Pricing Freight Rate Options

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1 Prcng rgh Ra Opon n Kokbakkr a Roar Adland b* gbørn ødal c a Agdr Unvry Collg rvcbox Kranand orway. Emal: [email protected] b Clarkon und anagmn Ld. 3 Lowr ham r London EC3R 6HE Und Kngdom. Emal: [email protected] c Agdr Unvry Collg rvcbox Kranand orway. Emal: [email protected] Abrac h vron: arch 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

2 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.

3 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

4 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

5 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

6 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 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 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 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 >

9 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

10 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 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

11 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% 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 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 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

12 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.

13 Rfrnc Adland R. Cullnan K h non-lnar dynamc of po frgh ra n ankr mark. ranporaon Rarch Par E: Logc and ranporaon Rvw Adland R. Cullnan K A m-varyng rk prmum n h rm rucur of bulk hppng frgh ra. Journal of ranpor Economc and Polcy Alzary B. Dcamp J. P. Kohl P A P.D.E approach o Aan opon: analycal and numrcal vdnc. Journal of Bankng and nanc Bmbndr H. 99. orward conrac and frm valu: nvmn ncnv and conracng ffc. Journal of nanc and Quanav Analy Bmbndr H. Lmmon.L. 00. Equlbrum prcng and opmal hdgng n lcrcy forward mark. Journal of nanc Black h prcng of commody conrac. Journal of nancal Economc Black. chol h prcng of opon and corpora labl. Journal of Polcal Economy Brnnan.J. chwarz E A connuou m approach o h prcng of bond. Journal of Bankng and nanc Clwlow L. rckland C Valung nrgy opon n a on facor modl fd o forward prc. Workng papr Unvry of chnology ydny. Clwlow L. rckland C Enrgy Drvav Prcng and Rk anagmn. Lacma Publcaon London. Dnwood J. orr J ankr forward frgh agrmn: h fuur for frgh fuur? arm Polcy and anagmn Dwynn J.. Wlmo P A no on avrag ra opon wh dcr amplng. IA Journal of Appld ahmac Duff D Dynamc A Prcng hory. Prncon Unvry Pr Prncon w Jry nd don. roo K. charfn D. n J Rk managmn: Coordnang corpora nvmn and fnancng polc. Journal of nanc Eydland A. Gman H Prcng powr drvav. RIK Ocobr

14 Gman H. Vack O. 00. Pluggng no lcrcy. RIK 4 Augu Gman H. Eydland A Domno ffc: Invrng h Laplac ranform. RIK arch. Gman H. Yor Bl proc Aan opon and prpu. ahmacal nanc Gbon R. chwarz E ochac convnnc yld and h prcng of ol conngn clam. Journal of nanc Hah D. Jarrow R. oron A. 99. Bond prcng and h rm rucur of nr ra: A nw mhodology for conngn clam valuaon. Economrca Hllard J.E. R J Valuaon of commody fuur and opon undr ochac convnnc yld nr ra and ump dffuon n h po. Journal of nancal and Quanav Analy 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 Kavuano.G. omko h forward prcng funcon of hppng frgh fuur mark. Journal of uur ark 98: Kavuano.G. omko. 000a. uur hdgng whn h rucur of h undrlyng a chang: h ca of h BIEX conrac. Journal of uur ark Kavuano.G. omko. 000b. Dynamc hdgng n h frgh fuur mark Journal of Drvav 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 Kavuano.G. omko Prc dcovry caualy and forcang n h frgh fuur mark. Rvw of Drvav Rarch 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

15 Kavuano.G. Vvk I. 004b. ark nracon n rurn and volal bwn po and forward hppng frgh mark. Journal of Bankng and nanc Kavuano.G. Vvk I. Bachlor R Ovr-h-counr forward conrac and po prc volaly n hppng. ranporaon Rarch Par E: Logc and ranporaon Rvw Kavuano.G. Vvk I. Goullmou.A 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 h unbadn hypoh n h frgh forward mark: Evdnc from congraon. Rvw of Drvav Rarch Kmna A. Vor A A prcng mhod for opon bad on avrag a valu. Journal of Bankng and nanc Kokbakkr. Adland R odllng forward frgh ra dynamc mprcal vdnc from m charr ra. arm Polcy and anagmn Lvy E 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 h valu of an Aan opon. Journal of Appld Probably chwarz E h ochac bhavour of commody prc: Implcaon for prcng and hdgng. Journal of nanc ulz R anagral dcron and opmal fnancng polc. Journal of nancal Economc ødal. Kokbakkr. Adland R ark wchng n hppng a ral opon modl appld o h valuaon of combnaon carrr Workng Papr Agdr Unvry Collg. alb Dynamc hdgng: anagng vanlla and xoc opon. Wly w York. gka I. gka D.. gka Opon prcng and rk managmn n hppng. In: Procdng for h AE ympoum on hp opraon managmn and conomc Ahn Grc ay

16 vd J Valuaon of a Europan fuur opon n h BIEX mark. Journal of uur ark vd J Valuaon of VLCC undr ncom uncrany. arm Polcy and anagmn Vack O An Equlbrum Characraon of h rm rucur. Journal of nancal Economc Yor rom planar Brownan wndng o Aan opon. Inuranc: ahmac and Economc Zhang J.E. 00. A m-analycal mhod for prcng and hdgng connuouly ampld arhmc avrag ra opon. Journal of Compuaonal nanc

17 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

18 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

19 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

20 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

21 gur : Aympoc bhavour of an A prc approxmaon. /3 / R /3 /

22 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 % % % % % b % % % % % ar % % % % % Apr % % % % % ay % % % % % Jun % % % % % Jul % % % % % Aug % % % % % p % % % % % Oc % % % % % ov % % % % % Dc % % % % % Cap % loor % $/day $/day 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 % % 5.7.7% % % b % % % %. 0.0% ar % % % %.9 0.0% Apr % % % %. 0.0% ay % % % %. 0.0% Jun % % % % % Jul % % % % % Aug % % % % % p % % % % % Oc % %. 0.0% % % ov % %. 0.0% % % Dc % %.6 0.0% % % Cap % loor % $/day $/day 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 mulad po ra pah