnsday ding h Janay o Dcmb 4 piod. W clan o daa sing h sam ls applid by Hson and Nandi (). h implid volailiis a calclad sing h Black-Schols fomla. 7

29 abl Pilo calibaion wih S&P5 indx opions Modl RM SE Paams () BS (.48) () (θ) (ν) VG (.8) -.65 (.9).79 (.6) () () () NIG (.5) (.5).47 (.8) () (λ) (p) (η) (η) JD-DE.6.-4 (.577).484 (.47).474 (.54) 4.85 (.47) 4.67 (.888) (C) (G) (M) (Y) CGMY..748 (.89).579 (.87) (.88) -6.7 (.6) (κ) (θ) () (ρ) (V) HS.7.69 (.).98 (.486).698 (.46) -.99 (.47).99 (.58) () () () (γ) (λ) HN(R) (.5-) (.8-) (.5-) (.8-) -.49 (.9-) () () () (γ) (λ) HN(U) (.) (.44-7) (.95-7).654 (.4) 7.96 (.5) (κ) (κ) (θ) (θ) () () () SC (.4) (.78) (.) (.869) (.) (.645) (.5) (ρ) (μj) (j) (λj) (.) (.8) (.46) (.879) B. (V) (Vba) () () (ρ) (λ) (μλ) (λ) (.477-4) (.74-) (.7) (.96) (.675) (.96) (.4) (.585) (ω) (b) (a) (c) (η) CH (.669-6) (.86) (.579) (.4-6) ( ) () () () (γ) () () () CFGNIG ( ).56-9 (7.84-7).56-9 (.4-6) (.48).74 (.49).958 (.9).46 (.94) Nos: Calibaion is caid o by applying h FRF appoach o pic opions. W consid opions add on Jly 5. Af claning h daa w hav 78 cods o consid on ha paicla day. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

30 abl 4A: Calibaion wih opions add ov h piod Janay -Jn Modl RMS E Paams () BS..678 (.59) () (θ) (ν) VG.8.44 (.48).9 (.).88 (.9) () () () NIG (.78) 584. (.77).74 (.) () (λ) (p) (η) (η) JD-DE (.6).8 (.9).9998 (.66) (.78) 5.7 (.7) (C) (G) (M) (Y) CGMY (.7759) (.7) 57.5 (.789) -.96 (.49) (κ) (θ) () (ρ) (V) HS (.897).47 (.48). (.8).99 (.).4 (.) () () () (γ) (λ) HN(R) (.579-6) (5.6-7) (.98-6) (.7-5) (.7-5) () () () (γ) (λ) HN(U) (.58) (5.69-7) (.966-6) 49. (.) -.56 (.7) (κ) (κ) (θ) (θ) () () () SC. 4.7 (.687) (.4).5 (.886).56 (.4).775 (.4) (.).87 (.97) (ρ) (μj) (j) (λj) (.8).6 (8.8-).9 (.4-).95 (.49) (V) (Vba) () () (ρ) (λ) (μλ) (λ) B.4.8 (.4).45 (.4).68 (.5).69 (.) (.6).748 (.4).7 (.7) (.49) (ω) (b) (a) (c) (η) CH (.9-7).47 (.) (.784) (.-4) () () () (γ) () () () CFGNIG (.59) (.99-7) (5.5) (.79) (.46) 9.84 (.677) Nos: W consid opions add on vy Wdnsday. Af claning h daa w hav 79 opion conacs. W applid h FRF appoach o pic opions which significanly dcs h calibaion im. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

31 abl 4B O-of-sampl valaion os fo call opions add in h scond half of RMSE AAE MOE RMSE AAE MOE RMSE AAE MOE Days o maiy < 4 4 Days o maiy < 7 7 Days o maiy.95<(s/k)<.99 BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<. BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<.5 BS HS HN(R) HN(U) SC B CH CFGNIG Nos: h modls a calibad on fis half of h sam ya. oal nmb of conacs availabl fo h scond half is 456. RMSE is h oo man sqa AAE is h avag absol o and MOE is h man osid o. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

32 abl 5A: Calibaion wih opions add ov h piod Janay -Jn Modl RMS E Paams () BS (.5) () (θ) (ν) VG.9.4 (.).997 (.84).46 (.9) () () () NIG (.89) (.8).47 (.69) () (λ) (p) (η) (η) JD-DE (.5).78 (7.4).9998 (8.856).666 (8.69) (8.48) (C) (G) (M) (Y) CGMY (.66) 879. (.45) 8.8 (.49).689 (.7) (κ) (θ) () (ρ) (V) HS.9.45 (.88).99 (.8).55 (.794).99 (.4).4 (.9) () () () (γ) (λ) HN(R) ( ) (.6-6).66-5 ( ) 49.4 (.444-4) (.76-4) () () () (γ) (λ) HN(U) (.8) (.44-7) (.5-6) (.7) -.84 (.78) (κ) (κ) (θ) (θ) () () () SC (.48).44 (.799) (7.58-).888 (.9).94 (.57).6595 (.6).887 (.54) (ρ) (μj) (j) (λj) -.5 (.9) -.79 (.48).5 (.).88 (.469) (V) (Vba) () () (ρ) (λ) (μλ) (λ) B.86.9 (.8). (.7).598 (.554).6 (.49) -. (.5).8 (.75).59 (.65) (.) (ω) (b) (a) (c) (η) CH (.5-7).84 (.48).6674 (.44974) (.759-) -. (.) () () () (γ) () () () CFGNIG (.9) (.58-7) (.55-7) -.5 (.5)) (.56).446 (.766) 8.4 (.8) Nos: W consid opions add on vy Wdnsday. Af claning w hav 67 opion conacs. W applid h FRF appoach o pic opions which significanly dcs h calibaion im. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

33 abl 5B O-of-sampl valaion os fo call opions add in h scond half of RMSE AAE MOE RMSE AAE MOE RMSE AAE MOE Days o maiy < 4 4 Days o maiy < 7 7 Days o maiy.95<(s/k)<.99 BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<. BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<.5 BS HS HN(R) HN(U) SC B CH CFGNIG Nos: h modls a calibad on fis half of h sam ya. oal nmb of conacs availabl fo h scond half is 66. RMSE is h oo man sqa o AAE is h avag absol o and MOE is h man osid o. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

34 abl 6A: Calibaion wih opions add ov h piod Janay 4 - Jn 4 Modl RMS E Paams () BS (.44) () (θ) (ν) VG.7.85 (.9) -.96 (.).55 (.7) () () () NIG (.4) (.9).6988 (.787) () (λ) (p) (η) (η) JD-DE.7.6 (.5) (.777).79 (.98) (7.964) (6.8777) (C) (G) (M) (Y) CGMY (.6) 5.98 (.8) (.4).7 (.6) (κ) (θ) () (ρ) (V) HS.7.97 (.76).45 (.95).4 (.698) -.99 (.8).49 (.8) () () () (γ) (λ) HN(R) (.4) (4.-7).66-5 (.6-6) 49.4 (.) (.4) () () () (γ) (λ) HN(U) (.55).5-6 (9.47-8) ( ) 49. (.54) (.59) (κ) (κ) (θ) (θ) () () () SC.65.9 (.68).8 (.74).57 (7.66-).5 (.94).794 (.69).645 (.58).98 (.485) (ρ) (μj) (j) (λj) -.54 (.9) -.87 (.5).66 (.6).84 (.4) B.66 (V) (Vba) () () (ρ) (λ) (μλ) (λ). (.9).85 (.9).74 (.664).4 (.588) (.6).5 (.88) (.69) (.9) (ω) (b) (a) (c) (η) CH ( ).56 (.69).5954 (9.854).89 (.9-7) -.6 (.4) () () () (γ) () () () CFGNIG (.5) (.4-7) (.7-7) (.7) (.8).94 (.66) (.6) Nos: W consid opions add on vy Wdnsday. Af claning w hav 578 opion conacs. W applid h FRF appoach o pic opions which significanly dcs h calibaion im. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

35 abl 6B O-of-sampl valaion os fo call opions add in h scond half of 4 RMSE AAE MOE RMSE AAE MOE RMSE AAE MOE Days o maiy < 4 4 Days o maiy < 7 7 Days o maiy.95<(s/k)<.99 BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<. BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<.5 BS HS HN(R) HN(U) SC B CH CFGNIG Nos: h modls a calibad on fis half of h sam ya. oal nmb of conacs availabl fo h scond half is 55. RMSE is h oo man sqa o AAE is h avag absol o and MOE is h man osid o. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

36 abl 7A: Calibaion wih opions add ov h piod Janay -Dcmb Modl RMS E Paams () BS.8.76 (.5) () (θ) (ν) VG.5.9 (.).78 (.44).94 (.4) () () () NIG (.445) (.84).4566 (.79) () (λ) (p) (η) (η) JD-DE.8.76 (.5).7968 (.84).47 (.9) 5.5 (.55) 5.5 (.47) (C) (G) (M) (Y) CGMY.7.7 (.9) (.55) 5.76 (.9).64 (.57) (κ) (θ) () (ρ) (V) HS.4.87 (.775).46 (.).79 (.744).99 (.9).4 (.9) () () () (γ) (λ) HN(R) (.9-4) ( ).54-5 (.987-6) (.6-4) (.55-4) () () () (γ) (λ) HN(U) (.7) ( ).85-5 (.688-6) (.7) -.78 (.46) (κ) (κ) (θ) (θ) () () () SC..9 (.588).6459 (.49).4 (.487).49 (.65).8455 (.6).57 (.55).59 (.4) (ρ) (μj) (j) (λj) (8.486-) -.89 (.77).54 (.594-).986 (.9) (V) (Vba) () () (ρ) (λ) (μλ) (λ) B..4 (.).9 (.4).5558 (.68).6 (.4) (.8).78 (.84).46 (.8).64 (.8) (ω) (b) (a) (c) (η) CH ( ).746 (.8).774 (9.547) (.588-7) -.6 (.) () () () (γ) () () () CFGNIG (.45) (.-7) (.9-7) -.56 (.5) (.5) -.9 (.5).6878 (.49) Nos: W consid opions add on vy Wdnsday. Af claning w hav 5848 opion conacs. W applid h FRF appoach o pic opions which significanly dcs h calibaion im. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

37 abl 7B O-of-sampl valaion os fo call opions add in h fis half of 4 RMSE AAE MOE RMSE AAE MOE RMSE AAE MOE Days o maiy < 4 4 Days o maiy < 7 7 Days o maiy.95<(s/k)<.99 BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<. BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<.5 BS HS HN(R) HN(U) SC B CH CFGNIG Nos: h modls a calibad sing 5 and 6 conacs. oal nmb of conacs availabl fo h scond half is 578. RMSE is h oo man sqa o AAE is h avag absol o and MOE is h man osid o. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

38 abl 8A: Calibaion wih opions add ov h piod Janay -Dcmb 4 Modl RMS E Paams () BS (.44) () (θ) (ν) VG (.45) -.6 (.97).8 (.9) () () () NIG (.85) (.4).566 (.55) () (λ) (p) (η) (η) JD-DE (.44).5 (.67).447 (.776).7655 (.8875) 6.76 (.44) (C) (G) (M) (Y) CGMY (.).956 (.76) (.66).8597 (.89) (κ) (θ) () (ρ) (V) HS (.4).5 (.49).97 (.65) -.99 (.49).8 (.9) () () () (γ) (λ) HN(R) (.65-5) (.69-7) (.557-6) (.7-4) (.65-5) () () () (γ) (λ) HN(U) (.44) (.87-8) (.887-7) (.9) (.9) (κ) (κ) (θ) (θ) () () () SC (.5547).598 (.99) (.884).4- (.74).4 (.7) 5. (.84).477 (.56) (ρ) (μj) (j) (λj) -. (.) -.9 (.5).548 (.58).9 (.68) (V) (Vba) () () (ρ) (λ) (μλ) (λ) B (.6).45 (.5).74 (.9).7 (.6) (.74).5 (.8).87 (.6).97 (.6) (ω) (b) (a) (c) (η) CH (.486-7).78 (.7).644 (7.4) (.-7) -.5 (.489-4) () () () (γ) () () () CFGNIG (.) (.55-7) (.7-7) -.54 (.54) (.4).48 (.4) (.479) Nos: W consid opions add on vy Wdnsday. Af claning w hav 696 opion conacs wih a man opion pic of 4. and avag implid volailiy of.9. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

39 abl 8B O-of-sampl valaion os fo call opions add in h fis half of 5 RMSE AAE MOE RMSE AAE MOE RMSE AAE MOE Days o maiy < 4 4 Days o maiy < 7 7 Days o maiy.95<(s/k)<.99 BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<. BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<.5 BS HS HN(R) HN(U) SC B CH CFGNIG Nos: h modls a calibad sing 6 and 7 conacs. oal nmb of conacs availabl fo h scond half is 94. RMSE is h oo man sqa o as dfind AAE is h avag absol o and MOE is h man osid o. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

40 abl 9A: Calibaion wih opions add ov h piod Janay -Dcmb 4 Modl RMS E Paams () BS (.46 () (θ) (ν) VG (.8).8 (.6). (.) () () () NIG (.445) (.84).568 (.79) () (λ) (p) (η) (η) JD-DE (.49).8 (.8).459 (.94) (.4) 9.79 (.4) (C) (G) (M) (Y) CGMY (.6) (.64) 5.76 (.).448 (.85) (κ) (θ) () (ρ) (V) HS (.44).54 (.49).8 (.68).99 (.49).7 (.9) () () () (γ) (λ) HN(R) ( ) (.-7).-6 (.6-6) 55.5 ( ) (.546-4) () () () (γ) (λ) HN(U) (.67) (.96-8).8-6 ( ) (.9) (.7) (κ) (κ) (θ) (θ) () () () SC (.594) (.454) (.6) (.559).96 (.7).897 (.99).994 (.58) (ρ) (μj) (j) (λj) -.7 (7.766-) -.78 (.9).687 (9.98-).67 (.488) (V) (Vba) () () (ρ) (λ) (μλ) (λ) B 5..7 (.4).5 (.4).6956 (.6).67 (.4) (.6). (.75).568 (.8).89 (.6) (ω) (b) (a) (c) (η) CH (.7-7).499 (.8).4874 (4.875) () () () (γ) () () () CFGNIG (.4).86-8 (.95-7)).68-8) (.9-7) (.58) (.4) 7.55 (.75) (.8) Nos: W consid opions add on vy Wdnsday. Af claning w hav 844 opion conacs. W applid h FRF appoach o pic opions which significanly dcs h calibaion im. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

41 abl 9B O-of-sampl valaion os fo call opions add in h fis half of 5 RMSE AAE MOE RMSE AAE MOE RMSE AAE MOE Days o maiy < 4 4 Days o maiy < 7 7 Days o maiy.95<(s/k)<.99 BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<. BS HS HN(R) HN(U) SC B CH CFGNIG <(S/K)<.5 BS HS HN(R) HN(U) SC B CH CFGNIG Nos: h modls a calibad sing opions add on -4. oal nmb of conacs availabl fo h scond half is 94. RMSE is h oo man sqa o as dfind AAE is h avag absol o and MOE is h man osid o. h modls in colmn () a dfind in abl wih hi isk-nal Chaacisic Fncions.

42 Appndix A Momns of NIG Lévy Innovaion ANIG() is infinily divisibl and h associad Lévy pocss has h disibion of incmns ov [s s] chaacid by a NIG(). Schon () shows ha h fis fo momns of h X ~ NIG() andom vaiabl a: [ ] X E (A) [ ] X V (A) [ ] 4 X Skw (A) [ ] 4 X K. (A4) Wih h momns of h NIG andom vaiabl as in (A)- (A4) h fis wo condiional momns of h log-ns bcom: [ ] E X E λ λ λ I I 4 (A5)

43 V [ X I ] V λ ( ) ( ) ( ) I. (A6) hs h dynamics of X a a scald and shifd vsion of I ~ NIG( ) condiional skwnss and kosis a: X 4 [ I ] ( ) ( ) Skw (A7). h K 4 I. (A8) [ ] X h xisnc of condiional skwnss and condiional kosis nss ha smil-skw pans can b modld whn log-n dynamics follow a GARCH wih NIG-Lévy innovaions.

44 Appndix B GARCH wih NIG Lévy Innovaion and Eqivaln Maingal Mas W slc an Eqivaln Maingal Mas (EMM) ha dpnds on finding a solion o h condiional Essch qaion (Gb al. 994; Shi al. 4) X X M M I I θ θ ˆ ˆ (B) wh s M X I is h condiional momn-gnaing fncion (MGF) dfind as [ ] I I sx X E s M. (B) In h cas of GARCH dynamics wih NIG-Lévy innovaions (7) h condiional Essch qaion (B) bcoms E E ˆ ˆ λ θ λ θ E E λ θ θ ˆ ˆ (B)

45 which simplifis o (B4) sing h consan c : [ ] [ ] E E c c λ θ θ ˆ ˆ (B4) Insing qaion () ino (B4) w obain: ˆ ˆ xp ˆ xp ˆ xp c c c c θ θ θ θ λ (B5) Givn h paams of h NIG-Lévy pocss and h GARCH volailiy pocss in qaion (8) h solion θˆ of (B5) can b sd o dscib h disibion of log-ns as X X X s s s s s s M l M l M θ θ ˆ ˆ ~ I I I. (B6) Givn o assmd disibion fo innovaions and h fac ha h GARCH on-piod-ahad volailiy is known (B6) bcoms [ ] [ ] [ ] [ ] I I & (5) (4) ˆ ˆ ˆ ˆ ˆ ˆ xp ˆ ˆ xp ~ ; ~ c lc c c c l E E NIG E E l M l l c c l l c c l X s s θ θ θ θ λ λ θ θ λ λ θ λ θ (B7) Compaing qaions () and (B7) w cogni ha nd-emm innovaions a NIG-disibd wih a nw chaaciaion c θ ˆ.

46 W also nd o s wha oh paams a inflncd by his nw chaaciaion. W sa wih h dynamics of h volailiy nd h maingal mas: [ ] V X V λ ~ ~ I I (B8) hs h mak and al mass a lad hogh h chaaciaion c θ ˆ and h mak and al volailiy pocsss a lad hogh. W nx idnify h maining paams ha qi nw chaaciaion o kp h n dynamics qivaln. Und h al mas w hav wh λ λ λ λ λ λ X X X X (B9) Hnc w inodc placing fo h dynamics o b chaacid by a maingal mas. W can achiv his fom h following: