Prcng rgh Ra Opon n Kokbakkr a Roar Adland b* gbørn ødal c a Agdr Unvry Collg rvcbox 4 4604 Kranand orway. Emal: n.kokbakkr@ha.no b Clarkon und anagmn Ld. 3 Lowr ham r London EC3R 6HE Und Kngdom. Emal: Roar_Adland@yahoo.com c Agdr Unvry Collg rvcbox 4 4604 Kranand orway. Emal: gborn.odal@ha.no Abrac h vron: arch 0 006 In h papr w up h horcal framwork for h valuaon of h Aan-yl opon radd n h frgh drvav mark. Aumng lognormal po frgh dynamc w how ha orward rgh Agrmn A ar alo lognormal pror o h lmn prod bu ha h lognormaly ubqunly brak down. W ugg approxma dynamc n h lmn prod for h A ha lad o clod-form opon prcng formula for Aan call and pu opon wrn on h po frgh ra ndc n h Black 976 framwork. In a on Carlo xprmn w how ha our formula gv vry accura prc n parcular for forward-arng frgh opon. Kyword: rgh ra Bulk hppng orward frgh agrmn Opon Rk managmn * Corrpondng auhor. Inroducon h frgh drvav mark ard wh h nroducon of frgh fuur on h Balc Inrnaonal rgh uur Exchang BIEX n ay 985. h BIEX conrac wa dgnd o facla hdgng of frgh ra n h dry bulk frgh mark bad on h Balc rgh Indx BI. h lraur on corpora rk managmn for nanc ulz 990; Bmbndr 99; roo carfn and n 993 argu ha frm can bnf from hdgng mark rk bcau xcv volaly ncra h xpcd co of fnancal dr and can lad o ubopmal nvmn. Dp amp o chang h ndx pcfcaon o ncra h aracvn of h BIEX mark for hdgng purpo Kavuano and omko 000a; Kavuano and omko 003 fald o arac uffcn radng volum and vnually cad o rad n Aprl 00. Howvr nc 99 h compng OC mark for orward
rgh Agrmn A had nabld hpownr and charrr o hdg hr phycal xpour o h po frgh mark on ndvdual rou. A ar ffcvly conrac for dffrnc bad on h avrag po frgh ra or charr hr ra ovr a pcfd prod of m. hr x a larg body of acadmc rarch on h characrc of boh h frgh fuur and A mark focung prmarly on hr prc dcovry mchanm Kavuano and omko 999; Kavuano Vvk and nachof 004 hdgng ffcvn Kauano and omko 000bc; Kavuano and Vvk 004a nformaon flow bwn po and forward mark Kavuano and Vvk 004b and h mpac of h nroducon of A on po mark volaly Kavuano Vvk and Bachlor 004. I worh nong ha h xng mprcal work on A condr only a mall ub of h mark namly hor-rm conrac up o hr monh forward n h Panamax dry bulk cor. nc h ncpon of h frgh drvav mark hr hav alo bn amp o ablh a frgh opon mark. h frgh opon currnly radd boh OC and clard ar conrac o l h dffrnc bwn h avrag po frgh ra ovr a dfnd prod of m and an agrd rk prc o-calld Aan opon. uch frgh opon can b ud by hpownr for nanc o cur a la om mnmum frgh rvnu for h duraon of h conrac wh h aocad rducon n h dfaul rk on hr loan oblgaon. h opon can n prncpl b radd for any of h ndvdual rou or compo ndc comprd by h Balc Capz Indx BCI h Balc Panamax Indx BPI h Balc upramax Indx BI h Balc Dry ankr Indx BDI or h Balc Clan ankr Indx BCI. Boh h chncal pcfcaon of h undrlyng rou and vl and h lmn mchanm ar dncal o h A mark. Prhap du o h llqud naur of h frgh opon mark unl now hr hav bn fw amp n h lraur o nvga h prcng of frgh opon. vd 998 propo an analycal oluon o h valuaon of h now-xnc Europan-yl BIEX fuur opon undr h aumpon ha h undrlyng po ra proc a log-normal man rvrng proc wh an aborbng lvl an xnon of h log-normal proc of Brnnan and chwarz 979. gka gka and gka 005 propo o prc hypohcal Europan opon on h po frgh ra ung an xnon of h andard Black and chol 973 framwork whr h drf of h po frgh ra proc drmnd by a ourr r. hr of h ud abov dal wh h prcng of h Aan opon acually radd n h frgh drvav mark oday. h prmary conrbuon of h papr hrfor h dvlopmn of a framwork for prcng uch opon. Valuaon of opon on forward conrac wa fr analyd n Black 976. W xnd h analy by condrng forward conrac ha ar ld agan h arhmc avrag ra of h undrlyng a and w propo a clod form oluon o h opon prcng problm n uch a horcal framwork. urvy of h u of frgh drvav Dnwood and orr 003; Kavuano Vvk and Goullmou 005 ypcally fnd ha famlary wh h produc povly corrlad wh parcpaon ra n h frgh drvav mark. Accordngly h framwork prnd hrn hould provd a uful praccal conrbuon o h fldglng frgh opon mark a an ducaonal ool and a an By compo ndx w hr rfr o h arhmc avrag of h rpcharr ra n h BCI BPI or BI rpcvly.
analycal bnchmark for frgh opon prcng n an ffor o rduc hgh bd-offr prad and boo mark lqudy. h rmandr of h papr rucurd a follow: con dcu h rucur of h frgh opon conrac. con 3 prn h horcal prcng framwork. con 4 conan a numrcal valuaon xampl and h rul of a on Carlo xprmn ang h prcng rror of our volaly approxmaon. nally con 5 conan concludng rmark and uggon for fuur rarch.. rgh opon rgh opon blong o h famly of pah-dpndn conngn clam calld Aan opon whch n gnral hav payoff bad on an avrag of om undrlyng varabl uch a prc or mpraur. Aan opon ar ofn ud n hnly radd commody mark o avod problm wh prc manpulaon of h undrlyng a nar or a maury. In om commody mark h naur of h commody naurally promo avrag-bad conrac. or xampl lmd pobly of orag n h naural ga and lcrcy mark lad o connuou purcha for nrgy conumr and Aan opon ar naural hdgng nrumn for rk managmn purpo Lvy 997 for vral ohr xampl. rgh ra ar alo non-orabl and o h logc appl o h frgh ra mark a wll. A charrr oprang n h po mark for nanc ypcally fac frgh ra xpour durng om prod of m. Drvav frgh ra conrac uch a forward and opon ar mor drc hdgng nrumn han Europan yp conrac dfnd on a parcular fuur m prod. Bu a frgh ra lf mplcly avrag bad nc rfr o a pcfc voyag. h frgh ra for a voyag whn h vl fxd on h po mark. I follow ha h frgh rvnu proc for a gvn hp n h phycal mark gvn by dcrly ampld prc a ochac nrval n h ordr of wk or vn monh wh ach fxng rprnng h rvnu durng ha nrval. In h am way h daly po frgh ra quo from h Balc Exchang rprn a andardzd voyag wh a andardzd duraon. In h n avragng alrady akn car of n h po frgh ra. oday mo frgh drvav ar ld agan an avrag of h po frgh ra. h arhmc avrag-bad lmn procdur for frgh opon nhrd from h A mark whr hr ha bn a gradual lnghnng of h avragng prod parly n rpon o concrn abou h pobly of mark manpulaon by larg parcpan n a hnly radd mark. or nanc Kavuano Vvk and nachof 004 no ha pror o ovmbr 999 voyag A wr ld on h avrag of h la fv radng day n h monh compard o h currn vn day. ankr A bad on h Balc ankr ndc launchd n Augu 00 ar ld on h avrag ovr all h radng day n a calndar monh. uch a lmn procdur wll alo br mmc h cah flow from a fl of hp and hu ponally mprov hdgng prformanc. Aan opon com n many flavour. An opon n whch h avrag frgh ra ld agan a fxd rk prc durng a pcfd prod calld an Avrag ra Aan opon. An opon n whch h frgh ra a a gvn fuur m pon ld agan h rk prc durng a pcfd m pror o lmn calld an avrag rk Aan opon. In ln wh mark pracc w only condr h fr yp n h 3
arcl and rfrrd o a an Aan opon for convnnc. Whl an Aan opon wh gomrc avragng ha a clod-form oluon n a andard gomrc Brownan a-prcng framwork xac prcng formula for arhmc avrag opon do no x nc h drbuon of h arhmc avrag of a lognormal proc unknown Kmna and Vor 990. h fac ha ruld n a larg rarch lraur on dffrn valuaon mhodolog arng wh h on Carlo mulaon approach of Kmna and Vor 990. umrcal oluon o h paral dffrnal quaon whch characr h prc of an Aan opon hav bn h focu of work by Rogr and h 995 Dwynn and Wlmo 995 Alzary Dcamp and Kohl 997 and Zhang 00. Yor 993 and Gman and Yor 993 dvlop analycal oluon o h Aan opon problm bu non-andard numrcal ngraon chnqu ar ndd o compu xplc prc Gman and Eydland 995 for a numrcal applcaon. vd 998 ugg ha BIEX fuur opon wr n pracc nformally valud ung h analycal prcng formula of Black 976 or Black and chol 973. Whl unufabl from a horcal and mahmacal pon of vw h wll-known opon prcng formula ar fa ay o u and famlar o radr. alb 997 conclud ha hy ar ud mor or l non-paramrcally by mark parcpan o lnk h mahmacal modl wh a ral-world daa gnrang proc. Boh A and frgh opon hav a h undrlyng a h po frgh ra of h ndvdual rou n h ca of voyag-bad conrac or h arhmc avrag rpcharr /C hr for h vl yp a publhd daly by h Balc Exchang. lmn prc ar calculad hr a h avrag po frgh ra or charr hr ovr all radng day of a calndar monh for all ankr voyag or avrag /C bad drybulk conrac or a h avrag po frgh ra or charr hr n h la 7 or 0 radng day of h monh for voyag-pcfc conrac n h Capz/Panamax and upramax drybulk mark rpcvly. or OC A and frgh opon of duraon longr han on monh uch a h quarrly and calndar yar conrac common for mo rou cah lmn occur a h nd of ach calndar monh for h duraon of h conrac 3. Accordngly a long-rm frgh opon rucurd a a floor a frgh pu opon or a cap a frgh call opon whr ach floorl or capl ld on a rollng monhly ba a an Aan opon. In pracc accordng o Clarkon cur Ld lqudy n h frgh opon mark ypcally focud on h narby calndar yar conrac on h compo mcharr ndc n h Panamax and Capz drybulk cor. h uaon rahr dffrn n h mall bu growng ankr frgh opon mark whr h grar volaly and mall avrag lo z ha crad lqudy for hor-maury monhly and quarrly frgh opon. 3. horcal framwork hr ar currnly wo ram o h gnral drvav prcng lraur Clwlow and rckland 999. h fr on ar from a ochac rprnaon of h rman h pracc oday a vdncd by h drbuon of mpld volaly numbr n opon rpor drbud n h mark by brokr uch a Clarkon cur Ld. and Y Ld. 3 h xcpon quarrly and calndar-yar voyag-pcfc conrac n h Capz mark ha ar ld only agan h avrag of h la vn day of h fr monh of ach quarr. 4
h po a and ohr ky varabl uch a h convnnc yld on h a and nr ra and drv h prc of conngn clam conn wh h po proc Gbon and chwarz 990; chwarz 997; and Hllard and R 998. h pcal ca of non-orabl commod for nanc Vack 977 whr h concp of convnnc yld and a co-of-carry rlaonhp lnkng h po and forward prc brak down ha bn xamnd mprcally by Eydland and Gman 998 Gman and Vack 00 Bmbndr and Lmmon 00 and Kavuano and Vvk 004b. odl of h po frgh ra proc hav bn propod for nanc by vd 997 and Adland and Cullnan 006. In ordr o apply uch po mark modl o drvav prcng ncary o pcfy h unobrvabl mark prc of rk or rk prmum n h frgh mark. Whl connu m o b ha h rk prmum m varyng a llurad mprcally n Kavuano and Alzadh 00 and argud from a horcal pon of vw n Adland and Cullnan 005 a uabl pcfcaon for h purpo of drvng an ndognou forward curv do no y x. Rahr h rk prmum uually aumd o b zro for analycal convnnc vd 997. h cond ram of h lraur modl h voluon of h nr forward or fuur curv n h framwork of Hah Jarrow and oron HJ 99. h only amp a applyng h framwork o h frgh mark Kokbakkr and Adland 004 who fnd ha h volaly rucur of h phycal forward curv on avrag bump hapd and ha h corrlaon bwn dffrn maur gnrally low and vn ngav. Howvr hy condr h forward curv of h phycal dry bulk mark a dcrbd by h rm rucur of mcharr ra rahr han h A curv. Whl arbrag acvy bwn h wo mark o h xn h fabl n pracc 4 hould manan a clo rlaonhp bwn h phycal and papr forward curv uncran o whch xn h rul of Kokbakkr and Adland can b appld o h lar. h framwork of h currn papr rd n h cond ram modllng opon prc condonal on h obrvd frgh forward curv and volaly. h horcal ng a andard connuou m conomy wh a fnancal mark conng of on radd rky a wh mark prc ; Duff 996 for chncal dal. Aum ha w can rad connuouly n h a n h prod [ ]. rconl borrowng and lndng pobl a h conan rkl ra r. Aum ha hr x an quvaln marngal maur Q quvaln o P and ha follow a local marngal undr h probably maur P. h po prc of frgh a m whch a non-radd a dnod. A fuur arhmc avrag of con of fxng a m pon < <... <. h bac orward rgh ra Agrmn A a cah-ld fnancal conrac ha gv h ownr of h conrac h dffrnc bwn h avrag and h prc mulpld by a conan D. Dpndng on whhr h prc and ar maurd n $/day or $/onn or h Worldcal quvaln for ankr h conan D wll rfr o h numbr of calndar day covrd by h A conrac or an agrd cargo z rpcvly. h valu of an A can b found by dcounng h cah flow rcvd a m and akng h 4 Dffrnc n phycal pcfcaon uch a z pd and ful conumpon bwn h phycal hp and h gnrc hp undrlyng h po ndc a wll a dffrnc n ar and nd da for h phycal and A conrac wll nd o mak ru arbrag rad dffcul. 5
condonal xpcaon undr h prcng maur Q. nc co nohng o nr no an A 5 no up-fron paymn w can h xpcd valu qual o zro: Q r 0 E D Rarrangng and olvng for h A prc w fnd ha mply h xpcd avrag po prc undr h prcng maur: Q E [ ] 3. po and A dynamc A n Black 976 w wll aum ha po and forward prc ar log-normally drbud. Whl h may b a rong aumpon for h po frgh ra proc n om cor of h bulk hppng ndury wha mar n our opon prcng framwork h applcably of h aumpon o A prc. L h po frgh ra dynamc b gvn by h gomrc Brownan moon d P µ d dw 3 undr h ral world probably maur a ndcad by h uprcrp of h Brownan moon. Hr µ a ral valud funcon. I may for nanc pck up aonal varaon n h frgh ra. W aum ha h volaly conan. rom h Granov horm h dynamc undr h rk-nural maur Q d whr µ γ Q d dw 4 and γ a ral valud funcon ofn nrprd a h mark prc of rk. If rprnd a radabl a.g. a non-dvdnd payng ock or a commody wh col orag would yld h rk fr ra of rurn undr h rknural maur and h mark prc of rk would b drmnd by γµr/. nc n our modl rprn a non-radabl po frgh ra h rlaonhp do no hold and w ar forcd o kp h mark prc of rk n Equaon 4. Upon comparng Equaon 3 and 4 w ha f w γ 0 hr no dffrnc bwn h proc of undr h wo probably maur. Whn h rk prmum zro h mpl ha forward prc ar unbad ma of h fuur po prc; cf. Kavuano Vvk and nachof 004. Howvr whl h rk prmum affc h prc of h A w 5 h valu of h conrac a naon hould no b confud wh h rqurmn for h dpo of a rfundabl nal margn and pobly ubqun varaon margn a collaral agan dfaul f h A clard hrough on of h clarng hou. 6
7 wll ha do no nflunc h prc of h opon. h oluon o Equaon 4 gvn by dw Q 5 L <. By ubung Equaon 5 no Equaon and aumng qudan obrvaon.. Appndx how ha h A prc can b wrn a 6 rom h xpron w can calcula A-prc from po frgh ra. Appndx alo how ha h dynamc of h A-prc dw d Q 7 whr < <... W no from Equaon 7 h abnc of a drf rm. Hnc undr h gvn prcng maur A prc ar gomrc random walk. h mak n horcally. Conrac ar prcd accordng o rk nuraly undr h prcng maur. I co nohng o nr an A conrac and rk nural prcng mpl zro xpcd payoff from a zro prc conrac. h mporan pon o noc from Equaon 6 h fac ha h dynamc chang whn h conrac nr h dlvry prod. h A conrac lognormally drbud pror o h dlvry prod < bu h lognormaly no longr appl nd h lmn prod <<. W wll propo a lognormal dynamc for h lmn prod blow. 3. A lognormal approxmaon of A dynamc or h purpo of opon prcng lognormal dynamc of h undrlyng a provd wll-known clod form oluon cf. Black 976. I hrfor ncary o com up wh a afacory lognormal approxmaon o Equaon 7. Condr h followng dffrnal undr h prcng maur: dw d Q 8 whr
8 < <... A conrac wh dynamc a dfnd n Equaon 8 lognormally drbud a ln ~ ln d d 9 In h ca of qudan obrvaon n h lmn prod Appndx prov ha h xpron for h varanc can b wrn a 6 3 d 0 By dfnon w hav / o Equaon 0 can b wrn alrnavly a R whr R 3 3 3 3 In hory no pracc w can conruc a conrac ha l agan a connuou avrag frgh ra by lng. hn Equaon mplf o 3 o ha h fr par of h approxmaon m from volaly pror o h lmn prod whl h cond rm m from h lar prod. rom Equaon 8 and h undrlyng characrzaon of h prc drbuon gvn by Equaon 7 clar ha h approxmaon br h horr h dlvry prod and h longr no h fuur dlvry ak plac. gur plong h funcon R dmonra h gnfcanc of h numbr of fxng prod n h approxmaon. h funcon convrg rapdly oward h ca wh connuou lmn a long a >5. In hppng mo drvav conrac hav mor fxng han 5.. h lmn prod ypcally 7 bun day or longr and o h on-hrd-varanc rul n h lmn prod provd uffcn accuracy for praccal prcng purpo. < Inr gur abou hr >
3.3 rgh ra opon Equaon and how ha h un A conrac can b nrprd a h prc oday a m o dlvr a m h valu of h arhmc avrag of h undrlyng po frgh ra durng h prod [ ]. Imporanly h man ha an Aan opon can b rnrprd a a Europan opon on h forward conrac. andard arbrag argumn mply ha h prc a m for h avrag a prc durng h nrval qual o h rald avrag. Ung Equaon h payoff of an Aan call opon wh rk prc K and maury can hrfor quvalnly b ad a D K D[ K ] 3 h mlar xpron for a pu opon D K D[ K ] 4 I wll-known from fnancal hory Duff 996 ha h valu of a conngn clam gvn by h xpcd payoff wh rpc o h prcng maur dcound by h rk fr ra. Accordngly h mark valu a m < of h Aan call opon C and pu opon P wh maury can b wrn a C r Q D E [ K ] 5 and r Q D E [ K ] P 6 A cap a drvav conrac ha provd frgh ra procon for h buyr abov a prdrmnd lvl h cap ra - for a prdrmnd prod of m. A floor guaran downd procon a a prdrmnd ra h floor ra durng a prdrmnd prod of m. 6 A hpownr oprang n h po mark and farng low fuur frgh ra can buy a floor for nuranc. A charrr oprang n h po mark ypcally buy a cap. In fac nc h pay off of h pu and call xpron n Equaon 5 and 6 ar conngn on h marald frgh ra durng a m prod hy f h dfnon of cap and floor. ngl payoff opon ar ofn calld floorl pu and capl call. Cap and floor old n h mark ar ofn dfnd on m prod longr han ndvdual opon conrac. uch rucur ar mply h um of opon bundld oghr. or nanc h valu oday of a calndar-yar frgh cap floor h um of h valu of ndvdual Aan call pu opon wh dncal 6 Clwlow and rckland 000 for ohr drvav conrac ud n nrgy mark. 9
rk prc ach corrpondng o a monhly lmn prod. ormally h valu of a -prod frgh cap opon can b wrn a Cap m m C 7 whl h mlar floor opon gvn by loor m m P 8 whr h m rprn h nd of ach pcfc dlvry prod. o compu h xpcaon n Equaon 5 and 6 w nd a ochac modl for hr xplcly or mplcly hough h dynamc of. Gvn h A dynamc dcrbd hrn ha ld o h approxmaon n Equaon 8 frgh opon prcng now bol down o applyng Black 976 andard opon prcng formula ung h volaly plug-n from Equaon or. hu h prc a m < for a call opon gvn by C r D φ d Kφ d 9 whr d ln K d d and φx h andard cumulav normal drbuon funcon. or h pu opon w can calcula h xpcaon n Equaon 6 drcly or alrnavly u h pu-call pary for fuur conrac and h ymmry propry of h normal drbuon o drv P r D K d φ d φ 0 wh d d and a dfnd abov 7. In ordr o prc frgh floor and cap.. frgh opon wh > monhly lmn prod Equaon 7 and 8 ar appld n conuncon wh Equaon 5 and 6. 4. umrcal xampl A dcud n con A and frgh opon ar ypcally ld agan h avrag po frgh ra ovr h la vn radng day n a monh or h avrag 7 W no ha n gnral can b nrprd a h oal varanc for h naural log of from o undr h aumpon of log-normaly. In h ca of conan volaly durng h nr prod h bcom h mor famlar. 0
acro all radng day n h monh dpndng on h undrlyng ndx. Whl h wo lmn procdur ar ypcally no ud concurrnly for any ndvdual ndx hough n an OC mark h par ar fr o agr on any lmn arrangmn hy wh uful o a h mpac of h choc of fxng prod on h frgh opon valu. W would alo lk o ablh h lvl of accuracy of h propod volaly approxmaon hrough a on Carlo C mulaon xprmn. In h on Carlo ngraon w u andard Europan opon a conrol vara o mprov ffcncy. Dal of h procdur can b found n h Appndx. Condr h followng numrcal xampl for a calndar-yar frgh opon wh monhly lmn. h currn po frgh ra $500/day h rk prc K $5000/day and h annuald volaly of h undrlyng po frgh ra 30% 0. 3. In a ral world prcng ca w would ypcally u h obrvd A prc a npu bu nc our xampl for llurav purpo only w mply compu A prc from Equaon 6. W h rk nural frgh ra drf o 0.03. A pov drf for h po prc undr h prcng maur mpl an upward lopng rm rucur of forward ra. h vdn from h fr column n abl. h conan D calndar day. Ung uprcrp o dno h monh of h yar w hav D 3 January D 8 bruary D 3 Dcmbr. or boh cap and floor w calcula h prc a h ar of h yar for boh a 7-day panl A and -day panl B lmn arrangmn. In h fr ca w hav m 5/5 *m and m /5 *m for m 0. or h lar lmn arrangmn w hav m /5 *m and m /5 *m for m 0. W aum qudan obrvaon gnorng wknd and holday and apply Equaon for h plug-n varanc n Equaon 9 and 0. h on Carlo prc for ach floorl/capl ar gvn n h column nx o h approxmad prc. h prcnag dffrnc bwn h wo ar alo rpord. vral pon ar worh makng abou h numrcal rul n abl. r h A prc n panl A ar margnally hghr han h corrpondng prc n panl B. h du o h fac ha h lmn prod n panl A ar h fr radng day of h monh whl n panl B ar on day 5. Hnc all A n panl A hav longr m o dlvry han ho of panl B. cond all floorl prc ar hghr han h corrpondng capl prc a wh K > ; h floorl ar n-h-mony. On pon worh nocng ha floorl prc do no ncra monooncally wh maury n boh panl A and B; hy drop n bruary pmbr and ovmbr. h du o h numbr-of-day-n-h-monh ffc a h prcng formula dpnd on h acual numbr of calndar day hrough h conan D. h floorl valu for pmbr lowr han Augu bcau pmbr ha on day l han Augu D 9 3 vru D 0 30. or bruary wh only 8 day h ffc qu rong. or capl n our parcular xampl m valu domna h numbr-of-day-n-h-monh ffc and hrfor capl ncra monooncally wh m. nally w ha boh h cap and h floor hav hghr prc n panl A han n panl B. hr ar wo ffc prn hr; dffrn volaly npu and dffrn A prc. h fr ffc h rongr. or h valuaon of h cap boh ffc pull n h am drcon. hor dlvry prod boh ncra h A prc and h varanc npu n panl A rlav o panl B. or h floor lookng a an ncra n A prc n olaon gv hghr floorl prc n panl B han n panl A. Howvr an A wh hor lmn prod ha hghr
varanc han a conrac wh longr lmn prod cf. Equaon 6 and a hghr varanc gv hghr capl/floorl prc. W from h abl ha h varanc ffc much rongr han h prc ffc rulng n a hghr floor prc n panl A compard o panl B. h mp.vol column provd h volaly npu ha gv h am prc of a pu/call n a andard Black modl wh maury a our corrpondng floorl/capl prc. h column dmonra h volaly ffc of h lmn prod for h conrac. h undrlyng frgh ra volaly 30%. rom Equaon 3 w know ha h A alo hav 30% volaly pror o h lmn prod whl dcra o zro n h lmn prod. Wh 30% a an uppr lm for h A volaly w no ha for h la floorl/capl n boh panl A and panl B volaly vry clo o 30% nc h lmn prod rlavly hor compard o h oal lf of h conrac. or hor rm conrac h dfnon of h lmn prod ha a rong ffc wh mpld volaly of 7.9% vru 6.9% for h conrac wh lmn h followng monh. o um up abl clarly how ha our analycal approxmaon farly accura and ha h prcng rror dcrang n m o maury. < Inr abl abou hr > 5. Concludng rmark In h papr w hav prnd h mahmacal framwork for frgh opon modllng. Aumng lognormal po frgh dynamc w how ha A ar lognormal pror o h lmn prod bu ha h lognormaly brak down n h lmn prod. W ugg approxma dynamc n h lmn prod for h A ha lad o clod form opon prcng formula for Aan call and pu opon wrn on h po frgh ra ndc n h Black 976 framwork. In a on Carlo xprmn w how ha our formula gv qu accura prc. h analy ung mpld volaly n h Black 976 framwork mgh b xndd o a mor ralc modl for h undrlyng a prc dynamc n h on Carlo xrc for xampl allowng for ump and ochac volaly. W lav h for fuur rarch. I alo vdn from marm conomc hory ha ohr ochac pcfcaon of h po and forward frgh ra proc may b mor appropra. or nanc ødal Kokbakkr and Adland 005 dvlop a ral opon bad valuaon modl for combnaon carrr whch hng on h aumpon of man rvron of h dry bulk and ankr frgh mark. an rvron n frgh ra mpl a rm rucur n h volaly. uur xnon of h work hould ncorpora h rm rucur of volaly ha x du o man rvron n h po frgh ra proc cf. Kokbakkr and Adland 004; vd 998. h pobl xnc of aonal volaly hould alo b nvgad.
Rfrnc Adland R. Cullnan K. 006. h non-lnar dynamc of po frgh ra n ankr mark. ranporaon Rarch Par E: Logc and ranporaon Rvw 43-4. Adland R. Cullnan K. 005. A m-varyng rk prmum n h rm rucur of bulk hppng frgh ra. Journal of ranpor Economc and Polcy 39 9-08. Alzary B. Dcamp J. P. Kohl P.. 997. A P.D.E approach o Aan opon: analycal and numrcal vdnc. Journal of Bankng and nanc 63-640. Bmbndr H. 99. orward conrac and frm valu: nvmn ncnv and conracng ffc. Journal of nanc and Quanav Analy 6 59-53. Bmbndr H. Lmmon.L. 00. Equlbrum prcng and opmal hdgng n lcrcy forward mark. Journal of nanc 57 347-38. Black. 976. h prcng of commody conrac. Journal of nancal Economc 3 67-79. Black. chol. 973. h prcng of opon and corpora labl. Journal of Polcal Economy 8 637-654. Brnnan.J. chwarz E.. 979. A connuou m approach o h prcng of bond. Journal of Bankng and nanc 3 33 55. Clwlow L. rckland C. 999. Valung nrgy opon n a on facor modl fd o forward prc. Workng papr Unvry of chnology ydny. Clwlow L. rckland C. 000. Enrgy Drvav Prcng and Rk anagmn. Lacma Publcaon London. Dnwood J. orr J. 003. ankr forward frgh agrmn: h fuur for frgh fuur? arm Polcy and anagmn 30 45-58. Dwynn J.. Wlmo P. 995. A no on avrag ra opon wh dcr amplng. IA Journal of Appld ahmac 55 995 67-76. Duff D. 996. Dynamc A Prcng hory. Prncon Unvry Pr Prncon w Jry nd don. roo K. charfn D. n J. 993. Rk managmn: Coordnang corpora nvmn and fnancng polc. Journal of nanc 48 69-658. Eydland A. Gman H. 998. Prcng powr drvav. RIK Ocobr 7-73. 3
Gman H. Vack O. 00. Pluggng no lcrcy. RIK 4 Augu 93-97. Gman H. Eydland A. 995. Domno ffc: Invrng h Laplac ranform. RIK arch. Gman H. Yor. 993. Bl proc Aan opon and prpu. ahmacal nanc 3 349-375. Gbon R. chwarz E.. 990. ochac convnnc yld and h prcng of ol conngn clam. Journal of nanc 45 959-976. Hah D. Jarrow R. oron A. 99. Bond prcng and h rm rucur of nr ra: A nw mhodology for conngn clam valuaon. Economrca 60 77-05. Hllard J.E. R J. 998. Valuaon of commody fuur and opon undr ochac convnnc yld nr ra and ump dffuon n h po. Journal of nancal and Quanav Analy 33 6-86. Kavuano.G. Alzadh A. 00. h xpcaon hypoh of h rm rucur and rk prma n dry bulk hppng frgh mark: An EGARCH- approach. Journal of ranpor Economc and Polcy 36 67-304. Kavuano.G. omko. 999. h forward prcng funcon of hppng frgh fuur mark. Journal of uur ark 98: 353-376. Kavuano.G. omko. 000a. uur hdgng whn h rucur of h undrlyng a chang: h ca of h BIEX conrac. Journal of uur ark 03 775 80. Kavuano.G. omko. 000b. Dynamc hdgng n h frgh fuur mark Journal of Drvav 8 40-58. Kavuano.G. omko. 000c. Conan v. m-varyng hdg rao and hdgng ffcncy n h BIEX mark. ranporaon Rarch Par E: Logc and ranporaon Rvw 369 9-48. Kavuano.G. omko. 003. Prc dcovry caualy and forcang n h frgh fuur mark. Rvw of Drvav Rarch 68 03-30. Kavuano.G. Vvk I. 004a. h hdgng prformanc of ovr-h-counr forward hppng frgh mark. In: Inrnaonal Aocaon of arm Econom IAE Confrnc Procdng Izmr urky 30 Jun July 004. 4
Kavuano.G. Vvk I. 004b. ark nracon n rurn and volal bwn po and forward hppng frgh mark. Journal of Bankng and nanc 83 05-049. Kavuano.G. Vvk I. Bachlor R. 004. Ovr-h-counr forward conrac and po prc volaly n hppng. ranporaon Rarch Par E: Logc and ranporaon Rvw 409 73-96 Kavuano.G. Vvk I. Goullmou.A. 005. An nvgaon of h u of rk managmn and hppng drvav: h ca of Grc. In: Procdng for h Annual Confrnc of h Inrnaonal Aocaon of arm Econom IAE Lmaol Cypru Jun 3 5. Kavuano.G. Vvk I. nachof D. 004. h unbadn hypoh n h frgh forward mark: Evdnc from congraon. Rvw of Drvav Rarch 78 4 66. Kmna A. Vor A. 990. A prcng mhod for opon bad on avrag a valu. Journal of Bankng and nanc 4 33-9. Kokbakkr. Adland R. 004. odllng forward frgh ra dynamc mprcal vdnc from m charr ra. arm Polcy and anagmn 39 39 335. Lvy E. 997. Aan Opon. In: Clwlow L. and rckland. Ed. Exoc Opon h a of h Ar Inrnaonal hompon Bun Pr London. Rogr L. C. G. h Z. 995. h valu of an Aan opon. Journal of Appld Probably 3 077-088. chwarz E.. 997. h ochac bhavour of commody prc: Implcaon for prcng and hdgng. Journal of nanc 58 93-973. ulz R. 990. anagral dcron and opmal fnancng polc. Journal of nancal Economc 6 3-7. ødal. Kokbakkr. Adland R. 005. ark wchng n hppng a ral opon modl appld o h valuaon of combnaon carrr Workng Papr Agdr Unvry Collg. alb. 997. Dynamc hdgng: anagng vanlla and xoc opon. Wly w York. gka I. gka D.. gka. 005. Opon prcng and rk managmn n hppng. In: Procdng for h AE ympoum on hp opraon managmn and conomc Ahn Grc ay - 4. 5
vd J. 998. Valuaon of a Europan fuur opon n h BIEX mark. Journal of uur ark 8 67-75. vd J. 997. Valuaon of VLCC undr ncom uncrany. arm Polcy and anagmn 4 59-74. Vack O. 977. An Equlbrum Characraon of h rm rucur. Journal of nancal Economc 5 77-88. Yor. 993. rom planar Brownan wndng o Aan opon. Inuranc: ahmac and Economc 3 3-34. Zhang J.E. 00. A m-analycal mhod for prcng and hdgng connuouly ampld arhmc avrag ra opon. Journal of Compuaonal nanc 5 59-80. 6
7 Appndx Proof of quaon 6 7 and 0 Proof of Equaon 6: A prc a a funcon of undrlyng po prc Equaon 6 follow from nrng Equaon 5 no Equaon and akng xpcaon. h p-by-p calculaon a follow: [ ] E E dw Q Q Q 0 A. h corrpond o Equaon 6 n h x. h hrd qualy follow from h xpcaon of a lognormal random varabl and h fourh from h aumpon of qudan obrvaon. Proof of Equaon 7: h A dynamc h dynamc for h A conrac chang a nr h lmn prod. In parcular h volaly of h conrac dcra a h lmn prod gradually maral. hrfor h dynamc of h A conrac mu b nvgad boh pror o h lmn prod < and nd h lmn prod < <. r howvr w mu lnk h A prc o h undrlyng po prc. h A prc wrn a a funcon of h undrlyng po prc gvn by A.. ow aum ha h fr < fxng hav alrady bn obrvd o ha < <. [ ] Q E A. ow w wan o apply Ió lmma o h xpron for h A prc. Ió lmma a horm of ochac calculu ha how ha cond ordr dffrnal rm of Wnr
8 proc bcom drmnc undr ochac ngraon. I omwha analogou o h chan rul n ordnary calculu. L x b a gnrald Wnr proc. ha l dw x b d x a dx Accordng o Ió lmma x f alo a gnrald Wnr proc a follow: dw x f x b d x f x b f x f x a x df A.3 Rcall h dynamc of from Equaon 4. o fnd h dynamc of for < w apply Ió lmma o A.. h gv dw dw d d A.4 h la qualy follow from Equaon A. and w hav ablhd h lognormal propry of pror o. o fnd h dynamc of for < < wh h fr < fxng obrvd apply Ió lmma o A.. h gv dw dw d d A.5 W obrv from A.5 ha h lognormal propry no longr hold. W hrfor nd boh A.4 and A.5 o dcrb h dynamc of h A. Dvdng ach of h quaon A.4 and A.5 wh prov Equaon 7 n h x. Proof of Equaon 0: h volaly plug-n rom Equaon 8 w know ha gvn by < <... h varanc durng h prod [ ] wh < dfnd by d. h can b calculad xplcly a
9 6 3 d d d d d L L L L A.6 h la qualy follow from h aumpon of qudan obrvaon. h prov Equaon 0 n h x. Appndx - h on Carlo procdur on Carlo calculaon ar ofn ud o chck h accuracy of approxma formula. Evn hough a condonal xpcaon canno b olvd n clod form w can compu h xpcaon numrcally by mulang h ochac proc a hand and compung h opon prc. h on Carlo ma h avrag opon prc from many mulaon. h accuracy of our maor h dffrnc bwn h on Carlo opon prc and h ru opon prc can b mad arbrarly mall by ncrang h numbr of mulaon. W dmonra h on Carlo procdur for floorl only. In ach draw w mula prc undr h marngal maur and collc h rul n a vcor [ ] u K. x w calcula h rald floorl u h : r K u h A.7 A mpl on Carlo maor fˆ for h floorl hn ˆ u h f A.8 whn h mulaon rpad m. h accuracy of h maor can b xprd by h andard rror ε ˆ A.9 whr
ˆ h u h u h mallr h andard rror h clor w g o h ru floorl prc. o rduc h andard dvaon w hav mployd h conrol vara chnqu. h nal ung h prc n ach random draw o compu a prc for a mlar conrac for whch w do hav a clod form oluon. * Dno by g h Black 976 clod form prc oday for a Europan pu opon ha pay off [ ] K a maury wh volaly and nr ra qual o h ca of h floorl. Dfn h rald pu prc a [ K ] g u A.0 r W know ha h analycal prc qual h on Carlo prc for a andard pu uch ha h xpcaon: Q * E g g u 0 A. h allow u o dfn a on Carlo maor for h floorl a f * h u g g u A. h andard rror of h maor can b compud h am way a bfor bu h maor wh a conrol vara ha lowr varably han h mpl on Carlo ma. W can from h xpron of f how varably rducd. If an opon nd ou of h mony h mulad prc zro. W nd mor mulaon u o g a pov valu of our opon. On h ohr hand f our mulaon ruld n a hug payoff for h opon w nd a lo of mulaon o g h avrag prc back o a raonabl lvl. h ob of h conrol vara o do h adumn for ach draw. In ordr for h o work afacorly h acual drvav mu b rongly corrlad wh h conrol vara. In our ca h man ha whn h u zro g u ypcally alo zro du o an upward drfng mulad frgh ra pah ladng o an upward adumn of h prc maor n Equaon A.. Convrly n ca whr h mulad frgh ra rongly downward drfng h adumn ngav. Ovrall h adumn cancl ou du o Equaon A.. h fnal rul ha f l varabl han fˆ mprovng h ffcncy of h maor. 0
gur : Aympoc bhavour of an A prc approxmaon. /3 / R /3 /6 0 4 6 8 0 4 6 8 0 4 6 8 30
abl : Cap and floor valuaon Clod form approxmaon and on Carlo ma Panl A: lmn prod la 7 radng day of ach monh Cap valuaon C rror loor valuaon C rror onh A Imp.vol Approx. C %-dff d. % Approx. C %-dff d. % Jan 548 6.9% 48 433-0.00 % 3.0 0.% 7879 789-0.06 % 7.3 0.0% b 605 8.5% 803 806-0.03 % 3.6 0.04% 7486 7488-0.003 % 6. 0.0% ar 66 9.0% 5696 5700-0.0 % 4.6 0.03% 87475 87484-0.00 % 6.5 0.0% Apr 78 9.3% 64 64-0.00 % 4.8 0.0% 8876 887 0.006 % 6. 0.0% ay 775 9.4% 765 766-0.003 % 5. 0.0% 95463 95467-0.004 % 6. 0.0% Jun 83 9.5% 396 3966-0.05 % 5.3 0.0% 9576 9579-0.04 % 5.7 0.0% Jul 889 9.6% 388 385 0.007 % 5.6 0.0% 0058 0056 0.00 % 5.8 0.0% Aug 946 9.6% 4974 4968 0.04 % 5.8 0.0% 04963 04970-0.007 % 5.7 0.0% p 3004 9.7% 46065 4606 0.006 % 5.7 0.0% 048 0475 0.006 % 5.4 0.0% Oc 306 9.7% 5033 5045-0.03 % 6.0 0.0% 06 060 0.00 % 5.5 0.00% ov 39 9.7% 5448 54474 0.0 % 6.0 0.0% 08877 08877-0.00 % 5. 0.00% Dc 377 9.8% 60408 6043-0.009 % 6.3 0.0% 4706 4708-0.00 % 5.3 0.00% Cap 4065 408-0.004 % loor 653 65357-0.003 % $/day 099 099 $/day 393 393 Panl B: lmn prod all radng day of ach monh Cap valuaon C rror loor valuaon C rror onh A Imp.vol Approx. C %-dff d. % Approx. C %-dff d. % Jan 59 7.9% 34 333-5.478 % 5.7.7% 76645 76677-0.04 % 3.0 0.0% b 586 4.7% 5484 5497-0.5 % 6.7 0.% 7630 7659-0.039 %. 0.0% ar 64 6.6% 83 86 0.049 % 8.4 0.07% 8590 859-0.00 %.9 0.0% Apr 699 7.5% 8586 8588-0.0 % 8.8 0.05% 86699 86708-0.00 %. 0.0% ay 756 8.0% 537 558-0.08 % 9.6 0.04% 93554 93556-0.00 %. 0.0% Jun 83 8.3% 9696 970-0.06 % 9.7 0.03% 940 94006 0.005 % 0.6 0.0% Jul 870 8.6% 359 3593-0.03 % 0.4 0.03% 0049 004 0.007 % 0.7 0.0% Aug 97 8.8% 40863 4086 0.005 % 0.7 0.03% 0348 0349-0.00 % 0.5 0.0% p 985 8.9% 4409 4435-0.060 % 0.6 0.0% 0783 0769 0.04 % 0.0 0.0% Oc 304 9.0% 5009 50080 0.0 %. 0.0% 08795 08788 0.006 % 0. 0.0% ov 300 9.% 5668 5653 0.07 %. 0.0% 0760 0765-0.004 % 9.7 0.0% Dc 358 9.% 58596 58594 0.003 %.6 0.0% 3469 3484-0.03 % 9.9 0.0% Cap 37487 374348-0.06 % loor 4543 45304-0.005 % $/day 05 06 $/day 338 338 Aumpon: po ra 500 mark prc of rk 0.03 rk prc K 5000 volaly 30% on Carlo rul ar n ach ca bad on 5000000 mulad po ra pah