VALUATION OF AN OIL FIELD USING REAL

Transcription

1 VALATION OF AN OIL FIELD ING REAL OPTION AND THE INFORMATION PROVIDED BY TERM TRTRE OF OMMODITY PRIE AHIER DE REHERHE D EREG N LATIER Delphe EREG versty Pars IX & ERNA ENMP Emal : lauter@cera.esmp.fr Tel: Address : EREG versté Pars IX Dauphe Place du Maréchal de Lattre de Tassgy Parx edex 6 Frace

2 VALATION OF AN OIL FIELD ING REAL OPTION AND THE INFORMATION PROVIDED BY TERM TRTRE OF OMMODITY PRIE ABTRAT: Ths artcle emphasses that the formato provded by term structures of commodty prces has a fluece o the real opto value ad o the vestmet decso. We exhbt frst of all the aalyss framework: the evaluato of a ol feld. We suppose that a sgle source of ucertaty - the crude ol prce - affects the vestmet decso. We also preset the two term structure models used to represet the dyamc behavour of ths prce ad to evaluate the et cash flows of the feld. The we preset the real optos valuato method. Lastly smulatos llustrate the sesblty of the real optos to the term structure of commodty prces ad we aalyse the vestmet sgals gve by the optoal method. Our prcpal coclusos are twofold. Frstly t s essetal to take to accout the formato provded by the term structure of futures prces to uderstad the behavour of the real opto. ecodly the vestmet sgal assocated wth the optoal method does ot dffer for some specfc prce curves from the oe gve by the et preset value. KEY WORD: coveece yeld stochastc models real opto to delay crude ol term structure et preset value.

3 ETION. INTRODTION The obectve of ths artcle s to hghlght the mpact of term structure of futures prces o real optos value. The real opto theory s based o a aalogy wth the facal optos. It ams to detfy the optoal compoet cluded most vestmet proects ad to evaluate t whe possble. Its ma advatage s that cotrary to the methods tradtoally used for the selecto of vestmet proects - lke the et preset value - t takes to accout the flexblty of a proect. The theory leads to the detfcato of dfferet famles of real optos ad uderles that most vestmet proects clude several optos. However ths artcle eve f the possesso of a ol feld mples the holdg of several optos we oly take to cosderato the opto to delay the explotato utl some useful formato arrves ad gves the sgal to vest. We also suppose that a sgle source of ucertaty affects the proect value: the crude ol prce. As a cosequece the formato gve by ths prce has a crucal fluece o the vestmet decso. Ths prce ca be represeted by a futures prce whch s at a specfc date ad codtoally o the formato avalable at that date a expectato of the future spot prce. The term structure of commodty prces coects all the futures prces for dfferet maturtes. Ths curve ca take dfferet shapes. Whe the spot prce s hgher tha the futures prces there s backwardato. Gve ths formato a decrease the future spot prces s expected. Whe coversely the spot prce s feror to the futures prces the term structure s cotago whch meas that oe ca wat for a crease the future spot prces. Naturally there are also more complcated prces curves wth for example a backwardato o the earest maturtes ad a cotago for log-term cotracts. Wth a term structure model t s possble to compute a futures prce for ay exprato date eve f t s very far away. Thus such a model eables the valuato of the et cash flows assocated wth the vestmet proect ad t gves all the formato eeded for the vestmet decso. Provded that we take to accout ths formato t s possble to uderstad the behavour of the real opto whch becomes cosstet wth the behavour of a facal opto. Ths paper proceeds as follows. ecto s devoted to the aalyss framework: we gve some precso o the ol feld characterstcs ad o the method used to represet the behavour of the crude ol prce. We also expla why we cocetrate o the opto to delay ad we preset ths opto. ecto 3 troduces the optoal method. The latter reles o two term structure models of commodty prces whch are preseted. ecto 4 aalyses the sesblty of the real opto to ts ma determats. ecto 5 s cetred o the vestmet decso ad shows that for certa term structures the optoal method does ot dffer from the et preset value. ecto 6 cocludes. A presetato of the real optos theory ca be fd opelad et Atkarov 00 Grblatt et Ttma 00 Lauter 003 Trgeorgs

4 ETION. THE ANALYI FRAMEWORK The aalyss framework presets two characterstcs. Frstly we reta oly the opto to delay the explotato. ecodly we oly cosder the ucertaty assocated wth the crude ol prce. Before we tackle the valuato tself we shall ustfy these two choces... A sgle real opto I ths study we oly cosder the opto to delay the developmet of a ol feld. Yet the real opto theory shows that most of the tme a vestmet proect cludes more tha oe real opto. The real optos the most frequetly evoked the lterature are the opto to delay the opto to abado the tme to buld opto the opto to alter operatg scale the opto to swtch use the growth opto ad the multple teractve optos. At least four of them are assocated wth the holdg of a ol feld. Amog them the opto to delay s udoubtedly the smplest. It represets the possblty for the ower of the feld to wat before vestg utl some useful formato arses. Naturally the vestor has other potetaltes. Oce the producto has begu t s for example always cocevable to reouce the proect: ths s the opto to abado. Lkewse the explotato ecesstates several successve vestmet steps: explorato ol developmet ad producto tself. Each step amelorates the formato o the level ad the qualty of the resource ad ca lead to pursut vestmet whe the formato s favourable or coversely to gve up: ths s the tme to buld opto. Fally t s possble to reduce or eve to temporarly shut dow the explotato: ths depcts the opto to alter operatg scale. I vew of ths profuso oe may cosder that t s smplstc to focus o a sgle real opto. To expla ths choce we argue that the valuato of real optos whch follows the methods developed for the facal assets presets some dffcultes arsg from the dffereces betwee real ad facal assets. However the am of ths artcle s ot to deal wth these dffcultes but to study the relatoshp betwee real optos ad the formato gve by term structure of prce. Therefore the most elemetary set up was retaed: the case of a sgle opto. Amog the dfferet optos cluded the proect we choose the opto to delay. everal reasos ustfy ths choce. Frstly t s qute smple to evaluate. ecodly amd the dfferet steps of the proect the developmet phase ecesstates the most mportat expedtures. Therefore the opto to delay s probably the most expesve oe. Thrdly we dd ot take to accout the possblty to shut dow temporarly the explotato because ths kd of operato s very harmful for the udergroud mes ad for the ol felds. The terrupto of the explotato causes deed the flood of the mes. Ad the reopeg cost s ot very far from the cost of a ew developmet. Fourthly the opto to abado s eglected because oce the explotato has begu the petroleum dustry t s geerally coducted utl the ed. The latter are dvsble fugble ad most of the tme actvely traded o a secodary exchage. Ths s ot the case for the real assets ad the valuato must be udertake carefully especally whe the proect cludes several optos. 4

5 .. The real opto to delay The opto to delay s the smplest real opto ad udoubtedly the most frequetly evoked the lterature 3. It represets the possblty to wat before vestg order to collect some useful formato. To preset the real opto to delay let us establsh a aalogy betwee the holdg of a call o a share ad the possesso of explotato rghts o a feld. The frst makes t possble to buy a share ad to eoy the dvdeds ad the captal ga or loss. The secod gves the possblty to explot the feld ad to beeft from the et preset value of the resource. Because a ol feld ca be dffcult to sale or to shut dow temporarly such a vestmet s regarded as rreversble. Ths real opto s Amerca because the vestor has the choce to develop wheever he wats utl hs rghts expre. The aalogy betwee a facal call ad the real opto to delay presets evertheless two lmts. Frst of all the real opto exprato date ca be very far away 4. ecodly the exercse of the facal call gves rse to the mmedate trasfer of the property rghts o the uderlyg asset. Ths s ot the case whe the real opto s exercsed. Ideed whe the feld s exploted oe must wat several weeks or moths utl the crude ol arrves to the cosumpto areas. These dffereces have a fluece o the real opto valuato..3. A sgle source of ucertaty Last partcularty of ths study: we suppose that the crude ol prce s the oly source of ucertaty havg a mpact o the vestmet value. uch a choce mples that we made several assumptos: the reserves the extracto costs ad the developmet costs are supposed to be kow; we gore the exproprato rsk; techologcal progress s eglected ad the volume of the reserves ad ther producto costs are depedet of the explotato date. The choce of ths framework ca be explaed as follows. Durg a log tme ths dustry the most mportat questo cocerg a feld s explotato was: how ca we develop at the lower cost? Today most of the tme the ewly dscovered felds are ot mmedately exploted. Oe wats for a hgher prce especally whe the reserves are substatal. Lastly our example the ucertaty s a purely exogeous factor o whch the operator has o power. Ths choce amouts to sayg that the vestor has o possblty to fluece the crude ol prce. uch a assumpto s ot ocuous for most operators the crude ol market. Thus ucertaty s oly due to the crude ol prce. Because oly the prce has a mpact o the proect s value ad o the vestmet decso we pay a specal atteto to the represetato of the prce dyamc. Ideed order to apprecate the value of the opto to delay the explotato the value of ts uderlyg asset must be kow. Yet the later depeds drectly o the hypothess cocerg the evoluto of the crude ol prce. I ths stuato the term structure models of commodty prces ca be useful tools. We frst of all preset the two models used. The we aalyse the mpact of each model o the proect s et preset value. 3 Ths real opto was also studed by McDoald ad egel 986 ad Brea et chwartz The legth of the explotato rghts o a petroleum feld s extremely varable: t ca stretch over 99 years or t ca be less tha 5 years. 5

6 .3.. The two term structure models of commodty prces We use two well-kow models: Brea ad chwartz developed the frst 985 ad chwartz proposed the secod 997. The prcpal dfferece betwee these two models s due to ther represetato of the prces dyamc behavour. Brea ad chwartz choose a geometrc Browa moto for the spot prce. However a few years later chwartz reouced to ths modellg ad referred to a mea revertg behavour. Brea ad chwartz s model Brea ad chwartz s model s the frst ad the smplest verso of a term structure model of commodty prces. I ths model the movemets of the futures prces deped oly o the spot prce. The dyamc of the latter s the followg: where: - s the spot prce d t µ t dt σ t dz - µ s the drft - σ s the spot prce s volatlty - dz s a cremet to a stadard Browa moto assocated wth. Ths formulato meas that the spot prce s varato at t s depedet of the prevous varatos ad the drft µ coducts the prces evoluto. The ucertaty affectg the prce s evoluto s proportoal to the level of : whe vetores are rare the spot prce s hgh; ths stuato ay modfcato the demad has a mportat mpact o the spot prces because the physcal stocks are ot suffcetly abudat to absorb the prces fluctuatos. The soluto of the model for a futures prce havg a exprato date T s 5 : FtT e r c τ where: - r s the rsk free terest rate assumed costat - c s the coveece yeld assumed costat - τ s the cotract s maturty: τ T - t Brefly defed the coveece yeld represets the comfort assocated wth the holdg of vetores. Its correlato wth the spot prce s postve. chwartz s model I Brea ad chwartz s model the spot prce dyamc gores that the operators the physcal market adust ther vetores to the evoluto of the spot prce ad to chages supply ad demad. Moreover ths represetato gores that a futures prce does ot deped oly o the spot prce but also o the coveece yeld. The latter s a parameter ad t s supposed to be costat. However 989 Gbso ad chwartz showed that such a aalyss s lmted. They proposed to troduce the coveece yeld as a secod state varable the latter havg a mea revertg behavour. chwartz s model cludes these recommedatos ad retas the followg dyamc: d t d t µ t κ α t t dt σ t dz dt σ dz 5 The soluto of the models ca be obtaed usg a Feyma-Kac soluto. For more detals see for example Lauter

7 wth : κ σ σ > 0 where : - s the spot prce - s the coveece yeld - µ s the drft of the spot prce - α s the log ru mea of the coveece yeld - κ s the speed of adustmet of towards α - σ s the spot prce s volatlty - σ s the volatlty of the coveece yeld - dz s a cremet to a stadard Browa moto assocated wth - dz s a cremet to a stadard Browa moto assocated wth. The spot prce ad the coveece yeld follow a ot dffuso process: ρ dt E t [dz dz ] where ρ s the correlato coeffcet lkg the two Browa motos. Ths represetato of the coveece yeld behavour supposes that there s a average level of physcal stocks whch satsfes the eeds of the dustry. Therefore the operators behave such as the volume of stocks ad cosequetly the coveece yeld coverge o ths level. Whe the coveece yeld s low stocks are abudat ad the operators susta a hgh storage cost compared wth the beefts assocated wth the holdg of the merchadse. If they behave ratoally they wll try to elmate the surplus stocks. oversely whe the coveece yeld s hgh they wll ted to recosttute ther stocks. The soluto of chwartz s model s the followg: [ H B τ ] F t T e wth : H e κ B τ κτ α α λ κ σ σσρ e r α τ σ 3 κ κ 4 κ κτ where λ s the rsk premum assocated wth the coveece yeld. σ e ακ σσρ κ κ Ths model s more realstc tha the prevous oe. It authorzes prce curves that are more vared ad closer to the curves observed o commodtes markets. However ths model s also more complex. Partcularly t cludes seve parameters whereas the prevous has oly two of them..3.. The represetato of the prces dyamc ad ts fluece o the et preset value. I order to show how the represetato of the spot prce s behavour flueces the value of the et future cash flows we study the sestvty of the et preset value. We make a addtoal hypothess for the calculato of the NPV: we suppose that the explotato leads to the producto of oe sgle crude ol barrel at the ed of each year durg a perod of N years. The NPV s: NPV N PV a T p T T N B a T I 0 κτ 7

8 where : - PVaT s the et preset value of a barrel produced T a s the dscout rate p s the cost of producto per barrel. It s supposed to be costat durg the proect s lfe. BaT s the preset value of oe dollar equal to e -at f a s costat I 0 s the tal vestmet. We made three assumptos cocerg the crude ol prce behavour. The frst oe reles o the practce commoly adopted the petroleum dustry. It cossts supposg that the prce wll be costat durg the whole lfe of the proect. The two others hypotheses are based o the term structure models prevously preseted. Each assumpto leads to a dfferet et preset value. Whe the Brea ad chwartz s model s retaed t becomes: wth : wth: N N N ct e P T T rt NPV t T e I β β ct rt β e ad β p e I0 T N T Whe chwartz s model s used the et preset value s: N N N H t A T e P T T 0 rt NPV t T t e I β β H τ [ κ ² α 0.5σ κρσ σ σ H A τ exp κ 4κ H t rt β A T e ad β p e I0 T N T less clearly specfed the data retaed for the calculatos preseted below are the followg: the proect lfetme N s fxed at 5 years. The producto cost P s establshed at 7 dollars per barrel. Ths level s realstc for a explotato the North ea. I order to determe I 0 we cosdered that a proect s proftable whe ts et preset value s aulled for a costat spot prce of dollars per barrel ad for a dscout rate of 5%. Hece I 0 was fxed at 5 dollars. These values ca be dscussed. The terest of the exercse s ot to work wth values correspodg to a specfc proect but to compare the valuato methods. I order to compute the futures prces correspodg to each term structure model oe must also precse the parameters ad the state varables parameters. These values are drectly extracted from emprcal tests prevously performed o the crude ol market Lauter ad Gall 00. Therefore the coveece yeld s set to 0.. Moreover for chwartz s model the parameters values are the followg: the log ru mea of the coveece yeld α s equal to 0.; the spot prce s volatlty σ s set to 0.3 ad the volatlty of the coveece yeld σ s supposed to be 0.4. The speed of adustmet κ s establshed at the correlato coeffcet ρ s fxed at 0.9 ad the rsk premum assocated wth the coveece yeld λ s set to 0.. Lastly the NPV dscout rate ad the terest rate are supposed to be equal to 5%. The study of the evoluto of the NPV computed wth the three methods leads to several coclusos. The frst are vald whatever the hypothess cocerg the prce behavour. As s show Fgures ad the NPV s a creasg fucto of the tal crude ol prce ad of the proect 8 0

9 lfetme. However t decreases wth the explotato cost wth the tal vestmet ad wth the dscout rate. The relatoshp betwee the NPV ad the tal spot prce s lear ad postve whatever the method retaed to compute the et cash flows assocated wth the proect as s show Fgure. Fgure. Net preset value as a fucto of the spot prce NPV $ pot prce $b pot prce Brea ad chwartz chwartz The study of the evoluto of the proect NPV as a fucto of ts lfetme Fgure shows that a crease the proect lfetme has a postve fluece o the NPV. However ths fluece dmshes as the exprato date rses. As far as Brea ad chwartz s model s cocered the relatoshp betwee the terest rate ad the coveece yeld fxes the shape of the etre term structure of prces ad t s essetal for the NPV. For chwartz s model the ma determats of the prce curve for the earest exprato dates are the tal level of the coveece yeld the speed of adustmet κ ad the log ru mea α. Whe the speed of adustmet s hgh all thgs beg equal the ma parameter s the log ru mea because the coveece yeld ca ot go very far from t. oversely whe the speed of adustmet s low the prces curve s strogly flueced by the tal dstace betwee the coveece yeld ad the log ru mea Lauter 00. Fgure. Net preset value as a fucto of the proect lfetme NPV $ Lfetme years ostat prce Brea ad chwartz chwartz Fgures 3 4 ad 5 llustrate that the et preset value whatever the hypothess retaed to represet the spot prce evoluto s a decreasg fucto of the producto costs the tal amout vested ad the dscout rate. 9

10 α NPV $ Fgure 3. Net preset value as a fucto of the producto costs Producto costs $b ostat prce Brea ad chwartz chwartz Fgure 4. Net preset value as a fucto of the tal amout vested NPV $ Ital amout vested $b ostat prce Brea ad chwartz chwartz NPV $ Fgure 5. Net preset value as a fucto of the dscout rate % 4% 6% 8% 0% % 4% 6% 8% D scout rate ostat prce Brea ad chwartz chwartz These smulatos also show that all the cases preseted above the NPV assocated wth chwartz s model s always the lowest. The comes the oe assocated wth Brea ad chwartz s model. Lastly the NPV assocated wth the costat prce hypothess s systematcally the hghest. Ths result depeds however o the parameter values whch correspod to a specfc stuato llustrated by Fgure 6: all the prces curves are backwardato especally for Brea ad chwartz s model. 0

11 Fgure 6. Term structures of prces wth 0$b r 5% ad Future prce $b Years ostat prce Brea-chwartz 85 chwartz 97 Aother set of parameters ca lead to opposte coclusos. Ths s the case for the curves represeted Fgure 7. I that stuato the term structure assocated wth Brea ad chwartz s model s cotago ad correspods to the hghest NPV: dollars. The ext hghest NPV s the oe assocated wth chwartz s model 7.3 dollars. The last s the oe lked wth the costat prce assumpto 70.8 dollars. The valuato methods used to represet the dyamc behavour of the crude ol prce have therefore a strog fluece o the proect et preset value. Ths wll ecessarly have a mpact o the results of the optoal method. Fgure 7. Term structures of prces wth 0$b r % ad Future prce $b Years ostat prce Brea-chwartz 85 chwartz 97 ETION 3. THE VALATION OF THE REAL OPTION TO DELAY The real optos valuato method s spred by the oe developed for the facal optos. I ths secto we preset ths method ad we show how to apply t our case. 3.. Prcples of real optos valuato A facal opto s cosdered as a asset whose prce movemets are totally specfed by a set of uderlyg factors. The latter represet the sources of ucertaty fluecg the prce evoluto

12 of the dervatve asset. For the smplest facal optos there s oe sgle uderlyg factor: the prce of the uderlyg asset. Therefore the frst step of a opto valuato cossts determg the dyamc behavour of the uderlyg asset prce order to represet how the prce wll vary betwee the acqusto ad the exercse date. Most of the tme ths behavour s charactersed by two elemets: a determstc the drft ad a radom compoets. The valuato method uses the arbtrage reasog ad leads to the costructo of a hedge portfolo. Brefly speakg the reasog s the followg: a complete market where the trasactos are cotuous a dervatve asset ca be duplcated by a combato of others exstg assets. If the latter are suffcetly traded to be arbtrage-free evaluated they ca costtute a hedge portfolo whose behavour replcates the dervatve behavour. Ther proportos are fxed such as there are o arbtrage opportutes ad the strategy s rsk-free. The equlbrum the retur of the portfolo must be the rsk free rate. The valuato s made a rsk eutral world: t does ot deped o the atttude toward rsk of the operators. 3.. The opto to delay the explotato ad the term structure models Two detals must be added before we apply the valuato method to the real opto to delay the ol feld explotato. Frst of all our example the uderlyg factor s the crude ol prce whose behavour s represeted by a term structure model. Therefore the choce of the model has a fluece o the real opto value. ecodly the case of the real opto to delay we do ot estmate how log t s ecessary to wat. We compute the crtcal exercse prce of the opto amely the prce correspodg to a opto value above or equal to the proect NPV The real opto to delay assocated wth Brea ad chwartz s model Whe we use Brea ad chwartz s model to represet the dyamc behavour of the uderlyg asset the real opto to delay VtT satsfes the followg equato: σ V r c V rv Vτ 0 The boudary codto of ths equato s: V t T max[ NPV t T 0] Ths codto meas that at the exprato date T f the NPV of the feld s egatve or equal to zero the real opto value s zero. If however the NPV s postve the opto value s equal to the NPV. Ths opto s a Amerca oe: the vestmet ca be udertake at ay momet. chwartz 997 proposed a aalytcal soluto to ths equato. upposg that the exprato date teds to the fty ths soluto s: N V * β β * ct rt wth: β e β p e I0 T r c r c r d σ σ σ N T d βd * β d where * s the crtcal prce.

13 Havg a aalytcal soluto s a huge advatage: the complexty ad the slowess of the umercal methods used for chwartz s model whch are preseted the ext paragraph s a good llustrato of that pot. However the aalytcal soluto presets a mportat drawback: t ca ot be used whe the coveece yeld s egatve or whe t s equal to zero. Yet the coveece yeld et of the storage costs ca be egatve whe the market s cotago The real opto assocated wth chwartz s model I chwartz s model the real opto value VtT satsfes the dfferetal equato : σ V σ V ρσ σ V V r V [ κ α λ] rv Vτ 0 The boudary codto s the same as the oe preseted before. Ths equato s more complex tha the prevous oe because chwartz s model supposes that two ucertaty sources have a mpact of the real opto: the spot prce ad the coveece yeld. Ths equato does ot have to our kowledge a aalytcal soluto. Therefore we used a fte dfferece method 6 : we replaced the partal dervatves by Taylor expasos. Amog the varous fte dfferece methods avalable we chose the rak-ncholso method for the stablty of ts solutos. The dscretzato cocers the state varables ad ad the maturty τ T-t. The latter s dvded to tme tervals of legth k. The opto value s recursvely computed at the date s-k ad t depeds o ts value s wth t s T. Whe s T the opto expres. The varato doma for ad s dvded to m tervals of legth h. The grd s therefore costtuted of the pots h h k the space t such as 7 : h for 0 m h for m m t k for 0 The opto value s represeted by a scheme three dmesos: Vt Vhhk V. Oce the dscretzato has bee completed the rak-ncholso scheme cetred space s: { T V W X Y Z } { V W X Y Z } wth : for σ k V σ W k ρσσk X 4R 4h R 8hR k r h k κ α h rk Y Z T 4R 4hR R rk R m ad m m A Alteratve Drectos Implct ADI method s the appled o ths scheme. Ths method s spred by the oe proposed by Mtchell ad Mc Kee 970. It eables the resoluto of the 6 Ths method s preseted the appedx. It s spred by Bellalah 990. chwartz 997 also used t. 7 The varato terval of the coveece yeld dffers from the oe that was retaed for the spot prce: deed the coveece yeld ca be egatve whch s ot the case for the spot prce. 3

14 dfferetal equato two steps despte the presece of a cross dervatve. The use of a ADI method cossts troducg a termedary term order to separate at each tme step the prevous equatos system to two sub-systems. Each of the latter s successvely solved at each tme step. Ths umercal method chose for ts rapdty s stll especally slow whe t s appled our cotext for two reasos. Frst of all the varato terval of the spot prce s totally dfferet of the oe of the coveece yeld. ecodly the opto exprato date s far away. 3.. The dffcultes of the real optos valuato The lmts of the method used for the valuato of real optos are twofold. Frstly the valuato of facal optos reles o hypotheses that are ot really respected the case of real assets. ecodly the approach leadg to the valuato s dffcult to udertake. The valuato of a facal opto reles o assumptos cocerg the market characterstcs ad the ature of the trasactos. The market s supposed to be perfect amely wthout trasacto costs or taxes. hort sales are ot restrcted ad borrowg ad ledg rates are equal. All the assets are perfectly dvsble ad the formato cocerg the prces ad assets characterstcs s freely avalable. The trasactos are cotuous. Therefore t s always possble to modfy the portfolos composto. The market s free of arbtrage opportutes. The trasposto of such a theoretcal framework the case of real optos s ot straghtforward. It s deed dffcult to mata that the perfect market hypothess s stll realstc. Frstly the world of real assets markets mperfectos ca be cosdered as a ecessary codto to the appearace ad the explotato of vestmet opportutes Myers 977. ecodly the real assets are sometmes ot dvsble. Beyod these hypotheses the valuato method of facal optos must be cautously used the case of real optos. The real assets are less traded tha the facal assets. Thus from the frst step of the valuato whch cossts determg the most mportat ucertaty source affectg the real opto prce some dffcultes arse. The exteral valuato of the uderlyg asset guarateed by the market mplct the case of facal optos ca be mpossble a real assets world whe the asset s specfc ad has o value except for the frm holdg t. I the most favourable stuato whe there s a market for the real asset t s probably arrow ad mperfect ad the trasacto wll be charactersed by ucompleted or asymmetrcal formato. Therefore the secod step of the valuato method whch supposes that the assets are traded wth o arbtrage opportutes must be cautously udertake. Partcularly the valuato wll probably ot be realsed a rsk eutral world or several rsk eutral probabltes wll coexst. The lmts of the valuato are therefore so mportat that the relevace of the aalogy betwee real optos ad facal optos becomes questoable. Does t mea that ths aalogy must be restrcted to a purely qualtatve aalyss? The study of the determats of the real optos to delay secto 4 gves a aswer to that questo. 4

15 ETION 4. ENITIVITY OF THE OPTION VALE The smulatos preseted ths secto are based o the term structure models of commodty prces preseted prevously. The models are used for the valuato of the et cash flows assocated wth the feld ad for the computato of the real opto value. The values of the parameters ad the state varables are the same tha those used the frst secto except for the proect s lfetme. The latter has bee reduced to te years because of the slowess of the umercal method. 4.. Real opto term structure of futures prces ad state varables The umber of state varables chages wth the term structure model. I Brea ad chwartz s model oly the spot prce has a fluece o the futures prce evoluto. I chwartz s model the coveece yeld s troduced. Table reproduces the evoluto of the value of the real opto to delay the ol feld explotato whe the spot prce ad the coveece yeld vary. As far as the spot prce s cocered the cocluso s the same for the two models: the opto s value s a creasg fucto of ths state varable. Ths result s cosstet wth the aalogy betwee facal ad real optos prevously made. Ideed the two models whatever the exprato date s cosdered the futures prces s a postve fucto of the spot prce. The latter affects the whole prces curve. Thus a rse the spot prce coducts to a mprovemet of the proect s et preset value eve f the prces curve s backwardato. It ust so happes that whe the prce of the uderlyg asset rses we should observe a augmetato of the call value. Table. Real opto value as a fucto of the state varables ad 8 B ch B ch $b ; 0. ; p 7$b ; I 0 5$ ; σ 0.3 ; r 5% ; α 0. ; κ ; σ 0.4 ; ρ 0.9 ;λ 0. ; The real opto value s o the other had a decreasg fucto of the coveece yeld because the two models backwardato rses wth the coveece yeld havg thus a egatve mpact o the proect et preset value ad fluecg therefore the value of the real opto to delay. The value of the real opto assocated wth chwartz s model s fally a decreasg fucto of the speed of adustmet of the coveece yeld as s show Table. Table. Opto value as a fucto of the speed of adustmet κ.5 ch $ ; 0. ; r 5% α 0. ; σ 0.3 ; σ 0.4 ; ρ 0.9 ; λ 0. ; P 7$ I 0 5$ 8 B s used for the Brea ad chwartz model ad ch for the chwartz model. 5

16 Ths result ca be smply explaed: the more the tedecy of the coveece yeld to retur to ts mea rses the more the ucertaty assocated wth the log-term cash flows of the proect dmshes. I ths stuato the behavour of the real opto value s cosstet wth what would have bee expected of a facal opto: the opto value creases whe the volatlty growths. Ad the more the speed of adustmet teds toward zero the more the results obtaed wth the two-factor model are close to those ssued from Brea ad chwartz s model. 4.. Real opto term structure of futures prces ad explotato codtos A secod seres of smulatos aalyses the mpact of the explotato codtos o the real opto value. These codtos arse from the crude ol market tself extracto cost per barrel ad tal vestmet expeses ad more geerally to the evromet level of terest rates. The study of the real opto value establshes that t s a decreasg fucto of the extracto cost per barrel P ad the tal amout vested I 0 Table 3. It s also a creasg fucto of the terest rate Table 4. These results are cosstet wth the behavour of a facal opto: the call value dmshes wth the value of the uderlyg asset ad the exercse prce. However t s a postve fucto of the terest rates. tudyg the way the producto cost per barrel flueces the opto value authorzes frstly smlar coclusos for the two models. Whe these costs rse the opto value dmshes. A crease the producto costs reduces deed the et futures cash flows assocated wth the proect ad therefore the value of the uderlyg asset. Ths leads aturally to a decle of the call value. Lkewse whe the producto cost rases the vestmet crtcal prce creases: the prce per barrel must asced to compesate for the egatve mpact of the producto cost o the cash flows. ecodly the results deped o the model. The value of the real opto assocated wth Brea ad chwartz s model s deed lower tha the oe assocated wth chwartz s model because backwardato s more proouced for the oe-factor model. osequetly the value of the uderlyg asset dmshes more rapdly. Table 3. Real opto value as a fucto of the explotato codtos P $b B ch I 0 $ B ch $ ; I 0 5$ ;0. ; r 5% ; α 0. ; κ ; σ 0.3 ; σ 0.4 ; ρ 0.9 ;λ 0. ; Whe we study the mpact of the tal amout vested the opto exercse prce we reach smlar coclusos. The real opto value decreases wth the tal expeses. Lastly the study of the sestvty to the terest rate shows that the more the terest rate s hgh the more the real opto value rses. The delay of explotato leads deed to the postpoemet of the tal expeses. The latter ca be vested the facal markets utl the opto s exercsed ad ths vestmet s more terestg whe the terest rate s hgh. 6

17 Table 4. Real opto value as a fucto of the terest rate r % B ch $ ; p 7$b ; I 0 5$ ;0. ; α 0. ; κ ; σ 0.3 ; σ 0.4 ; ρ 0.9 ;λ 0. ; omparg the two models leads however to the cocluso that Brea ad chwartz s model s more sestve to the terest rate. Whe the terest rate rses from 3 to 5% there s a 49% crease of the call value for the oe-factor model whereas the varato s oly of 9.7% for the twofactor model. Ths stroger sestvty of the oe-factor model appears also wth the two other parameters amely the explotato costs ad the tal expeses but t mafests tself more tesvely wth the terest rate. The study of the opto sestvty to ts ma determats leads to two coclusos. Frstly the real opto to delay behaves lke a facal call. Therefore the aalogy betwee real ad facal optos s relevat ot oly qualtatvely but also quattatvely. ecodly the real opto value s sesble to the choce of a specfc term structure model to evaluate the et future cash flows of the proect. The et preset value ad the real opto value deped strogly o the hypotheses retaed cocerg the dyamc of the crude ol prce. ETION 5. REAL OPTION AND INVETMENT DEIION Ths secto presets the results of smulatos amg to study the fluece of the formato provded by term structure of commodty prces o the vestmet decso. I order to determe whe the vestmet must be udertake we retaed two methods. The frst s the NPV. I that case the vestmet must be udertake whe the spot prce s suffcetly hgh to lead to a ull NPV. The secod method s optoal: t defes the prce leadg to the exercse of the real opto. 5.. The vestmet threshold ad the level of the coveece yeld Table 5 shows how the vestmet threshold evolves whe the coveece yeld rses 9. The results arouse three remarks. Frst of all the fgures obtaed wth the Brea ad chwartz s model whe the coveece yeld teds toward zero 0 are surprsg. The crtcal prce to vest that case has o ecoomcal sgfcace: dollars per barrel! I that stuato the et preset value method whch recommeds a mmedate vestmet must be preferred to the optoal approach. The latter s of lttle terest because t s based o a term structure model whch s ot suted for dstat maturty dates. Ideed wth ths model whe the coveece yeld s lower tha the terest rate the term 9 We dd ot study the fluece of the spot prce o the vestmet threshold because t s depedet of the spot prce. 0 I the table 5 for the Brea ad chwartz model we coducted the smulato wth a coveece yeld equal to because the model does ot accept ull or egatve values for the coveece yeld. 7

18 structure of prces s cotago ad the futures prces ca atta for dstat maturty dates values havg o ecoomcal sese. Table 5. rtcal prce as a fucto of the coveece yeld NPV 0 rtcal prce B ch B ch $b ; p 7$b ; I 0 5$ ; r 5% ; α 0. ; κ ; σ 0.3 ; σ 0.4 ; ρ 0.9 ;λ 0. ; ecodly whe the coveece yeld s hgher tha the terest rate 0. ad 0.3 the two vestmet methods lead to the same decso for Brea ad chwartz s model: the proect must be abadoed. Ideed the smulatos show that whe the coveece yeld s for example equal to 0. the crtcal prce s 0 dollars per barrel for the et preset value method ad t s 5.8 dollars for the optoal method. Yet the tal spot prce chose for smulatos s equal to 8 dollars per barrel ad the term structure of prces s backwardato. As a cosequece the future spot prces should decrease regularly ad the proect must be reected. Thrdly the threshold prces assocated wth chwartz s model are lower tha those obtaed wth the oe-factor model ad they are also closer to those assocated wth the NPV method. However f these results seem more reasoable the terpretato of the vestmet decso s ot really smple. It s possble oly f we take to cosderato the term structure of prces whch s backwardato Table 5 whatever the level s retaed for the coveece yeld. The crtcal prce s for example equal to 4. dollars per barrel for 0.. The tal spot prce whch s equal to 8 dollars s therefore hgher. There are a pror two ways to terpret ths result: ether t meas that oe must vest mmedately as suggested by the NPV method ether t meas that oe must wat that the prce lowers utl t reaches the level of 4. dollars. However the term structure of prces beg backwardato the future spot prce s supposed to decrease regularly. As a cosequece the frst terpretato must be retaed. 5.. The vestmet threshold ad the speed of adustmet If chwartz s model leads to more reasoable crtcal prces tha those assocated wth the oe-factor model t s maly due to the presece of a speed of adustmet whch has a mpact o the coveece yeld ad flueces drectly the spot prce gvg the latter a mea revertg tedecy. As a result the prce of the uderlyg asset evolves a more restrcted terval. Table 6 presets the smulatos wth varous speeds of adustmet. The valdty of our computato method has bee tested for the two models. We compared our results wth the oe obtaed by chwartz 997 o the copper market. Whe we use the same values for the parameters ad for the state varables we obta exactly the same results. 8

19 Table 6. chwartz s model crtcal prce as a fucto of the speed of adustmet κ NPV0 rtcal prce $ ; 0. ; r 5% α 0. ; σ 0.3 ; σ 0.4 ; ρ 0.9 ; λ 0. ; P 7$ I 0 5$ These smulatos show that the crtcal prce s a decreasg fucto of the speed of adustmet assocated wth the coveece yeld. Whe the magtude of the speed of adustmet s hgh the prce returs rapdly to ts log ru mea. I addto the results preseted Table 6 uderle aother advatage of chwartz s model. The latter authorzes deed more vared prce curves tha the oe-factor model. These curves ca be for example backwardato for the earest exprato date ad the cotago. Ths s the case the example above whe the speed of adustmet s set to. I that case eve f the crtcal prce of.4 dollars per barrel s hgher tha the tal spot prce of 8 dollars t s recommeded to wat before vestg. It s better to wat a decrease the prce because t wll later crease. The suggesto obtaed wth the optoal method s cotradctory wth the oe based o the NPV method whch ecourages vestg mmedately Ivestmet decso ad term structure of prces The study of the vestmet decsos gves rse to the followg remarks. Frst of all the qualty of the formato provded by Brea ad chwartz s model s lmted: ths model ca ot be used for all the possble values of the coveece yeld. Ad eve f we restrct the varato terval of ths varable the results are sometmes meagless. ecodly oe must woder about the mpact of the optoal valuato whe the term structure of prces s backwardato. I that case the opportuty to delay s of lttle terest because a decrease of the future spot prces s expected. Yet backwardato ca be the most frequet stuato certa markets lke the crude ol market. Therefore for these markets the optoal method leads to the same alteratve that the NPV method: vest ow or ever. oversely for markets charactersed by cotago the optoal method wll lead ether to the mmedate vestmet ether to the wat for a hgher prce. The proect wll be abadoed ay case. ETION 6. ONLION Ths study shows that the value of a real opto ad the vestmet decso strogly deped o the method used for the valuato of the et future cash flows assocated wth a vestmet proect. Ideed the smulatos dcate that the assumptos o the dyamc behavour of the state varables the term structure model have a cosderable fluece o the proect s value ad o the vestmet decso. The aalyss based o the two-factor model ca be cosdered as the most relable because the latter provdes a covcg represetato of the prces term structure for dstat maturtes whch s ot the case of Brea ad chwartz s model. 9

20 The results gve rse to certa prudece towards the use of the optoal method for the vestmet decso. Frstly the optoal method does ot seem to be more terestg tha the et preset value for markets that are most of the tme backwardato. Ideed that case the vestor s choce cossts vestg ow or ever whatever the crtero retaed. ecodly eve wth a model a pror pertet lke chwartz s model havg o market prces for very dstat maturtes oe ca ot be sure of the lablty of the aalyss for these exprato dates. Thrdly the dstortos of the term structure of prces ca sometmes be frequet ad rapd leadg to dfferet vestmet sgals at very close valuato dates. REFERENE BHAPP R.R. & GZMAN J. 995 «Meral vestmet decso makg : a study of mg compay practces» Egeerg ad Mg Joural pp July. BELLALAH M. 990 «Quatre essas sur l évaluato des optos sur dce et des optos sur cotrat à terme d dce» Ph D thess versty Pars IX. BELLALAH M. 00 «Le chox des vestssemets les optos réelles et l'formato : ue revue de la lttérature» forthcomg. BERNANKE B «Irreversblty ucertaty ad cyclcal vestmet» The Quarterly Joural of Ecoomcs pp February. BRENNAN M.J. & HWARTZ E «Evaluatg atural resources vestmets» The Joural of Busess vol 58. BRENNAN M.J. & TRIGEORGI L. 999 Proect flexblty agecy ad product market competto : ew developmets the theory ad applcato of real optos aalyss Oxford versty Press. OPELAND T. & ANTIKAROV V. 00 Real optos :a practtoer s gude Texere 30 p. ORTAZAR G. & HWARTZ E «Mote-carlo evaluato model of a udeveloped ol feld» Joural of Eergy Face ad Developmet vol. 3 pp ORTAZAR G. HWARTZ E.. & AA J. 00 «Optmal explorato vestmets uder prce ad geologcal-techcal ucertaty: a real optos model»; R & D Maagemet Aprl vol. 3. DIXIT A. & PINDYK R. 994 Ivestmet uder ucertaty Prceto versty Press. FRIMPONG. & J.M. WHITING J.M. 997 «Dervatve me valuato: strategc vestmet decsos compettve markets» Resources Polcy vol 3 4 pp GIBON R. & HWARTZ E «Valuato of log term ol-lked assets» Workg Paper Aderso Graduate chool of Maagemet versty of alfora Los Ageles. GRINBLATT M. & TITMAN. 00 Facal markets ad corporate strategy d edto Mc Graw Hll 880 p. The maturty s o loger tha 7 years the crude ol market whch s however the market wth the logest maturty. 0

21 LATIER D. 000 «La structure par terme des prx des matères premères : aalyse théorque et applcatos au marché pétroler» PhD Thess Pars IX versty. LATIER D. & GALLI A. 00 «modèle de structure par terme des prx du pétrole brut avec comportemet asymétrque du redemet d opportuté» Féco vol p LATIER A. 00 «Tros modèles de structure par terme des prx du pétrole brut : ue comparaso» Baque et Marchés mars-avrl 57. LATIER D. 003 «Les optos réelles ue dée sédusate u strumet facle à créer mas dffcle à valorser» Ecoomes et océtés ma. MAJD. & PINDYK R «Tme to buld opto value ad vestmet decsos» Joural of Facal Ecoomcs vol 8 pp 7-7. MDONALD R. & IEGEL D. 986 «The value of watg to vest» Quarterly Joural of Ecoomcs vol. 0 pp MYER «Determats of corporate borrowg» Joural of Facal Ecoomcs vol 5 pp PINDYK R. 980 «certaty ad exhaustble resource markets» Joural of Poltcal Ecoomy vol pp PINDYK R. 99 «Irreversblty ucertaty ad vestmet» Joural of Ecoomc Lterature vol 9 pp 0-48 eptember. HWARTZ E «The stochastc behavor of commodty prces : mplcatos for valuato ad hedgg» The Joural of Face vol LII 3 July. HWARTZ E.. & MITH J.E. 000 «hort-term Varatos ad Log-Term Dyamcs ommodty Prces» Maagemet cece July vol pp TRIGEORGI T. 999 Real optos MIT press. APPENDIX: NMERIAL REOLTION METHOD ED FOR THE VALATION OF THE REAL OPTION AOIATED WITH HWARTZ MODEL Ths appedx presets the method used for the resoluto of the valuato equato of the real opto assocated wth chwartz s model. Ths method authorzes the resoluto of a partal dervatves equato two steps despte the exstece of a cross partal dervatve. The partal dervatves equato s frst of all dscretzed. The we apply to t a alterate drecto method. The value of the opto to vest satsfes the followg dfferetal equato : σ V σ V ρσσv r V [ κ α ] V Vτ rv 0 α α λ κ wth: ad : V T T max[ NPV T T 0]

22 The soluto of ths equato s acheved wth a fte dfferece method: ts partal dervatves are replaced by approxmatos obtaed wth Taylor expasos. Amog the fte dfferece methods we retaed the rak-ncholso method for ts stablty.. DIRETIATION METHOD The grd s costtuted of the pots h h k the space t such as: h for 0 m h for m m t k for 0 The opto value s represeted by a scheme three dmesos:. V Vhhk Vt.. APPROXIMATION OF THE PARTIAL DERIVATIVE We obta the approxmatos applyg Taylor expasos aroud the pots k. At each perod s T - k the frst ad secod order dervatves wth respect to ad ad the dervatve wth respect to the tme are: h h h ² h 4 h k t We defe the followg operators o the varables: [ ] [ ] [ ] [ ] [ ] [ ].. APPROXIMATION OF THE PARTIAL DERIVATIVE EQATION Replacg the partal dervatves by ther approxmato the partal dervatves equato we obta for each k T s at the pots h ad h the followg expresso: { } 4 k r h h h r h h α κ σ ρσ σ σ ² 4 4 rk h h k h r k h k h k k z κ α σ ρ σ σ σ z I addto we have: :

23 z z Replacg z wth ts value we obta the system: σ k σk ρσ σk k 4 4h² 8h σ k σk ρσ σk k 4 4h² 8h r h k κ α h 4 µ h k κ α h 4 4h 4h rk rk The rak-ncholso scheme cetred space ca also be wrtte the followg way : { T V W X Y Z } { V W X Y Z } wth : V σ k 4R W k σ 4h R X ρσ σ k hr Y k r h 4R Z k α h 4hR R rk 8 κ T rk R for m ad m m. We the apply a Alteratve Drecto Implct method to ths scheme. It cossts troducg a termedary term the system R to separate t at each tme step to two subsystems. Each of the latter leads to a trdagoal system correspodg to oe dmeso space. The two sub-systems are successvely solved for each tme step. R. ALTERNATIVE DIRETION METHOD The ADI scheme for parabolc equatos requres the resoluto of two trdagoal systems at each tme step. The frst system s solved for a fxed the secod for a fxed. We spre us of the ADI scheme whch was proposed by Mtchell ad Mc Kee 970. It eables the resoluto of the parabolc equato: u t Lu wth : L a x y t b x y t c x y t x xy y Ths scheme authorses the resoluto of the parabolc equato two steps despte the presece of a cross dervatve. sg a separatg method close to the oe proposed by Douglas ad Rachford 956 the termedary term at the pot Y s defed such as: [ V Y ] [ V ] R R s the frst sub system. It authorzes the treatmet of o the dmeso. R s the mapulated order to obta the secod sub-system. If we elmate the termedary term each of the two sub-systems we retur to R. R s multpled by [ W Z ]: [ V Y ][ W Z ] [ W Z ] [ W Z ][ Y V ] R3 3

24 I a rak Ncholso scheme the approxmato errors are of order. Therefore : { } 0 { } 0 { } 0 { } 0 t k t k Let us troduce the followg term R: { } YW VZ VW YZ X we obta the system R4 : { } { } YW VZ VW YZ Z Y X W V T YW VZ VW YZ Z Y W V ubtractg R3 from R4 we have: 0 YW VZ VW YZ Z Y X W V T V Y Z W Z W Ths subtracto has permtted the elmato of the term. A smplfcato leads to the secod sub-system: Z Y X W V T Z W R5 Wth ths sub-system for each fxed value of t s possble to compute whe vares. We have also defed R whch authorzes for each fxed value of the computato of whe vares: [ ] [ ] V Y Y V R5 gves the values of as a fucto of the values -. R gves the values of as a fucto of the values - ad... FIRT TEP OF THE REOLTION The frst step of the resoluto correspods to the sub-system R5 whch ca be rewrtte the followg way: 4 X Z W Z W Y V Y V W V T Z W whch s a trdagoal system: d c b a wth : a Z - W b W c - Z - W 4

25 d X T 4V W V Y W Z W Z V Y We therefore have a relatoshp - betwee 3 values of - aroud for a fxed. These three values are expressed as a fucto of the passed value -. The relatoshp s mplct: the values of - ca ot be detfed separately or ca they be obtaed explctly wth ths passed value. At each tme step ad for each fxed we ca wrte: M - d where M s a trdagoal matrx wth the coeffcet a b c o each of ts dagoals ad where - ad d are vectors. Each vector d s expressed as a fucto of the passed values -. The values of the vectors - are obtaed by the resoluto of - M - d... EOND TEP OF THE REOLTION The secod step of the resoluto cossts solvg the followg sub-system: [ V Y ] [ Y V ] R Replacg each of the operators by ts value we ca wrte: V V Y V Y a ' Y V V V Y We obta the a secod trdagoal system : b ' wth : - a Y - V - b V - c - V - Y - c ' d ' d ' Y V V V Y Though we have a relatoshp betwee three values of aroud for a fxed. These values are expressed as a fucto of the passed values - computed durg the frst step of the resoluto ad of the values -. We ca wrte ths system at each tme step ad for each fxed the followg way: M d where M s a trdagoal matrx wth the coeffcets a b c o each dagoal ad where ad d are vectors. The latter are expressed as a fucto of the values of - computed durg the last step ad of the values of -. The values of the vectors are obtaed by the resoluto of M - d..3. BONDARY ONDITION The boudary codto whe the opto expres s the followg: [ NPV 0] V T T max : The use of a mplct fte dfferece method ecesstates the expresso of addtoal boudary codtos. We must express the codtos of the system at the lmts of ad at each tme step. For each maturty τ we reta the followg codtos: - V0τ 0 5

26 - V m τ - V mτ 0 - V -m τ Because the opto s Amerca we also troduce the followg codto that must be verfed at each tme step: V τ max [V τ NPV ] The codto V m τ must have a specfc treatmet. We ca approach ths value at the pot m wth a explct scheme : h{ m 4 m 3 m } h 4 m 3 m m Ths expresso s combed wth the scheme at the pot m- order to preserve the trdagoal structure of the system. The scheme m- s the followg: am m bm m cm m d m bm m cm m d m am m If we replace m- by ts value we obta : [ h 4 m 3 m ] b m m c m m d m a m [ b m 4a m ] m [ c m 3a m ] m d m ha m The coeffcets a b c ad d have therefore the followg values whe m : am bm-4am- bm cm- -3am- cm 0 dm dm- - ham- 6