The Valuation of Petroleum Lease Contracts as Real Options

Transcription

1 The Valuaion of eroleum Lease Conracs as Real Opions J. Graham Weir Kellogg College Universiy of Oxford A hesis submied in parial fulfillmen of he requiremens for he degree of MSc in Mahemaical Finance Augus 4

2 Table of Conens Chaper Inroducing he eroleum Lease Valuaion roblem.... eroleum Leases and he need for heir Valuaion.... The roducion Cycle of eroleum Companies.... A efiniion of a Real Opion and is Applicaion o Valuing eroleum Leases The Course of his Invesigaion...5 Chaper A erivaion of he BSM E o Value Real Opions...7. Inroducion...7. Sick s erivaion of he BSM E for Real Opions...8 Chaper 3 A Sochasic rocess o Model Crude Oil rices Inroducion Aribues of an Appropriae Sochasic rocess The Aribues of he Inhomogeneous Geomeric Brownian Moion ( IGBM ) rocess Selecion of an Appropriae Sochasic rocess...5 Chaper 4 Esimaion of arameers Inroducion Time-Series Approach Calibraion Approach Comparison of Esimaes and Selecion of arameers... Chaper 5 The Valuaion of eroleum Reserves as erpeuiies Inroducion Valuaion of roducing eroleum Reserves IGBM rices Esimaes of he Volailiy of Reurns from Holding Reserves...7 Chaper 6 The Valuaion of eroleum Leases Inroducion Soluions of he BSM Equaion for an IGBM Moion Valuaion of a erpeual American Call Valuaion of a erpeual American u Valuaion of a lan wih a Level Exracion Rae and Opions on Same Valuaion of Syncrude Exracion, evelopmen, and Exploraion Opions Valuaion of an Opion on a rojec wih a eclining Exracion Rae Valuaion of a Convenional eroleum Lease...45 Chaper 7 Conclusions Conclusions from This Invesigaion...46 Appendix Comparison of Incremenal Assumpions o erive he BSM E...49 Appendix II Comparison of Single-Facor Sochasic rocesses...5 Appendix III Saisics for Spo Crude Oil Time-Series aa...53 Appendix IV Esimaes of arameers by Calibraing Calls on Crude Oil Fuures...55 Appendix V Valuaion of a roducing eroleum Reserve GBM rices...56 References...59

3 Lis of Tables Table 4. Esimaes of IGBM arameers, Time-Series Approach...7 Table 4. Esimaes of IGBM arameers, Calibraion Approach... Table 4.3 Implied Volailiy of Crude Oil Fuures A November 3, 3... Table 6. Free Boundary Condiions for erpeual us and Calls on an IGBM Asse...3 Table 6. As a Funcion of and I...4 Table 6.3 re-income Tax Valuaion of Syncrude Leases a Oc. 3, Table A-II- Comparison of Single-Facor Sochasic rocesses...5 Table A-III- Summary Saisics for Spo Crude Oil Time-Series...53 Table A-III- -Saisics for Regression arameers...54 Lis of Figures Figure 4. ercenile vs ercenile (Monhly rices)...8 Figure 4. Monhly Reurns from WTI vs Equiy...9 Figure 4.3 Acual Vs Model Fuures rices... Figure 6. Comparison of Call and u Valuaions...33 Figure 6. Comparison of Syncrude roducion Opion and erpeuiy Valuaions...4 Figure 6.3 Comparison of Valuaions of he Syncrude rojec, Tau...4 Figure 6.4 Call Value for One BBL per Annum of Syncrude rojec...4 Figure 6.5 Value of Convenional evelopmen Opion...45 Auheniciy Saemen I hereby declare ha his disseraion is my own work and confirm is auheniciy. Name: Address: Signed: J. Graham Weir 694 Lamon Cour S.W., Calgary, AB T3E 6G5 Canada ae: Augus 3, 4

4 Symbol Glossary Uppercase Roman Leers Uppercase Greek Leers A, B Consans of inegraion in soluions of differenial equaion refix for finie bu small incremen (e.g., C Operaing cos of a projec Γ Gamma funcion F Value of he opporuniy o inves; value of a Fuures Conrac ϕ ayoff funcion G Value of an opion on an opion I Capial cos of invesmen Lowercase Greek Leers M, U Kummer Funcions α rif parameer of simple Brownian moion, or proporional growh M Minus, marginal uiliy of consumpion α, γ, ( x y) ) rae parameer of geomeric Brownian moion, Consans (Chaper 6) N, A normal disribuion wih mean x and variance y Variable in fundamenal quadraic. Is posiive and negaive roos are denoed by and, respecively Oupu price, crude oil ε Random variable disribued N(,) Mean reversion value of δ Rae of reurn shorfall or convenience yield Q Quaniy of resources or reserves η Mean-reversion rae R Revenue from a projec µ Risk-adjused (CAM) discoun rae S Scale funcion π rofi flow or saionary densiy (Chaper 3) T Terminal ime φ Marke price of risk V Value of asse in place ρ Correlaion coefficien X Value of uncerain asse a ime σ Variance parameer in Brownian moion τ One minus ax rae Lowercase Roman Leers ω The rae of decline in he oupu rae of a projec a, b Coefficiens of diffusion process d Infiniesimal incremen prefix (e.g. he differenial d ) Momen Operaors dz Incremen of sandard Wiener (Brownian moion) process ( ) q The oupu rae of a projec ( ) r Risk-free ineres rae ( ) Curren ime ( ) Adaped From ixi &indyck (994) Ε Expecaion operaor Cov Covariance operaor Sd Sandard deviaion operaor Var Variance operaor

5 Chaper Inroducing he eroleum Lease Valuaion roblem. eroleum Leases and he need for heir Valuaion eroleum companies acquire he righ o produce peroleum by means of several ypes of conracual arrangemens wih he owners of peroleum righs. A common ype of hese conracs is referred o as a peroleum lease. ursuan o he erms of a peroleum lease, an owner of peroleum righs grans a lessee he exclusive righs wihin a defined volume of he earh s subsurface o: () explore for peroleum during an iniial period of ime, referred o as he primary erm; and () develop and exrac peroleum during subsequen periods of ime, referred o as renewal erms. The enure of he primary erm is, ypically, five years for onshore leases and eigh o en years for offshore leases. If economic quaniies of peroleum resources have been discovered, he lessee may elec successive renewal erms o develop and exrac he peroleum. The consideraion paid by he lessee o he owner of he peroleum righs for a lease comprises: an iniial cash paymen; annual renal paymens; and a consan, or variable, percenage of he peroleum exraced from he leased volume of he subsurface, referred o as a royaly paymen. Normally, he renal amoun and he royaly percenage are seleced by he owner of he peroleum righs prior o offering he lease o prospecive purchasers. Compeiive negoiaions or enders of sealed bids in a compeiive aucion deermine he magniude of he iniial paymen. In eiher case, peroleum companies require a mehodology o deermine he fair value of a given peroleum lease. The valuaion of peroleum leases using cerain mahemaical finance echniques is he subjec of his hesis. This chaper opens wih an overview of he producion cycle followed by peroleum companies o explore for, develop and exrac peroleum. A definiion of a real opion and a framework for casing invesmen decisions by peroleum companies as real opion valuaions follows. A lis of he mahemaical finance echniques which will be developed in he subsequen chapers o value peroleum leases as real opions closes he chaper.. The roducion Cycle of eroleum Companies eroleum companies expend quaniies of capial o produce peroleum, via a sequenial producion process comprised of five phases. The aciviies comprising each phase follow, in chronological order.

6 rospec Generaion hase: The producion cycle begins wih geoscieniss prospecing specific geographic areas where he subsurface may conain accumulaions of peroleum subsances. Acquiring he exclusive righ o explore hese prospecs for peroleum, by means of peroleum leases, or oher conracs, complees he firs phase. Exploraion hase: rospecs are explored for peroleum resources in he second phase by conducing geophysical sudies, drilling exploraion wells, logging he resulan well bores and esing for he presence of hydrocarbons. The collecive cos of he exploraion aciviies, referred o as exploraion cos, end o be large. Typical exploraion coss for onshore and offshore prospecs would be on he order of magniude of $ o $ million and $ million, respecively. If he aciviies in he exploraion phase fail o discover resources, he case more ofen han no, a peroleum lease can be relinquished o he owner of he peroleum righs and he prospec abandoned. Alernaively, if a peroleum reservoir conaining sufficien resources has been discovered, hen a peroleum company owns he righ, wihou he obligaion, o develop he discovered resources ino reserves. evelopmen hase: The developmen phase comprises: he drilling, compleing, and equipping of developmen wells; he consrucion of gahering pipelines; and he fabricaion and insallaion of processing plans. The cos of developmen is he sum of he coss of he developmen aciviies and may be an order of magniude greaer han he cos of exploraion. A peroleum company holding developed reserves has he righ o elec o iniiae exracion, pursuan o he erms of is peroleum lease. Exracion hase: Exracing peroleum from he subsurface involves: lifing; gahering and processing; markeing; and paying he requisie royalies and axes. Generally, he collecive coss of exracion can be allocaed as fixed or variable, on he basis of 8% and %, respecively. As he exracion phase proceeds, he wihdrawal of volumes of peroleum from he subsurface reservoir will be accompanied, in mos cases, by a decline in reservoir pressure causing he rae a which he peroleum is produced o decline. The onse and rae of decline can, o some exen, be miigaed by he iniiaion of reservoir pressure mainenance and enhanced recovery schemes. Ineviably hough, he rae of peroleum producion will fall below ha necessary o recover boh fixed and variable exracion coss. Abandonmen hase: When he economic limi of he exracion phase is reached, abandonmen and reclamaion occur. Ne of salvage, expendiures in he abandonmen phase have, hisorically, been small relaive o hose in he preceding phases. The lengh of ime required o plan and implemen each phase runs o years for onshore, and decades for offshore peroleum projecs. Cumulaively, he ime required o complee he producion cycle can be a full generaion in human erms. Throughou he producion cycle, uncerainies from hree principal sources will challenge peroleum companies. Geologic risks, including: () he absence of a reservoir; and () an insufficien quaniy of peroleum, if any, recoverable from he reservoir, are significan in he exploraion and developmen phases. The produciviy of he reservoir will be definiively revealed, for he mos par, in he exracion phase. Technical risks because of logisical problems on he surface and unanicipaed condiions in he subsurface, in erms of rock hardness or pressure levels, can cause he coss of exploraion and developmen operaions o exceed budgeed levels. As a peroleum company explores - and

7 perhaps subsequenly develops a prospec - geological and echnical informaion is revealed ha reduces he levels of geologic and echnical risk. The volailiy of peroleum prices, in conras, conribues a high level of uncerainy ha pervades all phases of he producion cycle. In making capial budgeing decisions, peroleum companies primary valuaion meric has been he ne presen value ( NV ) rule. ursuan o he NV rule, an affirmaive decision o underake a projec is made if and only if ( IFF ) he presen value of he expeced fuure ne cash flows from he projec, discouned a a rae ha reflecs he sysemaic risk of he projec, is greaer han he capial cos of he projec. Implici in he NV rule are he assumpions ha: () all cash flows will happen exacly a he imes and in he amouns prescribed by he cash flow forecass; and () all invesmens are irreversible. The firs assumpion precludes managers from using new informaion, as i arrives, o revise heir sraegy. The sequenial, muliphase composiion of he peroleum companies producion cycle means here are many decision poins where managers can uilize he informaion learned in he previous and curren phases o selec an opimum course of acion for he nex phase. The assumpions of he NV rule and he sequenial, muliphase srucure of he producion cycle are no consisen. In he laer phases of he peroleum producion cycle, where mos of he informaion ha can arrive, has arrived, managerial flexibiliy is less valuable. uring he exploraion and developmen phases, when relaively lile informaion regarding geologic and echnical risks is available, flexibiliy is more valuable. The proper valuaion of his flexibiliy may provide a peroleum company wih he advanage i needs o succeed in he highly compeiive marke for peroleum leases. Is here anoher valuaion approach whose formulaion and underlying assumpions are more aligned wih he muliphase order of he peroleum producion cycle? An alernaive o he NV rule is o value a projec as a real opion and accep he projec IFF is value is greaer han is cos.. A efiniion of a Real Opion and is Applicaion o Valuing eroleum Leases Seppi () defines a real opion by firs considering how o define a commodiy. In his consrucion, a commodiy is formed from hree aribues: he naure of a good, G; he ime when i is presen, ; and he locaion where i is available, L, denoed by {G,, L}. This definiion enables commodiies o be a flow during a ime period or a sock a a specific poin in ime. Consider he example of he commodiy known as Wes Texas Inermediae ( WTI ) crude oil, which is defined as a good having a sulfur conen of less han.4% and a graviy of approximaely 38 degrees AI and is priced for delivery a Cushing OK. A real opion, hen according o Seppi, is defined as a echnology o physically conver one or more inpu commodiies {G,, L} ino an oupu commodiy {G,, L }. Many real opions can be idenified in he peroleum indusry by he applicaion of Seppi s definiion, including he following. The Syncrude rojec, which upgrades biumen o WTI crude oil, is a goods conversion opion. A pipeline ha ranspors crude oil from he field gae o a refinery represens an opion o change locaions. A crude oil sorage erminal is an opion o change he ime crude oil is available. Mos imporan of all, he phases of peroleum cycle represen goods conversion opions, as is shown below. 3

8 Exracion Opion: Exracion is he means o conver reserves ino above ground barrels of crude oil. A holder of peroleum reserves owns hree opions, subjec o he erms of is lease. The firs is he exracion opion. When peroleum revenues - he produc of he price of crude oil and he volume exraced - exceed he cos, exracion will occur. In his sae, he holder of he reserves receives he cash flow and reains a righ o suspend producion. Conversely, if peroleum revenues fall below lifing coss, he holder of he reserves can suspend exracion, and reains he righ o wai for higher crude oil prices in he fuure. In effec, he holder of shu-in reserves has an ou-of-he-money call on he price of crude oil. If reservoir condiions permi, a some high enough price of crude oil he holder may wish o exercise is second opion: o enhance he recovery of he reserves. A some very low crude oil price, a peroleum company may exercise he hird opion, when he expeced fuure ne cash ouflows will be greaer han he cos of abandonmen, by abandoning he reserves and reclaiming he surface sie. The value of he opion o exrac peroleum, denoed by e, is a funcion of hree sochasic sae variables wih heir associaed risk facors: he price of crude oil, () and he volailiy of crude oil prices, given he informaion available during he exracion phase, σ p (e); he quaniy of recoverable reserves, Q() and he volailiy of geological risk, given he informaion available in he exracion phase, σ q (e); he cos of exracion, C e () and he volailiy of exracion cos, σ c (e). The moions of he hree variables are specified by he sochasic differenial equaions ( SE ) d dce () α () e d σ () e dz() dq( ) α ( e) d σ ( e) dz( ) p () α () e d σ () e dz() c c. p The lengh of ime exracion proceeds is T e and he ime value of money is r. Symbolically, e e[ (), Q(), Ce () ; σ p ( e), σ q ( e), σ c( e), Te, r]. The quaniy of informaion revealed by exploraion and developmen regarding Q() and C e () will be large, meaning σ q (e) and σ c (e) will be small. The uncerain pah followed by crude oil prices will be he dominan sochasic variable in he valuaion of developed reserves. evelopmen Opion: The echnology used o conver resources ino reserves is ermed developmen. The righ o develop resources ino reserves, wihou he obligaion o do so, is a real opion, he value of which accrues o a holder of a resource. The holder of a resource will develop i when he value of a producing reserve, given by e, exceeds he cos of developmen, C d (). The developmen opion will have he boundary condiion Max[ e( ) Cd ( ), ] where is he opimal ime o develop he resource. The value of he developmen opion, denoed by d, will be given by he funcion: d d[ e, Cd ( ) ; σ e( d ), σ cd ( d ), Td, r] where T d is he renewal erm of he lease. Bu e is a funcion of, Q and C e, so: d d[, Q, Cd, Ce; σ p ( d ), σ q( d ), σ cd ( d ), σ ce ( d ), Td, Te, r] where σ q( d ) is he risk associaed wih Q based on he informaion available prior o developmen, σ cd ( d ) is he degree of uncerainy in he cos of developmen. Noe ha he risk facors for he quaniy of reserves, σ q( d ) > σ q( e), and he cos of exracion, σ ce ( d ) > σ ce ( e), are greaer in he developmen phase han he exracion phase. The volailiy of peroleum prices will coninue o affec he moion of he value of he underlying reserves. q q 4

9 Exploraion Opion: Exploraion is he echnology which convers prospecs ino peroleum resources. I follows ha a peroleum lease, which affords he holder he exclusive righ, wihou obligaion, o drill an exploraion well a any ime during he primary erm, can be valued as an American call opion ha will be referred o as he exploraion opion. The exploraion opion will be exercised when he value of he developmen opion, given by d, exceeds he exploraion coss, denoed by C x ( ). The boundary condiion for he exploraion opion will be Max[ d( ) Cx( ), ] where is he opimal ime o explore. The value of he exploraion opion is a funcion of he sae variables d and Cx wih associaed risk parameers σ d ( x) and σ c x ( x), x x[ d, Cx ; σ d ( x), σ cx ( x), Tx, r] where he enure of he primary erm of he lease is T x. However, d is a funcion of e and C d. Likewise, e is a funcion of, Q and C e. I follows ha x is a funcion of hese variables and parameers as well. So x x [, Q, C, C, C ; σ ( x), σ ( x), σ ( x), σ ( x), σ ( x), T, T, T r ] x d e p q cx cd ce x d e, Geologic risk, represened by σ q ( x) exploraion wells are abandoned han compleed. The quaniy of echnical risk, σ ( x), is significan in he exploraion phase, as more, will also be large due he uncerain cos of drilling rock no previously peneraed, in a remoe locaion. As he producion cycle proceeds, he geologic and echnical informaion revealed in he exploraion and developmen phases will significanly reduce he quaniy of uncerainy associaed wih Q and C, symbolically, σ q( x) > σ q( d ) > σ q ( q) and σ cx > σcd > σ ce. The opion of deferring exploraion or developmen unil some of he uncerainy can be resolved, say by compeiors drilling in same he area, will be valuable. The foregoing discussion has esablished he sequenial arrangemen of he real opions o explore, develop and exrac peroleum. Consequenly, a peroleum lease can be valued as a hree sage compound opion, subjec o hree sources of uncerainy. The valuaion of a peroleum lease is furher complicaed since he wo variables Q and C, represening he sae of he geology and echnical cos, are no raded and priced in capial markes. To simplify he valuaion of a peroleum lease i is assumed ha: () he variables Q and C are deerminisic; () he price of crude oil, which conribues a significan quaniy of uncerainy in all phases of he producion cycle, can be modeled by a single-facor sochasic process; and (3) consisen wih he long ime frames required o complee exploraion, developmen and exracion programs, he opions comprising he producion cycle have infinie lives.. c x.3 The Course of his Invesigaion To value a peroleum lease as a compound real opion, a number of mahemaical echniques are needed, including he following. () A mehod of deriving he Black-Scholes-Meron parial differenial equaion ( BSM E ) is developed in Chaper, consisen wih he fac ha while barrels of crude oil 5

10 on he surface are raded in a coninuous commodiy marke, barrels of resources and reserves in he subsurface are no. () A sochasic process, he aribues of which enable i o model crude oil prices, is seleced in Chaper 3, afer consideraion of six single-facor candidae processes. (3) Mehods o esimae he parameers of he sochasic process used o model crude oil prices are developed in Chaper 4 by employing wo approaches. In he firs, a regression equaion specific o he sochasic process seleced in Chaper 3 is derived and applied o a ime-series of spo crude oil prices. The second approach consiss of deriving he value of a fuures conrac for crude oil and using i o calibrae he parameers of he seleced sochasic process o he fuures marke. (4) The valuaions of cerain perpeuiies, where he cash flow is a funcion of he price of crude oil modeled by he sochasic processes, are derived in Chaper 5. The perpeuiy valuaions are used o demonsrae an imporan difference beween he sochasic behavior of barrels of crude oil on he surface and in he subsurface and as paricular soluions o he ordinary differenial equaions ( OE ) derived in Chaper 6. (5) erivaions of he BSM E using he seleced single-facor process are developed in Chaper 6 which when solved as OE s by repeaedly applying he boundary condiions known as value-maching and smooh-pasing value he compound real opions inheren in he peroleum producion cycle. Finally, in Chaper 7, cerain conclusions are presened. 6

11 Chaper A erivaion of he BSM E o Value Real Opions. Inroducion This chaper conains a derivaion of he BSM E appropriae o value a real opion. The valuaion of a real opion differs fundamenally from ha of a financial opion. Cerain of he assumpions ha underpin he derivaion of he BSM E o value financial opions canno be relied upon when valuing real opions. The assumpion ha he underlying asse is coninuously raded and can be used o hedge he real opion, is no consisen wih he valuaion problem described in Secion.. To idenify an appropriae derivaion of he BSM E, five alernae derivaions were reviewed, considering boh he number of assumpions required and he consisency of he assumpions wih respec o he valuaion problem. The assumpions necessary o obain a derivaion of he BSM E can be allocaed ino wo broad classes: hose common o all he derivaions reviewed; and he incremenal assumpions specific o each derivaion. Le he coningen claim o be valued, represening a firm or projec, be a funcion of he uncerain oupu price received, (), and ime,, and denoed by F(,). All of he derivaion mehods considered herein are based on hree common assumpions: () The underlying asse, (), follows a diffusion process specified by he SE, d( ) α (, ) d σ (, ) dz( ) (.) where: (a) α (,) is a deerminisic funcion ha specifies he expeced insananeous growh rae of, in dollars per uni-ime; (b) σ (,) is a deerminisic funcion ha specifies he annualized sandard deviaion of he reurns from ; and (c) dz () is he incremen of a Wiener process wih zero drif and uni variance per uni of ime. () F(,) is a leas wice differeniable, so ha Iô s Lemma can validly be applied o F. (3) The risk-free ineres rae, r, is known and consan across ime. Wha incremenal assumpions are necessary o complee he derivaion of he BSM E? The answer depends on he roue followed o derive he BSM E. The principal incremenal assumpions of four of he five mehods: ela Hedging, Replicaion, Spanning Asses, and ynamic rogramming, were found o be inconsisen wih he valuaion of real opions. This discussion is conained in Appendix I. The fifh mehod, he equilibrium mehod due o Sick (995), based on he Consumpion Capial Asse ricing Model ( CAM ), is parsimonious in erms of he number of incremenal assumpions required and heir consisency wih real opion valuaions. Sick s derivaion of he BSM E is exposied below. 7

12 . Sick s erivaion of he BSM E for Real Opions According o Sick (995), he discree ime version of he Consumpion CAM assers ha he value a ime of an uncerain asse X ~ ~ a ime, denoed by V [ X ], is a funcion of: he risk-free rae, r ; he ime,, expeced value of he asse a ime ; and he covariance of he asse price wih minus he ime marginal uiliy of consumpion a, ~ denoed by M,. This can be represened as ~ ~ ~ ~ [ X ]( r) E [ X ] Cov [ X, M ] V, (.) In (.) le: ~ ~, X δ ( ), ~ [ X ] V, ~ ~, and, where δ(,) is a deerminisic funcion ha specifies he rae, in dollars per uni ime, of he ~ income paid o he holder of he underlying asse,. Replace r wih r and M, wih M ~, o allow o approach zero, hen ~ ~ ~ [ ] Cov [ δ (, ), M ] ( r ) E (, ) δ. (.3) Since δ(,) and are known a ime hey can be reaed as consans. For random variables X and Y any consan b, Cov(Xb,Y) Cov(X,Y). Then δ (, ) can be removed and subraced from inside he covariance in (.3) wihou effec giving (.4), E ~ ~ ~ [ (, ) ] r Cov [, M ] δ. (.4) Taking he limi as d of (.4) yields [ d (, ) d] rd Cov [ d dm ] E δ,. (.5) The moions of and M are diffusions. Assume M follows he diffusion process ( M, ) d σ M ( M ) dzm dm α M,. (.6) Subsiuing (.) and (.6) in (.5) and using Cov ( ax b, cy d ) ac Cov( X, Y ) [ d δ (, ) d] rd σ (, ) σ ( M, ) Cov( dz ) E,. M dz M, hen The sandard deviaion of boh he Brownian Moions dz and dz M is d. Since Cov ( X, Y ) σ X σy ρ X, Y hen Cov( dz, dz M ) ρz, Z M d, where ρ Z,Z M is he correlaion coefficien beween dz and dz M, or beween reurns on M and. So E [ d δ (, ) d] rd σ (, ) σ ( M, ) ρ d. (.7) M Z, Z M 8

13 The holder of expecs o receive capial gains a he rae α(,) and income a he rae δ(,) so over ime d, [ d δ (, ) d] α(, ) d δ ( )d E,. (.8) Equaing (.7) and (.8) and eliminaing d yields (, ) δ (, ) r σ (, ) σ M ( M, ) ρ Z, Z M α. (.9) This, (.9), is he oal reurn equaion. I says rae of capial gain plus income equals he risk free reurn plus a risk premium. Repeaing he foregoing seps, (.) o (.8), leads o, [ df (, ) d] rfd Cov [ df dm ] E π,, (.) an expression for he expeced capial gain, df, plus income, denoed by he deerminisic funcion π(,), from F(,). F(,) was assumed o be a leas wice differeniable, in is firs argumen and differeniable in is second argumen, allowing an expansion of df by Iô s Lemma as follows, df F σ (, ) F α(, ) F d (, ) F dz σ. (.) Subsiue (.) on boh he RHS and LHS of (.), and he diffusion assumpion for dm, eliminae d from boh sides, and collec like erms o arrive a [ ρ ] F rf π (, ) F σ (, ) F α(, ) σ (, ) σ M (, ), M. (.) In (.9) i was shown: α(, ) σ (, ) σ (, ) ρ, r δ ( ) M M, F (, ) [ (, )] (, ) σ F r δ F rf π. (.3) If he risk facor for he marginal uiliy of consumpion is α r r r σ M σ σ M M ( M, ) φ M M which is he marke price of risk, hen (.) can be wrien as [ φρ ] F rf π (, ) F σ (, ) F α(, ) σ (, ), M. (.4) This general form of he BSM E was obained wih no reliance on assumpions of coninuous rading and hedging of he underlying asse. Furhermore, (.4) is in a general enough form o allow he use of he mos appropriae diffusion process o model he behavior of. The consideraion of cerain diffusion processes wih a view o selecing one o model crude oil prices is he subjec of he nex chaper., so 9

14 Chaper 3 A Sochasic rocess o Model Crude Oil rices 3. Inroducion In he previous chaper he deerminisic funcions ha define he moion of he underlying asse, or oupu price, via he diffusion process (.), were lef undefined. The ask in his chaper is o define funcions for he drif and diffusion erms in (.) appropriae o model he price of crude oil. The chaper opens wih a discussion of he aribues of an appropriae sochasic process followed by an examinaion of he behavior of a process ha mees he necessary crieria. A comparison of six sochasic processes and he selecion of an appropriae process close he chaper. 3. Aribues of an Appropriae Sochasic rocess Robel () posis ha o model commodiy prices usefully, a sochasic process, { () }, mus exhibi hree characerisics: () generaes posiive values, () >, for all ; () revers o a mean value,, over ime; and (3) if he process involves more han one uni of he commodiy, revers o he number of unis of he commodiy imes he reversion price of one uni. To Robel s lis a fourh characerisic is added: he propery ha as, () is no araced o eiher of he boundaries, or. The economic necessiy and mahemaical definiion of each of hese characerisics are discussed below. osiive rices: Commodiy prices mus be greaer han zero, oherwise suppliers would have no moivaion o vend. To demonsrae ha a paricular process will generae only posiive values, wo seps were uilized. Firs, a soluion of he SE ha defines he process was found. The soluion of he general linear single-facor SE is given in Klebaner (998) p. -3. Second, he range of he soluion funcion, (), was shown o be posiive over he funcion s domain. Reversion o a Mean: Mean reversion is he endency of a random variable if i is above (below) some normal level o drif down (up) over ime, owards he normal level. There are economic argumens ha commodiy prices should exhibi mean reversion. In a compeiive marke, wih no barriers o enry or exi, if a commodiy price rises above an equilibrium price, new supply will come forward and demand will fall, drawing he marke price back o he normal level. Conversely, supply will decrease and demand increase if he price of a

15 commodiy falls below an equilibrium level. In he long run, a commodiy s price should rever o is supply cos because of compeiion. In a marke where pure compeiion does no exis, he case for he reversion of prices is sronger. For he commodiy crude oil, a carel of suppliers, known as he Organizaion of eroleum Exporing Counries ( OEC ), has saed ha i will adjus he supply of crude oil o obain a price objecive beween $ and $8 per barrel ( BBL ), for he OEC baske marker crude. The saed inenions of OEC augmen he case for mean revering behavior in he price pah of crude oil, bu in no way guaranee i. In boh of he ersian Gulf Wars, 99 and 3, he price of crude oil spiked up, o over $4 per BBL. The opposie occurred in 998 when over producion by cerain members of OEC caused prices o fall o $ per BBL, briefly. To prove a given sochasic process revers o a mean value, as, wo approaches are employed. In he firs, an expression for he firs momen of he process is found, if i exiss and he limi aken as. The second roue requires he derivaion of: he process saionary densiy, is probabiliy densiy a, and he firs momen of he saionary densiy. Applicaion of boh approaches may be necessary, because no all processes have boh a firs momen and a saionary densiy. The saionary densiy is defined in Klebaner (998) p Homogeneiy Condiion: This characerisic will be required if he underlying asse is he produc of price and quaniy, so ha he price of n BBL s will rever o n. Robel () defines he homogeneiy condiion for an SE of he form () () [, ] d σ [ () ] dz( ) d α,. Then for any a >, he moion of he process Y() a() should be deermined by () Y () [, Y ] d σ [ Y () Y ] dz( ) dy α,, where Y a. The homogeneiy condiion will be saisfied if he drif and volailiy funcions are homogeneous funcions of degree one of he pair { ( ) },. Boundary Behavior: In he long run, commodiy prices end neiher o zero, nor o infiniy. This implies ha o realisically model commodiy prices, a sochasic process canno be araced o eiher of he boundaries, or. The behavior of a sochasic process a a boundary, wheher i is araced or refleced, is examined by deermining he convergence or divergence, respecively, of he scale funcion a he boundary. The inegral ha defines he scale funcion can be found in Klebaner (998) p The Aribues of he Inhomogeneous Geomeric Brownian Moion ( IGBM ) rocess The IGBM process is defined by he SE, (3.), wih ( ) p, d [ ] d σ () dz( ) () () η. (3.)

16 In (3.) η is he speed, or srengh of reversion o and σ is he volailiy of he diffusion. The IGBM process is sudied o see if i posses he four characerisics necessary o model commodiy prices, below. osiive rices: The direc applicaion of he general soluion for linear SE s provides he soluion of (3.) η ds. (3.) () σ () σ exp σ Z p η exp η s σ Z () s Since exp( x) for all x, hen ( ) > for all. Reversion o a Mean: The firs momen is found by aking he expecaion a on boh sides of (3.) and applying Fubini s Theorem o he LHS of (3.) d d E [ () ] η E[ () ] η. (3.3) Equaion (3.3) is a firs order, non-homogeneous OE in [ ( ) ] inegraing facor exp ( η) as follows, { e se[ ( s) ]} eηsds E ha can be solved wih he d η η. (3.4) Compleing he inegraion and subsiuing [ ( ) ] p E η [ () ] ( p ) e E in (3.4), resuls in he firs momen,. (3.5) Taking he limi of (3.5) as proves ( ) revers o. A second mehod o show ha () revers o involves finding he saionary densiy of ( ), denoed by π ( p), and deriving is firs momen. The saionary densiy of he diffusion (.5) and he IGBM is π ( p) ( y) ( y) C α exp ( ) dy σ p σ η C σ p σ exp η σ p (3.6) since for he IGBM process α ( y) η( y) and ( y) σ y σ. The consan C is deermined by seing he inegral of (3.6) from o equal o one. The IGBM s saionary densiy is η η η η σ ( ) σ σ p η π p p e / Γ. (3.7) σ σ In (3.7) Γ( X ) denoes he gamma funcion. The firs momen of ( p) π is defined as

17 E η η η η η σ σ σ ( p) pπ ( p) dp p σ e σ dp / Γ (3.8) η η η E σ σ σ ( p) Γ / Γ. Again, he IGBM process is shown o rever o as. Homogeneiy Condiion: Le () a( ) Y for a >, hen from (3.) ( Y (), Y ) η ( Y Y () ) η ( a a( ) ) aα( ( ), ) α and ( Y (), Y ) σy () aσ() aσ ( ( ), ) σ. The IGBM process is homogeneous of he pair { ( ) } condiion., saisfying he homogeneiy Boundary Behavior: The scale densiy for he diffusion process (.5) and he IGBM is α S ( p) exp σ ( y) ( y) dy η σ p η exp. (3.9) σ p The inegral of (3.9) yields he scale funcion for he IGBM S η η exp σ ( p) S ( y) dy yσ y dy. (3.) A he boundary exp η / σ will dominae he inegral and he limi of exp[ η / σ ] as will equal if η / σ >, so he inegral, S ( ) will no converge and he IGBM diffusion will no reach. Similarly, a he boundary, he erm η / σ he erm [ ] / will dominae he inegral and he limi η σ, if η / σ >. The inegral S ( ) will diverge and he IGBM diffusion will no reach. While possessed of he preceding four posiive aribues, he IGBM process does have one derimenal feaure. There are resricions on he values he IGBM s parameers, η and σ, can ake on, les he variance of he process become infinie! The deerminaion of he parameer resricions begins by deriving an expression for he second momen of he saionary densiy. E ( p ) p π ( p) dp η η η η σ η p σ e σ p dp / Γ σ σ 3

18 E η η η ( p ) Γ / Γ σ σ Var ( ) ( σ ) / ( η σ ) σ η η σ p (3.) If η σ hen he denominaor in (3.) will be zero and he variance infinie. The variance mus be posiive, implying η > σ, for η >. The resricions on he parameers of he IGBM process are explored furher by deriving is second momen. Applying Iô s Lemma o () shows d [ () ] () η () σ ( ) [ η ] d σ( ) dz. (3.) Again, on boh sides of (3.) he expecaion is aken a and Fubini s Theorem applied on he LHS, yielding d d [ () ] ( σ ) E () E [ ] η E () [ ] η. (3.3) [ () ] can be solved using he inegraing facor exp[ ( σ ) ] E is known from (3.5). As a resul (3.3) is a firs order, non-homogeneous OE ha η, [ ] { } d{ e( η σ ) S E () s } η e( η σ ) sds ( p ) e( ησ ) sds. (3.4) The inegrals on boh sides of (3.4) depend on he erms in he exponenial funcions. Variaions in he values of he exponens resuls in hree differen cases for he second momen. Case : If η σ, hen (3.4) becomes E E { e s ds } [ ) ] p η ds ( p ) ( η [ ( ) ] p ( p )( e η ) η. (3.5) The limi as of (3.5) will be because of he in he second erm on he RHS. Case : If η σ, hen (3.4) becomes { ds } [ () ] p eσ sds ( p ) σ E σ e E σ σ σ [ () ] p e ( e ) σ ( p ) e. (3.6) 4

19 The limi as of (3.6) is. While finie, he variance of he IGBM process in Case may be orders of magniude larger han σ, for he values associaed wih mos commodiy prices,. Var[( )] >> σ. Case 3: If neiher η σ nor η σ, hen (3.4) will be e ( σ ) E () E { sds } [ ] p e( η σ ) sds ( p ) e( η ) η η σ [ () ] e ( η σ ) ( η σ ) [ ( ) ] ( ) η σ η η e p e e ( η ) η σ [ ] η σ The limi of (3.7) as [ ( ) ] ( σ ) / ( η σ ) is η / ( η σ ). (3.7). I follows ha Var. Noe his expression for he variance of he IGBM process is he same as he variance of he saionary densiy, (3.), since boh are aken a. To summarize, he foregoing has esablished ha in order ha he variance of he IGBM process be finie and posiive he parameers of he process are subjec o he resricions η σ, η σ and η > σ. 3.3 Selecion of an Appropriae Sochasic rocess The mahemaical echniques employed above o discern he characerisics of he IGBM process were uilized o review he aribues of five oher single-facor sochasic processes: () Geomeric Brownian Moion ( GBM ), d α d σdz ; () Ornsein-Uhlenbeck ( OU ), d η ( ) d σdz ; (3) Exponenial OU, d η [ ln( ) σ / η ln( ) ] d σdz (4) Cox-Ingersoll-Ross ( CIR ), d η ( ) d σ / dz ; and (5) Sochasic Logisic Verhuls ( SLV ), d η ( ) d σdz. The resuls of he review are discussed and summarized in Appendix II. Of he six, singlefacor sochasic processes reviewed, only he IGBM process has all he aribues lised in Secion 3. necessary o model commodiy prices. The IGBM process: () generaes sricly posiive values; () revers o ; (3) nowihsanding is name, is homogeneous; and (4) will be refleced a eiher of he boundaries or, if η / σ >. For hese reasons he IGBM process is seleced o model crude oil prices. Wheher, or no, he parameer resricions found in Secion 3. will preven he IGBM process from modeling he price of crude oil is addressed in he nex chaper, wherein values of is parameers are esimaed. ; 5

20 Chaper 4 Esimaion of arameers 4. Inroducion In his chaper wo approaches o esimae he parameers necessary for he IGBM process o model crude oil prices are implemened. In he firs approach a regression equaion is derived and hen applied o an hisoric ime-series of spo crude oil prices. The second approach involves deriving formulas for fuures conracs and call opions on fuures conracs and calibraing hese formulas o he prices prevailing in he markes for each of hese conracs. The esimaes obained from he wo approaches are compared and a selecion of appropriae parameers is made in he closing secion. 4. Time-Series Approach erivaion of Regression Equaion: A coninuous ime soluion of he IGBM SE is derived and hen resaed in discree ime o obain a recursive form, below. d () η [ () ] d σ () dz( ) (4.) [ ] To solve (4.), le Y () ( ) dy () ηy () d ησ () dz( ) η and apply Iô s Lemma o obain. (4.) The resuling SE can be solved wih he inegraing facor, exp ( η), which gives d ηs ηs [ e Y () s ] η e σ () s dz() s Inegrae (4.3) and subsiue for Y ( ) o find () ( e η ) ( ) e η e η ( s) σ () s dz() s. (4.3). (4.4) Evenly subdivide he inerval [,T] ino N subinervals. Le i it/n for i,...,n and denoe each ime sep as i i. Then (4.4) recas in discree ime is i i i s i ( ) ( ) ( e η ) ( ) e η η s e i σ ( ) dz( ). (4.5) i By he Iô Isomery Theorem he inegral on he RHS of (4.5) will be a N(, σ ) random variable. So 6

21 ( ) ( e η ) ( ) e η σ ε ε, (4.6) i where ε is a random variable disribued (,) [ ] obain E ε ( i ) he muliplicaion rule, dz ( s) ds E i i i η ( ) N. Le ( ) e i s ε σ ( ) γ dz( s). To i i, and Fubini s heorem are uilized, so [ ] i η ( s e i ) ε ( ) E [ σ ( ) σ η ] ds ( i ) [ e ] For small i i i, ( ) i i η E i σ is approximaed by ( ) E [ ε ( )], hen he variance of ( ) σ i (4.7). Then from (4.6) ε i, ε i. (4.7) σ i in (4.7). Since, for he IGBM will be given by he las erm in ( ) ( ) i i i η e η e ( ) ( ) ( ) ( ) i Equaion (4.8) suggess he linear regression approach σ η e η ε. (4.8) ( ) ( ) i i a b ( ) ( ) i i ε, (4.9) where â is he inercep, b is he slope and σ ε is he sandard error of residuals. The parameers of he IGBM process are obained from he above saisics by η ln ( ) a, b / ( a ), and σ σε η η e. Applicaion o a Time-Series: A monhly ime-series comprised of spo WTI crude oil prices for he period from May 983 o Augus 3 was obained. Some basic saisics for he ime-series follow in Appendix III. The regression scheme specified by (4.9) was applied o he whole of he ime-series and a subse from Augus 998 o Augus 3. The subse of he ime-series was seleced o see wheher he parameer esimaes varied wih he sample period. The esimaes conained in Table 4. were compued. Table 4. Esimaes of IGBM arameers, Time-Series Approach arameer May 83 Aug. 3 Aug. 98 Aug. 3 Speed of reversion (η) Mean Reversion Value ( ) $.87 $9.633 Annual Volailiy (σ) 34.3% 38.79% While he speed of reversion and he volailiy have remained somewha consan over he sample period of weny years, he mean reversion value,, appears o have increased. The observed upward movemen in is consisen wih cerain supply policies announced by 7

22 OEC. The upward shif in is likely he reason for he deviaions from a sraigh line, beween $ and $3 per BBL, in he percenile versus percenile plo in Figure 4.. $45. IGBM Simulaed rices $4. $35. $3. $5. $. $5. $. $5. $5. $. $5. $. $5. $3. $35. $4. $45. Acual WTI rices(us $/BBL) vs Figure 4. ercenile vs ercenile (Monhly rices) In Figure 4. he perceniles for he monhly ime-series of boh hisoric and simulaed crude oil prices are compared. The simulaed prices were generaed using he Milsein Scheme for he IGBM given by (4.) using he esimaes of η,, and σ deermined by he regression for he May 983 o Augus 3 ime-series. The Milsein Scheme in (4.) was derived by he direc applicaion of (4.9) and (4.),. 33 in Jackel () o (3.). ) η ( i ) η σ ε σ [ ] (4.) ( i ε Ten ime-series of crude oil prices were generaed wih (4.) and he perceniles for each calculaed. The median percenile for he sample of en ime-series is ploed on he verical axis on Figure 4.. The poins in Figure 4. form an approximaely sraigh line hrough he origin, consisen wih he noion he IGBM can model crude oil prices. Noe here is lile deviaion a boh ends of he disribuion. Time-series daa is also used o esimae he risk premium for crude oil, ρm σ φ, as defined in Chaper. The correlaion beween he reurns from holding crude and he equiy marke is denoed by ρ M. Monhly reurns for each of crude oil and common equiy were compued using WTI spo prices and he Sandard & oors 5 ( S& 5 ) oal reurn index, respecively. The relaionship beween he monhly reurns for he S& 5 and WTI crude oil is illusraed on he previous page. The correlaion coefficien beween he wo reurn ime-series, for years of daa, is -.4. While he magniude of ρ is small, is sign has a significan implicaion: he risk premium ρm σ φ is negaive. Time-series daa was used in Weir () o esimae ranges for he hisorical equiy risk premium, µ M r, and equiy volailiy of 3.8% o 5.% and 3% o 3%, respecively. Single poin esimaes of 4.5% and 7% for he equiy risk premium and equiy marke volailiy, respecively, were seleced. This implies a marke price of risk of φ.65, since φ µ M r / σ. ( ) M 8

23 WTI Reurns (%) Equiy Reurns (%) Figure 4. Monhly Reurns from WTI vs Equiy 4. Calibraion Approach The calibraion approach o esimaing parameers enails wo seps. Firs, a valuaion model of a fuures conrac is derived, assuming he spo price evolves hrough ime as an IGBM. Second, a search is conduced for hose values of he parameers ha will calibrae a sequence of fuures valuaions o a sequence of prices of raded fuures conracs. The sequence of raded fuures conracs, ordered according o he increasing mauriy of he conracs, is referred o as he erm srucure, or fuures curve. Valuaion of Fuures Conracs: The value of a fuures conrac a ime ha seles a a laer ime T, denoed by F (T), according o Seppi () is equal o he ime risk-neural expecaion of he spo price a ime T, [ ( T )] Ε [ ( )] F ( T ) Ε T. (4.) The risk neural moion of, denoed by, is given by an IGBM process wih a risk-neural drif, α (,). Subracing he risk premium, ρ M σ ( )φ, from he real world drif resuls in α. The volailiy of an IGBM process is, σ ( ) σ, so (,) η [ ] ρσ ( ) φ ( η ρσφ) () ( ) η () α,. η ρσφ and η / ( η ρσφ) Le η η ρσφ so ha he risk-neural process for he spo price is d ( ) η ( ) d σ ( ) dz( ). (4.) Since he expeced value a ime of a sochasic variable following an IGBM is (3.5), hen, from (4.), (3.5) and (4.) he value of a fuures conrac is 9

24 F ( T ) Ε F [ ] ( T T η ( ) ( ) e ) η η ρσφ η e η ρσφ ( ) () ( η ρ σ φ )( T T ). (4.3) The above relaionship, (4.3), has also been derived as a soluion of he E for fuures prices by Bos, Ware, and avlov (3). The value of a fuures conrac, given by (4.3), can be inerpreed as having wo componens. The firs erm is a long run mean. The second erm is a revision in expecaions. The long run mean can be obained by aking he limi of (4.3) as T. Consider he abiliy of (4.3) o model he observed erm srucure of crude oil fuures prices. When he spo price, (), is greaer (less) han he long run mean, hen he second erm will be posiive (negaive) and will decline monoonically wih he mauriy of a conrac, since df (T)/dT is negaive. Fuures curves having a negaively, or posiively, sloped erm srucure are said o be in backwardaion or conango, respecively. Boh ypes of fuures curves can be modeled by (4.3). The expression (4.3) is, however, no capable of represening a erm srucure ha has boh a posiive and a negaive slope, a differen mauriies. Such erm srucures arise when he mid-mauriy conracs are higher, or lower, han boh he spo price and long daed conracs. The monoonic expression, exp [ ( η ρσφ)( T ) ] inhibis (4.3) from modeling erm srucures ha are convex, or concave. Esimaes by Calibraion o he Fuures Curve: Baker, Mayfield and arsons (998) and Bessembinder, Coughenour, Sequin, and Smolder (998) have observed ha when spo crude oil prices are subjec o a shock up, or downwards, he subsequen erm srucure ends o be negaively or posiively sloped, respecively. This observaion is consisen wih he view ha crude oil prices are mean revering and suggess ha he fuures markes view of he long run mean revering price is he price of he longes daed fuure. For crude oil, his implies F () 6, since crude oil fuures conracs are raded having mauriies of up o six years. Using he assumpion ha F ( 6) and he fuures curve F () o F () 6, he SOLVER ool in Excel is used o search for values of he parameers η and ρσφ ha calibrae (4.3) o he observed erm srucure of crude oil fuures, wih he minimum absolue error. The esimaes of parameers obained by calibraing (4.3) o he fuures curve a wo arbirary daes are shown in Table 4.. Table 4. Esimaes of IGBM arameers, Calibraion Approach arameer Ocober 3, 3 Ocober 3, 3 η ρσφ 3.5% 7.89% $6.9 $6.69 The calibraion procedure is illusraed in Figure 4.3 for he crude oil fuures curve on Ocober 3, 3.

25 Crude oil rice ($ / BBL) $9 $8 $7 $6 $5 $4 $3 $ $ $ Years o Mauriy Acual Model Figure 4.3 Acual vs Model Fuures rices The above graph illusraes he difficuly of replicaing a convex fuures curve wih (4.3). The parameers esimaed by he search algorihm place he model in he mids of he acual erm srucure. The relaive error in applying he model o he Ocober 3,3 curve is approximaely %. Noneheless, he monoonic propery of (4.3) means i canno fully capure he shape of he erm srucure of fuures conracs in all possible cases. Esimaes by Calibraion o Opions on Fuures: A direc esimae of he volailiy of crude oil is no available by calibraing (4.3) o he fuures curve. Insead, valuaion models of call opions exercisable for crude oil fuures conracs are calibraed o marke prices o exrac an esimae of he volailiy of crude oil fuures. Applying he fuures call opion model in Black (976) o a sequence of a-he-money calls resuled in he following esimaes of implied volailiy of he underlying fuures conracs. Table 4.3 Implied Volailiy of Crude Oil Fuures A November 3, 3 Call Expiry Feb. 4 June 4 ec. 4 ec. 5 Volailiy 3.7% 3.7% 5.9% 5.7% A rend of declining implied volailiy wih increasing mauriy of he fuures conracs is apparen in he Table 4.3. Wha does his say abou he volailiy of spo prices? The answer depends on he model of fuures prices used in he call valuaion. Black s 976 call valuaion model uilizes he cash and carry model of fuures prices, F(, T ) exp[ ( r δ )( T ) ], where dela is he convenience yield ne of carrying coss. Applying Iô s Lemma o find df and compuing reurns by dividing df by F, shows ha he volailiy of he fuures prices, σ F, equals ha of crude oil prices, σ. Which of he above lis of implied volailiies should we selec as being an esimae of σ? Since Black s 976 model is based on he assumpion ha he underlying is following a GBM, i canno model he aenuaion of volailiy wih increasing mauriy of he fuures conracs. Clewlow and Srickland () propose modeling he aenuaion of volailiy wih he increasing mauriy of fuures conracs explicily, using he SE df η ( ) ( T T e ) F ( T ) dz( ) σ. (4.4) There is no drif erm in (4.4). Clewlow and Srickland sae ha in a risk-neural world a fuures conrac ha had no cos can offer an invesor no reurn. Uilizing (4.4) o model he underlying forward conrac ha maures a ime s, Clewlow and Srickland () derive he

26 value, denoed by C, a ime of a European call opion wih srike price K ha expires a ime T, as follows C [, F(, s); K, T, s] (, T ) F(, s) N( h) KN( h ω ) log ( F(, s) / K ) ω σ η st η s ω ( e e ) h ( ) ( ) ω η, where. (4.5) The valuaions of call opions exercisable for fuures given by (4.5) are used o calibrae he marke prices of a se of a-he-money fuures opions o a corresponding erm srucure of prices of crude oil fuures conracs. Calibraion proceeds by searching for he magniudes of σ and η ha minimize he difference beween he model values and marke prices of he fuures call opions. The search is conduced using he SOLVER Tool in he Excel spreadshee locaed in Appendix IV. The esimaes for σ and η found by he search, using he marke prices prevailing on November 4, 3, are 35% and.64, respecively. 4.3 Comparison of Esimaes and Selecion of arameers To implemen valuaion models, where he price of crude oil is assumed o follow he IGBM process, esimaes of four parameers are required: he mean-revering price, ; he speed of reversion, η ; he volailiy of prices, σ ; and he risk premium, ρ M σ φ. The esimaes from he ime-series and calibraion approaches are compared and an appropriae selecion for each parameer is made, below. arameer : The ime-series approach esimaes for are $.83 and $9.63 per BBL for samples comprising he las weny years and five years, respecively. The assumpion ha has remained consan for he enire weny-year sample period is no consisen wih he $8 per BBL difference beween he wo esimaes. The dichoomy beween he quaniy and he relevance of he ime-series daa reduces he weigh ha can be given o he ime-series esimaes of. The fuures curve esimae, based on he longes daed fuures price of $6.4 o $6.69 per barrel, is he marke price oday for delivery in six years. As such i is he marke price ha will balance supply and demand in he fuure, bearing in mind he OEC range of $4 o $3 per BBL. lacing more weigh on he esimae drawn from he longes daed fuure, resuls in $7 per BBL being seleced as an esimae of. arameer η : For he speed of reversion, η, hree esimaes were acquired, based on: he ime-series regression approach, of.393 o.49; he calibraion o he fuures curve approach, of.369 o.46; and he calibraion o call opions on fuures approach, of.64. The calibraion o fuures esimaes are an order of magniude larger han hose obained from he oher wo approaches. On he wo days hese esimaes were compued, he crude oil fuures curve had a seep negaive slope. The calibraion o fuures esimae of η on hese days had o have sufficien magniude o bend he model fuures curve o fi he price of boh he firs and las monhs fuures. While here should be a relucance o selec parameers oher han hose consisen wih he fuures curve, more weigh is placed on he ime-series esimae, which is a direc measure of he speed of reversion of spo prices and he calibraion