Commodity market modeling and physical trading strategies

Transcription

1 Commodiy mare modeling and physical rading sraegies by Per Einar S. Ellefsen Ingénieur de l Ecole Polyechnique, 8 Submied o he Deparmen of Mechanical Engineering in parial fulfillmen of he requiremens for he degree of Maser of Science in Mechanical Engineering a he Massachuses Insiue of Technology June Massachuses Insiue of Technology. All righs reserved. Signaure of Auhor: Deparmen of Mechanical Engineering May 3, Cerified by: Paul D. Sclavounos Professor of Mechanical Engineering and Naval Archiecure Thesis Supervisor Acceped by: David E. Hard Professor of Mechanical Engineering Chairman, Deparmen Commiee on Graduae Sudens

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3 Commodiy mare modeling and physical rading sraegies by Per Einar S. Ellefsen Submied o he Deparmen of Mechanical Engineering on May 3, in parial fulfillmen of he requiremens for he degree of Maser of Science in Mechanical Engineering ABSTRACT Invesmen and operaional decisions involving commodiies are aen based on he forward prices of hese commodiies. These prices are volaile, and a model of heir evoluion mus correcly accoun for heir volailiy and correlaion erm srucure. A wo-facor model of he forward curve is proposed and calibraed o he crude oil, shipping, naural gas, and heaing oil mares. The heoreical properies of his model are explored, wih focus on is decomposiion ino independen facors affecing he level and slope of he forward curve. The wo-facor model is hen applied o wo problems involving commodiy prices. An approximae analyical expression for he prices of Asian opions is derived and shown o explain he mare prices of shipping opions. The floaing sorage rade, which appeared in he oil mare in lae 8, is presened as an opimal sopping problem. Using he wo-facor model of he forward curve, he value of soring crude oil is derived and analyzed hisorically. The analyical framewor for physical commodiy rading ha is developed allows for he calculaion of expeced profis, riss involved, and exposure o he major ris facors. This maes i possible for mare paricipans o analyze such physical rades in advance, creaes a decision rule for when o sell he cargo, and allows hem o hedge heir exposure o he forward curve correcly. Thesis supervisor: Paul D. Sclavounos Tile: Professor of Mechanical Engineering and Naval Archiecure 3

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5 Table of Conens. INTRODUCTION Commodiy mares Definiions and mares Moivaion Objecives Mehodology and ouline.... MARKET MODEING.... Raionale.... Exising lieraure Exploraory daa analysis Two-facor model of commodiy fuures Principal componens analysis Forward curve seasonaliy Mare calibraion Forward ris premia from he ris neural o he objecive measure Exension o hree facors...3. Model of he saic forward curve...3. Applicaions of he mare model ASIAN OPTIONS ON COMMODITIES Definiions and mares ieraure on Asian opions Approximae formulas under he wo-facor model Comparison o oher Asian opion models and mare prices Hedging of Asian opions Dependence of he Asian opion price on he parameers

6 4. THE FOATING STORAGE TRADE Inroducion The floaing sorage problem Soluion mehods Analyical properies of he soluion Profi and ris Resuls Origins of excess profis General commodiy rading problem CONCUSIONS Summary of resuls Suggesions for fuure research APPENDIX.... Traded volumes in commodiy derivaive mares.... Spo price process implied by he wo-facor model Principal Componens Analysis of he wo-facor model Evoluion of he consan-mauriy forward curve under he wo-facor model Impac of a hird facor on he consan-mauriy forward curve Blac volailiies of he Average price conrac Semi-analyical soluion o he opimal sopping problem Roues, cargoes and ships used in he floaing sorage rades REFERENCES

7 . INTRODUCTION. Commodiy mares On July 3 rd, 8, Bren crude oil fuures were rading a 46 US dollars per barrel. The TD3 Arabian Gulf Japan shipping roue was quoed a 4 Worldscale, and analyss were predicing crude prices a dollars wihin he nex monhs. On December 3 rd, Bren raded a 45 US dollars per barrel and TD3 a 7 Worldscale, drops of 69 and 7 percen respecively. The commodiy mares are among he mos volaile in he world, and heir volailiy is a source of boh profis and riss for he acors involved. In order o manage hese riss he physical spo mares have from an early sage been accompanied by forward mares, laer ransforming ino financial derivaives mares. The Chicago Board of Trade inroduced exchange-raded fuures conracs on agriculural producs in 848, and crude oil was raded forward from is beginnings in he 86s (Yergin, 8). Mos modern commodiy mares consis of wo inerwined mares: he physical and he financial mare. The physical or spo mare is made up of all mare paricipans selling or aing delivery of he commodiy produc. In he crude oil mare hese are oil companies, refiners and physical rading companies. Trading in he spo mare usually occurs hrough broers, maching sellers and buyers of cargoes a specific daes and locaions. The financial commodiy mare is he mare for derivaive conracs based on he spo. These derivaives ae he form of forwards, fuures and opions, and are used for ris managemen by companies involved in he physical mare and speculaion by oher players. Imporanly, he derivaives sele agains he physical mare, hereby lining he wo. In some cases he derivaives are physically seled, i.e. he buyer receives he acual commodiy. In ohers, he derivaives sele financially agains a spo index published daily based on ransacions in he physical mare. The relaive volumes of he financial and physical mares depend on he level of developmen of he derivaives mare. As seen in Appendix, in 9, he volume of derivaives (fuures and opions) raded on crude oil was 33 billion barrels, compared wih an annual world producion of 33 billion barrels (CIA, 9), maing he derivaives mare nine imes he size of he physical mare. In aner shipping, he derivaives mare raded 34 million onnes of oil cargo in 9, compared o 45 million aner deadweigh onnes raded spo in 6 (Sopford, 9), which evaluaes he size of he derivaives mare o wice he physical mare. There is sill a large growh poenial in he freigh derivaives mare, which will happen hrough sandardizaion and changes in he convenions for physical price seing, similar o wha has occurred in he oil mare since he 98s. The linage beween he spo and forward mares for commodiies will be he main opic of his hesis, and in paricular how he financial mare can be used o gain greaer insigh ino physical rading decisions. While he focus will be on crude oil and aner shipping, we will presen resuls in a general seing and he same principles apply for dry commodiies such as coal. 7

8 . Definiions and mares Definiions In his hesis we will be considering a commodiy mare where he commodiy is rading a a spo price S( ) on dae. This is he mare price for delivery as soon as possible, which can be he nex day for elecriciy or during he nex monh for crude oil. Associaed wih his mare are forward prices F(, T ) on dae. These are he prices in he mare for delivery of he commodiy a he dae T, which is in he fuure. The disincion is ofen made beween forward and fuure conracs, he laer being more sandardized and mared-o-mare daily, bu we will no mae such a disincion here. In he case when he forward is financially seled, a long posiion in he forward conrac enered on dae will pay off S( T ) F(, T ) on he selemen dae T. As long as he physical mare is liquid, enering a physical or financial forward conrac is herefore equivalen wih respec o mare ris boh give a fixed purchase price of F(, T ). The se of forward conracs rading in he mare allows us o consruc a forward curve F(, T ). Usually he mauriies T are monhly, bu hey can be more granular in he shor end. We will also index his curve by he ime-o-mauriy τ = T, which is he ime o selemen of he forward conrac: f (, τ ) = F(, + τ ). Crude oil mare The crude oil derivaives mare is by far he larges commodiy mare, wih a volume of 33 billion barrels raded in 9 (ICE, 9 and CME, 9). I is, however, no a single commodiy he price of crude oil depends on is grade (mainly specific graviy and sulphur conen) and locaion. There are however wo reference grades of crude oil: BFOE (Bren, Fories, Oseberg and Eofis) in he Norh Sea and ligh swee crude oil a Cushing, Olahoma in he Unied Saes, also nown as Wes Texas Inermediae (WTI). Mos grades of oil a oher locaions are priced a a differenial o hese marer crudes. WTI fuures conracs rade on he New Yor Mercanile Exchange (NYMEX) and are physically deliverable ino he pipeline sysem in Cushing, Olahoma, during he monh of he conrac. This maes he fronmonh WTI conrac he spo conrac of crude oil a ha locaion. However Cushing is inland and only reachable by pipeline, while impored crude oil will generally arrive by aner a he ouisiana Offshore Oil Por (OOP) in he Gulf of Mexico. We will herefore also be considering he spo price of ouisiana igh Swee (S) which is a ligh swee crude priced a S. James, ouisiana. BFOE is he mos complee crude forward mare. The spo price, nown as Daed Bren, is assessed daily by Plas from rades during he Plas rading window, and corresponds o cargoes delivered beween and days forward. Saring a monh ou here are fuures conracs rading on he Inerconinenal Exchange (ICE) and seling financially agains he ICE Bren Index. Beween he wo, raders can hedge prices during more specific ime windows using over he couner Bren Conracs For Difference (CFDs). 8

9 Using hese differen prices, a very precise forward curve can be consruced for BFOE, especially in he shor end. In addiion o fuures conracs, here is a liquid mare for opions on hese crude oil marer prices. The exchange-raded opions are mosly American and deliver a fuures conrac when exercised. Taner shipping mare While oil has been ranspored on ships since 86 (Yergin, 8), is ransiion from a logisical exercise conrolled by he oil majors o a spo mare is relaively recen, wih 7% of spo charering in he 99s versus only % in 973. In he spo mare, aners are charered for a single voyage (e.g. Sullom Voe OOP) hrough broers, wih all coss included in he price. There are a number of reference roues for diry and clean aners, numbered TD hrough TD8 (diry, i.e. crude) and TC hrough TC (clean, i.e. producs). A he end of each rading day he Balic Exchange polls broers and publishes an assessmen of he price level for each of hese roues, maing up he Balic Exchange diry and clean aner indices. This is he recognized spo price in he aner mare. Taner raes are usually published in a uni called Worldscale (WS). This uni, specific o each roue, is updaed yearly by he Worldscale Associaion and represens a reference price for a reference aner on he specific roue, in US dollars per deadweigh onne. A spo price of WS will hen be equal o % of his price, while WS5 would be 5% of his price. These aner raes for voyage charers include all coss, i.e. fuel, por and canal coss. I is useful o bac ou a TimeCharer Equivalen (TCE) price for he ship, in US dollars per day, corresponding o he daily price of hiring he ship ne of hese coss. This requires nowing deails abou he disance covered, ship speed, fuel consumpion and fuel prices. Based on his assessmen he Balic Exchange also publishes daily TCE prices for VCC, Suezmax, Aframax and MR aners. The aner mare has also seen he relaively recen developmen of a Forward Freigh Agreemen (FFA) mare. While BIFFEX fuures were raded as early as 985 hey los populariy and have since been replaced by roue-specific FFAs. FFAs sele financially a he end of heir conrac monh on he arihmeic average of he daily values of he underlying Balic Exchange index during ha monh. FFAs are raded hrough broers, wih he Inernaional Mariime Exchange (Imarex) having he larges mare share in aner FFAs. iquidiy is concenraed in a few ey roues, such as TD3 (VCC Arabian Gulf Japan), TD5 (Suezmax Wes Africa US Alanic Coas) and TC (MR produc aner Roerdam New Yor). In addiion o broer prices, he Balic Exchange publishes a daily assessmen of FFAs obained by polling broers. An imporan specificiy of he shipping mare is ha he commodiy being raded, onne-miles, is a service, no a physical commodiy ha can be sored. I is similar in his respec o elecriciy mares. While i is no impossible o sore onne-miles, i can be done by slow seaming or laying up ships for example, i is more difficul and his lac of invenories induces higher spo price volailiy and lower correlaions beween forward conracs of differen enors. There is also a nascen mare in freigh opions, spearheaded by Imarex. These are of Asian syle and, lie he FFAs, sele on he average of a spo index during a monh. Balic Index Freigh Fuures Exchange, a Balic Exchange iniiaive, exised from 985 o 9

10 3. Moivaion The volailiy of commodiy prices exposes mare acors o considerable ris. All invesmen decisions involving commodiies expose he invesors o he forward curve. Such decisions include buying a coal mine, operaing a power plan, ordering and canceling a new ship, rading commodiies beween differen locaions and wriing opions on a commodiy. As sressed in Dixi and Pindyc (994), hese decisions should no be aen based solely on forecass of prices. The 4% yearly volailiy of he crude oil spo price will have subsanially more impac on invesmen decisions in crude oil asses han a forecased growh of %. The year 8 was a paricularly volaile year in he oil mare. I was also mared by he ransiion of he forward mare o a seep conango afer years of bacwardaion as he spo price plummeed. A he same ime, aner raes fell 7%. This led o an array of aners being used as floaing sorage faciliies, anchored up near delivery pors o sore he unused crude oil and ae advanage of he conango, a phenomenon no seen since 973. Deciding when o release crude from such a floaing sorage rade also depends on he forward mare and price volailiy. While many such invesmens and rades are being execued, hey are no, in general, evaluaed using a proper framewor. The correc valuaion and operaional decision-maing for such invesmens or rades requires he use of a simple and correc model for he commodiy forward curves involved. Such a model opens furher possibiliies of managing he firm s ris correcly and maing informed choices abou differen possibiliies. 4. Objecives The main objecive of his hesis is o develop a simple and efficien framewor for he opimal physical rading of commodiies. Such a framewor will allow us o undersand and analyze he floaing sorage rade ha appeared in lae 8 and coninued ino 9. The quesions we will aemp o answer in his hesis are: Can a simple wo-facor forward curve model explain he hisorical volailiies and correlaions of raded forward conracs, in differen commodiy mares? Wha is he consequence on he spo price process for such a wo-facor model? How should commodiy Asian opions, as raded in he shipping mare, be inerpreed, priced, and hedged by mare paricipans? Wha is he meaning of implied volailiy for such opions? When has he cross-alanic crude oil arbirage window been open? When were here floaing sorage opporuniies in his rade?

11 Wha is he opimal floaing sorage sraegy o follow o maximize profis for he rader? Is here value o eeping exposure o he forward curve by no selling he cargo forward immediaely, and how can we undersand his value? Wha is he opimal ship rouing sraegy o follow when a general physical rading problem is considered? When should a ship be re-roued from is iniial desinaion? 5. Mehodology and ouline The general framewor we will be woring under is ha of coninuous-ime financial mares using Iô s sochasic calculus as formulaed in Musiela and Ruowsi (8). Securiies prices will generally be assumed o follow diffusions of he ype ds( ) = µ ( ) d + ( ) dw ( ) S( ) where W ( ) is a Brownian moion, µ ( ) will be called he insananeous drif and ( ) he insananeous volailiy of he sochasic process S( ). This hesis is boh heoreical and pracical. We presen new models and new heoreical resuls. Each ime we presen a new model or resul, however, we will also presen is calibraion o mare daa or hisorical performance and analyze hose resuls. In Par we presen a wo-facor model of commodiy forward curves and show ha i reproduces he main hisorical feaures of he forward curves of four differen commodiies. We also explore is heoreical properies and reformulae i in erms of mean-revering facors shocing he consan-mauriy forward curve. Using his parameric sochasic model of he forward curve we are able o derive an approximae evoluion of average price conracs such as FFAs in Par 3. This hen allows us o find approximae bu closed-form formulas for Asian opions ha ae ino accoun he main feaures of commodiy fuures: he erm srucure of prices and he erm srucure of volailiy, as well as relaively shor averaging periods. We hen compare he prices obained o mare prices of shipping opions and find a very good fi o mare daa. In Par 4 we use a crude oil forward curve model, daa on shipping mares and sochasic dynamic programming o formulae he opimal rouing and floaing sorage problem. Having formulaed he opimal sopping problem for rading crude oil across he Alanic we examine he empirical resuls of his rade during 7-9 and idenify is ey feaures: wha condiions mus be saisfied for i o be ineresing, when i performs well and wha he origins of he profis are.

12 . MARKET MODEING. Raionale The media and commodiy mare analyss end o focus on he rends of prices based on expeced supply and demand evoluion. This is an imporan as, bu commodiy mares are volaile and an expeced growh of wo percen will be dwarfed by a price volailiy of fory percen as is he case for crude oil. The mare expecaions of fuure supply and demand balances are refleced in he fuures mares for he differen commodiies. Mos commodiy mares now have liquid forward curves wih long mauriies and hese complee forward curves should be guiding long-daed invesmen and operaional decisions, no only he spo price. Wih his in mind, a model of commodiy prices needs o provide a realisic model for he evoluion of he complee forward curve and he volailiy of he differen conracs. Such a model can hen be used in a variey of applicaions, such as pricing oher derivaives or real asses wih operaional flexibiliy. Such a model mus also have a small number of parameers and correspond o realiy when calibraed o mare prices. Wih a realisic parameric model, analyical expressions for he prices of opions and real asses can be obained easily, as will be shown in Pars 3 and 4.. Exising lieraure Early sudies of commodiy mares have focused on modeling he spo price, as i has been he only observable mare price. Following wor in equiy mares he spo price has been modeled as geomeric Brownian moion wih consan growh rae, such as Brennan and Schwarz (985) and Paddoc e al (988) for crude oil. Observing ha price-based decisions on he supply or demand side will have a endency o bring prices bac o an equilibrium level, oher auhors such as Dixi and Pindyc (994) have favored modeling he spo price as a process mean-revering o a nown and consan mean value. Ådland (3) develops a mean-revering spo price model for freigh raes, arguing for he use of a spo price model because of he absence of liquidiy in he forward mare. These one-facor models of he spo price give a good inuiion abou he behavior of prices, bu fail o capure imporan effecs, mos noably ransiions of he forward curves from conango o bacwardaion and he decreasing volailiy of fuures conracs wih respec o mauriy. ongsaff, Sana-Clara and Schwarz (999) deail how failing o accoun for several facors leads o subopimal exercise sraegies in he swapions mare. In order o accoun for his Gibson and Schwarz (99) inroduce a mean-revering sochasic convenience yield. In heir model here are hus wo facors shocing he forward curve: he spo price, affecing levels, and he convenience yield, affecing slope. This wo-facor model can be reinerpreed in erms of long-erm and shor-erm shocs, such as in Baer, Mayfield and Parsons (998) and Schwarz and Smih (). In his model he spo price is shoced by a mean-revering shor-erm facor and a persisen long-erm facor.

13 These models are all spo price models: hey see o explain he behavior of he spo price, which is radiionally he observable and mos liquid price. They hen price fuures from his process by inroducing a mare price of ris and arbirage-free pricing, and derive he process for he forward curve. The converse approach consiss in aing he complee forward curve as he primary process. Milersen and Schwarz (998), Clewlow and Sricland () and Sclavounos and Ellefsen (9) develop such a model inspired by he muli-facor Heah, Jarrow and Moron (99) model for he erm srucure of ineres raes. I consiss in decomposing he covariance marix of he forward curve ino a small number of orhogonal principal componens. The spo price process is hen derived as he fron price of he forward curve. I is his approach ha we will adop, bu we will mae parameric hypoheses abou he principal componen shapes and calibrae hese o he covariance marices. 3. Exploraory daa analysis In order o ge an idea of he main feaures of he commodiy forward mares we will begin by an analysis on he hisorical prices of differen commodiies. Spo price In Figure we presen he spo price of differen commodiies over recen ime periods. In many mares, such as crude oil, his spo price is undersood o be he price of he fron-monh fuures conrac wih physical delivery. In oher mares, such as shipping, he spo price is an index compiled daily using spo fixings from differen broers, on which he financial fuures conracs sele Crude oil Heaing oil Naural gas TD3 shipping Figure. Daily spo prices of crude oil, heaing oil, naural gas and he TD3 shipping roue from January 5 o 9. Index = on January s, 5. 3

14 Forward curves Crude oil Naural Gas 6 Spo Forward price 6 Spo Forward price 4 4 Price ($/bbl) Price ($/mmbu) Figure. Spo and forward prices of crude oil and naural gas on differen daes Figure presens he forward curves of crude oil and naural gas on differen daes. We observe ha he level of he forward curves shifs wih he spo price and ha he curves ransiion beween conango and bacwardaion. Furhermore, he forward curves for naural gas have a seasonal paern embedded in hem. Volailiy erm srucure 9 Crude oil Heaing oil Naural gas TD3 shipping Volailiy (%) Time-o-mauriy τ (monhs) Figure 3. Volailiy erm srucure of fuures conracs on crude oil, heaing oil, naural gas and TD3 shipping Figure 3 presens he erm srucure of volailiies for crude oil, heaing oil, naural gas and TD3 shipping. This is he hisorical volailiy of he conracs wih fixed ime o mauriy. We observe ha for all hese commodiies, he volailiy of near-erm conracs is higher han he volailiy of conracs furher ou on he curve, consisenly wih Samuelson s (965) hypohesis. This is an imporan feaure of commodiy mares and happens because hey are more inelasic in he shor run han in he long run. If he aner mare is sauraed i is impossible o add new ships wihin a monh, bu new ships can be buil o accommodae he increasing demand in he nex years. 4

15 Correlaion srucure The forward prices for a given commodiy do no move independenly. Observing he correlaion marix of conracs wih differen mauriies quanifies he relaionship beween hese movemens. Figure 4 presens he correlaion marix for crude oil conracs. The correlaion marix shows a srong correlaion beween differen conracs, wih an 84% correlaion beween he fron-monh and 6-monh conrac. However he correlaion beween he fron-monh conrac and oher conracs decays more rapidly han he correlaion beween he 6-monh conrac and neighboring conracs..95 Correlaion τ (monhs) 3 τ (monhs) Principal componens analysis Figure 4. Correlaion srucure of crude oil fuures To ge more insigh ino he srucure of he co-movemens of he forward prices we can perform a principal componens analysis (PCA) of he price series. This consiss in finding he eigenvalues and eigenvecors of he covariance marix. The eigenvalues can be inerpreed as he volailiies of each of he facors and he eigenvecors as he weighs wih which he principal componens shoc he forward curve. We presen resuls of a PCA of he crude oil mare in Figure 5. As can be seen from hese resuls, he dominan facor is he firs facor, which accouns for 96.9% of he variance. This facor is he parallel shif facor, shifing forward prices in he same direcion. The second facor, explaining.8% of he variance, affecs he slope of he forward curve by shocing he fron end and long end of he forward curve wih differen signs. This accouns for ransiions from conango o bacwardaion. The hird facor affecs he convexiy of he forward curve by shocing he fron and long ends posiively and he middle of he curve negaively. 5

16 6 5.5 PC PC PC 3 4 Volailiy (%) 3 PC weigh u(τ) Principal componen Time-o-mauriy τ (monhs) Volailiy of he firs five principal componens Principal componen weighs Figure 5. Volailiies and weighs for he firs principal componens of he crude oil mare 4. Two-facor model of commodiy fuures Consider a commodiy forward mare where we on each dae observe a forward curve F(, T ) seling on he spo price S( ) a dae T. S( ) could represen he spo price of some radable commodiy a ime (e.g. a specific grade of crude oil a a specific locaion), or he daily published value of an index. If a long forward posiion is enered a dae, i will receive he difference S( T ) F(, T ) a dae T. Absence of arbirage ells us (Musiela and Ruowsi, 4) ha under he ris-neural measure, E [ B(, T )( S( T ) F(, T ))] = * F T E S T * (, ) = [ ( )] (.) where B(, T ) he ime price of he zero coupon bond ha is maures a ime T. The forward price of S a ime is he expecaion of he spo price a ime T, under he ris-neural measure and given he informaion a ime. In some mares where he spo is sorable, such as equiies or currencies, here is a igh arbirage enforcing he relaionship beween spo and forward prices. In mares where sorage is limied, such as crude oil or shipping, he forward price is deermined by supply and demand. I is no our goal here o impose a parameric model for he shape of he iniial forward curve, which we ae as given, bu o give a model of is fuure sochasic evoluion. Following Baer, Mayfield and Parsons (998) and Schwarz and Smih (), we sugges a wo-facor model for he sochasic evoluion of he forward curve. We presen his model as a forward curve model raher han a spo price model, considering ha he commodiy derivaive mares are generally more liquid han heir physical counerpars, and conain more informaion o calibrae on han he spo price. 6

17 We sugges he following wo-facor model for he forward curve under he ris-neural measure: df(, T ) α ( T ) = Se dws ( ) + dw ( ) F(, T ) dw dw S = ρd (.) This is a four-parameer model and as we will show he parameers can be inerpreed as follows: S is he volailiy of shor-erm shocs o he forward curve, is he volailiy of long-erm shocs, α is he mean-reversion speed, quanifying how fas shor-erm shocs dissipae, ρ is he correlaion beween shor-erm and long-erm shocs. Covariance and correlaion This model implies a covariance marix beween conracs ha can be calculaed as a funcion of he parameers Covariance marix: Σ ( T, T ) = Cov, d F(, T ) F(, T ) ( T ) ( T ) S S df(, T ) df(, T ) α α = ( e + ρ )( e + ρ ) + ( ρ ) = Σ( τ, τ ) (.3) where τ = T is he ime o mauriy of he conrac Fuures insananeous volailiy funcion: = + ρ + ρ = τ (.4) ( T ) ins (, T ) ( Se α ) ( ) ins ( ) Spo volailiy: = ( S + ρ ) + ( ρ ) (.5) 7

18 Correlaion marix: df(, T ) df(, T ) Σ( T, T ) ρ ( T, T ) = Corr, = F(, T ) F(, T ) ins (, T ) ins (, T ) ( e + ρ )( e + ρ ) + ( ρ ) = ( ρ ) ( ρ ) ( ρ ) ( ρ ) ρ( τ, τ ) = α ( T ) α ( T ) S S α ( T / / ) α ( T ) Se + + Se + + (.6) All hese quaniies depend only on he ime-o-mauriies τ = T of he conracs involved, and no on ime. Implied spo price process In Appendix we show ha he spo price model consisen wih his forward curve model is: dlog S( ) = α ( µ ( ) log S( )) d + dw ( ) + dw ( ) dµ ( ) = m( ) d + dw ( ) S (.7) i.e. he spo price is mean-revering o a sochasic mean. This is equivalen o he Schwarz and Smih () model which can be rewrien as µ ξ dlog S = κ + ξ log S d + dz + dz κ dξ = µ d + dz ξ ξ ξ χ χ ξ ξ (.8) From equaion (.7) we can see ha α can be inerpreed as he speed of mean-reversion and as he volailiy of he long-erm shocs. 5. Principal componens analysis As discussed in Sclavounos and Ellefsen (9), fuures mares can be analyzed and modeled in a nonparameric way hrough principal componens analysis (PCA) of he covariance marix, leading o a mulifacor Heah-Jarrow-Moron model of he form d df(, T ) = (, T ) dw, dw dwl = δld (.9) F(, T ) = Given he parameric model presened here, we can perform a PCA of he model s covariance marix and deduce he shape of is principal componens. This will allow us o reformulae he model in erms of independen facors ha can be inerpreed in erms of heir acions on he forward curve. 8

19 In he coninuous seing we perform he Karhunen-oève decomposiion of he process following Basilevsy (994). e f (, τ ) = F(, + τ ) be he consan-mauriy forward wih ime-o-mauriy τ. We wan o decompose is evoluion ino: Where: The z are independen Brownian moions df (, τ ) = µ (, τ ) d + λ u ( τ ) dz (.) f (, τ ) The funcions uare he eigenvecors of he covariance marix Σ ( τ, τ ) wih associaed eigenvalues λ : for some arbirary maximal enorτ max, = τ τ max max Σ ( τ, τ ) u ( τ ) dτ = λ u ( τ ) u ( τ ) dτ = (.) We solve his eigenvecor problem analyically in Appendix 3, and show ha here are only wo disinc funcions u (because i is a wo-facor model), and hey can be wrien in he form ατ u ( τ ) = A e + B (.) where ( A, B ) and λ are soluions of he wo-dimensional eigenvalue problem A λ = B τ τ max max ατ ατ ατ ( Se + ρ S ) e dτ ( Se + ρ S ) dτ max max ατ ατ ατ ( ρ S e + ) e dτ ( ρ S e + ) dτ τ τ A B (.3) The volailiy of facor is hen relaed o λ by = λ. The shape of he eigenfuncions is given in Figure 6 in he case of crude oil fuures. We can noice ha u corresponds o parallel shifs of he forward curve, whereas u corresponds o ils. This is consisen wih he wo firs facors observed doing a PCA of he hisorical covariance marix (Sclavounos and Ellefsen, 9). 9

20 .5 u (τ) u (τ) TTM (m) Figure 6. Shape of he eigenfuncions u ( τ ) and u ( ) τ for = 8.%, = 3.3%, α =.84 yr,.95 ρ = and τ max = 5 years This allows us o reformulae he evoluion of he individual forward conrac expiring a dae T, in he risneural measure: df(, T ) = u ( T ) dz( ) + u( T ) dz( ) (.4) F(, T ) S Consan-mauriy forward curve In Appendix 4 we show how his ranslaes o he evoluion of consan-mauriy forward curve. We show ha he consan-mauriy fuures price f (, τ ) = F(, + τ ) can be wrien as: log f (, τ ) = log F(, + τ ) + ψ (, τ ) + ψ (, τ ) + g ( ) + g ( ) + u ( τ ) f ( ) + u ( τ ) f ( ) (.5) where: df ( ) = α f ( ) d + dw ( ) dg ( ) = B α f ( ) d dψ (, τ ) = u ( + τ ) d (.6) Thereby we have decomposed he forward curve s shape a ime ino Is iniial shape F(, + τ ), which under he ris-neural measure is also is expeced shape A deerminisic ris-neural drif ψ(, τ ) + ψ (, τ ) ensuring ha E * [ F(, T)] = F(, T )

21 A sochasic drif g( ) + g( ), independen of he mauriy τ Two independen mean-revering facors f ( ) and f ( ) (volailiies and mean-reversion speeds α ), giving rise o a parallel shif and a il, according o he shape of he facor weighs u ( τ ) The spo price process S( ) is given by he zero ime-o-mauriy price f (,). This allows us o express he evoluion of he forward curve as he resul of shocs from wo independen mean-revering facor values. f, he parallel shif facor, affecs he average level of he forward curve and is he dominan facor. f, he il facor, affecs he slope of he forward curve, as seen in Figure 7. u (τ)f u (τ)f u (τ)f Figure 7. Effec of a posiive parallel shif (lef) and il (righ) on he forward curve 6. Forward curve seasonaliy A number of commodiies have seasonal prices. I appears because demand or supply is seasonal, and invenories are no sufficien o smooh his seasonaliy ou over he year. Examples of seasonal commodiies are heaing oil and naural gas (winer heaing demand), gasoline (summer driving season and differen volailiy requiremens during summer and winer) and agriculural producs (seasonal supply). This seasonaliy in spo prices is refleced in he forward prices because of mare expecaions. The difficuly when analyzing such forward prices is ha he seasonaliy mass he underlying shifs in level and il ha we are ineresed in. When considering such a seasonal commodiy, he forward curve can be decomposed ino a rend componen and a seasonal componen: log F(, T ) = log F (, T ) + log F (, T ) (.7) T S

22 The rend componen FT (, T ) represens he underlying non-seasonal forward curve, whereas he seasonal componen FS (, T ), which for a given is -year-periodic in T, represens he seasonal aspecs of he curve. Pilipovic (7) suggess a funcional form ha we have successfully applied o he naural gas and heaing oil mares. In Secion. we will show ha his form is also consisen wih he saic shape of our hree-facor model. For he rend componen, log F (, T ) = ( A e + B ) f + ( A e + B ) f + ( A e + B e + C ) f (.8) T α ( T ) α ( T ) α3 ( T ) α3 ( T ) This funcional form is flexible enough o reproduce he shapes of he underlying forward curve. For he seasonal componen, we use sinusoidal seasonaliy wih wo harmonics (ime mus be measured in years) log F (, T ) = a cos( π ( T )) + b sin( π ( T )) + a cos(4 π ( T )) + b sin(4 π ( T )) (.9) S The only es of his model is how good he fi o he forward curve is. We find he parameers by leassquares minimizaion for each day in he daa se. A selecion of forward curves is presened in Figure 8. While he fi is no perfec, he rend componen seems o correcly capure he underlying rend, and ha is wha we are ulimaely ineresed in. We hen use his rend as he new forward curve, and carry ou he res of he calibraion procedure on i Forward curve Fied curve Trend componen Forward curve Fied curve Trend componen May, 4 May 5, 8 Figure 8. Fied Naural Gas forward curves on differen daes

23 .8.8 ρ.6.4 ρ τ (monhs) 4 3 τ (monhs) τ (monhs) 3 τ (monhs) Before deseasonalizing Afer deseasonalizing Figure 9. Correlaion surfaces of Naural Gas fuures before and afer deseasonalizing he curves Figure 9 shows he effec of he procedure on he correlaion surface. The resuls of calibraing he wo-facor model o his new correlaion surface are discussed below. 7. Mare calibraion To be successful he model needs o correcly reproduce he volailiies and insananeous correlaions of he raded insrumens. We show ha here is a good fi o he crude oil, aner shipping, naural gas and heaing oil mares. Mehod : eas squares fi of he covariance marix In order o calibrae he model, we perform he following seps:. If he forward curve is seasonal (such as naural gas, gasoline, heaing oil), deseasonalize i using he echnique described above (Secion.6), and eep only he non-seasonal par F (, T ). From he available se of conrac prices F(, T j ), consruc consan-mauriy prices f (, τ j ) by linear inerpolaion using τ ( + τ T )log F(, T ) + ( T τ )log F(, T ) j j j+ j+ j j log f (, j ) =, Tj < + j < Tj+ Tj+ Tj T τ j (.) 3. From observaions of f (, τ j ) a daes,..., M +, consruc logarihmic reurns ne of roll yield and heir mean value 3

24 f (, ) log (, ) M i τ j f i τ j R( i, τ j ) = log ( i i ), R( τ j ) = R( i, τ j ) f ( i, τ j ) (.) τ M i= 4. Calculae he hisorical covariance marix M Σ ɶ ( τ, τ ) = ( R(, τ ) R( τ ))( R(, τ ) R( τ )) (.) j i j j i M i= 5. Find he parameers,, α, ρ ha minimize he squared error: S N min Σ,,, (, ) (, ),,, S α ρ τ j τ Σɶ τ j τ S α ρ (.3) j, = The resuls are presened in Table for he crude oil, shipping, naural gas and heaing oil mares. The resuls indicae ha a saisfacory fi o he volailiy erm srucure and correlaion surface can be obained using he wo-facor model presened here. The bes calibraion resuls are obained for crude oil fuures, which is arguably he mos liquid mare of he four. I is also ineresing o noe he differences beween he values obained. The shor-erm volailiy of shipping fuures is exremely high, a 43%, reflecing he high spo price volailiy, bu is long-erm volailiy is comparable o he oher mares, a 8.7%. 4

25 Table. Calibraion resuls for differen commodiy mares Crude oil (Nymex WTI) Period: April 5 Ocober 8 Taner shipping (Imarex TD3) Period: January 5 March 9 Daa Conracs: NYMEX WTI fuures Frequency: daily Conracs: Imarex TD3 fuures Frequency: weely Source: Thomson Daasream Source: Imarex Parameers S α ρ 8.% 3.3% S α ρ 43% 8.7% Hisorical Model Hisorical Model 3 Volailiy Vol (% annual) Vol (% annual) TTM (m) TTM (m) Volailiy erm srucure of crude oil fuures Volailiy erm srucure of TD3 fuures Hisorical Model Hisorical Model Correlaion τ l (m) τ (m) τ l (m) 5 4 τ (m) 6 8 Correlaion surface of crude oil fuures Correlaion surface of TD3 fuures 5

26 Principal Componens.5.5 u (τ) u (τ) TTM (m) u (τ) u (τ) TTM (m) Model principal componens for crude oil Model principal componens for TD3 fuures Naural Gas (Nymex Henry Hub) Period: Ocober Augus 9 Heaing oil (Nymex New Yor Harbor) Period: May Augus 9 Daa Conracs: NYMEX NG fuures Frequency: daily Conracs: NYMEX HO fuures Frequency: daily Source: Reuers Source: Reuers Parameers S α ρ 53% 7.3% S α ρ 7.6% 6.4% Hisorical Model 4 38 Hisorical Model 5 36 Volailiy Vol (% annual) Vol (% annual) TTM (m) TTM (m) Volailiy erm srucure of naural gas fuures Volailiy erm srucure of heaing oil fuures 6

27 Hisorical Model.98 Hisorical Model Correlaion τ l (m) τ (m) τ l (m) 5 τ (m) 5 Correlaion surface of naural gas fuures Correlaion surface of heaing oil fuures Principal Componens u (τ) u (τ) TTM (m) u (τ) u (τ) TTM (m) Model principal componens for naural gas Model principal componens for heaing oil 7

28 Mehod : Calibraion of he individual facors The firs wo principal componens have a simple expression in his model, and can be used for calibraion. The mehod is he same as above, bu we replace seps 4 and 5 wih: 4. Calculae he PCA of he hisorical covariance marix and exrac he firs wo facor loadings uɶ ( ) τ j, uɶ ( ) τ j 5. Calibrae he exponenial funcional form on each of he facors by leas squares: N α min ( ) ( τ j u τ j A e B ) A, B, α ɶ + (.4) j= We presen he resuls of his mehod for crude oil fuures in Table and Figure. Table. Principal componen parameers for crude oil fuures, using wo calibraion mehods Principal Componen Principal Componen Mehod Mehod Mehod Mehod 54.9 % 54.9 % 9.53 % 9.5 % A B α Hisorical Mehod Mehod.4. Hisorical Mehod Mehod TTM (m) TTM (m) Principal Componen Principal Componen Figure. Fi of he shape of he wo principal componens using he wo differen calibraion mehods 8

29 We see ha he wo mehods give very close resuls, excep ha he second mehod allows for a differen value of α which gives a slighly beer fi o he second principal componen. I should be noed ha Mehod adds one exra free parameer by allowing α and α o be differen. 8. Forward ris premia from he ris neural o he objecive measure The presen model has been formulaed under he ris-neural measure. The prices evolve under he real measure. The change of measure from he ris-neural o he real measure involves inroducing a rispremium λ for each of he Brownian moions W. We assume his ris premium o be consan. dw dw + λ d (.5) This will affec he facor processes f ( ) and g ( ) sudied in Secion.5: And for he drif process g ( ) : α ( s) ( ) = ( ( ) + λ ) f e dw s ds α ( s) ( ) = ( ) + λ + ( α )( ( ) + λ ) df dw d e dw s ds (.6) df ( ) = ( λ α f ) d + dw ( ) = ( µ α f ) d + dw ( ) α ( s) ( ) = ( )( ( ) + λ ) g B e dw s ds α ( s) dg ( ) = B dw ( ) + λd dw ( ) + λd + ( α ) e ( dw ( s) + λds) d (.7) dg ( ) = B α f ( ) d Hence he facor process f ( ) follows an Ornsein-Uhlenbec process mean-revering o µ = λ / α insead of. The definiion of g( ) does no change. We le µ = λ be he drif erm for he facor. The sochasic evoluion of he forward price wih enor T can hen be wrien as df(, T ) = ( µ u ( T ) + µ u( T )) d + u ( T ) dw + u( T ) dw (.8) F(, T ) These resuls show how o incorporae drifs of he forward curve ino he model. These can be based on hisorical evidence of drifs in prices or subjecive evaluaions of he expeced fuure prices. This allows 9

30 valuaion models of physical asses o ae ino accoun forecass of fuure price evoluion. Financial derivaives, however, will be valued under he ris-neural measure. 9. Exension o hree facors The model we have considered is sufficien o reproduce he volailiy and correlaion erm srucures of mos forward mares. However, i only allows for cerain movemens of he forward curve, i.e. parallel shifs and ils. As shown previously in he Principal Componens Analysis, he forward curve does have oher movemens, and he hird principal componen is generally undersood o correspond o changes in curvaure. Cerain sraegies, such as a buerfly rade, are especially sensible o his ind of change. We sugges modeling he hird principal componen as τ = + + (.9) ατ ατ u( ) Ae Be C As is shown in Figure i gives a good fi o he hird principal componen calculaed from a hisorical covariance marix. Wih parameers A and C posiive and B negaive he funcion u( τ ) will ae posiive values for small imes-o-mauriy, negaive values for inermediae τ, and hen posiive values again, hereby affecing he convexiy of he curve. Figure. Fi of he parameric hird PC o he hird PC from he covariance marix (crude oil) 3

31 u 3 (τ)f 3 u 3 (τ)f 3 u 3 (τ)f 3 Figure. Effec on he forward curve of a posiive shoc from he hird principal componen In order o sudy is inerpreaion we will consider is effec on he consan-mauriy forward curve, as we did in Secion.4 for he firs wo componens. In Appendix 5 we show ha he consan-mauriy forward curve can be wrien as log f (, τ ) = log F(, + τ ) + ( ψ (, τ ) + g ( ) + u ( τ ) f ( )) (.3) = + ψ (, τ ) + B e g ( ) + C h ( ) + u ( τ ) f ( ) α 3 τ where df ( ) = α f ( ) + dw ( ) (Ornsein-Uhlenbec process) α3 ( s) 3 = α = α3 3 dg ( ) ( f ( ) g ( )) d g ( ) e f ( s) ds dh ( ) = α f ( ) d h ( ) = α f ( s) ds (.3) The process f ( ) 3 is an Ornsein-Uhlenbec process mean-revering o zero wih mean-reversion speed α 3 and volailiy 3. The processes g ( ) and h ( ) are sochasic drifs inegrals of f ( ) wih differen weighs.. Model of he saic forward curve While he saring poin of our modeling is ha he iniial forward curve F(, T ) is given, here are siuaions where one would wan o model his curve wih a small number of parameers. Using he facor 3

32 model presened in his par, we can express he possible shapes of he curve when saring from an iniial forward curve: N log f (, τ ) = log F(, + τ ) + ψ (, τ ) + g ( ) + u ( τ ) f ( ) (.3) If we assume he iniial forward curve o be fla, F(, τ ) = F, his formulaion simplifies o: = N log f (, τ ) = A( ) + ψ (, τ ) + u ( τ ) f ( ) (.33) Where A( ) is a ime-dependen scalar no depending on ime-o-mauriy τ and = (, ) = (, + ) ψ τ s τ ds (.34) This gives he possible shapes ha can be aen by he forward curve given an iniially fla curve. We can furher simplify his by remaring ha he firs facor, he parallel shif facor, has a funcion u ( ) τ ha is almos consan, such ha he consan erm can be merged ino he firs facor value. Thereby he forward curve can be wrien as N f (, τ ) = exp ψ (, τ ) + u ( τ ) f ( ) (.35) = Hence i can be described by N + sae variables:, f,..., f N. Their iniial values can be calibraed on he iniial forward curve by calculaing max ( ) f () = u ( τ )log F(, τ ) dτ (.36) τ This formulaion also allows us o relae he average level of he forward curve and he firs facor value by forming he geomeric average weighed by u ( τ ) : τ max F( ) = exp w( τ ) log f (, τ ) dτ, w( τ ) = τ max u ( τ ) u ( τ ) dτ (.37) 3

33 such ha N τ max f( τ ) + u( τ ) ψ (, τ ) dτ = F( ) = exp τ max u( τ ) dτ (.38) We can also examine he iniial slope of he curve, ha we will use o deermine if he curve is in bacwardaion or conango: f τ log f = f (,) τ τ = τ = (.39) and N log f ψ u = + f ( ) (.4) τ τ τ = If we are considering a wo-facor model, he firs facor is almos fla such ha is derivaive is zero. In ha case he only conribuion comes from he second facor: log f ψ u = + f( ) τ τ τ (.4) such ha he iniial slope of he forward curve is f ψ u τ τ τ = f (,) + f( ) τ = τ = τ = (.4) Thus he value of f ( ) deermines he slope of he forward curve.. Applicaions of he mare model Derivaives pricing The main applicaion of sochasic models of forward curves is in derivaives pricing. The sochasic model ha we have derived and calibraed allows for simple pricing of paper derivaives depending on he volailiy of prices, such as European or Asian opions wrien on he forward or spo price. In Par 3 we will derive analyical prices of commodiy Asian opions using he wo-facor model derived here. 33

34 Real asse valuaion and operaion There are a number of physical asses whose value depends on commodiy prices and forward curves. Oil or gas reservoirs are a simple example, bu more complex asses such as refineries, power plans or oil in ransi depend on hese prices in a more complex way. Their value depends no only on he spo price bu on he complee forward curve, and operaional decisions should be made aing ino accoun he possible fuure evoluions of he complee curve. The value of such an asse can be wrien as V (, f ( τ )) where f ( τ ) is he curren forward curve. If a wofacor model such as he one in his hesis is adoped, f ( τ ) is a funcion of he iniial forward curve F ( τ ), ime and he facor values f and f, such ha he value can be wrien V (, f ( τ )) = V (, f, f ) (.43) The sochasic evoluion of his value funcion can hen be derived, using Io s formula and he independence of he facors, as V V V V V dv = d + df + df + df + df f f f f V V V V V V V = + ( µ α ) + ( µ α ) f f f f f f f f d dw dw (.44) Associaed wih appropriae boundary condiions his allows for he calculaion of he value of he real asse and he hedging of is value using he facors. In Par 4 we presen he resuls of his mehodology for a physical crude oil rade involving he shipmen and possibly sorage of crude oil. Ris evaluaion Once a porfolio of paper and real asses has been valued, he ris of he porfolio can be evaluaed using he mare model presened here. We assume ha given a forward curve f ( τ ) and a dae he porfolio has a value V (, f ( τ )). If we assume a wo-facor model his value can be re-wrien as V (, f, f ) and is sochasic evoluion as V V dv = µ (, f, f ) d + dw + dw f f (.45) Thereby V V V ( ) µ ( s, f ( s), f ( s)) ds = + dw ( s) + dw ( s) Hence he expeced value of V a a horizon is (.46) f f 34

35 E V ( ) = ( s, f ( s), f ( s)) ds (.47) [ ] µ and is sandard deviaion V V Sd[ V ( )] = E ds + ds f f (.48) These values can be calculaed if he value of V as a funcion of he facors and ime is nown explicily. Alernaively Mone Carlo simulaion can be used, using he wo independen processes f and f, o esimae he complee disribuion of V a he horizon ime. This Mone Carlo simulaion will only require he simulaion of wo independen sochasic variables and no of each forward price separaely. This informaion abou he disribuion of he porfolio value can be used o evaluae he ris of he posiion and calculae ris measures such as value-a-ris. Hedging As seen above a porfolio ha depends on forward prices has, according o he wo-facor model, a sochasic evoluion ha can be wrien / V V dv = µ (, f, f ) d + dw + dw f f = µ (, f, f ) d + δ dw + δ dw (.49) In order o hedge he ris relaed o facor he porfolio mus be complemened wih a posiion of he facor. In ha case he hedged porfolio V ɶ has he sochasic evoluion δ in dvɶ f f f d dw (.5) = ( µ (,, ) δ ( µ α )) + δ j j j j Such a posiion in a specific facor can only be esablished wih he raded fuures F(, T j ), j =,..., N. The fuure wih enor T has he insananeous evoluion ( ) df(, T ) = F(, T ) u ( T ) dw + u ( T ) dw (.5) Consider a porfolio wih w conracs F(, T ), such ha j j N dw dv = wjf(, Tj ) u ( Tj ) d (.5) j= 35

36 For his porfolio o hedge he facor equaions mus be saisfied: f while being unaffeced by he facors f, l, he following l N j= N j= w F(, T ) u ( T ) = j j j w F(, T ) u ( T ) =, l j j l j (.53) If he number of facors is smaller han N here are several soluions o he equaions. If here are only wo facors, his can be accomplished using wo disinc conracs F(, T ) and F(, T ). To hedge facor : u( T ) w = w F(, T ) u ( T ) + wf (, T ) u ( T ) = F(, T ) D(, T, T ) w F(, T ) u( T ) + w F(, T ) u( T ) = u( T ) w = F(, T ) D(, T, T ) (.54) D(, T, T ) = u ( T ) u ( T ) u ( T ) u ( T ). where The porfolio wih w conracs expiring a T and w conracs expiring a T replicaes he sochasic par of he facor f. Similarly, he porfolio replicaing he sochasic par of f wih hese conracs is u ( T ) w = F(, T ) D(, T, T ) w u ( T ) = F(, T ) D(, T, T ) (.55) I should be noed ha using only wo conracs maes he hedge very sensiive o hese wo conracs. If a coninuous forward curve F(, T ) is available, a hedge of f can be formed using all he conracs if he following condiions are saisfied: τ τ max max w( τ ) F(, + τ ) u ( τ ) dτ = w( τ ) F(, + τ ) u ( τ ) dτ = l l (.56) A soluion o his equaion is hen: w( τ ) = u ( τ ) F(, + τ ) 36 (.57)

37 3. ASIAN OPTIONS ON COMMODITIES. Definiions and mares Mos liquid commodiy fuures raded on exchanges sele on a specific day. For example, Bren fuures rading on he InerConinenal Exchange sele on he ICE Bren index price on he day following he las rading day of he fuures conrac. Fuures wih physical delivery, such as NYMEX WTI fuures, do no have cash selemen bu he opions rading on hem sele on heir value on a specific day. In he case of freigh derivaives he spo indices, published daily by he Balic Exchange, are no considered liquid enough o be used for derivaives selemen. Given ha here are relaively few spo ransacions on a paricular day, a big mare paricipan migh be able o manipulae he mare o his favor over a period of a couple of days. To avoid his, he forward conracs sele on he average spo price over a monh. This srucure can also be found in over-he-couner swaps in oher mares, such as crude oil or meals. Given a se of selemen daes T,..., T N (generally he rading days of a given monh), he selemen price of he average conrac seling on hese daes will be N F ( T ; T,..., T ) = S( T ) (3.) A N N N = This selemen price is also used for seling Asian opions wrien on he same commodiy. For example, he payoff of an Asian call opion wih srie K seling on he spo fixings on he daes T,..., T N is N C( TN, K; T,..., TN ) = max S( T ) K, N (3.) = Asian opions are very common in commodiies indeed hey firs appeared hrough commodiy-lined bonds (Carr e al, 8). They are popular no only because hey avoid he problems of mare manipulaion as deailed above, bu also because hey are less expensive han heir European counerpars. The Asian opions we will consider are arihmeic average opions wih European exercise. Given a se of fixing daes T,..., T N, he opion will pay off a dae T N he value N C( TN, K; T,..., TN ) = max S( T ) K, N = (for a call) N P( TN, K; T,..., TN ) = max K S( T ), N = (for a pu) (3.3) 37

38 . ieraure on Asian opions The exising lieraure on Asian opions focuses on Asian opions wrien on soc or foreign exchange raes. In his case he main effec of he averaging is in reducing he sandard deviaion of he payoff funcion. However, he disribuion of he average of log-normal variables is no log-normal, and his is he main obsacle o pricing Asian opions using he sandard Blac-Scholes framewor. To acle his, several echniques have been developed. Mone Carlo simulaion can be used, such as in Kemna and Vors (99), Hayov (993) and Joy e al. (996). A parial differenial equaion depending on he spo price and he observed average price can be derived and solved numerically: Dewynne and Wilmo (995) and Rogers and Shi (995). Geman and Yor (99) derive a semi-analyical expression for a spo price following geomeric Brownian moion. Turnbull and Waeman (99) and evy (99) derive approximae expressions by maching he momens of a log-normal disribuion wih he momens of he average price disribuion. A closed form expression is derived in Geman and Yor (99) for a spo price following geomeric Brownian moion. Approximae expressions have been obained by Turnbull and Waeman (99) and evy (99). Haug (6) presens hese and oher approximaions for Asian opions on fuures. Koeebaer, Ådland and Sødal (7) find an approximae expression for he Asian opions rading in shipping, assuming he spo price follows geomeric Brownian moion. Koeebaer and Ollmar (5) use a one-facor forward curve model wih ime-varying volailiy and derive an approximae process for he shipping forward freigh agreemen. A major issue in using hese formulas for commodiy fuures opions is ha hey assume geomeric Brownian moion for he spo price, which is no consisen wih a muli-facor model wih mean-revering facors. They also ignore he exisence of a forward curve which gives he ris-neural expecaions of he spo price. 3. Approximae formulas under he wo-facor model For opion pricing we will wor in he ris-neural measure. The wo-facor model of he forward curve is, as formulaed in Par, df(, T ) α ( T ) Se = dws + dw, dwsdw = ρd (3.4) F(, T ) 38

39 Consider an Asian forward conrac The average price conrac saisfies: F A seling on he average of he daily conracs F(, T ),..., F(, T N ). N FA ( ) = F(, T ) N = N dfa ( ) = df(, T ) N = (3.5) where ( τ ) = when τ < (he conrac has already seled, so is price is fixed). Then N N α ( T ) Se F T F T = = dw N S N (, ) (, ) dfa ( ) = + dw FA ( ) F(, T ) F(, T ) = = (3.6) Assumpion : we assume he forward curve o be fla hrough he selemen period of he Asian conrac: F(, T ) = F ( ) (all he daily conracs have he same price) A If we mae Assumpion hen he above equaion simplifies o he lognormal evoluion Tha is, he facor volailiies of dfa ( ) = FA ( ) + N N α ( T ) Se dws dw (3.7) = F A are he average of he facor volailiies of he individual conracs. Assumpion : e us assume ha he fixing daes are equally disribued: T = T ( N ) h. Denoe by c T T = N he conrac lengh. Then for T < (pre-selemen): N α ( TN ) N dfa ( ) Se αh = e dw + dw F ( ) N A = S e = + N( e ) α Nh α ( TN ) Se dw h S dw α (3.8) N α ( T N T ) N α ( T ) N e Se dw ( TN T )/( N ) S dw α ( ) = + N e α ( N )/( ) Assumpion 3: The number of fixing daes N is large enough ha ( T N e T N ) αc. Then αc dfa ( ) α ( T ) N e Se dws + dw (3.9) F ( ) αc A 39

40 Dynamics inside he selemen period An essenial issue of Asian conracs is wha happens inside he selemen period. As he conrac eners he selemen period, is consiuen prices are progressively discovered, and he uncerainy on is price a expiraion diminishes. For example, he day before expiraion, he only uncerainy is on he las price s evoluion during one day, which only conribues a small par o he average conrac price. Furhermore, as par of he conrac is priced, our Assumpion of a fla erm srucure F(, T ) = FA ( ) becomes wrong. Indeed on dae, TM < T M +, he spo prices S( T ),..., S( T M ) have been observed and hey will no be equal o FA ( ). We will herefore consider he observed average A( ) and he adjused average conrac price F ' A( ) : M A( ) = S( ), T < T M = m m+ M F ( ) F ( ) A( ) F(, T ) N ' A = A = N N = M + (3.) If we assume a fla erm srucure ahead, N F T F N M ' (, ) = A( ) for m + hen N ' dfa ( ) = df(, T ) N = M + N N = F e dw + dw N N M = M + ' α ( T ) A( )( S S ) ' N dfa ( ) = e dw dw ' FA ( ) N M + = M + α ( T ) S S (3.) The adjused average conrac price F ' ( ) has an approximaely lognormal evoluion wih volailiies equal o A he average of he volailiies. This, however, is only valid for TM < T M + and assuming a fla erm srucure. The exac evoluion of FA ( ) from = T o = TN is complex, and an exac derivaion would have o follow he lines of Geman and Yor (993). We do however ge a good idea of he resul by assuming ha he already seled prices S( T ) and he daily forward prices F(, T ) are all equal, in which case he average conrac follows he evoluion: N N dfa ( ) = ( T ) dw ( T ) dw FA ( ) + N N (3.) = = By definiion ( τ ) = for τ <. In ha case he average conrac sill has a lognormal disribuion inside he selemen period, bu wih volailiy decaying o as shown in Figure 3. 4

41 5 day monh 3 monhs Vol (%) Time o mauriy (m) Figure 3. Insananeous volailiy of he Asian conrac as a funcion of ime o mauriy, for differen conrac lenghs We wan o quanify his conrac s sochasic evoluion unil mauriy a T N. Consider a dae such ha T T < T. Then: M + + ' α ( TN ) N dfa ( ) Se α ( N ) h N = ' e dws + dw FA ( ) N M = + N M e T + α ( TN ) α ( TN ) N Se dw ' S ' dw αcm cm (Assumpion 3) (3.3) where c = T T is he residual conrac period. ' M N M Pricing Asian opions As discussed previously, in some mares he Asian forward conrac/swap is raded in he mare, and also seles on he average of he spo: Therefore we can rewrie he payoff of he opion as N F ( T ; T,..., T ) = S( T ) (3.4) A N N N = 4

42 N max ( ), max (,,..., ), N = S T K = ( FA TN T TN K ) (3.5) This payoff is a European call opion wrien on he forward conrac F, which simplifies he problem considerably. We will denoe by C(, F (, T,..., T ), K ) (3.6) A N A he price, a ime, of a call opion seling on he average of he spo prices a imes T,..., T N, wih srie K. Similarly P(, FA (, T,..., TN ), K) denoes he price a ime of a pu. Pu-Call pariy: because he opions are sandard European opions wrien on a swap, pu-call pariy holds in he form C(, F ( ; T,..., T ), K) P(, F ( ; T,..., T ), K) = B(, T )( F ( ; T,..., T ) K) (3.7) A N A N N A N where B(, T N ) is he discoun facor. We will henceforh focus he discussion on calls. Blac s formula under ime-dependen volailiy We will begin by esablishing Blac s (976) formula for fuures opions under deerminisic ime-dependen volailiy. We will follow he derivaion of Musiela and Ruowsi (8). Consider a fuures process F(, T ) wih ime-varying volailiy: df(, T ) = µ (, T ) d + (, T ) dw (3.8) F(, T ) We consider he fuures opion seling a dae T on F( T, T ) wih srie K: C( T, F) = ( F K ) + (3.9) Consider a self-financing fuures sraegy φ = ( g( F, ), h( F, )). Since he replicaing porfolio φ is assumed o be self-financing, he wealh process V (, F, φ ), which equals saisfies V (, F, φ ) = h( F, ) B = C(, F ) (3.) dv = g( F, ) df + h( F, ) db = rv (, F ) d + µ (, T ) F g( F, ) d + (, T ) F g( F, ) dw (3.) 4

43 If we assume ha he funcion C is sufficienly smooh, we find by Io s lemma ha dc(, F ) (, T ) F (, T ) F d F dw F F F C C C = + µ C + + (3.) Equaing he values of V and C we find ha we mus have C g( F, ) = (, F ) (3.3) F And C C + (, T ) F rc = F + C( T, F) = ( F K) (3.4) The soluion of his parial differenial equaion is where [ ] C(, F(, T )) = B(, T ) F(, T ) N( d ) KN( d ) (3.5) ln( F(, T ) / K) + ( T ) Blac (, T ) d =, d = d Blac (, T ) T (, T ) T Blac (3.6) Blac T (, T ) = ( s, T ) ds T (3.7) Hence he formula for a call wrien on a fuures conrac wih ime-varying volailiy is he same as Blac s (976) formula for a fuures opion, excep ha he volailiy is replaced wih he roo-mean-square of he insananeous volailiies during he period. Insananeous volailiies e us now calculae he insananeous volailiies of FA ( ) pre- and in-selemen, based on he sochasic evoluion derived in equaions (3.9) and (3.3). For < T αc α ( T ) N e A(, TN ) = Se + + ( ) ρ ρ αc (3.8) and in-selemen, for T TN : 43

44 α ( T ) N ' e TN TN A(, TN ) = S + ( ) ' ' + ' αcm cm cm ρ ρ (3.9) Blac volailiies Given hese insananeous volailiy funcions, we can calculae he Blac volailiy, i.e. he sandard deviaion of he conrac price a expiraion: T Blac (, T ) A(, ) T = s T ds (3.3) In-selemen Consider he case when daes T,..., T M have been priced, such ha we are in fac considering he adjused conrac In Appendix 6 we show ha is Blac volailiy is: Such ha he square of he Blac volailiy is given by: F ( ) F ( ) S( T ) F(, T ) M N ' A = A = N = N = M + (3.3) T Blac (, T ) = A(, ) T α ( T ) α ( T ) S e e = ' + + α cm α T α T α ( T ) α ( T ) ρ S T e e ' + + αcm α α T ( T ) 3 c ' M s T ds (3.3) In he case when αc his simplifies o Blac (, T ) S + ρ S + (3.33) 3 Pre-selemen In Appendix 6 we show ha 44

45 T T T T Blac (, T ) = A( s, T ) ds = A( s, T ) ds + Blac ( T, T ) T T (3.34) T where T αc α ( T T ) α ( T ) αc α ( T T ) α ( T ) e e e e e e A( s, T ) ds = S + ρ S + ( T ) αc α αc α (3.35) and ( T, T ) is given by equaion (3.3). If αc, Blac α ( T ) α ( T ) c e c e c Blac (, T ) S + + ρ S + + T α c 3 T αc 3 3 T (3.36) Opion price Once he Blac volailiy is nown, pricing fuures opions is simply a maer of applying he equaion (3.6) o he process F (, T ). If he opion is pre-selemen, A [ A ] [ ] C(, F (, T ), K) = B(, T ) F (, T) N( d ) KN( d ) A P(, F (, T ), K) = B(, T ) KN( d ) F (, T ) N( d ) A A (3.37) where log( FA (, T ) / K) + ( T ) Blac (, T ) d =, d = d Blac (, T ) T (, T ) T Blac (3.38) If i is in-selemen, inroduce he already priced average. The payoff a expiraion can be rewrien as A( ) = S( ), T < T M M M M + (3.39) = ' M ' T max ( FA ( T ) K,) = max FA ( T) K A( ), max FA ( T ) K A( ), N c (3.4) such ha he opion can be considered o be wrien on he log-normal process F ' A( ) wih an adjused srie K A( ) M / N. If K A( ) M / N. The prices of he opions are herefore: 45

46 T T C(, FA(, T ), K) = B(, T ) FA (, T ) A( ) N( d) K A( ) N( d) c c T T P(, FA (, T ), K) = B(, T ) K A( ) N( d) FA (, T ) A( ) N( d) c c (3.4) where: T FA (, T ) A( ) ' log c + ( T ) Blac (, T ) T K A( ) d = c d = d T T ', ' Blac (, ) Blac (, T ) T (3.4) ' and (, T ) is given by equaion (3.3). Blac If K A( ) M / N <, he average over he conrac period will always be larger han he srie, such ha he call opion will always be exercised and he pu opion will never be exercised. Thus C(, F (, T), K) = B(, T )( F (, T ) K) A P(, F (, T ), K) = A A (3.43) 4. Comparison o oher Asian opion models and mare prices We examine wo oher Asian opion models ha are commonly used in he freigh opions mare: he evy (99) approximaion and he Koeebaer, Adland and Sodal (7) formula. These models boh assume ha he spo price follows geomeric Brownian moion under he ris-neural measure: ds( ) = λd + dw (3.44) S( ) e T = T + ( ) h be he selemen period daes and define he observed running average A( ) as in m equaion (3.) above. Also le M ( ) = A( TN ) A( ) and N [ ] v E M E M * * ( ) = ln ( ) ln ( ) * m ln E ( ) ln ( ) M K A N d =, d = d v( ) v( ) (3.45) 46

47 Then he evy (99) approximaion is, for Tm < T m +, m C S A e E [ M ] N d K A N d N rτ * ( ( ), ( ), ) = ( ) ( ) ( ) ( ) m P S A e E M N d K A N d N [ ][ ] [ ] rτ * ( ( ), ( ), ) = ( ) ( ) ( ) ( ) (3.46) Expressions for E * [ M ( )] and * E [ M ( ) ] are given in he appendix of he original aricle. Koeebaer, Ådland and Sødal (7) arrive a an opion price along he following lines:. Assume geomeric Brownian moion for he spo under he ris-neural measure. Derive wha he average forward price should be, consisen wih his spo process N λ ( TN ) λn * e e A( ;,..., N ) = [ ( i )] = λ N i= N e F T T E S T S (3.47) 3. Derive he forward s approximae evoluion: dfa ( ; T,..., TN ) = ( ) dw where F ( ; T,..., T ) A N T T S λ( T i ) e N i = M + T M M F (, T, ) M < < T < < + M T + N N N ( ) = N M T T (3.48) 4. Use Blac (976) wih ime-varying volailiy o price he opion The resul is as follows: for < T (pre-selemen) le F = ( T ) + R( N)( T T ) (3.49) N 3 + R( N) = N N 3 3 N (3.5) For = T (in selemen) M 47

48 M M F = ( TN TM ) + 3 N N N / (3.5) Then [ ] [ ] C(, F ( ; T,..., T ), K) = B(, T ) F ( ; T,..., T ) N( d ) KN( d ) A N N A N P(, F ( ; T,..., T ), K) = B(, T ) KN( d ) F ( ; T,..., T ) N( d ) A N N A N (3.5) where B(, TN ) is he discoun facor and FA ( ; T,..., TN ) ln F K + d =, d = d (3.53) F We now evaluae opion premia based on hese models. The volailiy inpus o he models are evaluaed based on he hisorical volailiies esimaed from ime series of prices of shipping fuures conracs. We compare hese o he opion premia quoed by Imarex in heir weely Imarex Freigh Opions repors for he TD3 roue. The main parameers are given in Table 3.. Table 3. Parameers on December 8, 8 Dae December 8, 8 3-monh IBOR.9% Spo price (WS) 9.73 F Observed average spo / /5 (WS) The ey inpu o each of he models is he volailiy. We use For evy and KAS, he hisorical volailiy of he spo, using weely log reurns over he year 8. This is evaluaed o 48.4%. We calibrae he wo-facor forward curve model o he esimaed hisorical covariance marix for he year 8. 48

49 Vol (% annual) Hisorical Model Parameers: = 7.4% S = 34.8% α = 3.45 ρ = TTM (m) Figure 4. Two-facor model fied o hisorical volailiies Table 4. Model Call opion ATM prices on December 8, 8 compared o mare Conrac FFA Imarex Premium ATM Model opion premium ATM (WS) Year Period WS WS evy KAS Two-facor 8 Dec Jan Feb Mar Apr Q Q Q CA The resuls are lised in Table 4. We noe ha he evy and KAS prices are very similar. This is no surprising as hey are based on he same model and parameers for he underlying, only wih differen ways of approximaing he opion premium. The opion premia obained wih he erm srucure of volailiy from Figure 4 are lower han he evy and KAS premia. This is obvious from Figure 5: he difference beween KAS and he presen sudy is he 49

50 volailiy inpu o he Blac formula - F and Blac respecively which are he roo-mean-square of he ime-dependen insananeous volailiy ( ). Since he hisorical volailiy of fuures wih increasing enors is lower han ha of he spo, he call premium from he presen sudy will be lower han he KAS premium. 8 6 Two-facor KAS & evy 4 Vol (%) Time o mauriy (m) Figure 5. Asian volailiy erm srucure for he wo-facor model and he KAS & evy models Comparing hese premia o mare prices quoed by Imarex, he mare prices are lower han he prices obained from he presen model supplied wih he hisorical volailiies namely he mare implies a lower fuure volailiy han he hisorical volailiy over 8. Ye, 8 was a paricularly volaile year in he freigh mare, giving a high hisorical volailiy, whereas he mare migh be expecing 9 o be calmer. Implied volailiies calibraion of he models o mare prices Volailiy is he mos imporan parameer of an opion pricing formula. Considering ha he oher parameers are observable, a quoed mare price for an opion implies a value for his parameer. I is herefore imporan o be able o bac ou his parameer from he opions premia observed in he mare, resuling in he implied volailiy. This follows by solving he equaion C(, F (, T ), T, K, r, ) = C (, T, K) (3.54) A mare for or possibly several parameers ha ener he definiion of. evy (99) The formula (3.46) has a complicaed dependence in and a numerical echnique mus be used o bac ou he volailiy of he spo from he opion price. As his is a single parameer, o each opion price C( T, K ) 5

51 here corresponds an implied volailiy ( T, K ). The volailiy of he spo and he opions implied evy volailiy are no he same because he mare doesn follow he assumpions of he model. In paricular, here is a erm srucure of volailiy which is no consisen wih he geomeric Brownian moion model for he spo price. Imarex quoes hese implied volailiies in heir Freigh Opions repors. Koeebaer, Adland and Sødal (7) The formula (3.5) is jus Blac (976) wih a weaed volailiy inpu. Exracing implied volailiies from he Blac formula is sandard, giving rise o ( T, K ). The implied volailiy of he spo is hen Blac KAS T ( T, K) = Blac ( T, K) g (, T,, T ) N (3.55) where, consisenly wih equaions (3.49) and (3.5), T + R( N)( TN T ) < T g(, T,, TN ) = M M (3.56) ( TN TM ) + T = TM 3 N N N As wih he evy model, differen implied volailiies are obained for each mauriy and srie. Two-facor model The inpu o he Blac formula is he ime dependen volailiy ( A, T ) modeled according o (3.8) and (3.9). The parameers in he model need o be esimaed from he erm srucure of opions prices. Rebonao () discusses in deph he calibraion of he IBOR mare model o raded opions in he ineres rae mares, and much of his discussion applies here. ( T ) for a-he-money opions is firs obained from Blac opions mare prices, and he parameers of he model are esimaed by nonlinear leas squares: Model min Blac ( T, S,, α, ρ) Blac ( T ) S,, α, ρ (3.57) T where he summaion is over all available liquid opion mauriies. I should be noed ha unlie he evy and KAS models which esimae one implied volailiy per opions conrac, he presen model esimaes four parameers from all liquid opions prices by a nonlinear leas squares echnique, which is more parsimonious, bu can lead o inaccuracies if he model is no suiable. 5

52 Noe on calibraing a muli-facor model o implied volailiies The wo-facor model ha we presen in his aricle is inended o reproduce no only he erm srucure of volailiies bu also he correlaion surface beween he conracs. When pricing vanilla opions, however, only he erm srucure of volailiies maers and he correlaions are irrelevan. Thus, when calibraing he four parameers of he model o implied volailiies alone one canno expec o correcly reproduce he correlaion srucure. However, considering ha we have observed hisorical esimaes of he parameer ρ o be close o, a simple soluion consiss in fixing his parameer o and calibraing,, α o he implied volailiy erm srucure. As long as his produces a good fi, vanilla opions will be priced correcly. Resuls Based on he prices published in he Imarex Freigh Opions repor on Dec 8, 8, we exrac implied parameers for he differen models. The resuls are presened in Table 5. We fi he wo-facor model (.) o he mare prices using he procedure described above, and he resul of he opimizaion is displayed in Figure 6. We can see ha he wo-facor model gives a very good fi o he opion mare prices. S Figure 6. Blac implied volailiies from mare quoed opions and calibraed wo-facor model. Mare prices are from he Imarex Freigh Opions repor on December 8, 8 5

53 Table 5. Implied volailiies for he differen models on December 8, 8 Conrac Imarex Premium Implied volailiies Year Monh WS Imarex repor evy KAS Blac ( T ) ( T ) ( T ) evy KAS Blac Two-facor 8 Dec % 69.% 5.53% 67.96% S 77.5% 9 Jan 8.8 5%.88% 3.3% 97.43% 47.68% 9 Feb 9. % 99.6%.3% 87.5% α Mar 8. 9% 89.9% 9.3% 8.3% ρ 9 Apr 8. 8% 79.5% 79.76% 73.86% 9 May % 74.98% 75.6% 7.79% 9 Jun 8.7 7% 69.6% 69.74% 66.4% 9 Jul % 65.98% 66.% 63.3% 9 Aug % 66.7% 66.38% 63.76% 9 Sep % 66.5% 66.4% 63.8% 9 Oc.8 63% 63.44% 63.5% 6.49% 9 Nov. 6% 6.46% 6.53% 6.7% 9 Dec.5 6% 6.% 6.6% 6.4% 5. Hedging of Asian opions Grees of Asian opions When wriing opions, he seller may be ineresed in dela-hedging his porfolio wih he underlying o consruc a dela-neural posiion. The quesion is wha posiion o ae in he underlying o hedge he opion: he dela. The hedge raio changes wih he price of he underlying, ime and volailiy and mus be adjused regularly, leading o dynamic hedging sraegies. When hedging a number of quesions mus be addressed: wha is he underlying? Wha insrumens are we going o hedge wih? The opions are wrien on spo, bu in shipping he spo is on-miles ha are no be possible o buy and hold, nor shor, since i is a service. In shipping he forward conracs ha are rading in he mare, he Forward Freigh Agreemens, sele on he average of he spo, and should herefore be used as hedging insrumens. 53

54 The Grees follow upon differeniaion of he Blac formula. e d and d be defined as in Secion 3., hen we have for a call opion: = = C e r( T ) N d FA ( ) C n( d ) Γ = = F F T r( T ) e A A B dc C C F θ = = + = rc e n( d ) + Vega d C r( T ) Vega = = FAe T n d B C ρ = = ( T ) C r B r( T ) A B B B T ( ) (3.58) Because he volailiy is ime dependen, he hea θ of he opion also depends on he emporal variaion of he Blac volailiy: ( ) (, ) T B (, ) Differeniaing his wih respec o and rearranging we ge T T = s T ds (3.59) T T T = ( T ) (, T ) B (, ) B (, ) (, ) B (3.6) which can be calculaed using he formulas in Secion 3. Dela-hedging an opion posiion When a forward conrac wih he same selemen period as he Asian opion is available, ha conrac should be used o dela-hedge he opion posiion o avoid basis ris. The posiion o be aen in his conrac is hen given by he previous formula. The posiion o ae in he conrac F (, T ) o hedge a shor call posiion seling on he same period is = ( ) (3.6) r( T ) C e N d In oher mares such a conrac is no available. In he crude oil mare, for example, he fuures conracs sele on a single dae while Asian opions will sele on he rading days wihin a monh. ucily, hese 54 A

55 conracs are highly correlaed and we can quanify he required number of conracs using he wo-facor model. The insananeous evoluion of he call price is C C dc = d + df F A A A A ( ) = θd + F (, T ) (, T ) dw + (, T ) dw FA A S S (3.6) Where FA (, T) = FA ( ; T,..., TN ) pre-selemen and F T = F T T in-selemen, and ' A(, ) A( ; M +,..., N ) e A αc S (, T ) = e < A (, T ) = T ' cm αc α ( T ) Se < T α ( T ) α ( T ) Se T ' M TM + αcm < T M M + T < T (3.63) A daily conrac seling on he dae T has he sochasic evoluion df(, T ) α ( T ) = Se dws + dw (3.64) F(, T ) We can use wo of hese conracs o hedge he call. Assume we ae a posiion w in conrac F(, T ) and w in conrac F T (, ), hen he hedge of a shor call mus saisfy: α ( T ) α ( T ) A w F(, T ) Se + wf (, T ) Se = F F (, ) (, ) A A T S T A ( w F(, T ) + wf (, T )) = F F (, ) (, ) A A T T (3.65) which yields w w = A A S (, T ) (, T ) e F (, T ) A S FA α ( T ) α ( T ) F(, T ) e e = A A S (, T ) (, T ) e F (, T) A S FA α ( T ) α ( T ) F(, T ) e e α α ( T ) ( T ) (3.66) 55

56 6. Dependence of he Asian opion price on he parameers Carr, Ewald and Xiao (8) esablish ha in he Blac-Scholes framewor, he premium of an arihmeic average Asian call opion wrien on soc increases wih volailiy. They also show ha his is no a rivial resul and does no hold ouside he Blac-Scholes assumpion, for example using a binomial model. In he model presened here here is no a single volailiy parameer bu four parameers governing he erm srucure of volailiy. We will sudy he dependence of he opion premium on hese four parameers. This has an imporan impac on opion ris managemen, given ha he implied erm srucure of volailiy can change sochasically over ime, hereby affecing prices. The dependence of he opion premium on he volailiy parameers is hrough Blac, herefore we can wrie C C B C C B =, =,... S B S B (3.67) We calculae he sensiiviy of he Blac volailiy on he four parameers, in he simplified case when αc. In-selemen, for TM < T M +, B (, T ) S + ρ S + (3.68) 3 B + S ρ B ρ S B B S, + = =, =, = 3 3 α ρ 3 S B B B (3.69) And pre-selemen, < T, α ( T ) α ( T ) c e c e c B (, T ) S + + ρ S + + T α c 3 T αc 3 3 T (3.7) α ( T ) α ( T ) B S c e ρ c e = S B T α c 3 B T αc 3 α ( T ) B ρ S c e T = B T αc 3 B c 3 B S c T T = S + e + ρ + e α B α( T ) c c α ( T ) B S c e = + ρ B T αc 3 α ( T ) α ( T ) (3.7) 56

57 e us examine he sign of hese quaniies. In-selemen, we have, on he condiion ha, ρ,, all he derivaives are non-negaive. For he pre-selemen values, α where ( ( T α ) ) B c S ρ = g( α) g( α) T + (3.7) S B B g( ) = e / ( αc) + / 3 is decreasing in α, such ha S B S c S + ρ g( α) T B (3.73) For, B c ρ S c S = g( α) + g() g( α) ρ + S T B B T B (3.74) For α we can see ha (, T ) is decreasing wih α. The Blac volailiy is increasing in ρ wihou Blac any condiions on he volailiies. Hence we have proven ha, if we assume, ρ,, he Blac volailiy and he call opion premium are increasing in he parameers S S, and ρ, and decreasing in α. 57

58 4. THE FOATING STORAGE TRADE. Inroducion Afer he collapse of oil and shipping prices in mid 8, floaing sorage, i.e. soring crude oil or producs in idle aners, became a viable opporuniy. This was made possible by a seep conango of he crude oil forward curve combined wih low freigh raes. This is only one example of wha inernaional oil rading consiss of: if a price difference in ime or space is higher han he shipping and capial cos involved, hen here is an arbirage opporuniy ha can be exploied. Two very basic examples are: Shipping crude oil from Nigeria o he US Soring crude oil in sorage ans in Cushing, OK when he WTI forward curve is in conango. Ofen several opporuniies presen hemselves o an oil rader, and he volailiy of he associaed forward curves maes i possible ha hese opporuniies could evolve during he voyage. For example, a ship leaving Nigeria wih crude oil has he opion of going eiher o he Unied Saes or o Europe, and he rader doesn necessarily have o mae ha choice immediaely he can choose o say on a norhward course in he mid- Alanic and defer he choice of desinaion por o a laer dae, when one opion will be significanly more ineresing han he oher. The ship can also choose o speed up or slow down o conrol is fuel consumpion and arrive a an opimal dae. When choosing o eep his opions open, he oil rader is exposed o movemens in he forward curves. The exisence of liquid fuures and opions mares a several ey locaions maes i possible o hedge his exposure parly, hereby reducing riss. Our aim is o creae a framewor for analyzing such rading sraegies and derive he opimal roue ha a ship should follow. This framewor can hen be used o evaluae expeced reurn and ris beforehand, o find he opimal roue ha he ship should follow, and o derive hedge raios o hedge he exposure o he dominan ris facors. We will concenrae on a simple problem: he cross Alanic crude oil arbirage wih possibiliy of floaing sorage, and presen he resuls for his. We will hen proceed o generalize he framewor o a general opimal rading problem.. The floaing sorage problem We consider a ship a locaion X a ime. The characerisics of he ship and he shipping roue are given in Table 6. The problem we are considering is as follows:. A dae = he decision is made o load he aner wih a cargo a he spo price S 58

59 . The cargo is loaded a dae τ load. 3. The ship hen sails a he consan speed u o he desinaion por. 4. Upon arrival, and unil he dae when he cargo is acually delivered, he ship is anchored a he desinaion por X X p Waiing for cargo o load Sailing, speed u A desinaion X Buy cargo, price S p d( X, X ) τ + u τ load load τ* : deliver cargo a delivery por The daily cos paid for he shipping is hen, a dae, τ load P g( ) = H+ B FC( u) τ load τ load + d( X, X ) / u P H + B FCa τ load + d( X, X ) / u (4.) The cargo is no necessarily sold ino he mare immediaely. A any locaion X and ime he decision can be made o sell he cargo for delivery τ days forward, a he price F(, τ ). Exercise profi We define he exercise profi Ω (, F( τ )) as he profi ha can be earned on he cargo if he ship is a locaion X() and he forward curve is F( τ ), by commiing o a specific delivery price someime in he fuure and sailing o deliver a ha ime. In effec, he rader gives up he possibiliy of changing delivery ime. A exercise, one chooses a ime-o-delivery τ. For one choice of his parameer he profi is τ sail P = d( X ( ), X ) / u ω( τ ) = F( τ ) S Hτ B ( FC( u) τ + FC ( τ τ )) BH, τ τ sail a sail sail (4.) Forward price ($/bbl) oaded spo price ($/bbl) Ship imecharer ($/bbl/day) Buner price ($/m) Fuel consumpion (m/day/bbl) Cos of bachaul rip ($/bbl) 59

60 The exercise profi consiss in maximizing ω over all possible imes-o-mauriy: Ω (, F( τ )) = max ω( τ ) (4.3) sail τ τ Table 6. Ship and roue parameers Uni Typical value Uni Typical value ocaion X Nm Ship Time days Type Very arge Crude Carrier (VCC) Cargo size m 7 m Roue Sullom Voe OOP DWT m 3 m Disance d Nm 4535 Nm Speed u nos 5 nos Cargo Bren Fuel consumpion sailing: FC(u) anchor: FCa m/day m/day 87.5 m/day (laden) 74 m/day (ballas) 85 m/day (pumping) 5 m/day (anchor) Barrel facor bbl/m bbl/m Timecharer price H USD/day VCC average imecharer equivalen (Balic Exchange) oading por X Sullom Voe Delivery por XP OOP oading price S USD/bbl Daed Bren - days Delivery price F(,τ) USD/bbl S forward curve oading delay τload Days 5 days IFO price B USD/m Fuel Oil 3.5% CIF NWE (Plas) I should be noed ha if he locaion of he ship X is he loading por, hen Ω is he arbirage profi from ha por and if i is posiive, he arbirage is said o be open. Furhermore, if he loading and desinaion pors are he same and Ω is posiive, hen here is a floaing sorage opporuniy a ha por and Ω is he profi ha can be made from i. This profi is also risless a leas mare-wise considering ha he profi is loced in by selling he cargo forward. Valuing he expeced profi of he voyage When he aner is loaded, he arbirage profi Ω can be loced in wihou ris. However, he large number of opionaliies available o he rader hroughou he voyage means ha he expeced profi is someimes higher. Typical values correspond o he modern double-hull VCC from Clarsons (9) 6

61 e us define V (, F( τ )) as he expeced profi from he cargo when he aner is a he locaion X ( ) and he forward curve is given by F( τ ). This is a real opion value as deailed in Dixi and Pindyc (994). When he ship is a a locaion X ( ), and he maximum exposure ime T has no been exceeded, he rader has wo choices: eiher exercise and sell he cargo forward, hereby earning he exercise profi Ω (, F( τ )) or choose o coninue speculaing during a ime d wihou exercising. The expeced profi is: F ( τ ) [ ] V (, F( τ )) = E V ( + d, F( τ ) + df( τ )) g( ) d (4.4) C This gives he coninuaion value V C. The forward curves are evolved during he ime period d using he wo-facor model from Par. Hence he expeced profi a locaion X ( ) given he forward curve F( τ ) is [ ] V (, F( τ )) = max Ω (, F( τ )), V (, F( τ )) (4.5) We noice ha we always have V Ω because he possibiliy of obaining Ω is included in V. If he maximal exposure ime = T is reached, he cargo mus be sold and V (, F( τ )) = Ω (, F( τ )). This value funcion V conains all he informaion needed o evaluae and run he physical rade: he value V (, F ( τ )) is he expeced profi from following he opimal rading sraegy he a priori ris of he sraegy and is exposure o he principal ris facors can be evaluaed hrough V, as seen in Secion 3.5. a a dae, given he forward curve F( τ ), compare he exercise value Ω (, F( τ )) and he coninuaion value VC (, F( τ )). o If Ω VC hen he delivery of he cargo should be specified. The opimal dae a which o deliver i is obained from he calculaion of Ω o If V C > Ω hen he ship should coninue sailing or anchoring wihou specifying when delivery will ae place. Simplificaion in he case of a wo-facor model If he dynamics of he forward curve are described by a simple wo-facor model as described in Par, he forward curve F( τ ) can be expressed in erms of he facor values f and f and he iniial forward curve F ( τ ) : log F ( τ ) = log F ( + τ ) + ψ (, τ ) + u ( τ ) f ( ) + u ( τ ) f ( ) (4.6) 6 C

62 Therefore he funcions Ω and V only depend on he values of f, f and ime since he beginning of he rade: Ω (, F( τ )) = Ω(, f, f ) V (, F( τ )) = V (, f, f ) (4.7) Opimal sopping formulaion Deermining V can alernaively be seen as an opimal sopping problem. The value funcion can equivalenly be wrien as τ V (, f, f ) = max E ( ) (, ( ), ( )) ( ), ( ) g s ds τ f τ f τ f f f f τ ST [, T + Ω = = (4.8) ] where ST [, T ] is he se of all sopping imes in [, T ]. The opimal sopping ime corresponds o he ime when V becomes equal Ω, i.e. and he iniial expeced profi from he rade is { s T V s f s f s s f s f s } * τ = min, (, ( ), ( )) = Ω (, ( ), ( ) (4.9) V = E g s ds + Ω f f * τ ( ) * ( τ,, ) (4.) 3. Soluion mehods Solving an opimal sopping problem such as he one ha has been formulaed for he floaing sorage rade is ain o calculaing he value of an American opion. A number of numerical mehods have been suggesed o his effec and we will review some of hem here. Dynamic programming The concepually simples mehod of solving an American opion problem is by dynamic programming. Discreizing ime ino daes =,,..., N = T, he value a dae i can be wrien as 6

63 V (, f, f ) = Ω(, f, f ) N N [ + ] [ ] V (, f, f ) = E V (, f + f, f + f ) g( ) C i i i V (, f, f ) = max Ω(, f, f ), V (, f, f ) i i C i (4.) The expecaion in he calculaion of VC ( i, f, f ) is calculaed using he ransiion probabiliies of f and f. When a binomial disribuion is assumed his yields he binomial ree mehod for American opions, as deailed in Clewlow and Sricland (998). Using he wo-facor model presened here we can evaluae i using ransiion probabiliies. The facor value space is discreized ino a N N recangular grid: ( f,..., f ) ( f,..., f ). The N N expecaion is evaluaed numerically using he previously calculaed values of V ( j+, f, fl ) and he join probabiliy densiy of ( df, df ) : E V (, f df, f df ) p V (, f, f ) p f, fl N N j+ + l + ' l ' j+ ' l ' NF ' = l ' = ' l ' l l ' l ' = exp π NF = ', l ' p ' l ' ( f f ( µ α f ) ) ( f f ( µ α f ) ) (4.) Parial Differenial Equaion In coninuous ime, he dynamic programming formulaion for V combined wih Io s formula yields a parial differenial equaion for V. e us assume ha V C > Ω, i.e. we are in he coninuaion region, such ha V = V C. We develop V using Io s formula: V V V V i f f i, j fi f j V ( + d, f + df, f + df ) = V + d + df + df + df df V V V V V = V + + ( µ α f) + ( µ α f) + + d (4.3) f f f f V V + dw + dw f f j Using he definiion of V and V C we find ha [ ] V (, f, f ) = V (, f, f ) = E V ( + d, f + df, f + df ) g( ) d (4.4) C 63

64 V V V V V + ( µ α f ) + ( µ α f ) + + = g( ) f f f f (4.5) This is valid for (, f, f ) in he coninuaion region such ha V (, f, f) > Ω (, f, f). For (, f, f ) in he exercise region we have V (, f, f) = Ω (, f, f). If we define ime, his enails he following boundary condiions on V S * ( ) as he coninuaion region a V f f f f f f S * (,, ) = Ω(,, ), (, ) ( ) (coninuiy) V f f = Ω f f f f S * (,, ) (,, ), (, ) ( ) (smooh pasing) (4.6) Using hese equaions for V we can calculae V using finie differences or a semi-analyical formulaion. Semi-analyical soluion Following Albanese and Campoliei (6), he parial differenial equaion for V can be solved in closed form if we assume he boundary of he coninuaion region o be nown. In Appendix 7 we show ha he soluion of (4.5) can be wrien as V (, f ) = p( f ', T; f, )( Ω( T, f ') G(, T )) df ' + p( f ', s; f, )( ψ ( s, f ')) df ' ds R R (4.7) * \ S ( s) = V (, f ) + V (, f ) eur early T where T is he maximum exposure ime, and * f S ( τ ) ψ (, f ) = Ω + * Ω f S ( τ ) τ Ω Ω Ω Ω Ω = ( µ α f ) + ( µ α f ) + + g( ) f f f f (4.8) The coninuaion region S * ( ) is defined as { } * S f f V f f f f ( ) = (, ), (,, ) > Ω (,, ) (4.9) The boundary second facor S * ( ) of his domain has o be deermined for each dae. We wrie i as a funcion of he { } S * ( ) = ( f * (, f ), f ), f R (4.) 64

65 such ha he equaion o be solved by f (, f ) is * V (, f ( τ, f ), f ) = p( f, f, T; f (, f ), f, )( Ω( T, f ') G(, T )) df ' * ' ' * R T p ( f, f, s; f (, f ), f, )( ψ ( s, f ')) df ' ds * f ( s, f ) = Ω(, f (, f ), f ) * ' ' * (4.) In Appendix 7 we deail he numerical procedure used o find his exercise boundary, which can be found recursively beginning a = T. Mone Carlo simulaion mehods The mehods discussed above, while suiable for a wo-facor model, become impracical if he number of facors is higher, for example if several pors are being considered. In his case a Mone Carlo mehod should be employed. Mone Carlo mehods are no ideally suied o American opion problems, because of heir bacward-recursion properies. However, ongsaff and Schwarz () sugges a leas-squares Mone Carlo mehod wih projecion of he value funcion ono a small basis, allowing for efficien pricing of American opions. A similar mehod could be employed in his case. 4. Analyical properies of he soluion We will examine some of he properies of he expeced profi funcion V using he analyical expression obained above. We will decompose he soluion as follows: V (, f ) = p( f ', T; f,)( Ω( T, f ') G(, T )) df ' + p( f ', s; f,)( ψ ( s, f ')) df ' ds R R (4.) * \ S ( s) = Ω (, f ) + EP + EP + EP drif convexiy early T where he excess profi componens EP drif, EP convexiy and EP early are defined as EP = Ω( T, Ef ( T ), Ef ( T )) G( T ) Ω(, f, f ) drif EP = V ( T, f, f ) ( Ω( T, Ef ( T ), Ef ( T )) G( T)) convexiy eur EP = V ( T, f, f ) early early (4.3) 65

66 Model parameer dependence We wan o examine he dependence of he value funcion on he model parameers α, and µ. e us consider firs he drif componen Remembering ha we can derive is dependence on he drif and on he model parameers: EP drif. Is value does no depend on he volailiies of he facors. αt µ αt Ef ( T ) = fe + ( e ) (4.4) α αt EPdrif e Ω = ( T, Ef( T ), Ef( T )) µ α f EPdrif Ef ( T ) Ω = ( T, Ef( T ), Ef( T )) α α f EP drif = (4.5) The convexiy componen can be wrien as EP = p( f ', T; f,)( Ω( T, f ') Ω( T, Ef ( T), Ef ( T ))) df ' convexiy R R R ( ) = p( f Ef ( T), f Ef ( T )) Ω( T, f, f ) Ω( T, Ef ( T ), Ef ( T )) df ' ' ' ' ' ( ) = p( f, f ) Ω ( T, Ef ( T ) + f, Ef ( T) + f ) Ω( T, Ef ( T ), Ef ( T )) df ' ' ' ' ' (4.6) ' ' (in he las line we change variables from f o f + Ef ( T ) ). We wan o show ha his convexiy premium does no depend on he expeced value of he facors. EP Ω Ω convexiy ' ' ' ' ' = p( f, f) ( T, Ef( T ) + f, Ef( T ) + f) ( T, Ef( T ), Ef( T)) df Ef f f R This erm is zero if Ω is a mos quadraic in he facors. In he general case we can wrie (4.7) Ω Ω Ω f f f ' ' ( Ef + f ) ( Ef ) = f ' + O( f ) (4.8) such ha: 66

67 α α EP e e T T convexiy ' ' ' ' ' C p( f, f)( f f ) df C Ef + = + α α R (4.9) For sufficienly small values of T his erm is small, of he order O( T ). Hence we have esablished ha he excess profis coming from convexiy are indeed independen of he expeced values of he facors, and herefore also of he drifs. Early exercise in a bacwardaed mare As we will see in Secion 4.6, i is generally opimal o sell he cargo immediaely when he forward curve ne of freigh is in bacwardaion. We will show his resul here. We assume ha he loading and delivery por are he same, such ha he rade is purely a floaing sorage rade. The forward curve ne of freigh is in ne bacwardaion if F ( τ ) ( H + FC ) τ < F () (4.3) a In his case i is opimal o sell he cargo spo, such ha a (4.3) Ω( ) g( s) ds Ω () = F ( ) ( H + FC ) F () < The iniial expeced excess profi is, as seen in equaion (4.), * τ * V (,,) = E g( s) ds + Ω( τ, f, f) (4.3) such ha ( ) * * * * * V Ω = E F ( τ ) ( H + FCa ) τ F () + E F ( τ ) exp( u() f( τ ) + u() f( τ )) (4.33) * Condiional on τ = τ (independen of he values of f and f, we hen find ha given he disribuions of f ( τ ) and f ( τ ), ( ) * * * E F ( τ ) exp( u() f( τ ) + u() f( τ )) = * * ατ α τ * * * u() ( e ) u() ( e ) F ( τ ) exp u() E[ f( τ )] + u() E[ f( τ )] + + α α (4.34) We now inroduce he iniial slope of he curve F * a = and consider only small values of τ, hen: τ τ = 67

68 a g * * V Ω = F () + u () µ + u() µ + ( u() + u() ) τ + O( τ ) F () (4.35) Hence, if a g + u() µ + u() µ + ( u () + u() ) < (4.36) F () here is no value o exercising laer, such ha V = Ω. The oher parameers being fixed, his can always be achieved for a srong enough ne bacwardaion. 5. Profi and ris The calculaion of he funcions V and Ω defines a physical rading sraegy ha can be applied in pracice. In order o assess how ineresing his sraegy is, we would lie o assess a priori is expeced reurn and ris. Furhermore we would lie o assess he dependence of he profis on he differen ris facors, o define a financial hedging sraegy using fuures or opions. Expeced and realized profi on he rading sraegy As discussed above, he values V ( =, f =, f = ) and Ω ( =, f =, f = ) are respecively he expeced profi and arbirage profi ha can be obained on he iniial dae. These numbers are expressed in US dollars per barrel ($/bbl) for crude oil or US dollars per gallon ($/gal) for gasoil. When he rading sraegy is execued he realized profi is no necessarily equal o he expeced profi, given ha he disribuion of forward curves is sochasic. The realized profi of he rip is, in he simple case of a single por, * * * * ( ) (, ( ), ( )) (4.37) W = g d + Ω f f where * is he exercise dae. This can be calculaed a poseriori o ge he realized profi. Bu seen a = his is a random variable wih a cerain disribuion. Is expeced value is V: V ( =, f, f ) = E W ( τ, f ( τ ), f ( τ )) f ( = ) = f, f ( = ) = f * * * (4.38) * where τ is he opimal sopping ime, which is a random variable depending on he realized values of f and f. Expeced ris and Sharpe raio There is no mare ris ied o he arbirage profi Ω, because he cargo is sold forward and he profi is fixed a he momen he decision is aen. However here is financial ris ied o he physical rading sraegy wih 68

69 expeced profi V: he forward curves will change before he decision o deliver he cargo is aen. This ris is refleced in he disribuion of he realized profi W. We have seen ha his disribuion is cenered on V: V ( =, f, f ) = E W ( τ, f ( τ ), f ( τ )) f ( = ) = f, f ( = ) = f * * * (4.39) Furhermore, a exercise, * * * * * = + Ω = + (4.4) W (, f, f ) g( ) d (, f, f ) g( ) d V (, f, f ) We define he process U represening he expeced profi and loss (P&) on he rade a ime by U (, f, f ) = g( s) ds + V (, f, f ) (4.4) The value of his process a exercise equals W, he realized P& of he rade. To find is disribuion we differeniae U using Io s formula: V V V V V V V du = g( ) + + ( µ α f) + ( µ α f) + + d + dw + dw f f f f f f (4.4) U U = dw + dw f f We find ha he process U has zero drif. This reflecs he fac ha V was correcly priced iniially. The insananeous volailiy of U over a ime period d is / / U U ( ) ( ) f f U = + = + (4.43) where is he dela of he value funcion wih respec o facor. * Furhermore, he disribuion of U given he sopping ime τ is V = (4.44) f * * τ τ * U U = + + f f (4.45) U ( τ, f, f ) U (, f, f ) (, f ( ), f ( )) dw ( ) (, f ( ), f ( )) dw ( ) We can calculae he firs momens of U: 69

70 E[ U ( τ, f, f )] = U (, f, f ) = V (, f, f ) (4.46) * * * τ τ * V V = + f f (4.47) Var U ( τ, f, f ) E (, f ( ), f ( )) d E (, f ( ), f ( )) d The variance depends on he sopping ime and can bes be evaluaed hrough Mone Carlo simulaion, simulaing he pahs of ( f, f ) and using he value funcion already calculaed. If we mae he simplifying assumpion ha he delas of he value funcion are consan, he variance can be approximaed as: ( ) * * Var[ W ] = Var U ( τ, f, f) + E[ τ ] (4.48) Alernaively, he complee disribuion of W can be evaluaed using Mone Carlo simulaion. I should be noed ha his Mone Carlo simulaion is simpler han he leas squares Mone Carlo echnique used for finding he opimal sopping ime. If we now he expeced profi and he sandard deviaion of he realized profi, we can calculae he annualized Sharpe raio of he sraegy a priori: SR = Expeced profi Sd. deviaion = E[ τ ] * / V ( + ) / * * τ τ V V E (, f( ), f( )) d + E (, f( ), f( )) d f f / * E[ τ ] V (4.49) 6. Resuls Arbirage resuls In his secion we presen he resuls from he calculaion of he funcion Ω a differen daes. This funcion, evaluaed a rade iniiaion ime ( = ), gives he arbirage profi ha can be obained from he shape of he forward curve a he curren dae. By sudying is dependence on he facor values f and f, we can also evaluae is dependence on he level and slope of he curve. 7

71 Arb profi (Ω) Spread Freigh cos 5 $/bbl 5-5 6/7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Dae Figure 7. Arbirage profi per barrel on he Sullom Voe-OOP roue 5 Time o delivery (days) 5 5 6/7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Dae Figure 8. Time o delivery for saic arbirage on he Sullom Voe OOP roue 7

72 6 Sullom Voe spo OOP spo 4 Price ($/bbl) /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 9. Spo prices of Sullom Voe (blue) and OOP (green) OOP forward curve slope 5 Slope ($/bbl/monh) /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure. Slope of he S forward curve, in US Dollars per barrel per monh, measured on he fron wo monh conracs 7

73 On each day in he sample period (Augus 7 o Ocober 9) we calculae he arbirage profi Ω ha can be obained from he observed forward curve and freigh prices on ha dae. The arbirage profi is calculaed as Ω = max ω( τ ) where τ τ sail τ sail P = d( X, X ) / u ω( τ ) = F( τ ) S Hτ B ( FC( u) τ + FC ( τ τ )) BH, τ τ sail a sail sail (4.5) The arbirage profi is obained by buying BFOE crude a Sullom Voe on he rade iniiaion dae and delivering i ino OOP in he opimal ime * τ, sailing a speed u o ge here, and anchoring up for a ime τ τ sail o wai for delivery. The cargo is bough a he spo price S and sold forward a he price * F( τ ). We decompose his arbirage profi ino a geographical spread Sp( τ ) = F( τ ) S and a freigh cos Fc, Fc( τ ) = Hτ + B ( FC( u) τ + FC ( τ τ )) + BH (4.5) The physical arbirage is said o be open when Ω >, i.e. sail a sail τ > τ : he spread ha can be earned on * * Sp( ) Fc( ) he crude oil is higher han he cos of ransporaion and sorage. In he opposie case i is said o be closed. When he arbirage window is open i is profiable o ship a cargo of oil on he considered roue. We presen he resuls for he Sullom Voe-OOP roue in Figure 7. The geographical spread is shown in green, he freigh cos in red, and he ne arbirage profi Ω in blue. We observe ha he arbirage window is open during large pars of he ime period under consideraion. In * Figure 8 we show he value of τ, he ime beween he curren dae and he opimal dae o exercise he opion of delivering he cargo. We observe ha he large profis from lae 8 and early 9 came from he large opporuniies in floaing sorage creaed by a seep conango and low imecharer raes. In Figure 9 we show he spo prices of crude oil a he loading and delivery pors. In Figure we show he slope of he forward curve a he delivery por. Figure 9 confirms wha maes his physical arbirage possible: he spread in spo prices beween European and American crude. However, he arbirage profis seem o be uncorrelaed wih he general level of crude prices. This sems from he fac ha inernaional crude prices largely move ogeher, parly because of such arbirage aciviy. There does, however, seem o be some relaion beween he forward curve slope and he arbirage profi. This is winessed in Figure where we regress he arbirage profi on he forward curve slope. The relaionship is sronger for a seeper conango. In Figure we presen daa colleced from differen research repors on he acual crude oil in floaing sorage worldwide alongside he opimal ime o delivery for he arbirage rade. We see ha here is a subsanial increase in he amoun of crude oil sored a sea saring in Ocober 8. This coincides wih he appearance of floaing sorage opporuniies according o our model. Furhermore, he shor disappearance of floaing sorage opporuniies according o our model in June 9 was accompanied by a clear downward rend in he number of aners soring crude in he Goldman Sachs and Gibson Research daa. 73

74 The same analysis can be performed for differen mares and differen roues. As a poin of comparison we presen he resuls for he gasoil arbirage beween Europe and he Unied Saes. The produc being raded is No. fuel oil, also nown as gasoil or heaing oil. The loading por is he Amserdam-Roerdam-Anwerp (ARA) region, Europe s major refining hub. The desinaion is New Yor harbor (NYH), which is he main delivery poin of refined producs on he eas coas of he Unied Saes. Deails of he roue and cargo are presened in Appendix 8. The arbirage profi, decomposed as described above, is presened in Figure 3. The opimal delivery ime is presened in Figure 4 and hese resuls should be compared o he spo prices in Figure 5 and he forward curve slope in Figure 6. We noe ha he arbirage window is open less frequenly han was he case for crude oil and he spread has been negaive on occasions, maing he inverse arbirage (U.S. o Europe) ineresing. However, here have been significan floaing sorage opporuniies since he end of 8 as winessed in Figure 4, and hese profis have been very ineresing: around cens per gallon for a gallon cosing less han dollars. 6 4 Daa inear regression Arbirage profi (Ω) y = x Rsq = Forward slope ($/bbl/monh) Figure. Relaionship beween he forward curve slope a he delivery por OOP) and he arbirage profi 74

75 Crude floaing sorage (mn bbl) Morgan Downey 35 (crude + resid) IEA/Goldman Sachs 3 Gibson research Time o delivery Time o delivery (days) 6//8 9/9/8 /8/8 3/8/9 7/6/9 /4/9 // 5// Figure. Crude oil in floaing sorage worldwide (lef axis) and opimal ime o delivery of he floaing sorage rade (righ axis). Sources: IEA/Goldman Sachs Global ECS Research, Gibson Research, Morgan Downey.5.4 Arb profi (Ω) Spread Freigh cos.3.. $/gal /3/4 7/7/5 /9/6 4//8 8/5/9 /8/ Dae Figure 3. Arbirage profi per gallon for gasoil rade beween ARA and NYH 75

76 6 5 Time o delivery (days) 4 3 /3/4 7/7/5 /9/6 4//8 8/5/9 /8/ Dae Figure 4. Time o delivery for arbirage on he ARA-NYH roue 4.5 ARA spo Heaing NYH spo Price ($/gal) /3/4 7/7/5 /9/6 4//8 8/5/9 /8/ Figure 5. Spo prices of No. fuel oil a Amserdam-Roerdam-Anwerp (blue) and New Yor harbor (green), in US Dollars per gallon 76

77 .8 Heaing NYH forward curve slope.6.4. Slope ($/gal/monh) /3/4 7/7/5 /9/6 4//8 8/5/9 /8/ Figure 6. Slope of he heaing oil forward curve a New Yor Harbor, in US Dollars per gallon per monh Expeced and excess profis The resuls presened for Ω were saic arbirage resuls. We now consider he opimal rading sraegy presened in Secion 4.. This rading sraegy yields a value funcion V which is he expeced profi of he physical rading sraegy. These resuls are obained using he semi-analyical formulaion presened in Secion 4.3, calculaing he exercise boundary numerically as described in Appendix 7. The wo-facor model used is calibraed on he crude oil fuures mare as described in Secion.4, and we mae he assumpion ha drifs are zero: he expeced spo price is herefore equal o he forward price. Furhermore, rades are limied o a maximal exposure ime T equal o days. We sudy he shape of Ω and V wih he iniial dae se o December 8, 8. As seen in Figure 7 he forward curve on ha dae was in conango. We plo Ω and V as funcions of he facor values f and f a rade iniiaion. 77

78 6 F (τ) F (τ) - Fc(τ) (ne of freigh) F(τ) ($/bbl) 5 45 Profi ($/bbl) 4 V τ (days) Ω f f.5 Figure 7. Forward curve and forward curve ne of freigh on December 8, 8 Figure 8. Ω and V as funcions of he facors f and f a rade iniiaion (December 8, 8) V(f,), au = days.9.8 V(,f ), au = days Ω V V eur V early Profi ($/bbl).6.5 Profi ($/bbl) Ω V V eur V early f * f F level ($/bbl) f Fwd curve slope ($/bbl/monh) Figure 9. Cross-secion of Ω and V a rade iniiaion as a funcion of f (lef) and f (righ) 78

79 Excess profi from coninuaion, V - Ω, τ = days Profi ($/bbl) S * (T) Fwd curve slope ($/bbl/monh) f f F level ($/bbl) Figure 3. Expeced excess profi from coninuaion V - Ω a rade iniiaion, in USD/bbl, and exercise boundary (red line) In Figure 8 and Figure 9 we show he dependence of V and Ω on f and f. The values corresponding o he iniial forward curve are V (,, ) and Ω (,, ), valued respecively a 6.55 $/bbl and 5.8 $/bbl. The oher values correspond o a forward curve ha has been shoced by he facor values f and f. We can see ha, predicably, a posiive parallel shif ( f > ) yields a higher expeced profi. The slope wih respec o he second facor is lower. We can also see ha V and Ω are he same a he maximal exposure ime: his is he erminal condiion ha we impose. Furhermore, a rade iniiaion V is higher han Ω, and more so for low values of Ω. Thus here is value o eeping he opions open. For negaive values of Ω i is sill possible o have posiive values of V: here is a chance ha prices will rebound enough o yield a profi during he rade period. Figure 9 decomposes he value funcion V ino wo componens: he European exercise value V eur and he early exercise premium V early. The European exercise value corresponds o he expeced profi ha would be earned if he cargo was held unil he maximal exposure ime T ( days in his case), and hen sold ino he mare. This value largely depends on he drifs of he facors. The mean-revering model has a large impac in his respec. When he value of f is negaive, i is expeced o increase, which pushes he expeced value up compared o he arbirage value. When he value of f is posiive, is expeced value is lower, pushing he expeced value down. 79

80 This is couneraced by he early-exercise premium, which is posiive for high values of he firs facor and for high absolue values of he second facor. This corresponds o siuaions where i is close o opimal o exercise. The decision o exercise is made based on he difference beween he expeced profi V and he exercise profi Ω. We plo his difference in Figure 3. The dares zone, where V = Ω, is he exercise region. If ( f, f ) falls in his zone i is opimal o specify delivery of he cargo and harves he profi Ω. Ouside his region i is opimal o coninue sailing and delay he decision abou delivery ime unil laer. The excess profi is seen o depend on he shape of he forward curve hrough he facor values f and f. The excess profi is seen o be highes when he firs facor is lowes: because i is mean-revering, eeping he posiion open gives more upside exposure han downside exposure. Dependence of expeced profis on model parameers The resuls above are presened for a wo-facor model ha has been calibraed on he crude oil mare as deailed in Par. We have seen ha he inerpreaion of he excess profis is lined o he model parameers α, and µ which deermine he disribuion of possible forward curves. I is herefore ineresing o examine he dependence of he profis on he values of hese parameers. We vary he parameers wihin reasonable ranges around he reference values ha have been used before, and plo he dependence of V, V, V and Ω on hese parameers. The rade iniiaion dae is December 8, 8. eur early The resuls are presened in Figure 3. The dependence on he mean-reversion parameer is raher wea compared wih he dependence on he oher parameers. This can be explained by he fac ha he maximal exposure ime, days, is raher shor, and ha he excess profi we are considering is aen a f f = =, such ha he expeced value of he facors is no affeced by he mean-reversion parameer. The dependence on he volailiies of he facors is very srong, wih a doubling of from 6% o 5% aing he excess expeced profi from.68 $/bbl o.9 $/bbl. The effec is larger in he firs facor because he magniude of he excess profis coming from he firs facor are much larger. Bu in relaive erms, doubling from.86% o 3.7% aes he excess profi aribuable o he second facor, i.e. he difference beween he profi for a larger han zero and he profi for =, from 7.78 c/bbl o 6.4 c/bbl, which is a significan increase. The effec of he drif parameers µ and µ is o change he expecaions abou wha he forward curve will loo lie in he fuure. In paricular, a negaive value for µ means ha he rader is aing a sharply negaive view on he fuure level of prices. In ha case i is more ineresing o exercise early o ae he profis given he curren level of prices. A posiive value for µ is a posiive view on levels and i will be preferable o wai o ae advanage of rising prices. A negaive value for µ corresponds o a view of a sharper conango, which is beneficial o he rade, while more bacwardaion ( µ > ) is derimenal. 8

81 Figure 3. Dependence of expeced profis on he model parameers 7 Dependence of profis on α 7 Dependence of profis on α 6 6 Profi ($/bbl) Ω V V eur V early Reference Profi ($/bbl) Ω V V eur V early Reference α α Dependence of profis on α. Reference is α =.84 yr - Dependence of profis on α. Reference is α =.84 yr - 9 Dependence of profis on 7 Dependence of profis on Profi ($/bbl) Ω V V eur V early Reference Profi ($/bbl) 4 3 Ω V V eur V early Reference Dependence of profis on. Reference is = 6% (annualized) Dependence of profis on. Reference is =.86% (annualized) Dependence of profis on µ Dependence of profis on µ 7 6 Profi ($/bbl) Ω V V eur Profi ($/bbl) Ω V V eur V early Reference V early Reference µ µ Dependence of profis on µ. Reference is µ = yr - Dependence of profis on µ. Reference is µ = yr - 8

82 Dependence of profis on he ship speed In some circumsances, i can be beneficial for he rade o sail he ship slowly across he Alanic in order o save on fuel coss. Inuiively, his will be especially useful when he forward curve is in a sligh conango. The speed will affec profis in hree ways: A faser ship will be able o deliver is cargo earlier, which is imporan in a srong bacwardaion A faser ship will be charered for less ime, such ha is oal ime charer cos will be lower A faser ship will consume more fuel. The fuel consumpion funcion FC( u ) is approximaely cubic in he speed u. In Figure 3 we examine he variaion of he profis wih he speed of he ship u for a rade beginning on Augus 3, 8 and April 8, 9. We noice ha he speed has a small influence on profis, of he order for c/bbl for a speed varying from 8 o 7 nos. The speed is fixed during he voyage. When he forward curve is in bacwardaion, here is incenive o deliver he cargo as soon as possible. A higher speed allows he rader o deliver he cargo earlier, bu a he cos of higher fuel consumpion. There is an opimal speed of around 3 nos yielding he bes radeoff. When he forward curve is in conango, he rade will involve some amoun of floaing sorage a desinaion, such ha fuel savings can be ineresing. The excess profi V Ω, however, is no affeced by he speed..6 Profi ($/bbl) Ω Freigh cos Spread Profi ($/bbl) Ω V Freigh cos (Ω) Spread (Ω) u (nos) u (nos) Augus 3, 7 (bacwardaion) April 8, 9 (conango) Figure 3. Variaion of profis wih vessel speed (fixed during he voyage) on wo differen daes, when he forward curve was in bacwardaion (lef) and conango (righ) 8

83 Time series of expeced profi, ris and Sharpe raio For each wee in he sample period, we perform he above calculaions and derive: The arbirage profi Ω, he expeced profi V and he excess profi V Ω The expeced ris, i.e. he sandard deviaion of W The a priori Sharpe raio We plo hese values as a funcion of ime. 5 Ω V V eur V early 5 Profi ($/bbl) -5-6/7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 33. Arbirage profi Ω and expeced profi V, decomposed ino V eur and V early, for weely loading daes from Augus 7 o Augus 9 83

84 .4 V - Ω. Excess profi ($/bbl) Sandard deviaion ($/bbl) /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 6/7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 34. Expeced excess profi V Ω for differen loading daes Figure 35. Expeced sandard deviaion of realized profis. The zero values correspond o daes when exercise is immediae 4.7 SR on V - Ω Exposure ime (days) 5 5 Sharpe raio /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 6/7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 36. Expeced exposure ime * E[ ] of he rade. The maximal exposure ime is T = days Figure 37. A priori annualized Sharpe raio of he excess profi V Ω The average long-erm Sharpe raio of he S&P5 is abou.4. The Sharpe raios in Figure 37 are calculaed on profis over he risless arbirage profi Ω. They are on he order of 6 when calculaed over he ris-free rae. 84

85 .4 V - Ω. Excess profi V - Ω ($/bbl) Forward curve slope ($/bbl/monh) Figure 38. Excess expeced profi V Ω vs. Forward curve slope a delivery por In Figure 33 we plo he arbirage profi Ω and he expeced profi V as a funcion of he rade iniiaion dae. I.e. each wee we examine he forward curve, spo price and shipping cos and deermine wha he arbirage profi would be for a cargo loaded wihin he loading windowτ load (5 days), as well as he expeced profi V from loading he cargo and execuing he opimal rading sraegy. We see ha hese profis are always a leas as grea as he arbirage profis. We have already sudied he behavior of he ime series of Ω, so we will concenrae here on he excess profi, V Ω. We plo his expeced excess profi as a funcion of ime in Figure 34. Is value varies beween and $/bbl, averaging 74 c/bbl in he period when he excess profi is posiive. We noe ha he period under consideraion can be separaed in wo: from 7 o mid-8 he coninuaion value is zero, while afer he mare crash in 8 he excess profi jumps o values around 75 c/bbl. The firs period corresponds o a bacwardaed forward curve, while he second period corresponds o a period of srong conango and low freigh raes following he crisis. In Figure 38 we show he relaionship beween he forward curve slope a he delivery por and he excess profi from coninuaion. Consisenly wih wha was proved in Secion 4.4, we find ha a forward curve in bacwardaion or in sligh conango yields a zero excess profi, while all he posiive excess profis are associaed wih a forward curve in conango. The sandard deviaion of he profis over he rade, presened in Figure 35, is significan, averaging 6.66 $/bbl in he period when he excess profi is posiive. This is he ris associaed wih eeping he exposure o he forward curve open, and is accordingly zero when he cargo should be sold forward immediaely, i.e. V = Ω. The expeced ime over which his exposure is held 85 * E[ ] is presened in Figure 36, and averages 7 days.

86 We noe ha he exposure ime never reaches he maximal exposure ime T ha is se o days here. Combining expeced profi and ris we can calculae he annualized Sharpe Raio associaed wih he sraegy, which is presened in Figure 37. We consider his Sharpe raio in excess of he risless profi Ω. I averages.4 during he period. In Secion 4.6 we examine he deail of hese ime series and aemp o explain he appearance of excess profis. Realized profi and sandard deviaion The funcions Ω (, f, f) and V (, f, f ) define a physical rading sraegy ha can readily be pu ino pracice. Given hisorical ime series of he acual moves in he forward curve we can calculae he profis ha would have been realized by following his sraegy, and compare hem o he expeced profis and riss presened above. Profi ($/bbl) Realized profi W ($/bbl) Expeced profi V ($/bbl) 95% C.I. for W Saisics E[W V].74 $/bbl Sd[W V] 6. $/bbl SR on W 5.95 SR on W Ω /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 39. Expeced profi V and realized profi W on differen daes Saisics are no calculaed on he enire period, bu on he period when V > Ω 86

87 Realized exposure ime * (days) Expeced exposure ime E[ * ] (days) 95% C.I. for * E Saisics * * [ realized E[ ]] * * Sd[ realized E[ ]] -.5 days 5 days 8 Exposure ime (days) 6 4 6/7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 4. Expeced and realized exposure imes * On each wee in he sample period, having calculaed he funcions rade iniiaion dae, we execue he rading sraegy defined by: V (, f, f ) and Ω (, f, f ) wih if V (,,) >, he rade is expeced o be profiable, so iniiae he rade by buying he cargo and charering he vessel A each ime from rade iniiaion >, observe he forward curve F(, τ ) and calculae he facor values using he orhogonaliy condiion in (.), τ max F(, τ ) f ( ) = u ( τ )log dτ F ( τ ) (4.5) If < T (maximal exposure ime, days in his case), compare he exercise and coninuaion profis: o o If V (, f ( ), f ( )) > Ω (, f ( ), f ( )), hen coninue sailing a speed u If V (, f ( ), f ( )) = Ω (, f ( ), f ( )), i is opimal o exercise, so sell he cargo forward and collec Ω (, f ( ), f( )) 87

88 If = T, sell he cargo forward and collec Ω ( T, f ( T ), f( T )) If he exercise ime is *, he realized profi on his rade is hen * * * * * * * = + Ω W (, f ( ), f ( )) g( ) d (, f ( ), f ( )) (4.53) As can be seen from Figure 39, he realized profi is highly variable bu i says wihin he bounds of he 95% confidence inerval for W based on he expeced ris calculaed previously. The sandard deviaion of W V calculaed over he period when here are excess profis is 6. $/bbl, close o he average sandard deviaion seen in Figure 35. The exposure ime, presened in Figure 4, varies widely The annualized Sharpe raio of he sraegy over his period is.99 if calculaed on he profis in excess of Ω, and 5.95 if considered in excess of he ris-free rae. Realized profis and sandard deviaion wih hedging The significan sandard deviaion of he realized profis W versus he expeced profis V comes from he exposure of he rade o he ris facors f and f. Using he hedge raios compued from he expeced profi funcion V we can simulae wha he realized profi is when he profi is dela-hedged wih respec o he firs or second facor. A ime ino he rade, assuming he cargo has no been sold, he value funcion has delas δ and δ wih respec o f and f : V δ = (, f ( ), f ( )) (4.54) f In order o eliminae he ris from facor, for example, we ae a posiion δ in he facor f. How o achieve his wih he available fuures conracs is explained in Secion.9. The impac of his posiion on he evoluion of he expeced porfolio P& U ɶ is duɶ = δ dw + δ dw δ df = δ ( α f µ ) d + δ dw (4.55) Hence he realized P& a he end of he rade is * * τ τ * = + + Uɶ ( τ, f, f ) V (, f, f ) δ (, f ( ), f ( ))( α f µ ) d δ (, f ( ), f ( )) dw ( ) (4.56) 88

89 The ris ied o he firs facor has herefore been eliminaed bu his also reduces he expeced profi. The same approach can be applied o he second facor, hedging ou ils. A common pracice is o hedge ou he parallel shif facor, which is he major ris facor, and eep he exposure o ils. Profi ($/bbl) Realized profi W ($/bbl) Expeced profi V ($/bbl) 95% C.I. for W Saisics E[W V] -.68 $/bbl Sd[W V] 3.3 $/bbl SR on W 6.48 SR on W Ω /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 4. Expeced profi V and realized profi W when hedging he firs facor Profi ($/bbl) Realized profi W ($/bbl) Expeced profi V ($/bbl) 95% C.I. for W Saisics E[W V]. $/bbl Sd[W V].4 $/bbl SR on W 9.5 SR on W Ω /7/7 9/5/7 /4/7 4//8 7//8 /9/8 /7/9 5/7/9 8/5/9 /3/9 Figure 4. Expeced profi V and realized profi W when hedging he firs and second facors 89

90 As can be seen in Figure 4 and Figure 4, he hedging does indeed diminish he ris of he sraegy. The hisorical sandard deviaion of W V is 3.3 $/bbl when hedging he firs facor.4 $/bbl when hedging he firs and second facor This should be compared o he unhedged sandard deviaion of 6. $/bbl. I is ineresing o noe ha even hedging boh facors does no render he sraegy risless, conrary o heory. There are wo reasons for his: The dela-hedging is only daily and no coninuous, and high-ampliude movemens (jumps) in he facors will no be hedged perfecly The forward curve does no only move in shifs and ils, and only hose movemens have been hedged ou 7. Origins of excess profis We have shown ha in addiion o significan arbirage profis o be made on arbiraging crude oil beween Europe and he Unied Saes, following an opimal sorage and selling sraegy could lead o significan excess profis. I is ineresing o undersand he origin of hese profis in order o undersand in wha fundamenal siuaions hey migh appear. We will mae a disincion in wha follows beween The origin of excess expeced profis The origin of realized profis, i.e. when he rading sraegy performs well Origin of excess expeced profis We have esablished in Secion 3.5 ha he period Augus 7 Augus 9 can be decomposed ino wo periods: Augus 7 o Ocober 8, when he forward curve for crude oil was in bacwardaion and here were no expeced excess profis, and Ocober 8 o Augus 9, when he forward curve was in conango and here could be found excess profis in eeping exposure o he forward curve open. We will concenrae on he second period here. We have already esablished ha he forward curve (ne of freigh cos) being in conango is a necessary condiion for he excess profi o be posiive. We can gain more insigh ino he origins of excess profis by decomposing he excess profi as follows 9

91 EP = V (, f, f ) Ω (, f, f ) = EP + EP + EP oal drif convexiy early EP = Ω( T, Ef ( T ), Ef ( T )) G( T ) Ω(, f, f ) drif EP = V ( Ω( T, Ef ( T ), Ef ( T )) G( T )) EP convexiy eur early = V early (4.57) When considering he iniial expeced profi, f = f = such ha Ef( T ) = Ef( T ) = and EPdrif = Ω( T,,) G( T ) Ω (,,) (4.58) This expeced profi will generally be zero for a forward curve in conango. I can, however, be significan for non-zero facor values because of heir mean-revering propery. The excess profi from convexiy can be wrien as [ ] EP = E Ω( T, f ( T ), f ( T )) Ω ( T, Ef ( T ), Ef ( T )) (4.59) convexiy and capures he non-lineariy of Ω. As for he early-exercise premium, i capures he possibiliy of selling he cargo before he dae T. We presen he ime series of he excess expeced profis and is decomposiion in Figure 43. We noice ha he major par of he excess profi comes from he convexiy, averaging 84% of he oal excess profi. The convexiy and early-exercise premia are raher regular..5.5 Excess profi ($/bbl) Toal Drif Convexiy Early - /9/8 /8/8 /7/9 3/8/9 5/7/9 6/6/9 8/5/9 /4/9 Figure 43. Decomposiion of expeced excess profis as a funcion of ime 9

92 Based on hese observaions we can conclude ha The exisence of an excess profi is condiional on he forward curve ne of shipping cos being in conango When he conango condiion is saisfied, he expeced excess profis are fairly sable. Trade performance and origin of realized profis We have idenified in wha siuaions excess profis are expeced. However, in a rading siuaion i is imporan o now in wha cases he rade will succeed and in which cases i will yield a loss, in order o undersand he expeced profis and ris manage he posiion..5 Profi ($/bbl) f f, f 5 U(), Ω() ($/bbl) Ω U f f f Days from rade iniiaion Figure 44. Evoluion of f, f, expeced P& U and exercise profi Ω during he rade saring on December 8, 8. ef, he pah of (f (), f ()) during he rade overlaid on he expeced profi V. Righ, hese funcions as a funcion of days from rade iniiaion. The delivery of he cargo is specified afer 9 days. In Figure 44 we presen he evoluions of he facor values and he expeced and exercise profis U and Ω during he physical rade iniiaed on December 8, 8. In his paricular case, he cargo is exercised afer 9 days, when he expeced profi and exercise profi are seen o converge. The realized profi W a he end of he rade is 3.6 $/bbl. We also presen he evoluions of he facor values f and f on he same figure. As we have already seen, V has he sronges dela wih respec o he firs facor, and he realized profi is highly correlaed wih he value of f during he rade. When hedging he firs facor, however, he realized profis are more correlaed wih he second facor, as is seen in Figure 45. 9

93 .. 8 f, f 6 U(), Ω() ($/bbl) -. Ω U f 4 f Days from rade iniiaion Figure 45. Evoluion of he facor values and profis during he rade saring on December 8, 8, when hedging he firs facor In order o assess how he rade will perform based on he evoluions of he wo facors i is useful o recall he shape of he payoff funcion Ω as a funcion of boh facors V(f,), au = days.9.8 V(,f ), au = days Ω V V eur V early Profi ($/bbl).6.5 Profi ($/bbl) Ω V V eur V early f * f F level ($/bbl) f Fwd curve slope ($/bbl/monh) Figure 46. Cross-secion of Ω and V a rade iniiaion as a funcion of f (lef) and f (righ) As can be seen in Figure 46 he dependence of he rade has he following characerisics wih respec o f and f: I is direcional wih respec o f, similar o a forward exposure I is a volailiy rade wih respec o f : he payoff is convex and has higher payoff for large movemens of f in eiher direcion. This is closer o a sraddle opion. 93

94 Figure 47. Realized profis as a funcion of realized drifs and volailiies during he rade period, for a rade saring on December 8, No hedging Facor hedged Facors & hedged Ref. value W 4 3 W 4 3 No hedging Facor hedged Facors & hedged Ref. value Realized µ Realized No hedging Facor hedged Facors & hedged Ref. value W 3.5 W 3.5 No hedging Facor hedged Facors & hedged Ref. value Realized µ Realized This inuiion is confirmed by he resuls in Figure 47. In his figure we presen he realized profis as funcions of realized drifs and volailiies, imposed in a Mone Carlo simulaion, differen from he a priori drifs and volailiies used when valuing he floaing sorage opporuniy. We can clearly see he direcional naure of he posiion in f, wih realized profis ha are linear in he drif µ. These are also increasing in he volailiy because of he sligh convexiy of he payoff funcion. On he oher hand, he realized profis are independen of he drif µ of he second facor, bu srongly relaed o is realized volailiy. 94

95 8. General commodiy rading problem The problem we have been considering is limied o a single delivery por and a single aner speed. The only choice ha is lef o he rader is ime of delivery. In general, an oil (or oher commodiy) cargo ha has no ye been sold forward can be reroued o a differen por. The ship can also sail slower in order o save fuel. These opionaliies mae he cargo more valuable o a rader han wha has been calculaed previously. A decision model for opimal ship rouing should ae ino accoun he forward prices a differen poenial delivery pors and open for he possibiliy of delaying he choice of delivery por o a laer dae. For example, a cargo of Bonny igh crude oil loaded in Nigeria could poenially be delivered o Europe or he Unied Saes. Insead of choosing a delivery locaion immediaely he oil rader could choose o roue he ship norhbound in he mid-alanic, and waiing o see if he spread evolves. We will formulae he sochasic conrol equaions governing how he ship should be roued o maximize profi. The noaions are he same as previously, bu we now inroduce: A se of desinaion pors X a which he cargo can be delivered, each wih a forward curve F ( τ ) P The speed of he ship u can be varied wihin bounds [ u, u ], usually beween 8 and 6 nos The insananeous direcion of he ship is he uni vecor d Exercise profi We define Ω ( X, F ( τ )) o be he profi ha can be earned on he cargo if he ship is a locaion X and he forward curve in por is F ( τ ), by commiing o a specific delivery price someime in he fuure and sailing o deliver a ha ime. In effec, he rader gives up he possibiliy of changing delivery por and ime. A exercise, one chooses a delivery por, a ime-o-delivery τ and a sailing speed u. For one choice of hese parameers, he profi is τ sail P (, u) = d( X, X ) / u ω(, τ, u) = F ( τ ) S Hτ B ( FC( u) τ (, u) + FC ( τ τ (, u))) BH ( τ τ (, u)) sail a sail sail (4.6) The exercise profi is obained by maximizing ω(, τ, u) over all possible pors, speeds and imes o delivery: Ω ( X, F ( τ )) = max ω(, τ, u) (4.6), u τ τ sail (, u) 95

96 Expeced profi and opimal roue e us define V ( X, F ( τ )) as he expeced profi from he cargo when he aner is a he locaion X and he forward curves are given by F ( τ ). e g( X, u ) be he daily cos of sailing a speed u when he ship is a locaion X, i.e. g( X, u) = H + B FC( u) if he ship is sailing P g( X, u) = H + B FC if he ship is a anchor a por a (4.6) When he ship is a a locaion X, he rader has wo choices: eiher exercise and sell he cargo forward, hereby earning he exercise profi Ω ( X, F ( τ )) or choose o coninue speculaing during a ime d wihou exercising. If he ship is a sea he can choose he opimal speed u and direcion d and he expeced profi is: V ( X, F ( τ )) = max E V ( X udd, F ( τ ) df ) + + g( X, u) d C u, d (4.63) If he ship is in por (floaing sorage), he expeced profi is [ τ ] P P V ( X, F ( τ )) = E V ( X, F ( ) + df ) g( X,) d (4.64) C l l This gives he coninuaion value V C. The forward curves are evolved during he ime period d using he wo-facor model. Hence he expeced profi a locaion X given he forward curves F ( τ ) is Hamilon-Jacobi-Bellman equaion [ ] V ( X, F ( τ )) = max Ω ( X, F ( τ )), V ( X, F ( τ )) (4.65) C When we assume ha he underlying facors follow diffusions we can derive a coninuous-ime equaion o evaluae V. This equaion is nown as he Hamilon-Jacobi-Bellman (HJB) equaion for he sochasic conrol problem, see Morimoo () and Chang (4). We assume ha each of he forward curves F ( τ ) is governed by a wo-facor model, such ha where: log F ( τ ) = log F ( + τ ) + ψ (, τ ) + ψ (, τ ) + u ( τ ) f ( ) + u ( τ ) f ( ) (4.66) 96

97 df ( ) = α f ( ) d + dw ( ), j =, j j j j j dψ (, ) ( ( )) j τ = j u j + τ d (4.67) For he sae of simpliciy we renumber he facors f as a sequence ( f i ) i =,..., M. While he wo facors for a j single forward curve are uncorrelaed, facors for differen forward curves will be correlaed, such ha in general df df = ρ d (4.68) l l l Consider a locaion X, ime and facor values region, s.. V = V. We develop V using Io s formula: C f i and assume ha V C > Ω, i.e. we are in he coninuaion V ( X + d dx, + d, f + df,..., f + df ) = V + d + d dx + df + df df M M M V V V V M M i i j X i= fi i= j= fi f j = V + + u d + f + d + dw M M M M V V V V V ( µ i αi i ) ρij i j i i X i= fi i= j= fi f j i= fi (4.69) Taing expecaions in he definiion of V C and simplifying, we finally ge he equaion: = max g( X, u) + + u d + ( f ) + u, d M M M V V V V µ i αi i ρij i j X i= fi i= j= fi f j (4.7) This is a ypical example of a Hamilon-Jacobi-Bellman equaion. I is valid in he coninuaion region, i.e. * ( f,..., fm ) S ( ) boundary S * ( ) :. The boundary condiions are given by he smooh pasing condiion on he free V (, X, f,..., f ) = Ω(, X, f,..., f ) * M M ( f,..., fm ) S (, X ), M = Ω V (, X, f,..., f ) (, X, f,..., f ) M (4.7) The erminal condiion is ha a he maximal exposure ime T, he cargo should be delivered: V ( T, X, f,..., f ) = Ω ( T, X, f,..., f ) (4.7) M M Solving he general problem This problem can in principle be solved numerically by dynamic programming or finie differences. However, he poenially large number of sae variables can mae i challenging o solve using hese mehods. The preferred mehod for such a problem would be a leas-squares Mone Carlo simulaion as presened in ongsaff and Schwarz (). This requires wor on finding appropriae basis funcions o projec he soluion on. 97

98 5. CONCUSIONS. Summary of resuls In Par we have developed a wo-facor model and given evidence ha i is sufficien for modeling he erm srucure of volailiy and he correlaion surface of a number of commodiies. We prove ha i is easily formulaed as a model involving wo independen and mean-revering facors ha represen he change in level and slope of he forward curve. We find ha he firs facor is he dominan facor and he majoriy of variance of forward prices comes from he firs facor. However, oher facors canno be ignored as hey will affec porfolios ha are weighed differenly. We also show ha he spo price process implied by his wo-facor model is consisen wih he Schwarz and Smih () formulaion wih shor-erm and long-erm shocs driving he spo price. Furhermore, we show ha he shapes of forward curves consisen wih he wo-facor model are exponenials of he facors weighed by heir facor loadings. This allows for a simple calibraion of forward curves o he mare model and an inerpreaion of he facor values in erms of mean level and iniial slope of he curve. The applicabiliy of his model o a number of forward mares, as well as is simple analyical formulaion, maes i useful in differen valuaion seings involving commodiy prices. In Par 3 we address he pricing of Asian opions wrien on commodiy forwards. We show ha by undersanding he erm srucure of volailiy correcly, as well as he effec of he averaging on he volailiy of he payoff, Asian opions can be priced approximaely bu analyically in a simple way. Comparing our heoreical prices o mare prices, we find ha i correcly reproduces he erm srucure of implied volailiies. The undersanding of his should increase liquidiy in he freigh opions mare. The undersanding of volailiy and is value also has a profound impac on valuaion and operaional decisions ha involve commodiies. In Par 4 we sudy he floaing sorage rade involving crude oil and aners using he wo-facor model. This rade can be viewed as he sum of a cross-alanic and emporal arbirage rade arbiraging crude oil beween Europe and he Unied Saes and beween now and he fuure and of a sorage rade where he rader can choose he opimal ime o release he oil ino he mare. We show ha while he arbirage window has been open for mos of he ime during 7-9, he sorage rade has only exised in he second half of his period. The floaing sorage opporuniy is associaed wih a forward curve in conango when need of freigh coss. When i is open, here is addiional value involved in no selling he cargo immediaely and aing advanage of he possibiliy of higher prices. The framewor ha we presen allows us o evaluae he profis from such a sraegy, he decision rules for running he rade, and is exposure o he wo ris facors hrough hedge raios. The excess value is undersood as a combinaion of he drifs in he facors, of he payoff convexiy and of an early exercise premium. 98

99 . Suggesions for fuure research As perains o he mare modeling, an essenial improvemen ha is no performed in he presen hesis would be o allow he model o be easily calibraed o mare implied volailiies as well as he hisorical correlaion surface. The crude oil mare, for example, has a very liquid opions mare ha can be used for such calibraion. The mare modeling framewor presened in his hesis can be applied o a number of problems relaed o commodiy rading. I would be very ineresing o see empirical resuls for he general commodiy rading problem presened in Secion 4.8, and undersand wha value is associaed wih he possibiliy of swiching desinaions. I can also be applied o iquefied Naural Gas cargoes ha are currenly being reroued from heir long-erm conrac desinaions in he Unied Saes o Europe or Asia. Furhermore, his mare-based rouing problem should be inegraed wih he opimal weaher rouing problem developed in Avougleas and Sclavounos (9). An underlying assumpion in our formulaion is ha roues are deerminisic and fuel consumpion only depends on speed. In pracice, ship rouing and fuel consumpion depends srongly on weaher, and using forecass and dynamic programming one can deermine he opimal roue o follow. Inegraing his uncerainy wih our model would give a much more precise evaluaion of he commodiy rade, especially when profis come from geographical spreads and no floaing sorage. However, his general problem involves a large number of sae variables and is difficul o solve using dynamic programming. Developing soluion mehods adaped o such a large-scale problem would grealy enhance is applicabiliy. One promising mehod, applied for American opions, is he ongsaff and Schwarz () leas squares Mone Carlo mehod. This would require finding suiable basis funcions on which o projec he soluion. In his hesis we view he shipping problem from he poin of view of a physical oil rader who has he possibiliy of charering a ship for one rade, before reurning i o he mare. Anoher direcion would be o see he problem from he poin of view of a shipowner or long-erm charerer who can operae he ship coninuously on several rades. In ha case, he decision aen on one rade, such as soring oil, will have consequences for he nex one. In some cases i migh be more profiable o sell he oil, reurn o he loading por and ae advanage of a beer geographical spread. The same framewor can be used, bu he problem is of longer-erm naure, of years raher han wees. 99

100 6. APPENDIX. Traded volumes in commodiy derivaive mares From ICE (9) and CME (9): ICE NYMEX (Fuures) NYMEX (Opions) Conrac Daily volume ( bbl) Yearly volume ( bbl) Bren Crude Fuures 87,355 74,37,75 Bren Crude Opions 83,34 WTI Crude Fuures 79,8 46,393,67 WTI Crude Opions 7 8, Crude oil physical 545,35 4,79,6 Crude oil 4,5,75,46 Miny WTI 3,369,737,97 Bren Financial Fuures,997 59, Dubai Crude oil Calendar 3,665 95,9 WTI Calendar 4,98,9,48 Bren Calendar Opions 546 4,96 Bren las day 74 9,4 Crude oil mo,857 74,8 Crude oil APO,73 3,35,38 Crude oil physical 3,3 9,458,5 Toal,7,66 3,698,7 From Imarex (9). One lo is meric ons. Period # rades # los Dec Nov Oc Sep Aug Jul Jun May Apr Mar Feb Jan Toal

101 . Spo price process implied by he wo-facor model Using he forward curve process and he spo-forward relaionship S( ) = F(, ), we ge: Such ha: df(, T ) = e dw + dw = (, T ) dw + (, T ) dw F(, T ) α ( T ) S S S S log S( ) = log F(, ) S (, ) S (, ) S ( ) (, ) (, ) ( ) s ds + s dw s s ds s dw s + d S s ds dw s log F(, ) S ( s, ) S ( s, ) log ( ) = S (, ) S (, ) + S ( ) ( s, ) ( s, ) (, ) (, ) ( ) s ds + dw s d + (, ) dw ( ) + (, ) dw ( ) S S We have: S ( s, ) ( s, ) = α S ( s, ), = Such ha S ( s, ) S = α S S e: dw ( s) ( s, ) dw ( s) = α + + log S( ) log F(, ) S (, ) (, ) (, ) ( ) s ds s ds s dw s µ α α log F(, ) α ( ) = S (, ) + ( S ( s, ) ) ds + log F(, ) + ( W ( ) W ()) α

102 log F(, ) α S α µ ( ) = ( S + ) + ( e ) + α log F(, ) + α W ( ) α α log F(, ) S α = + log F(, ) ( + e ) ( + α) + W ( ) α 4α α Then: d log S( ) = α ( µ ( ) log S( )) d + dw + dw dµ ( ) = m( ) d + dw S S where: log F(, ) log F(, ) m( ) = + + α α ( Se ) 3. Principal Componens Analysis of he wo-facor model We wan o find he funcions eigenvalues u ha are eigenvecors of he covariance marix Σ ( τ, τ ) wih associaed λ. For his we mus choose some arbirary maximal enor T, and find eigenvalues eigenvecors u ( τ ) saisfying: λ and τ τ τ max max max Σ ( τ, τ ) u ( τ ) dτ = λ u ( τ ) u ( τ ) dτ = u ( τ ) u ( τ ) dτ = δ l l Given he parameric form of Σ( τ, τ ) we find ha τ max ρ ρ ρ τ τ λ τ ατ ατ ( Se + )( Se + ) + ( ) u ( ) d = u ( ) eaving ou he index and developing his equaion we find ha τ max τ max ατ ατ ατ = S + S + S + λu( τ ) ( e ρ ) u( τ ) dτ e ( ρ e ) u( τ ) dτ

103 ατ Thus we see ha u( τ ) can be wrien in he form u( τ ) = Ae + B where A and B are consans. We replace his expression ino he equaion o find ha: τ max τ max ατ ατ ατ ατ ατ Se S Ae B d e S e Ae B d λ ( τ ) = ( + ρ )( + ) τ + ( ρ + )( + ) τ Equaing he consan and exponenial erms we ge he marix eigenvecor equaion: A λ = B τ τ max max ατ ατ ατ ( Se + ρ S ) e dτ ( Se + ρ S ) dτ max max ατ ατ ατ ( ρ S e + ) e dτ ( ρ S e + ) dτ τ τ A B This shows ha λ is an eigenvalue and ( A, B ) an eigenvecor of he wo-dimensional marix M. Thus here are only wo disinc eigenfuncions u( τ ) and eigenvalues λ - as expeced for a wofacor model. 4. Evoluion of he consan-mauriy forward curve under he wo-facor model α ( ) If we le (, T ) u ( T ) ( A e T = = + B ), we have: log f (, τ ) = log F(, + τ ) = log F(, + τ ) + ( s, τ ) ds ( s, τ ) dw ( s) = e: (, τ ) = (, τ ) (, τ ) ( ) s s ds s dw s Expand he sochasic componen (dropping he index for now): e: ατ α ( s) ( s, + τ ) dw ( s) = Ae e dw ( s) + B dw ( s) ατ ( Ae α ( s) B) e dw ( s) B ( α ( s) e ) dw ( s) = + + 3

104 α ( s) ( ) = ( ) f e dw s α ( s) ( ) ( ) ( ) g = B e dw s (, ) = (, + ) ψ τ s τ ds Then s(, τ ) = ψ (, τ ) + g( ) + u( τ ) f ( ) Differeniae his df ( ) = α f ( ) d + dw ( ) dg( ) = B( dw + α f ( ) d dw ) = Bα f ( ) d dψ (, τ ) = (, + τ ) ( s, τ ) ( s, τ ) ds d µ (, τ ) d + + = T We recognize ha f ( ) is an Ornsein-Uhlenbec process mean-revering o, g( ) is an inegral of f and ψ (, τ ) is a deerminisic drif. We can calculae µ (, τ ) explicily (, ) = (, + ) (, ) (, ) + + T µ τ τ s τ s τ ds ατ α ( + τ s) α ( + τ s) ( ) α ( ) = A e + B + A e + B A e ds α τ α τ α ατ α = ( A e + B ) + A e ( e ) + A Be ( e ) α ( + τ ) α ( ) + τ = A e A B e B α ( ) + τ µ (, τ ) = ( A e + B ) = u ( + τ ) Thus he consan-mauriy fuures price can be wrien as: log f (, τ ) = log F(, + τ ) + ψ (, τ ) + ψ (, τ ) + g ( ) + g ( ) + u ( τ ) f ( ) + u ( τ ) f ( ) where: 4

105 df ( ) = α f ( ) d + dw ( ) dg ( ) = B α f ( ) d dψ (, τ ) = u ( + τ ) d 5. Impac of a hird facor on he consan-mauriy forward curve We have, as in Appendix 4, ha 3 log f (, τ ) = log F(, + τ ) = log F(, + τ ) + ( s, τ ) ds ( s, τ ) dw ( s) = e: (, τ ) = (, τ ) (, τ ) ( ) s s ds s dw s We consider only he hird facor and will assume = 3 in wha follows. e us consider firs he sochasic par: e: α ( + τ s) α ( + τ s) ( s, + ) dw ( s) = ( Ae + Be + C) dw ( s) τ α ( s) ατ α ( s) α ( s) ( ) ( ) ( ) ( ) = u τ e dw s + Be e e dw s + C e dw s α ( s) ( ) ( ) α ( s) α ( s) α ( s) α ( s) ( ) = ( ), ( ) = ( ) ( ), ( ) = ( ) ( ) f e dw s g e e dw s h e dw s Then df ( ) = α f ( ) + dw ( ) (Ornsein-Uhlenbec process) ( ) = α( ( ) ( )) ( ) = α α ( s) ( ) dg f g d g e f s ds dh( ) = α f ( ) d h( ) = α f ( s) ds 5

106 The process f ( ) is an Ornsein-Uhlenbec process mean-revering o zero wih mean-reversion speed α and volailiy. The processes g( ) and h( ) are sochasic drifs inegrals of f ( ) wih differen weighs. 6. Blac volailiies of he Average price conrac In-selemen We consider a dae TM < T M +. T T α ( T s) e T s T s A( s, T ) ds = S + ρ ( ) ' ' + ρ ' αc M cm cm T S α s ρ S α s s = ( e ) + ( e ) s + ' ' ' ds α c M αcm cm e us calculae each of he erms separaely: T α ( T ) S α s S α ( T ) e ( e ) ds = T ' ' ( e ) + α c M α cm α α T T α s T α s T ρ S α s ρ S s e e ( e ) sds = s ' ' + αc M αcm α α α ( T ) ρ S ( T ) ( T ) e = ' + αcm α α T s ( T ) c Such ha he square of he Blac volailiy is given by: 3 ' ds = ' M 3 cm ds α ( T ) ( e ) 6

107 T Blac (, T ) = A(, ) T α ( T ) α ( T ) S e e = ' + + α cm α T α T α ( T ) α ( T ) ρ S T e e ' + + αcm α α T ( T ) 3 c ' M s T ds e us consider he case when αc and simplify his expression Blac ρ (, T ) ( T ) ( T ) ( T ) S S + + ' ' ' 3 cm 3cM 3 cm T (, T ) + ρ + Blac ' S S 3 cm and c = T T T such ha ' M N M (, T ) + ρ + 3 Blac S S Pre-selemen T T A = A + Blac ( s, T ) ds ( s, T ) ds ( T T ) ( T, T ) The second erm is nown from he calculaions above. e us calculae he firs erm. s T ds e ds T T αc α ( T s) e A(, ) = S + ρ + ( ρ ) αc T αc T αc α ( T s) e e α ( T s) Se ds ρ S e ds ( T ) αc αc = + + αc α ( T T ) α ( T ) αc α ( T T ) α ( T ) e e e e e e S ρ S = + αc α αc α + ( T ) If we assume ha αc, and noicing ha T T = c 7

108 αc α ( T ) αc α ( T ) e e e e c Blac (, T ) S + ρ S + α( T ) α( T ) T c ( + S + ρ S + ) T 3 α ( T ) α ( T ) c e c e c Blac (, T ) S + + ρ S + + T α c 3 T αc 3 3 T We chec ha when = T (i.e he conrac eners selemen): and when T, ( Blac, T ) ρ ( T, T ) S Blac S 7. Semi-analyical soluion o he opimal sopping problem We begin by presening he analysis in he simple case of one facor. The coninuaion region is hen given by * * S f ( ) = (, ( )). The equaion saisfied by he value funcion V is V V V * + ( µ α f ) + g( ), f < f ( ) f f V f = Ω f f f * (, ) (, ), ( ) V ( T, f ) = Ω( T, f ) e such ha V f V f V = ( µ α f ) + g( ) < V + V = ψ (, f ), ψ (, f ) = Ω + Ω * f f ( ) * f f The ransiion densiy for he Ornsein-Uhlenbec process is ( ) 8

109 µ f fe e α p( f ', τ, f, ) = exp α ( τ ) / α ( τ ) e e π α α This ransiion densiy saisfies he equaion The value funcion can hen be wrien as p + ( µ α f ) p + p = f f α ( τ ) α ( τ ) ' + ( ) V (, f ) = p( f ', T; f, )( Ω( T, f ') G(, T )) df ' + p( f ', s; f, )( ψ ( s, f ')) df ' ds = V (, f ) + V (, f ) eur early where G(, T ) is he cos of shipping beween imes and T : T G(, T) g( s) ds = e us verify his resul by differeniaing he above formula. Consider firs he European value Veur (, f ) : T Veur p G = ( f ', T; f, )( Ω( T, f ') G(, T )) df ' p( f ', T; f, ) df ' and he early-exercise premium: V V = + f f ( µ eur eur α f ) g( ) + T + Vearly p( f ', s; f, ) = p( f ', ; f, ) ψ (, f ') df ' + ( ψ ( s, f ')) df ' ds T + p p = ψ (, f ) + ( µ α f ) ( ψ ( s, f ')) df ' ds f f V V = + f f early early ( µ α f ) ψ (, f ) Furhermore, V eur and V early saisfy he erminal condiions 9

110 V ( T, f ) = p( f ', T; f, T)( Ω( T, f ') G( T, T )) df ' = Ω( T, f ) eur T V ( T, f ) = p( f ', s; f, T )( ψ ( s, f ')) df ' ds = early T such ha V = Veur + Vearly solves he equaion. The early exercise premium can be wrien in erms of he sopping boundary as: T V (, f ) = p( f ', s; f, )( ψ ( s, f ')) df ' ds early f * ( s) This formulaion gives a closed form expression of V. However, i involves he values of s T. These are deermined by he coninuiy condiion f * ( s ) for Ω = * * (, f ( )) V (, f ( )) T * * ( ', ; ( ), )( (, ') (, )) ' ( ',, ( ), )( (, ')) ' = p f T f Ω T f G T df + p f s f ψ s f df ds * f ( s) The value funcion a mauriy and he sopping boundary f * ( ) can be deermined recursively as follows: discreize he daes as =,,..., N = T. If f ( ),..., f ( + ) have been calculaed, le * * N N * * * = Ω + j ψ j j= + * f ( j ) F( f ) p( f ', T, f, )( ( T, f ') G(, T )) df ' p( f ',, f, )( (, f ')) df ' + p f f ψ f df * f * ( ', ;, )( (, ')) ' Finding he sopping boundary f * ( ) a ime involves finding, numerically, { } f ( ) = min f, F( f ) = Ω (, f ) * * * * Once his sopping boundary has been locaed he value funcion can be calculaed for all f using V (, f ) = p( f ', T, f, τ )( Ω( T, f ') G(, T )) df ' + p( f ',, f, )( ψ (, f ')) df ' N j= * f ( j ) j j We can exend his analysis o wo facors, by using he fac ha hey are independen. The ransiion densiy funcion for he join Ornsein-Uhlenbec process is

111 p( f, τ ; f, ) = p ( f, τ; f, ) p ( f, τ ; f, ) ' ' ' where p and p are he ransiion densiies for he one-dimensional Ornsein-Uhlenbec processes. This wo-dimensional ransiion densiy funcion solves he parial differenial equaion The equaion for he value funcion is where p + ( µ α ) p + p + ( µ α ) p + p = f f f f f f V + V = ψ (, f ) ψ (, f ) = Ω + Ω * f S ( ) * f S V V V V V = ( µ α f ) + ( µ α f ) + + g( ) such ha he value funcion can be wrien as f f f f ( ) V (, f ) = p( f ', T; f, )( Ω( T, f ') G(, T )) df ' + p( f ', s; f, )( ψ ( s, f ')) df ' ds * R R \ S ( s) = V (, f ) + V (, f ) eur early T In he wo-dimensional case he coninuaion region S * ( ) is defined as { } * S = f f V f f > Ω f f ( ) (, ), (,, ) (,, ) The boundary second facor S * ( ) of his domain has o be deermined for each dae. We wrie i as a funcion of he { } S * ( ) = ( f * (, f ), f ), f R such ha he equaion o be solved by f (, f ) is *

112 V (, f (, f ), f ) = p( f f, T; f (, f ), f, )( Ω( T, f ') G(, T )) df ' * ' ' * R T p ( f, f, s; f (, f ), f, )( ψ ( s, f ')) df ' ds * f ( s, f ) = Ω(, f (, f ), f ) * ' ' * To find he funcion * j f (, f ) we proceed recursively as in he one-facor case. Having deermined * f (, f ) for j >, we calculae F( f, f ) = p( f, f, T; f, f, )( Ω( T, f ') G(, T )) df df * ' ' * ' ' R N + + p( f, f, ; f, f, )( ψ (, f ')) df df j= + * ' f ( j, f ) + τ ( ψ (, f, f )) * ' ' * ' ' j j and we find he exercise boundary by varying * f : { } f (, f ) = min f, F( f, f ) = Ω (, f, f ) * * * * Once his exercise boundary has been locaed he value funcion can be calculaed for all f using V (, f, f ) = p( f ', T; f, )( Ω( T, f ') G(, T )) df ' + τ p( f ', ; f, )( ψ (, f ')) df ' j j * R j= f ( j, f ) N +

113 8. Roues, cargoes and ships used in he floaing sorage rades Crude oil Sullom Voe OOP Roue Sullom Voe OOP Ship Disance d 4535 Nm Type Very arge Crude Carrier (VCC) Cargo Bren Cargo size 7 m Barrel facor bbl/m DWT 3 m oading por Sullom Voe Speed u 5 nos oading price S Daed Bren - days Fuel consumpion sailing: FC(u) anchor: FCa 87.5 m/day (laden) 74 m/day (ballas) 85 m/day (pumping) 5 m/day (anchor) oading delay τload 5 days Timecharer price H VCC average imecharer equivalen (Balic Exchange) IFO price B Fuel Oil 3.5% CIF NWE (Plas) Delivery por OOP Delivery price F(,τ) S forward curve Heaing oil ARA NYH Roue ARA NYH Ship Disance d 3383 Nm Type Very arge Crude Carrier (VCC) Cargo No. fuel oil Cargo size 7 m Barrel facor 3.63 gal/m DWT 3 m oading por Amserdam-Roerdam-Anwerp Speed u 5 nos oading price S ICE Gasoil fron monh price Fuel consumpion sailing: FC(u) anchor: FCa 87.5 m/day (laden) 74 m/day (ballas) 85 m/day (pumping) 5 m/day (anchor) oading delay τload 5 days Timecharer price H VCC average imecharer equivalen (Balic Exchange) IFO price B Fuel Oil 3.5% CIF NWE (Plas) Delivery por New Yor Harbor Delivery price F(,τ) Nymex heaing oil forward curve Corresponds o he modern double-hull VCC from Clarsons (9) 3

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