Margrabe Formulas for a Simple Bivariate Exponential Variance-Gamma Price Process (II) Statistical Estimation and Application

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1 Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Volum I Issu I Augus- PP Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco W Hülm Wols luw Fcl Svcs Sldsss 69 CH-88 Züch Swzld. w.hulm@wolsluw.com Rcvd: -6- Rvsd: -7- Accpd: 4-7- Asc: A mulv mom mhod o h smpl mulv vc-gmm dsuo s cosdd. I s o ccou h s poducs o oh h coswss d couoss sos. h modl pms dpd upo h soluo o sc quo d h covc m s ucolly dpd upo coswss d couoss. h mhod ls smulous smo o h pms gv smpl sms o h m vco coswss vco d couoss m. Applco o h smo o h v vc-gmm modl o h Sdd & Poos 5 d NASDAQ soc m dcs s ud. h sscl g suls usd o comp h ogl Mg omul wh vc-gmm chg opo pcg omul dvd hough pplco o h spc dlo ppoch. ywods: mulv vc-gmm coswss couoss chg opo soc m.. INRODUCION I h s p o h ps sudy smpl mulv pol vc-gmm pc pocss hs cosdd s lv o h usul mulv pol Guss pocss. Bsd o s-pc dlo hs cosucd d ppld o dv pcg omuls o h chg opo wh v pol vc-gmm dld pc pocss. I hs osvd h h smpl modl s sy o wo wh u hs som sous dwcs. Fo mpl l colo co d oc h mgs d. Moov g dculy hs coud wh h o clo o opo pcs o h mgs. o lm som o hs dsdvgs s possl o dsg lv smo mhod. hs s h m pupos o h ps ollow-up. I pcul sscl smo o h mulv vc-gmm modl s dscussd d pplco o soc m dcs s psd. A ovvw o ou ppoch d h od m suls ollows. Bsd o h s ou moms o h commo gmm suodo d h hom o Issls o h hd d ouh od mulv oml moms w o hom. h plc pssos o h m covc coswss d couoss pms o h smpl mulv vcgmm modl. h mulv mom mhod s dsgd whch s o ccou h s poducs o oh h coswss d couoss sos. h mhod s o-pm so o h ovl ppoch psd []. wo us o mgh mod: h covc m o h mulv vc-gmm dsuo s ucolly dpd upo coswss d couoss. ARC Pg

2 Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco h pms dpd upo h soluos o sc quo. h cosdd mulv mom mhod ls smulous smo o h pms gv smpl sms o h m vco coswss vco d couoss m. o dmos h pccly o h w ppoch l-l cs sudy s psd Sco. I cocs h sscl smo o h gh pm v vc-gmm modl o h Sdd & Poos 5 d NASDAQ soc m dcs. h modl s succssully d o sv v dly d ss ov d m pods. h goodss-o- o h mgs s opmzd d compd wh h goodss-o- o h v oml whch us ou o h poo. h suls usd Sco 4 o comp h ogl Mg omul wh h vc-gmm chg opo pcg omul dvd []. Sco 5 s dvod o uh dscusso d coclusos. h Appd dvs smpl wy psso o h vc-gmm dsy h us ou o mo symmcl h h ogl dsy omul y []. I s usd o compu h Adso-Dlg goodss-o- sscs Sco.. A MULIVARIAE MOMEN MEHOD A dom vco... hs - dmsol smpl mulv vcgmm VG dsuo wh pm vcos... pm m d pm o sho ~ VG s cumul gg uco cg s gv y C u u l{ u u u}. o ll vlus o u u u... u o whch h psso. ss. h spcl cs s h h-pm clss o h symmc Lplc AL o whch mom mhod hs dsgd y [4]. h dom vco sss h sochsc pso G G Y. wh G ~ / / s gmm dom vl wh m o d vc Y ~ N s mulv oml wh vshg m vco d Y G dpd. I s sp w dm h m vco E ] E[ ] E[ ]... E[ ] vd [... d h m o -h od cl moms dod y M [ ] 4. Fo h m M ] D[ ] V s h covc m wh lms V E[ ]. h [ m [ ] S M cosss o h coswss lms S E[ ] d h m M ] cosss o h couoss lms 4[ E[ ]. I gl o hs h loshps S E[ V E[ V ] - V V V ] S V S V S V S V.. h ollowg sul glzs Poposo. [] spcl cs. hom. Moms o h smpl mulv VG h m covc coswss d couoss pms o h mulv VG dom vco ~ VG gv y Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 4

3 W Hülm V S..4 Poo: O uss h pso.. h psso o h m vco s mmd. Fo h cl moms sucs o cosd h cs. Wh. h vco compos o ssy h pso G G Y... wh G ~ / / s dpd o Y N. h mom loshps ~ 4 E[ G] E[ G ] E[ G ] E[ G ] 6 6 wll usd pdly whou uh mo. O hs / E[ ] E[ G Y Y G YY G] whch mpls h psso o h covc. Smlly o hs E[ Y Y ] E[ G Y Y Y Y G Y Y Y Y G Y Y G Wh h c h E[ Y YY ] hom o Issls o ss h / ]. 5/ E[ ]. Is hs d h c h V o h s lo o. o o h coswss omul.4. Pocdg h sm wy o shows h E[ Y Y Y Y Y ] E[ G Y Y Y Y Y Y Y 4 Y Y Y Y Y Y Y Y Y Y Y G Y / Y Y Y Y G Y Y Y G ]. Y G 7/ Sc E[ Y Y YY ] hom o Issls o gs E[ ] 6 6 Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 5 Is hs V d h coswss lo.4 o h scod p o. o o gm d coug o qul ms h couoss omul. W dy o h glzo o h mom mhod [] Sco. Fo y d d gv h m pms h coswss d couoss pms S d w dm h mg pms V ms o hm. I pcul s show h h covc m V o h mulv VG dsuo ucolly dpds upo coswss d couoss. Fs o ll gv h m d ssumg hs dmd s cl h s od om h m vco quo s. Smlly oc V hv dmd h pm m s od om h covc quo s V. N l us m h.

4 Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco coswss quos. Fo hs cosd h coswss vco S S... S dvd om h s poduc S M [] such h S S h ollowg sho hd oo o sums o covcs d pms s usd: V V V V V M..6 h vluo o.5 sd o h coswss omul.4 ylds h loshps { V M } MV S S uh S S d dd h quos.7 o g h quo M V : VM M S..8 Cosd ow h couoss quos d d h couoss m usg h s poduc M 4[ ] such h A clculo o.9 sd o h ls quos.4 ylds h loshps { V M V V } { M V V V} V M..... Fuh summg. wh h sho hd oo... o gs { V M } M { M V} V..... Wh o os hough ddo o. uh quo M V mly 4 M 6 VM V.. h sulg sysm o o-l quos h uows M V V V c solvd y pplyg h-sg pocdu s ollows. Sp : solv h quos.8 d. o h pms M V Fom.8 o gs V M S. M. Is hs psso o. d mulply wh ollowg sc quo h pms S : o s h M sss h 9 M 4 6 M 4 SM M S..4 Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 6

5 W Hülm hs quo s solvd smlly o h spcl cs []. Sp : solv h quos.7 d. o h pms V... Fom.7 o gs S MV { V M } Is hs o. o s h V MS V M V s uco o h pms M V S : I ollows h s uco o h sm pms mly S MS M { V M } V M V M wh o mus ssum h V M V M..8 Sp : h uows V od om h quo. wh o mus vy h h covc m V d s ssocd colo m posv sm-d. h dscd mom mhod s usul o pm smo. Gv smpl... N o sz N wh ch s osvo o h dom vco... o cosds h ollowg smpl sms o h coswss vco d couoss m: S Sˆ... Sˆ N ˆ N N N..9 Smpls sms o h qus S od hough summo s S ˆ Sˆ ˆ ˆ ˆ... ˆ ˆ.. Isg hs sms o h dvd omuls o os o y d sms o h mulv VG pms ms o h smpl m vco coswss vco d couoss m. h Sco lluss l-wold pplco o hs pocdu.. ESIMAION OF BIVARIAE LOGARIHMIC REURNS W cosd ow wo soc m dcs o whch ll h m coswss d couoss qus c smd. Ru osvos sm om h ollowg sv d ps o v d om h Sdd & Poos 5 SP5 d h NASDAQ ND d ss: SP5/ND/Y: 754 dly closg pcs ov ys om 4.. o.. SP5/ND/5Y: 59 dly closg pcs ov 5 ys om..8 o.. SP5/ND/Y: 56 dly closg pcs ov ys om.. o.. SP5/ND/5Y: 77 dly closg pcs ov 5 ys om..998 o.. SP5/ND/Y: 59 dly closg pcs ov ys om o.. SP5/ND/5Y: 6 dly closg pcs ov 5 ys om o.. Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 7

6 Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco SP5/ND/7Y: 688 dly closg pcs ov 7 ys om..986 o.. hs d ss ypcl s hy co sho o mdum hgh voll pods c d 5 ys mod log m pods d 5 ys d log m pods 5 d 7 ys. h ls d s hs cludd cus cos h hghs d lows dly chgs osvd so dop.9% d 6.% o SP5 spcvly ND o cs o 7.% o ND o... h ollowg l. lss h qud smpl mom sms o h v loghmc us od om ch o hs comos. l. Smpl mom sms o v log-us om wo soc m dcs mom sms u ^-4 ^-6 ^-6 SP5/ND μ μ S S S = Y Y Y Y Y Y Y Up o h 5Y d Y pods h coswss vco hs lwys gvly swd compos. h cpo s h ND. I h 5Y cs o hs lso S S S. Ov h logs pod o 7Y h coswss compos h hghs gv vlus. Up o h shos Y pod h ovll couoss coc cds 5 d s hghs o h 5Y d 7Y pods. Fo spcc d vlus o h v VG s d o h d ollowg h mom mhod dscd Sco. hough h smpl mulv VG s sy o wo wh d hs hoclly usd s [] Sco hs md h l colo co d oc h mgs d. Howv h poposd mom mhod dos o h mgs sply u povds ovll psmoous o ll s pms gdlss o h mgs d h dpdc sucu. Fo hs so s mpo o dscuss s goodss-o- cpls s compd o h v oml whch udls Mg s ogl pcg omul. I pcul s mpo o lyz h goodss-o- o h smd mgs. o do so ou goodss-o- GoF msu s sd o sscs whch msu h dc w h mpcl dsuo ucos F d h smd mgl dsuo ucos F. W us h Cmé-vo Mss mly o sscs dd y.g. [5]-[7] F F w df. wh w s sul wghg uco. I w / F F o gs h A Adso-Dlg [8] ssc. Cosd h od sscs o h u d such h... d l Fˆ... h smd vlus o mgl dsuo uco. h o hs A l Fˆ F ˆ.. h vlus Fˆ od umclly y go o h VG dsy. Ud h so-clld ll gmm pso hs s psso A.8 h Appd. h Adso-Dlg Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 8

7 W Hülm ssc ylds o o h mos powul s h d dsuo dps om h u dsuo h ls.g. [5] d s commdd hs suo. Now h osvd smpl u mgl d s swd d hs much hgh uoss h s llowd y oml dsuo whch dcs h h h ls ms d uss h us o h GoF sscs.. Ndlss o sy h poposd mom mhod s oly sg po o mpovd GoF smo mhods. Howv mo compl d lyss s yod h scop o h ps sudy. o wgh h luc o h mgs w us h Eucld dsc o d ovll GoF msu s A A A wh A h Adso-Dlg sscs o h mgs. o clcul h GoF sscs o h vc-gmm mgs s smpl o us h ll gmm pso o h mgs dd y.g. [9] [] G G G ~ dpd. wh h o-oo pm somo.4 wh h chgd pm oos / usd. h smd vlus o h mgl dsuo ucos. od umclly y go o h psso A.8 o h VG pd. h smd pms d GoF sscs summzd d compd h l.. As s osvo o ms h h v oml ylds h poo GoF ms o. whv h cosdd m pod o lyss s. o lm h possl oupu o dms y l d o vlus o o whch h ovll GoF msu A s smlls old ums h l. I s sg o comp h opml s wh h os om v symmc Lplc dsuo spcl cs. Ov h smlls Y m pod h l povds lmos h s ou ss u dps om ll oh css. Som commo d dvgg us c od. Ov h m pods up o Y h dpdc pm whch dms h colo coc w h mgs s h l l 4. ms sl oud.95 whl ov h log m pods om 5Y o hs vlu s ppomly.7. hs dvgg u lgs wh h c h h smd colo cocs F w h mgs dcs y csg m pod s s lso h cs o h smpl colo coc S s h l l 4.. h smd dsuos usd o comp h Sco h ogl Mg chg opo pcg omul sd o h v oml wh h o om smpl v vcgmm s dvd [] hom BLAC-SCHOLES VERSUS VARIANCE-GAMMA MARGRABE OPION PRICES I s sg o comp h clsscl closd-om Mg chg opo pcg omul wh h cospodg omul o h pol v VG modl. Rcll h h v Blc-Schols modl h uu pcs o u m dscd y h quos S S p Z 4. BS wh h Z s cold sdd oml wh colo coc. I h spcl cs S Mg s omul ds M BS E[ D S S ]. 4. BS Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 9

8 Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco l. Pm sms d GoF sscs o h smpl v VG mly pm sms GoF sscs u ^- ^- SP5/ND ν ξ ξ θ θ τ τ ρ A² A² A Y v oml Y v oml Y v oml Y v oml Y v oml Y v oml Y v oml I h uu pcs pol v vc-gmm S p wh ~ VG o uss h VG dlo D p d ss Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 4 o o h pcg omul s Hülm hom 6. M E[ D S c / z VG S ] c z z z c / z dz c c. 4.

9 W Hülm W hv vlud 4. o h sv ps Sco d compd h suls wh 4.. W BS ssum h 4. s clculd wh h h smpl covc m.. S s h BS smpl colo coc o h smd covc m.. F s h d colo coc. Sc C s [] omul 4. h omul 4. cos o 4. dpds upo h s- o u. Whou loss o gly sucs o comp h cs. l 4. low lss h pcg sgd lv dvo o h VG Mg omul om h ogl Mg omul h Blc-Schols modl. O os h ollowg pops. h symmc Lplc Mg omul s lwys o h s sd whh h smpl o smd covc mcs usd Blc-Schols modl. Ecp o h 5Y d Y pods h ogl Mg omul wh smpl covc m udsms h chg opo pc vlud wh h smd VG modl. Usg oh omuls h smd covc m o osv h hy dvg cosdly ov h smlls Y m pod >4% d >8% lv dvo d ply o h symmc Lplc spcl cs. Ohws h VG d h ogl Mg pcs soly clos wh chgg sgs dpdg o h pod o lyss. Fuhmo plll o h dvgg u osvd Sco ou h colo coc h VG opo pcs o pods up o Y m sl h vl.97.8 whl ov h log m pods om 5Y o hs pcs much hgh d vy h vl RESULS FURHER DISCUSSION AND CONCLUSIONS Ou sg po hs h osvo h h cosdd mulv vc-gmm modl s sy o wo wh u hs som dwcs. Fo mpl l colo co d oc h mgs d. o ccumv hs dsdvg w hv dsgd mulv mom mhod h dos o h mgs sply u povds ovll psmoous o ll s pms gdlss o h mgs d h dpdc sucu. h lgohmc sucu o hs ovl sscl mhod s smpl. Gv h coswss vco d h couoss m ollows h-sg pocdu. Fs o solvs h sc quo.4 h ovll VG pm M sum o h mgl VG pms d h pm o h gmm suodo. I sp o dms h mgl VG d covc pms V... usg h quos I h hd sp o os h covc m V whch cssly mus posv sm-d. h mg modl pms h sghowd ucos o h od qus. h dvo o hs mulv mom mhod uss h plc pssos o h m covc coswss d couoss pms o h mulv VG modl hom. s wll s h s poducs o h coswss d couoss sos. I s mpo o o h h s poduc hs lso ppld ld sscl mhods y []-[5]. W hv llusd h usulss o hs mom mhod wh cs sudy mly h sscl smo o h gh pm v vc-gmm modl o h Sdd & Poos 5 d NASDAQ soc m dcs. h modl s succssully d o sv v dly d ss ov d m pods d h goodss-o- o h mgs hs opmzd. h g suls hv usd Sco 4 o comp h ogl Mg omul wh h vc-gmm chg opo pcg omul dvd []. As mod o coclusos l us gv ouloo o som uh possl suds hs. Fs o ll h s d o mpovd goodss-o- smo mhod Sco h s o oly scd o h mgs. Scod w pc pplcos o h s-pc dlo ppoch o h pcg o mo compl cl dvvs d suc poducs wh mddd opos. Flly s possl o d h ps wo o ld mulv o-guss vc-m mu modls h clud mulv oml vs Guss d mulv oml mpd sl dsuos. h l so wll psd lswh. Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 4

10 Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco l 4. Compso o v BS d VG Mg opo pcs cs m colo Mg pc pcg sgd u ucy coc ^- lv dvo SP5/ND ν ρs ρf MBS/S MBS/F MVG S F Y Y Y Y Y Y Y APPENDI: Spcl uco pso o h vc-gmm dsy A v pm ll gmm BG dom vl s dd y G G ~ BG wh dpd G ~ G ~ sddzd gmm s wh scl pm. I sucs o sc h o o h BG wh vshg loco. h BG pd dod y ; s h covoluo o h wo gmm pd s:. A. h ollowg glzd gmm uco pso sms w. I s quvl o h pso A.6 low ms o h colu hyp-gomc uco o h d d. hom A. Glzd gmm uco pso h poly dsy uco o BG s gv y h ll gmm Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 4

11 W Hülm Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 4 A. wh h glzd gmm uco d. A. Poo: Usg h symmy lo ; ; sucs o cosd h cs. hough lmy go chg o vls y o os d dy y y I I dy y y y h somo u c wh c ylds uh c du u u c c I u. Is o h s gl psso o o g A.. I vu o h lmg popy lm d h mg o h gl A. s usd. Fuhmo o hs lso vlly. Aoh usco ss om h c h wh o h pd covgs o l- d gh-l gmm pd spcvly s should. Moov clos loo h colu hyp-gomc uco o h d d oducd y [] d lso clld com uco shows h loshp U A.4 wh h com uco s dd y.g. [] 48::6 d 48:.7 d d U. A.5 h glzd gmm uco s somd com uco d A. ws s. U U A.6 I h vc-gmm spcl cs BG VG h lv com uco ducs o Mcdold uco modd Bssl uco o h d d hypolc Bssl uco o h d d Bss uco modd Hl uco o h yp [] 48:4: d 48::6 U. A.7 Isg hs pssos o h com pso A.6 o os h VG pd p. A.8 hs closd-om psso hs s dvd [] o h pmzo

12 Mg Fomuls o Smpl Bv Epol Vc-Gmm Pc Pocss II Sscl Esmo d Applco. A.9 Howv s ogl om h VG pd s h lss symmcl om 4 p REFERENCES. A. [] Hülm W. A mom mhod o h mulv symmc Lplc dsuo Sscs & Poly Ls [] Hülm W. Mg omuls o smpl v pol vc-gmm pc pocss I hoy Apps IJSIMR. [] Md D. C P. d Chg E. h vc gmm pocss d opo pcg. Euop Fc Rvw [4] Vs H. O h pm smo o h symmc mulv Lplc dsuo. Comm. Ss. hoy d Mhods [5] D Agoso R.B. d Sphs M.A. Goodss-o-F chqus Mcl D 986. [6] Cz P. Hädl W. d Wo R. Sscl ools o Fc d Isuc Spg 5. [7] Buc. Mso A. d Wo R. Loss Dsuos MPRA Pp No. 6. [8] Adso.W. d Dlg D.A. Asympoc hoy o c goodss-o- c sd o sochsc pocsss A. Mh. Ss [9] C P. Gm H. Md D.B. d Yo M. h sucu o ss us: mpcl vsgo Joul o Busss [] üchl U. d pp S. Bll gmm dsuos d pocsss cl Mhmcs Sochsc Pocss. Appl [] com F.G. Sull uzo pgomch colu Il Al d Mmc Pu d Applc S Qu 6: [] Oldhm. Myld J. d Sp J. A Als o Fucos wh Equo h Als Fuco Clculo d d. Spg Nw Yo 9. [] ollo. d Svsv M.S. Esmo d sg o pms mulv Lplc dsuo Comm. Ss. hoy d Mhods [4] ollo. Mulv swss d uoss msu wh pplco ICA Joul o Mulv Alyss [5] Vs H. O h pm smo o h symmc mulv Lplc Dsuo Comm. Ss. hoy d Mhods AUHOR S BIOGRAPHY W Hülm hs sudd mhmcs d physcs EHZ wh h od hs PhD 98 wh hss hgh lg. A posdocol llowshps Yl Uvsy d h M Plc Isu Bo h cm 984 cuy Whu L d Psos so cuy Ao R d IRMG Swzld -6 so cosul IRIS Züch 6-8 d s culy mployd FRSGlol Swzld Wols luw Compy. H hs vsg ssoc posso cul scc h Uvsy o ooo dug h cdmc y Mo omo s oud hs hompg hps://ss.googl.com/s/whulm/. Iol Joul o Scc d Iovv Mhmcl Rsch IJSIMR Pg 44