Global Journal o Pur and Appld Mahac. ISS 973-768 Volu 3, ubr (7), pp. 83-86 Rarch Inda Publcaon hp://www.rpublcaon.co Modlng and Prcng o Wahr Drvav Mark Dr Sllauhu Prabakaran and Dr. J. P. Sngh Aoca Proor, School o conoc and Bun Adnraon Dparn o Accounng & Fnanc Ponca Unvrdad Javrana Cal.Cal, Coloba. Proor, Dparn o Managn Sud Indan Inu o chnolog, Roork 47667, Inda. Abrac In rcn ar, on o h acor ha had a gncan pac on h conoc dvlopn wa rprnd b clac chang. A nrnaonal lvl, h wahr rk anagn and or a pror or Govrnn, nuranc copan and copan whn h ndur acd b h wahr varabl. h an objcv o h ud o odllng and Prcng o Wahr Drvav Mark. h an goal o h ud ourold: ) Fr, w bgn our approach o conruc h praur odl undr Ornn Uhlnbck proc whch drvn b a Lv proc rahr han a andard Brownan oon nvgad. ) hn w xn our approach o brl prn h wahr drvav ark odl. 3) x w conruc h Modlng and prcng o wahr drvav. 4) Fnall, how wahr orcang and aonal orcang can ponall prov our valuaon o wahr drvav conrac. In addon, h papr nd wh concluon. KWord: Wahr Drvav, praur, Ornn Uhlnbck Proc Hang Dgr Da (HDD), Coolng Dgr Da CDD and Arbrag Prcng hor. IRODUCIO vn n our chnolog-bad chnolog, connu o lv largl a h rc o. I nlunc our lv and choc, and ha a hug pac on corpora arnng and pro. Unl rcnl, vr w nancal ool ord corpora procon agan cla rk. Howvr, h bgnnng o h drvav o - akng h a a radabl good - ha changd all h. Alo prcn o h US cono ad
84 Dr Sllauhu Prabakaran and Dr. J. P. Sngh o b drcl acd b cla and h nco and nco o vruall vr ndur - agrculur, nrg, nrann, conrucon, ravl, and ohr: "Cla no ju an nvronnal probl, a ajor conoc acor, and a la a rllon dollar o our cono cla nv. Wahr drvav ar nancal nrun ha can b ud b organzaon or ndvdual a par o a rk anagn rag o rduc h rk aocad wh advr or unxpcd wahr condon. Wahr drvav ar ndx-bad nrun ha uuall u orologcal daa obrvd n a wahr aon o cra an ndx on whch a pan a b bad. h ndx could b h oal prcpaon durng a rlvan prod - whch a b rlvan or a hdrolcrc gnraon copan - or h nubr whr h nu praur all blow zro, whch could b rlvan or a arr who proc agan ro daag. Unlk nurancbad covrag o "copnaon, hr no nd o prov ha a lo ha bn urd. h ln objcv, bad on h nal valu o h wahr ndx chon durng h chon prod. I a pan du, uuall don n a ar o a w da wh h ln prod ha dnd n h conrac: unlk nuranc conrac, hr no "lo adjun" proc. h rk ha copan ac du o h cla ar unqu. Wahr condon nd o ac volu and u or han h drcl ac prc. An xcponall war wnr, or xapl, can lav ul and nrg copan wh xc uppl o ol or naural ga (bcau popl nd l o ha hr ho). Alrnavl, an xcponall cold ur can lav hol and arln a p. Alhough prc a chang owha du o unuuall hgh or low dand, prc adjun do no ncarl copna or lo rvnu bcau o unl praur. Fnall, wahr rk alo unqu n ha hghl localzd, can b conrolld and dp gra advanc n orologcal cnc, ll canno b prdcd prcl and connl. Unl rcnl, nuranc wa h an ool ud b copan or procon agan unxpcd clac condon. Howvr, h nuranc provd procon onl agan caarophc daag. Inuranc do nohng o proc agan h rducd dand ha copan xprnc du o a cla ha warr or coldr han xpcd. In h la 99, popl bgan o ralz ha h quand and ndxd h cla n r o onhl or aonal praur, and addd a dollar aoun o ach ndx valu, h could n a n "pack" and Corcal cla. In ac, h p o rad would b coparabl o radng h varabl valu o ock ndc, currnc, nr ra and agrculural produc. h concp o cla a a radabl good, hror, bgan o ak hap. "Unlk h varou prpcv provdd b h govrnn and ndpndn orca, radng n wahr drvav gav ark parcpan a quanabl vw o ho propc," ad Agbl Ako, anagng parnr o nrg and orca r nr Ca. In 997, h r OC drvav rad
Modlng and Prcng o Wahr Drvav Mark. 85 ook plac, and h ld o cla rk anagn wa born. Accordng o Valr Coopr, orr xcuv drcor o h Wahr Rk Managn Aocaon (WRMA), an $ 8 bllon cla drvav ndur wa dvlopd whn a w ar o craon. In gnral, wahr drvav covr vn o low rk and hgh probabl. On h ohr hand, wahr nuranc uuall covr hgh rk and low probabl vn, a dnd n a hghl adapd or cuozd polc. In h ca, h copan know ha h p o cla wll ac rvnu. Howvr, h a copan would probabl bu an nuranc polc o proc agan daag caud b a lood or hurrcan (hgh rk and low probabl vn). In 999, h Chcago Mrcanl xchang (CM) ook h drvav o a p urhr and nroducd uur and opon on uur radd, h r produc o knd. OC drvav ar prval ngoad, ndvdualzd ln ad bwn wo par. Howvr, CM uur and uur opon ar andardzd publcl radd conrac n h opn ark n an lcronc aucon nvronn, wh connuou prc ngoaon and ull prc ranparnc.. SOCHASIC MODL FOR MPRAUR IDX In h Scon, w dcu drn ochac odl or praur varaon. W ugg an Ornn-Uhlnbck proc drvn b L v no o odl praur lucuaon, bu alo prn n dal ohr odl propod n h lraur. Or h Ornn-Uhlnbck (hroughou: OU) proc wa propod b Uhlnbck and Ornn (93) n a phcal odllng conx, a an alrnav o Brownan Moon, whr o knd o an rvrng ndnc calld or n ordr o adqual dcrb h uaon bng odlld. A Brownan oon a b a good odl or a parcl ovn. Ar a h h parcl do no op ar changng poon, bu ov connuoul wh dcrang pd. h Brownan oon no drnabl anwhr. In ahac, h Ornn Uhlnbck proc (OU) a ochac proc ha, roughl pakng, dcrb h vloc o a av Brownan parcl undr h nlunc o rcon. h proc aonar Gau Markov proc (whch an ha boh a Gauan and Markovan proc), and h onl nonrval proc ha a h hr condon, up o allowng lnar ranoraon o h pac and varabl. Ovr, h proc nd o dr oward long-r an: uch a proc calld an rvrng. hr ar a nubr o an-rvrng odl propod n h lraur or dal avrag praur, varng prarl n hr dcrpon o h rando `no' o praur varaon. I h unprdcabl coponn o h dal avrag
86 Dr Sllauhu Prabakaran and Dr. J. P. Sngh praur ha conu wahr rk. W can condr hr drn prcrpon o h rando no r: a raconal Brownan oon, a andard Brownan oon and a Lv Proc. h objcv o h Ornn Uhlnbck rlad ochac proc o a wd ahacal audnc wh a od prparaon n ochac anal. Bnh' papr [] propo a non-gauan Ornn-Uhlnbck odl or h voluon o praur, wh aonal an and volal, and wh rdual gnrad b a Lv proc. For a parcular locaon, l. dno h dal avrag praur on da... v. b a drnc uncon, dcrbng h aonal varaon. h praur rvr on da o h an lvl. dno h ra o an-rvron on da. dno h volal o dal praur lucuaon. I a aonal uncon, akng no accoun h varaon n volal hroughou h ar. For oral dnon o Lv proc, w rr h radr o h nroducon papr []. h odl wa r nroducd b [3] wh Brownan oon. W ocu on odl o h or d d d dx () Whr X() a ochac proc on h probabl pac, F, P. hu dal avrag praur ollow an Ornn-Uhlnbck proc, rvrng o an lvl,our aonal coponn. h pd o an rvron,, conan n h odl. Fro q () w can drcl u Io orula or arngal o g h xplc oluon u u dx u () For h purpo o ng h odl o our dal avrag praur, w rorula q () b ubracng ro, ku u dx u (3)
Modlng and Prcng o Wahr Drvav Mark. 87 Wh h noaon X X approxad b Addng :. h ochac ngral could b X (4) : o boh d (5) I w dn p b p: c :, w wll hav h ollowng hr par or odllng, (6) Whr h aonal coponn, c h cclcal coponn drvd ro q (5), and h ochac par. ow, w wll buld h OU proc drvn h gnral Lv proc. L a hoognou Lv proc, or, Ornn Unlnbck L L (OU) p proc ha X X X dl dl (7) I unqu rong oluon blow SD, dx X d dl, X x (8) Whr dno h ar o dca. h nr h aonar oluon o OU proc. h lad dcul oluon o SD. W can rov h dcul b a pl chang o n h ochac ngral [4]. W can rwr OU proc a ollow X X I Y Y dl an OU proc wh argnal law D, hn w a ha Y a D OU proc. Whn gvn a on dnonal argnal law D and onl D l dcopoabl [5]. W hav h rul ha [4], (9)
88 Dr Sllauhu Prabakaran and Dr. J. P. Sngh X X dl () An Ornn Uhlnbck proc, quaon: dx x d dw x, a h ollowng ochac drnal () Whr, and ar parar and W dno h Wnr proc. h abov rprnaon can b akn a h prar dnon o an Ornn Uhlnbck proc [6] or o alo nond a h Vack odl. [7]. h probabl dn uncon x h Fokkr Planck quaon, o h Ornn Uhlnbck proc a x () x x h Grn uncon o h lnar parabolc paral drnal quaon, whr D, and h nal condon conng o a un pon a a locaon x x, xp D D (3) Whch Gauan drbuon wh an and varanc. h aonar oluon o h quaon h l or ndng o nn whch a Gauan drbuon wh an and varanc. x x (4) h Ornn Uhlnbck proc a proop o a no rlaxaon proc. Condr or xapl a Hookan prng wh prng conan k who dnac hghl ovrdapd wh rcon cocn. In h prnc o hral lucuaon wh praur, h lngh x() o h prng wll lucua ochacall around h prng r lngh x ; ochac dnac dcrbd b an Ornn Uhlnbck proc wh: k, x, k. B
Modlng and Prcng o Wahr Drvav Mark. 89 Whr drvd ro h Sok nn quaon D kb or h cv duon conan. In phcal cnc, h ochac drnal quaon o an Ornn Uhlnbck proc rwrn a a Langvn quaon x Whr k corrlad a x x (5) B wh Gauan no B x x x x xp k. Flucuaon ar k k (6) Wh corrlaon k. A qulbru, h prng or an avrag nrg accordanc wh h quparon hor. k x x kb h Ornn Uhlnbck proc on o vral approach ud o odl (wh odcaon) nr ra, currnc xchang ra, and cood prc ochacall. h parar rprn h qulbru or an valu uppord b undanal; h dgr o volal around caud b hock, and h ra b whch h hock dpa and h varabl rvr oward h an. On applcaon o h proc a radng rag known a par rad [8]. n Fgur Fro g, hr apl pah o drn OU proc wh,. and. 3 : h Boo: Inal valu a= (a.)
8 Dr Sllauhu Prabakaran and Dr. J. P. Sngh op: Inal valu a = (a.) Cnr: Inal valu norall drbuon o ha h proc ha nvaran aur. h Ornn Uhlnbck proc an xapl o a Gauan proc ha ha a boundd varanc and ad a aonar probabl drbuon, n conra o h Wnr proc; h drnc bwn h wo n hr "dr" r. For h Wnr proc h dr r conan, whra or h Ornn Uhlnbck proc dpndn on h currn valu o h proc: h currn valu o h proc l han h (long-r) an, h dr wll b pov; h currn valu o h proc grar han h (long-r) an, h dr wll b ngav. In ohr word, h an ac a an qulbru lvl or h proc. h gv h proc norav na, "an-rvrng." h aonar (long-r) varanc gvn b var x (7) h Ornn Uhlnbck proc h connuou- analogu o h dcr- AR() proc. Apoc drbuon o h ML or h Ornn-Uhlnbck proc: 4, ˆ ˆ ˆ d n n n n (8). WAHR DRIVAIVS MARK MODL Wahr drvav ar gnrall rucurd a opon, orward, uur and wap bad on drn undrlng o wahr ndx. Hr, nroduc o ndx rqunl ud on h wahr drvav ark, whch h undrlng o h praur. Gvn a wahr aon, l u no b ax and n, rpcvl, h axu and h nu praur (gnrall n dgr Clu) aurd n on da. n ax (9) A abov nond, on poran undrlng varabl or wahr drvav h dgr da. For a gvn locaon, h dgr da h praur valu o h
Modlng and Prcng o Wahr Drvav Mark. 8 drn bwn h praur o gvn da and a praur hrhold. h quan dnd blow. L dn a h an praur o a da. W dn Hang Dgr Da ( HDD - aur o cold n wndr) and Coolng Dgr Da ( CDD - aur o ha n ur), gnrad on ha da a r HDD ax, () CDD ax, () Whr r r a h rrnc praur (n gnral bwn 8 C and prdrnd praur lvl and h avrag praur calculad a n (9) or a gvn da. Fro h q. (), w can wr cuulav HDD (CHDD): CHDD HDD Whr HDD calculad a n () and h horzon, whch gnrall a onh or a aon. Bad on abov xpron [9] o praur odl an OU proc drvn b a Lv proc ha conan ndpndn proc a Brownan oon and wo an rvrng copound Poon proc. h odl rprnd a ollow: () d d dl d b (3) d whr a cclcal proc o praur and rprnd n (4). Addonall, b h an-rvron parar, and ubcrp rprn. A B Cn w (4) Whr w, h pha angl. h drnal o h drvng Lv proc dl 365 dnd a ollow: dl dw dy dz (5) h Brownan coponn o L wll b approxad b h ARCH () odl. o rprn h drn jup rucur n praur n h or o a ngl jup and a r o jup, dy and dz ar dnd a a and low an-rvrng OU proc drvn b copound Poon proc wh nn o λy and λz, and α and β bng
8 Dr Sllauhu Prabakaran and Dr. J. P. Sngh an-rvron parar, rpcvl []. dy Y d dq dz Z d dr (6) Whr Q Y U, U ar d rando varabl, U ~ Y, Y, and R V, V ar d rando varabl V ~,. Z Y h oluon o h non Gauan proc ar [] h ollowng: Z Y Z dq, z dr (7) h oluon o (3) gvn a b b bu dlu. (8) o nd h valu o a praur-bad drvav, on nd h drbuon o h undrlng praur gvn n (8). Howvr, h do no hav a clod-or oluon. On wa o addr h probl o u a characrc uncon o h praur and appl nvron chnqu o nd h valu o an HDD, an approxad drbuon o CHDD, and an approxad drbuon or praur l. o nd h [] characrc uncon gvn n (8). Fr, ung L, h characrc xponn o (5) wll b drnd, whr characrc xponn (u) dnd a ul u. h oluon o L L udwu dq z dr (9) hn, h characrc xponn o h Lv coponn wll b BM u u C, (3) Whr C dwu du
Modlng and Prcng o Wahr Drvav Mark. 83 And or jup proc dr u u r Y r u u Y (3) u Z can b wrn larl. I no pobl o valua h ngral n (3). hn, b ung lnar approxaon,. u u u u Y Y (3) Agan, (u) can b wrn larl. Fnall, h characrc uncon o h praur odl can b wrn xplcl.. xp b b u d u u (33) In xplc or,. xp C b u z b u u u z z b z b b u (34) ow, w ocu on how o aur o HDD and nvron chnqu wll b ud o nd h valu o an HDD and drbuon and hnc h CHDD valu. o nd an approxang dn uncon o praur, nvron orula wll b appld o h characrc uncon o h praur dnd n (34). Bor applng h nvron orula, h ollowng horcu ar drvd ro (34). L (x) and z b h dn uncon and characrc uncon o praur, rpcvl. b (35) z b M z z b (36) M M (37) C b V Y z Y b (38)
84 Dr Sllauhu Prabakaran and Dr. J. P. Sngh zx hn, b nvron orula x zdz x xm V V h rul Bcau wahr drvav ar dnd on CHDD, hr drbuon wll b dnd. Clarl, HDD valu a how auocorrlaon. In addon, du o h naur o h propod praur odl n r o h ndpndnc o h ncludd proc and ovaon o kp h proc pl, h odl au h ndpndnc o h HDD. Wh h aupon, h approxad drbuon o CHDD can b ound ung CHDD ~ PB PM P, PV (4) HDD ar clarl conngn cla on how praur dva ro a ba praur. A on wa o nd h xpcd valu o an HDD, h ud wll r nd Fourr ranor. hn, h nvr Fourr ranor wll b appld o boh h HDD Fourr ranor and h characrc uncon o praur []. L z wx wx zc x, (39) HDD pao uncon gvn n (), ba (B) and wˆ F, gnralzd Fourr ranoraon. hn wˆ z xp zx wx zb wˆ z z, I z. dx. hn ow, h nvron wll b appld o wˆ z z, whr ŵz (4) dnd n () and h characrc uncon dnd n (34). L praur n (8) b dnd n horhand noaon a whr dnd a n (35) and b bu dl h characrc uncon o u can b oband ro (34) and wrn a (4)
Modlng and Prcng o Wahr Drvav Mark. 85 u xp u b b z b zz u b C Y z z (43) v v z hn, HDD wz dz, u DD wˆ u u R ˆ du whr rprn xpcaon (44) Fnall, rul ndcad ha h ngral qual o B M ; hror, HDD B M (45) 3. PRICIG OF WAHR DRIVAIVS h ark or wahr drvav a pcal xapl o an ncopl ark, bcau praur l canno b radd, and o on canno or a paralll bwn pra and qu. hror, w hav o condr h ark prc o rk, n ordr o oban unqu prc or uch conrac. Snc hr no a ral ark ro whch w can oban prc, w au or plc ha h ark prc o rk conan. Furhror, w au ha w ar gvn a rk r a wh conan nr ra r and a conrac ha or ach dgr Clu pa on un o currnc. hu, undr a arngal aur Q, characrzd b h ark prc o rk, our prc proc alo dnod b a h ollowng dnac: d dv d d a (46) d whr, V, a Q Wnr proc. Snc h prc o a drvav xprd a a dcound xpcd valu undr arngal aur Q, w ar b copung h xpcd valu and h varanc o undr h aur Q. Indd, a a Granov ranoraon onl chang h dr r, h varanc o Q h a undr boh aur. hror,
86 Dr Sllauhu Prabakaran and Dr. J. P. Sngh Var au F u du (47) h ochac proc dcrbng h praur w ar lookng or hould hav a an rvrng propr. Pung all h aupon oghr, w odl praur b a ochac proc oluon () o h ollowng SD d dw d a (48) Whr ar drn h pd o h an rvron. h probl wh (48) ha acuall no rvrng o n h long run [3]. o oban a proc ha rall rvr o h an (4) and w hav o drnaon wh rpc o, hn w can g d B wc co w (49) o h dr r n (48). A h an praur no conan h r wll adjud h dr o ha h oluon o h SD ha h long run an. Sarng a x, now w go h ollowng praur odl d dw d d a, (5) d Who oluon a x Whr A B C w n Morovr, ollow ro (5) ha P a F Hnc, n vw o q (5) w u hav Q P au F F u du dw (5) (5) (53) valuang h ngral n on o h nrval whr conan, w g ha Q P a F F And h varanc a (54)
Modlng and Prcng o Wahr Drvav Mark. 87 Var a F a ow, w nd o copu h covaranc o h praur bwn wo drn da. Indd, or u au F Var F Cov, (56) u Suppo now ha and n dno h r and la da o a onh and ar h proc a o ro h onh bor [, n]. o copu h xpcd valu and varanc o n h ca, w pl h ngral n (53) and (47) no wo ngral whr conan n ach on o h. W hn g Q P a a F F And h varanc Var a a a F a j j (55) j (57) a a a j a h gnralzaon o largr nrval bco now obvou. x, a nond bor, o wahr drvav nvolvng h praur ar bad on hang or coolng dgr-da. W wll conruc how o prc a andard hang dgr-da opon. W awar o ha opon conrac and h bur o a HDD call, or xapl, pa h llr a pru a h bgnnng o h conrac. In rurn, h nubr o HDD or h conrac prod grar han h prdrnd rk lvl h bur wll rcv a paou. h z o h paou drnd b h rk and h ck z. h ck z h aoun o on ha h holdr o h call rcv or ach dgr-da abov h rk lvl or h prod. On h opon ha a cap on h axu paou unlk, or xapl, radonal opon on ock. h parar o a pcal wahr opon ar: h conrac p (call or pu), h conrac prod (.g. Januar ), h undrlng ndx (HDD or CDD), An ocal wahr aon ro whch h praur daa ar oband, h rk lvl, h ck z and h axu paou ( hr an). o nd a orula or h paou o an opon, l K dno h rk lvl and h ck z. L h conrac prod con o n da. hn h nubr o HDD and CDD or ha prod ar HDD n n n (58) HDD, and C CDD (59) n ow w can wr h pan o an uncappd HDD call a
88 Dr Sllauhu Prabakaran and Dr. J. P. Sngh ax H n K, (6) h paou or lar conrac lk HDD pu and CDD call/pu ar dnd n h a wa. whr, or plc un o currnc/hdd and n H n ax 8, (6) h conrac (6) a p o an arhc avrag Aan opon. In h ca o a lognorall drbud undrlng proc, no xac analc orula or h prc o uch an opon known. Hr w hav an undrlng proc, whch norall drbud, bu h axu uncon coplca h ak o nd a prcng orula. W hror r o ak o or o approxaon. W know ha, undr Q, undr gvn noraon a, ~, (6) whr gvn b (57) and b (58). ow uppo ha w wan o nd h prc o a conrac who paou dpnd on h accuulaon o HDD durng o prod n h wnr. 4. FORCASIG OF WAHR PRICIG In h chapr, how wahr orcang and aonal orcang can ponall prov our valuaon o wahr drvav conrac. Dnacal odl o h aophr known a aophrc gnral crculaon odl (AGCM) produc o wahr orca. h odl ar bad on dcr nurcal hod or apng o olv h connuou quaon ha ar blvd o govrn larg-cal aophrc low. W wll ar b condrng h pl ca o ung wahr orca n wahr drvav prcng, whch h calculaon o h ar prc o a lnar wap conrac on a parabl and lnar ndx (.g. a CA ndx). ow w wll dcu ha h, onl orca o h xpcd praur nd o b ud and probablc orca ar no ncar. In h r ca, w condr h calculaon o h ar prc o a lnar wap conrac on a parabl non-lnar ndx (.g. HDD). Probablc wahr orca u now b ud, bu hr no parcular dcul n rgng horcal daa and h probablc orca. hn, w condr h gnral ca, whch nclud h calculaon o h ar prc or all ohr conrac (non-lnar wap and opon) and h calculaon o h drbuon o ouco or all conrac. h h o dcul ca, and w wll prn hr chnqu or olvng h probl.
Modlng and Prcng o Wahr Drvav Mark. 89 W now condr h aon o h ar rk o a lnar wap on a parabl lnar ndx bad on dal praur. h nclud lnar wap on CA ndc and lnar wap on HDD and CDD n ho ca whr hr no chanc o h praur crong h baln. B dnon, h ar rk o a lnar wap gvn b h xpcaon o h ndx drbuon Far rk = (x) (63) Snc w ar condrng a parabl ndx, w can wr h aggrga conrac ndx x n r o dal ndc z: x d z and hnc h xpcd ndx h u o h an dal ndc: d x z For a CA ndx (64) (65) z (66) I w ar par wa hrough h conrac hn valuang (x) nvolv ung aurd praur, orca, and xpcaon ro horcal daa. I w ar ung a orca wh valu n, hn on da o h conrac x d (67) h c d cl (68) whr h ar h known horcal praur, xpcd praur ovr h orca prod, and praur ro horcal daa. c ar ngl orca gvng h cl ar claologcal an ow, w nclud HDD and CDD ndc or whch hr o chanc ha h praur wll cro h baln. W can no longr xpr h an o h dal ndx n r o h an praur, bu rahr bco a uncon o h whol drbuon o dal praur, (): z z d (69) For norall drbud praur, h ngral can uuall b valuad n r
8 Dr Sllauhu Prabakaran and Dr. J. P. Sngh o h an and h andard dvaon o praur. For HDD z z whr d d I h cuulav noral drbuon o praur on da (7), h dn o praur on da, and and ar h an and andard dvaon o praur on da. Far valu or h wap conrac or an arbrar drbuon o praur now gvn b d h x z z d z d (7) h r r h accuulad ndx du o horcal praur. h cond r h xpcd conrbuon du o orca, and h hrd r h xpcd conrbuon du o praur bond h nd o h orca. h drbuon o praur n h cond r can b akn ro a probablc orca and n h hrd r ro horcal daa. h da n h hrd r could b rad oghr a on block, and h an o h aggrga ndx ad ro horcal valu o h aggrga ndx (.. ndx odllng, bu or par o h conrac onl). For norall drbud praur and or HDD h quaon bco h c c x ax, d cl cl í (7) c c Whr c and cl l h probablc orca co n va h an and h andard dvaon o c c praur on ach da durng h orca ( and ), and h horcal daa ud n h hrd r co n va h an and andard dvaon o horcal praur cl cl and. h andard dvaon o h ndx h quar roo o h varanc o h ndx. For a parabl ndx, h varanc h u o h r n h covaranc arx o dal
Modlng and Prcng o Wahr Drvav Mark. 8 ndx valu durng h conrac prod. For a CA ndx (h xapl w wll u or lluraon) h u o h r n h covaranc arx o h dal praur. For a conrac covrd parl b orca and parl b horcal daa w can pl h r n h covaranc arx no ho ha nvolv h orca onl, ho ha nvolv horcal daa onl, and ho ha nvolv a x o horcal daa and orca. x d j j j d d d j j j j c j c j j d d d cj j c j j j c j j d d d cl cl j j j c cl c pc cov (73) c dpnd on orca varanc and corrlaon bwn praur durng h orca prod. hn, pc rprn claologcal praur varanc and corrlaon. And, nall, cov rprn orca varanc, claologcal praur varanc, and corrlaon bwn praur durng h orca and h po-orca prod. How wahr orca can b cobnd wh hod bad on dal odllng. Cobnng orca wh dal odl prhap h o lgan wa o ncorpora orca no wahr prcng bcau o h naural wa ha dal odl cop wh h u o dpndnc and h valuaon o h cov r n quaon (63). Howvr, a w wll h hod alhough lgan ar raonabl coplx. Iplnng h hod probabl onl juabl conocall an organaon radng opon vr rqunl bad on orca. ow dcrb long conrac bad hod bad on dal odllng and nroduc Prunng hod [3-8]. Fr, w u a dal praur odl o a larg nubr o praur rack, whch covr h copll ranng prod o h conrac. h rack hould b nald ro h o rcn horcal daa. W alo calcula h probabl dn aocad wh ach rack ro h dal praur odl. Scond, w calcula anohr probabl dn or ach rack ung a probablc orca. h probabl j
8 Dr Sllauhu Prabakaran and Dr. J. P. Sngh conan h orca noraon. hrd, w convr ach rack no an ndx valu. h ndx valu ar wghd ung a wgh ha proporonal o h cond (orca) dn dvdd b h r (horcal) dn. h wghd ndx valu hn dn h ndx drbuon. In ac, h wghng o ulaon n h wa calld poranc aplng [9]. h ahacal ba or h prunng hod h ollowng. L p() b h pa-o du o praur rack, () b h claologcal probabl o, and g() b h orca probabl o. hn h claologcal c l xpcd pa-o p gvn b Τ ΤdΤ cl p (74) p whr h ngral ovr all pobl rack or. h orca xpcd pa-o gvn b Τg ΤdΤ c p (75) p o valua c p c l p w choo a o rack ha ar quall pacd along h claologcal CDF, F(). In ohr word, all h valu o Τ ΤdΤ o df Τ df whr h nubr o rack. h ngral bco df ar qual, cl p pdf p (76) whr h u ovr all h rack n a dcr o pobl rack. I w valua c p p p g w c p ung h a o rack, d df whr h wgh p g p w df w ar gvn b p g g w (78) In ohr word, w u h pa-o or all pobl rack, bu wh wgh. h wghng o b ud h orca probabl dn o a cran rack dvdd b h df (77)
Modlng and Prcng o Wahr Drvav Mark. 83 claologcal probabl dn o ha rack. 5. COCLUSIO Wahr rk ha o pcc copard o ohr ourc o conoc rk: n parcular, a local gographcal rk, whch canno b conrolld. h pac o wahr alo vr prdcabl: h a cau wll alwa lad o h a c. Morovr, wahr rk on rrrd o a a volurc rk, ponal pac bng anl on h volu and no (a la drcl) on h prc. h xplan wh hdgng o wahr rk va h radng o cood uur d cul and prc. Wahr drvav ar nancal nrun who valu and/or cah low dpnd on h occurrnc o o orologcal vn, whch ar al aurabl, ndpndnl auhnabl, and ucnl ranparn o ac a rggrng undrlng or nancal conrac. pcall, locaon clarl dnd and aurn provdd b ndpndn and rlabl ourc. h undrlng orologcal vn can b condrd a no caarophc. U o wahr drvav gnralz andard nancal rk anagn pracc bad on hdgng orgn xchang and nr ra rk va nancal drvav. h an objcv o h ud o odllng and Prcng o Wahr Drvav Mark. h an goal o h ud wa dcud our objcv. W bgn our approach o conruc h praur odl undr Ornn Uhlnbck proc ha drvn b a Lv proc rahr han a andard Brownan oon nvgad. hn, w xn our approach o brl prnd h wahr drvav ark odl. In addon, w ollowd o conrucd h Modlng and prcng o wahr drvav. Fnall, w dcud how wahr orcang and aonal orcang could ponall prov our valuaon o wahr drvav conrac. In addon, h papr ndd wh concluon. RFRCS [] F.. Bnh and J. Sal_-Bnh. Sochac odllng o praur varaon wh a vw oward wahr drvav. Appld Mahacal Fnanc, ():53-85, 5. [] A. Papapanolon, An Inroducon o Lév Proc wh Applcaon n Fnanc, Lcur o, Brln, 8. hp://pag.ah.ubrln.d/~papapan/papr/nroducon.pd. [3] F. Dornr and M Qurul, Cauon o h Wnd, nrg and Powr Rk Managn,, pp. 3 3. [4] O..Barndor-ln,. Shphard. on-gauan Ornn-Uhlnbck-
84 Dr Sllauhu Prabakaran and Dr. J. P. Sngh bad odl and o o hr u n nancal conoc (wh dcuon). Journal o h Roal Sacal Soc, Sr B 63, 67-4. (a). [5] K.I.Sao, Lv proc and nnl dvbl drbuon. Cabrdg unvr pr, 99. [6] Bbbona,.; Panlo, G.; avlla, P. (8). "h Ornn-Uhlnbck proc a a odl o a low pa lrd wh no". Mrologa. 45 (6): S7 S6. do:.88/6-394/45/6/s7. [7] Chan, K. C.; Karol, G. A.; Longa, F. A.; Sandr, A. B. (99). "An prcal coparon o alrnav odl o h hor-r nr ra". Journal o Fnanc. 47 (3): 9 7. do:./j.54-66.99.b4.x. [8] Opal Man-Rvron radng: Mahacal Anal and Praccal Applcaon. World Scnc Publhng Co. 6. [9] P. Alaon, B. Djhch, and D. Sllbrgr, On odllng and prcng wahr drvav, Appld Mahacal Fnanc, vol. 9, no., pp.,. [] A. Haav and I. alal, Sochac ulacor odlng o po lcrc prc, Journal o Copuaonal and Appld Mahac, vol. 59, pp. 434 44, 4. [] R. Con and P. ankov, Fnancal Modllng wh Jup Proc, Chapan & Hall/CRC Fnancal Mahac Sr, Chapan & Hall/CRC, Boca Raon, Fla, USA, 4. [] A. L. Lw, Opon Valuaon Undr Sochac Volal, Fnanc Pr, wpor Bach, Cal, USA,. [3] Jwon, S., and R. Caballro.. Mulvara long-or odllng o dal urac ar praur and h valuaon o wahr drvav porolo. hp://rn.co/abrac=458. [4] Jwon, S., F. Dobla-R and R. Hagdorn. 3b. h an and calbraon o nbl aonal orca o quaoral Pacc Ocan praur and h prdcabl o uncran. arxv:phc/3865. [5] Jwon, S., J. Haln and D. Whhad. 3c. Movng aon and akng on. nvronnal Fnanc, ovbr. [6] Jwon, S., and D. Whhad.. Wahr rk and wahr daa. nvronnal Fnanc, ovbr. [7] Jwon, S., and M. Zrvo. 3a. h Black Schol quaon or wahr drvav. hp://rn.co/abrac=4368. [8] Jwon, S., and C. Zhann. 3. Ung nbl orca o prdc h z
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86 Dr Sllauhu Prabakaran and Dr. J. P. Sngh [34] Cao, M. and W, J. (4) Wahr drvav valuaon and ark prc o wahr rk, Journal o Fuur Mark 4(): 65 89. [35] Clark,.. (4) Can ou-o-apl orca coparon hlp prvn ovrng?, Journal o Forcang 3(): 5 39. [36] Coon, W. R. and Plk, R. A. (7) Huan Ipac on Wahr and Cla, nd dn, Cabrdg Unvr Pr, Cabrdg. [37] Dchl, B. (998) A la: A odl or wahr rk, nrg & Powr Rk Managn (3):. [38]. Bjork, Arbrag hor n Connuou, Oxord Unvr Pr, w York, Y, USA, 998. [39] Prrong, C. and Jrakan, M. Valung Powr and Wahr Drvav on a Mh Ung Fn Drnc Mhod. Workng Papr, Oln School o Bun, Wahngon Unvr. S. Lou, Mour. Jun 999.