Exotic electricity options and the valuation of electricity generation and transmission assets

Transcription

1 Decision Suppor Sysems wwwelseviercomrlocaerdsw Exoic elecriciy opions and he valuaion of elecriciy generaion and ransmission asses Shi-Jie Deng a,), Blake Johnson b, Aram Sogomonian c a IE and OR Deparmen, UniÕersiy of California a Berkeley, CA 94720, USA b EES and OR Deparmen, Sanford UniÕersiy, CA 94305, USA c Risk Managemen, Pacificorp, Porland, OR 97232, USA Absrac We presen and apply a mehodology for valuing elecriciy derivaives by consrucing replicaing porfolios from elecriciy fuures and he risk-free asse Fuures-based replicaion is made necessary by he non-sorable naure of elecriciy, which rules ou he radiional spo marke, sorage-based mehod of valuing commodiy derivaives Using he fuures-based approach, valuaion formulae are derived for boh spark and locaional spread opions for boh geomeric Brownian moion and mean revering price processes hese valuaion resuls are in urn used o consruc real opions-based valuaion formulae for generaion and ransmission asses Finally, he valuaion formula derived for generaion asses is used o value a sample of asses ha have been recenly sold, and he heoreical values calculaed are compared o he observed sales prices of he asses q 2001 Elsevier Science BV All righs reserved Keywords: Elecriciy derivaives; Spark spread; Mean reversion; Exchange opion; Elecriciy fuures conrac; Real opions; Capaciy valuaion 1 Inroducion Wih deregulaion sweeping hrough he US elecric power indusry and a fully compeiive markeplace for elecriciy aking shape, elecric uiliies and heir cusomers accusomed o a cos-recovery pricing srucure for elecriciy mus adap o marke-based pricing Risk managemen needs his ransiion has generaed have made elecriciy derivaives one of he fases growing derivaives markes, ) Corresponding auhor Curren address: School of Indusrial and Sysems Engineering, Georgia Insiue of echnology Alana, GA , USA addresses: deng@isyegaechedu URL:rrwwwisyegaechedur ;deng S-J Deng, blakej@lelandsanfordedu B Johnson, aramsogomonian@pacificorpcom A Sogomonian as financial insiuions, uiliies and oher energy marke paricipans work o provide he ools necessary o manage he price and invesmen risks associaed wih compeiive markes While many of he risk-managemen ools and mehods now well esabw1,2,5,8x can be readily rans- lished in oher markes ferred o he elecriciy markes, he unique characerisics of elecriciy and elecriciy markes also presen new challenges o he risk-managemen discipline he mos imporan of hese are he challenges ha he non-sorable naure of elecriciy presens o he radiional mehods of modeling price processes and valuing derivaives w11 x Specifically, due o he non-sorable naure of elecriciy, he radiional sorage-based, no-arbirage mehods of valuing commodiy derivaives are unavailable In addiion, elecriciy prices can and do demonsrae r01r$ - see fron maer q 2001 Elsevier Science BV All righs reserved PII: S

2 384 S-J Deng e alrdecision Suppor Sysems properies such as srong mean reversion over shor ime horizons ha would be inconsisen wih an efficien marke for a sorable good A second riskmanagemen challenge ha elecriciy markes presen is he need o value a range of cross-commodiy ransacions, such as spark and locaional spreads wx 3 In his paper, we presen ools o address hese unique properies of elecriciy and elecriciy derivaives Firs, we develop a mehod o value elecriciy derivaives by replicaing hem wih fuures conracs raher han by aemping o sore or borrow elecriciy in he spo marke his allows us o apply radiional no-arbirage-based mehods of derivaives valuaion and o proceed wihou requiring he assumpion ha elecriciy is sorable We hen presen closed-form expressions for he value of a range of cross-commodiy derivaives, including spark and locaional spread opions, boh for he case in which he underlying price processes follow geomeric Brownian moion, and for he more plausible case in which prices are mean revering hese resuls are closely relaed o hose of Shimko s wx 9 analysis of fuures spread opions and Margrabe s wx 7 analysis of exchange opions Margrabe s work is relevan since an exchange opion can be hough of a spread opion wih a zero srike price Shimko s resuls, however, are for a fuures price process derived from a model of he spo price and convenience yield of a sorable good, while Margrabe s are exclusively for geomeric Brownian moion processes Afer deriving he valuaion formulae, we demonsrae how hese resuls can be used o value boh generaion and ransmission asses see w4,10x for a review of real opions and decision analyic approaches o capaciy valuaion, and presen a preliminary comparison beween he values hese models generae and he acual prices a which hese ypes of asses have recenly been sold he remainder of he paper is organized as follows In Secion 2, we inroduce he se of crosscommodiy derivaives we will consider in he paper, and idenify some of heir basic characerisics In Secion 3, we describe how hese derivaives can be replicaed and hus valued by arbirage using fuures conracs, and presen he principal valuaion resuls of he paper In Secion 4, we use hese resuls o develop a real-opions-based mehodology for valuing generaion and ransmission asses, and presen he resuls of our preliminary empirical evaluaion of he effeciveness of he mehodology 2 Cross-commodiy elecriciy derivaives here are wo principal caegories of cross-commodiy elecriciy derivaives; spark spread, or hearae-linked derivaives, and locaional spread derivaives We consider each below 21 Spark spread, or hea rae-linked deriõaiões he primary cross-commodiy ransacion in elecwx 6, which is based riciy markes is he spark spread on he difference beween he price of elecriciy and he price of a paricular fuel used o generae i he spread beween he price of elecriciy and a fuel ha can be used o generae i is of ineres since i is his spread ha deermines he economic value of generaion asses ha can be used o ransform he fuel ino elecriciy his idea is analogous o he concep of he Acrack spreadb used in he oilrrefining indusry In ha case he relevan spread beween crude oil and refined producs like diesel or gasoline see Shimko wx 9 he amoun of fuel ha a paricular generaion asse requires o generae a given amoun of elecriciy will of course depend on he asse s efficiency his efficiency is summarized by he asse s hea rae, which is defined as he number of Briish hermal unis Bu of he inpu fuel mea- sured in millions required o generae 1 megawa hour MWh of elecriciy hus, he lower he hea rae, he more efficien he faciliy he spark spread associaed wih a paricular hea rae is defined as he curren price of elecriciy less he produc of he hea rae and he curren fuel price hus, he lower he hea rae, he lower he fuel price, and he higher he elecriciy price, he larger he spark spread In a deregulaed marke, presumably only asses ha have a posiive spark spreads under prevailing marke condiions will be operaed his leads naurally o he definiion of he prevailing marke implied hea rae as: snumber of MM Bu needed for a marginal generaing plan o generae 1 MW h of elecriciy S E s MM BurMW h 1 S G

3 S-J Deng e alrdecision Suppor Sysems where SE is he spo price of elecriciy per MWh and SG is he spo price of he generaing fuel per MM Bu Wih he noion of a hea rae esablished, we define European spark spread pu and call opions Definiion 1 A European spark spread call opion wrien on fuel G a a fixed or AsrikeB hea rae K gives he opion holder he righ bu no he obligaion o pay K imes he uni price of fuel G a he opion s mauriy and receive he price of 1 uni of elecriciy Le SE and SG be he uni spo prices of elecriciy and fuel a ime, respecively Denoe he value of he opion a ime by C1 S E, S, G hen he payoff of he opion a mauriy ime is: C S, S, K, smax S yk S,0 2 1 E G E G Definiion 2 A European spark spread pu opion wrien on fuel G a a fixed hea rae K gives he opion holder he righ bu no he obligaion o pay he price of 1 uni of elecriciy and receive K imes he uni price of fuel G a mauriy ime Denoe he value of he opion a ime by P1 S E, S, hen he payoff of he opion a ime is: G P S, S, K, smax K S ys,0 3 1 E G G E he following example provides a simple illusraion of how spark spread opions can be used o manage elecriciy price risk v A power markeer in a region where he marginal generaing fuel is naural gas would like o buy power a ime a a marke implied hea rae no o exceed K An agreemen providing such a hea rae cap would ensure he markeer power a ime a a price given by S = min, K G Assuming he markeer sells he power ino he spo marke a ime, his payoff will be S y S = min, K E G, which is equal o S =max yk, 0 G Bringing S G inside he brackes, he payoff is max SE yks G, 0, which is exacly he same as ha of a European spark spread call opion wih srike hea rae K he power markeer can herefore achieve his goal by purchasing his spark spread call hroughou he remainder of he aricle, we make he following assumpions Assumpion 1 A complee se of fuures conracs for elecriciy and for he relevan generaing fuels are raded Assumpion 2 he risk-free ineres rae r is consan wx Following Shimko 9, we nex provide a pu-call pariy relaionship beween he spark spread pu and call opions, as well as upper and lower bounds on heir values We delay making specific assumpions abou he price processes ha elecriciy and he generaing fuels follow unil Secion 3 Proposiion 1 Pu-Call Pariy Le FE and FG denoe he fuures prices of elecriciy and he generaing fuel, respeciõely he following pariy relaionship holds for European spark spread pu and call opions wih he same fixed hea rae K and expiraion dae C sp qe yr F yk F E G Proof A ime, he payoff of a long posiion in 1 uni of spark spread call opion C1 is max SEy K S, 0 G ; he payoff of 1 uni of spark spread pu opion P is max K S y S, 0 1 G E Consequenly, he payoff of C1yP1 a mauriy ime is SEy yr KS G he presen value of SEyKSG is e F yk F herefore, E G C yp se yr F yk F E G Proposiion 2 ( No-arbirage lower r upper bounds) Le FE and FG denoe he fuures prices of elecriciy and he generaing fuel, respeciõely hen he Õalue of a spark spread call opion C1 can be bounded aboõe and below as follows: e yr max F yk F,0 F C F e yr F 6 E G 1 E Proof he firs inequaliy is by pu-call pariy 4 and he fac P1 G0 he second inequaliy is due o he fac max SEyKS G,0FSE a ime and he presen values of max SEyKS G, 0 and SE are C1 and e - r FE, respecively B B

4 386 S-J Deng e alrdecision Suppor Sysems Locaional spread opions Due o ransmission coss and consrains noe ha he impac of ransmission consrains is compounded by he non-sorabiliy of elecriciy, which forces real ime delivery, subsanial differences frequenly exis beween he price of elecriciy a differen locaions We refer o hese differences as locaional spreads, and define call opions on hem as follows Definiion 3 Locaional spread : a European call opion on he locaional spread beween locaion one and locaion wo wih mauriy gives is holder he righ bu no he obligaion o pay he price of 1 uni of elecriciy a locaion one a ime and receive he price of K L unis of elecriciy a locaion wo ransmission cos from locaion 1 o 2 can be incor- poraed by seing K L being less han 1 Le Si be he uni price of elecriciy a locaion i is1, 2 a ime Denoe he value of he opion a ime by C S, S, K, L hen he payoff of he opion a ime is: C S,S, K, smax S yk S, L 1 L 2 A European locaional spread pu opion can be defined in a similar way 3 Valuaion of elecriciy derivaives In his secion, we presen a fuures-based mehod of replicaing elecriciy derivaives, and illusrae he mehod by using i o derive explici expressions for he value of he spark spread and locaional spread opions defined above Valuaion equaions are provided for hese insrumens for boh geomeric Brownian moion price processes and mean-revering price processes In boh cases we explicily derive only he value of he call opions he value of pu opions can hen be derived using he pu-call pariy relaionship presened in Secion 2 31 Fuures-based replicaion of elecriciy deriõaiões As noed above, because elecriciy is non-sorable, he radiional sorage-based mehods of consrucing replicaing porfolios for commodiy derivaives canno be used o value elecriciy derivaives In place of he sorage-based mehods, we presen a mehod for replicaing elecriciy derivaives by dynamically rading fuures conracs of he appropriae mauriy Since, a mauriy, he price of a fuures conrac mus converge o he hen curren spo price, he mehodology permis exac replicaion Since he precise naure of he replicaing sraegy will naurally depend on he specific derivaive being replicaed, o illusrae he mehod we use i o derive he replicaing sraegy for spark and locaional spread opions We do so firs under he assumpion ha he relevan fuures price processes follow geomeric Brownian moion processes, and hen under he more reasonable assumpion ha hey follow mean revering processes 32 Geomeric Brownian moion price process We firs consider he case in which he fuures price processes of elecriciy and he appropriae generaing fuel of he relevan mauriy, FE and F G, follow geomeric Brownian moion processes d F rf sm dqs d B 1 E E E E d F rf sm dqs d B 2 G G G G 8 where B 1 and B 2 are wo Wiener processes wih insananeous correlaion r m E, m G, s E, and sg are assumed o be consans for he momen he valuaion resuls are herefore he same as hose in Ref wx 7 excep ha he underlying are fuures conracs he more general case where he volailiy and correlaion parameers can be funcions of ime is considered in he mean-reversion model 321 Valuaion of spark spread opions Denoe he ime- value of a spark spread call opion which maures a ime by V x, y, ',,, C1 F E, F G, K, y and le F) represen he price a ime of he commodiy fuures conrac wih mauriy dae By consrucing an insananeously risk-free porfolio using he elecriciy and generaing fuel fuures conracs and he riskless asse, i follows ha C 1, normalized by he value of

5 S-J Deng e alrdecision Suppor Sysems he risk-free asse, mus saisfy he parial differenial equaion PDE : yvq x Vxxsx q2 rsxsyxyvxyqy Vyysy 2 s0 9 wih boundary condiions V x, y, smaxxyy, 0,V x,0, sx, and V 0, y, s0 Proposiion 3 ( Value of a spark spread call opion) he closed-form soluion for C1 is: C F,, F,, K, y 1 E G yr y,, se FE N d1 ykfg N d2 10 where ln FE, r KFG, qõ 2 y r2õ d 1 s ' y d sd yõ' 2 1 y Õ ssey2 rsesgqsg,, Proof Verify VsC 1 F E, F G, K, y e ry solves PDE 9 wih he corresponding boundary condiions B 322 Valuaion of locaional spread opions he value of he locaional spread call opion can be derived in exacly he same way he value of he spark spread call opion was derived above Specifically, defining FE,1 and FE,2 o be he geomeric Brownian moion price processes ha he fuures prices of elecriciy a locaions 1 and 2 follow, we have: Proposiion 4 ( Value of a locaional spread call opion) he Õalue of C2 is giõen by C2 FE,1,, FE,2,, K L, y yr y,, se F N d yk F N d 11 E,1 1 L E,2 2 where ln FE,1, rklfe,2, qõ 2 y r2 d 1 s Õ ' y d sd yõ' 2 1 y Õ sse,1y2 rse,1se,2qse,2 33 Mean-reÕering price process In his secion, we assume ha he fuures price processes of elecriciy FE and of he relevan generaing fuel F follow he mean-revering processes G d FEskE me yln FE FEdqsE FEd B 1 d FGskG mg yln FG FGdqsG FGd B 2 12 where s and s E G are funcions of ime, m and m E G are he long-erm means, k E and kg are he mean-revering coefficiens, and B 1 and B 2 are, as above, wo Wiener processes wih insananeous correlaion r he mean-revering assumpion on he fuures price processes pu resricions on he choice of volailiy funcions s and s E G Among many oher forms, he following is a feasible one s ss e yk E ), s ss e yk G ) E E G G Valuaion of spark spread opions Denoe he ime value of a spark spread call opion which maures a ime by V x, y, ',, C F, F, K, y 1 E G Applying he same replicaion argumens applied above, i follows ha C 1 e ry mus saisfy he PDE: yvq x Vxxsx q2 rsx sy xyvxy qy V s s0 14 yy y wih boundary condiions V x, y, smaxxyy, 0,V x,0, sx, and V 0, y, s0 Proposiion 5 ( Value of a spark spread call opion) he closed-form soluion for C1 is: C F,, F,, K,y 1 E G yr y,, se FE N d1 ykfg N d2 15 where ln FE, r KFG, qõ 2 y r2 d 1 s Õ ' y d sd yõ' 2 1 y 2 Õ s 2 2 E E G G s s y2 rs s s s qs s d s y

6 388 S-J Deng e alrdecision Suppor Sysems Fig 1 Value of spark spread call under mean-reversion and GBM,, Proof Verify VsC 1 F E, F G, K, y e ry solves PDE 14 wih he corresponding boundary condiions B Fig 1 illusraes he value of spark spread opions under boh he geomeric Brownian moion GBM price process assumpion and he mean-revering price process assumpion When he underlying price process is acually mean-revering, he geomeric Brownian moion assumpion leads o he overvaluaion of spark spread opions, especially hose wih long mauriies Several comparaive saic properies of he spark spread call opion value can be derived by invesigaing he sign of he parial derivaives of C1 wih respec o heir parameers Proposiion 6,, As FE p increases, or FG o decreases C1p increases, F, E FG p and p C1p, F G ro or ro C 1 p Valuaion of locaional spread opions Defining FE,1 and FE,2 o be he mean-revering price processes ha govern he fuures prices of elecriciy a locaions 1 and 2 and following he derivaion above, we have: Proposiion 7 ( Value of a locaional spread opion) he Õalue of C2 is giõen by C2 FE,1,,FE,2,, K L, y yr y,, se F N d yk F N d 17 where E,1 1 L E,2 2,, ln FE,1rFE,2 q Õ 2 y r2 d 1 s Õ' y d s d y Õ' 2 1 y 2 Õ s 2 2 E,1 E,1 E,2 E,2 w x s s y2 rs s s s qs s d s y 4 Real opions valuaion of generaion and ransmission asses he righ o operae a generaion asse wih hea rae ha uses generaing fuel G is clearly given

7 S-J Deng e alrdecision Suppor Sysems by he value of a spark spread opion wih AsrikeB hea rae wrien on generaing fuel G Similarly, he value of a ransmission asse ha connecs locaion 1 o locaion 2 is equal o he sum of he value of he locaional spread opion o buy elecriciy a locaion 1 and sell i a locaion 2 and he value of he opion o buy elecriciy a locaion 2 and sell i a locaion 1 in boh cases, less he appropriae ransmission cos his equivalence beween he value of appropriaely defined spark and locaional spread opions and he righ o operae a generaion or a ransmission asse can be easily used o value such asses In his secion we illusrae his approach by developing a simple spark spread-based model of he value of a gas-fired generaion asse Once esablished, we fi he model and use i o generae esimaes of he value of several gas-fired plans ha have recenly been sold he accuracy of he model is hen evaluaed by comparing he esimaes consruced o he prices a which he asses were acually sold In he analysis we make he following simplifying assumpions abou he operaing characerisics of he generaion asses under consideraion Assumpion 3 Ramp-ups and ramp-downs of he faciliy can be done wih very lile advance noice Assumpion 4 he faciliy s operaion eg, saruprshudown coss and mainenance coss are consan hese assumpions are reasonable, since for a ypical gas urbine combined cycle cogeneraion plan he response ime ramp uprdown is several hours and he variable coss eg, operaion and mainenance are generally sable over ime o consruc a spark spread-based esimae of he value of a generaion asse, we esimae he value of he righ o operae he asse over is remaining useful life his value can be found by inegraing he value of he spark spread opions over he remaining life of he asse Specifically, Definiion 4 Le 1 uni of he ime capaciy righ of a naural gas-fired elecric power plan represen he righ o conver K unis of naural gas ino 1 uni of elecriciy by using he plan a ime, where K is he plan s hea rae he payoff of 1 uni of ime- capaciy righ is max SEyKS G, 0, where SE and SG are he spo prices of elecriciy and naural gas a ime, respecively Denoe he value of 1 uni of he ime capaciy righ by u Definiion 5 Denoe he virual value of 1 uni of capaciy of a gas-fired power plan by V hen, V is equal o 1 uni of he plan s ime- capaciy righ over he remaining life w0, x of he power plan, ie Vs ud 0 Wihou making any disribuional assumpions abou he price processes ha SE and SG follow, we have he following proposiion Proposiion 8 he Õalue of 1 uni of capaciy V of a plan ha has a useful life of has boh a lower bound and an upper bound: yr yr E G E 0 0 e max F yk F,0 dfvf e F d Proof By definiion and Proposiion 2 18 If we furher assume ha he price processes of elecriciy and naural gas spo and fuures prices follow he mean-revering processes as given by 12, hen we have us C where C 1 1 is given by Proposiion 5 he value of a gas-fired power plan wih lifeime is herefore V s C d 19 gen 1 0 Similarly, if we assume ha he price processes of he elecriciy fuures prices a wo differen locaions follow mean-revering processes, he value of a ransmission line connecing he wo locaions a and b in a nework is V s C qc d 20 ran 2,1 2,2 0 where C and C 2,1 2,2 represen he locaional spread opion value a ime from a o b and from b o a, respecively Eqs 19 and 20 are he wo B

8 390 S-J Deng e alrdecision Suppor Sysems Fig 2 Elecriciy and naural gas fuures price curve fundamenal valuaion formulae we propose for he valuaion of generaion and ransmission asses in a compeiive elecriciy marke 41 Applicaion of he model o recen generaion asse sales o evaluae he accuracy of 19, we fi he model and use i o consruc esimaes of he value of several generaion asses ha have been recenly sold For purposes of comparison, we also esimae he value of each asse using a sandard discouned cash flow DCF calculaion In order o fi he model, we firs esimae he volailiies of he price processes of he relevan fuures conracs Le fn be he price of he fuures conrac ha maures in n monhs, and assume ha fn follows a mean-revering process of he kind considered above Le R n' ln fn, hen d R sk ˆ m yr dqs d B 21 n 1 n n n We esimae sn using he New York Mercanile Exchange NYMEX elecriciy fuures hisorical price daa he naural gas volailiy erm srucure and he gas-o-elecriciy price correlaion are also esimaed using hisorical daa on he NYMEX naural gas enry ub fuures conracs Once esimaed, hese parameers along wih implied volailiy from he marke raded call opions are used o calibrae he parameers in he volailiy funcions 13 We will use 13 in he valuaion formulae derived in Secion 3 o consruc real-opions-based esimaes of he value of he asses in quesion o calculae he discouned cash flow value of he asses we use he relevan elecriciy and naural gas fuures curves he sample of generaion asses considered consiss of four gas-fired power plans which Souhern California Edison recenly sold o ouson Indusries A presen, no all of he individual plan dollar invesmens have been made public As a proxy, we use he oal invesmen made by ouson Indusries US$237 million o purchase four plans Coolwaer, Ellwood, Eiwanda and Mandalay, divided by he oal number of megawas MW of capaciy 2172 MW o ge approximaely US$110,000rMW or US$110rkW of capaciy for he enire package of plans owever, he Coolwaer Plan 1 Dagge, CA is he mos efficien wih an average hea rae of 9500 of he four plans in he package and hus 1 he Coolwaer Plan is made up of four unis: wo 256-MW combined cycle gas urbines wih a seam urbine, and wo convenional urbines wih capaciy 65 MW and 81 MW each Some repower work has been done on he larger unis

9 S-J Deng e alrdecision Suppor Sysems Fig 3 Capaciy value of a gas-fired plan should have a higher value per MW We herefore assume ha he implied marke value for Coolwaer could range from US$110,000 o US$220,000 per MW, or equivalenly, US$110rkW o US$220rkW Using he NYMEX elecriciy and naural gas fuures price daa on Ocober 15, 1997, see Fig 2, we compue boh he opion value and DCF value using a risk-adjused discoun rae of 10% of a gas-fired plan wih various possible hea raes assuming a remaining operaing life of 15 years for he plan I is also assumed he plan is operaed during peak hours only Fig 3 shows he plo of he opion values and he DCF values of a plan of various possible hea raes using forward curves a differen imes and a differen rading hubs We can see ha he opion values of capaciy are significanly higher han he DCF values For hea rae higher han 9500 BurkW h, he DCF values of capaciy are close o zero For example, a he hea rae level of 9500 and using he elecriciy forward curve a Palo Verde PV, he heoreical opion-based capaciy value of a plan comparable o he Coolwaer Plan 5 is around US$185rkW, while he DCF valuaion is only US$28rkW Remark he naural elecriciy rading hub for Coolwaer o sell ino is he Mead hub owever, due o he liquidiy of he Palo Verde financial fuures conracs we use he PV fuures conracs as a proxy for he elecriciy price informaion for Coolwaer In addiion, he basis differenial associaed wih PVrMead is ypically no large 5 Conclusions his aricle has presened a mehodology for valuing elecriciy derivaives by consrucing replicaing porfolios wih fuures conracs and he risk-free asse Fuures-based replicaion is made necessary by he non-sorable naure of elecriciy, which rules ou he radiional spo marke, sorage-based mehod of valuing commodiy derivaives Once developed, he mehodology was used o derive valuaion formulae for boh spark and locaional spread opions when he prices of he underlying asses follow eiher geomeric Brownian moion or mean revering processes hese valuaion resuls were in urn used o consruc real-opions-based valuaion formulae for generaion and ransmission asses Applicaion of he generaion asse valuaion formula o a sample of recen asse sales suggess ha he spark spread analysis generaes reasonable esimaes of he acual marke value of he asses, and cerainly more accurae esimaes han hose which radiional DCF mehods provide he esimaes generaed could be

10 392 S-J Deng e alrdecision Suppor Sysems improved by incorporaing a greaer level of deail abou he plans, paricularly heir hourly and daily operaing opionaliy Analysis of his kind presens a naural avenue for fuure research References wx 1 F Black, he pricing of commodiy conracs, Journal of Financial Economics , Sep wx 2 F Black, M Scholes, he pricing of opions and corporae liabiliies, Journal of Poliical Economy , May June wx 3 S Deng, Financial mehods in compeiive elecriciy markes, PhD Disseraion forhcoming, Universiy of California a Berkeley, CA, 1998 wx 4 AK Dixi, RS Pindyck, Invesmen Under Uncerainy, Princeon Univ Press, NJ, 1994 wx 5 D Duffie, Dynamic Asse Pricing heory, 2nd edn, Princeon Univ Press, NJ, 1996 wx 6 M su, Spark spread opions are ho! he Elecriciy Journal , Elsevier Sciences, March wx 7 W Margrabe, he value of an opion o exchange one asse for anoher, Journal of Finance wx 8 RC Meron, heory of Raional Opion Pricing, Bell Journal of Economics and Managemen Science , Spring wx 9 D Shimko, Opions on fuures spreads: hedging, speculaion and valuaion, Journal of Fuures Markes w10x JE Smih, RF Nau, Valuing risky projecs: opion pricing heory and decision analysis, Managemen Science 41 no w11x ES Schwarz, he sochasic behavior of commodiy prices: implicaions for valuaion and hedging, Journal of Finance versiy Dr Blake Johnson is an Assisan Professor of Engineering Economic Sysems and Operaions Research a Sanford Universiy is research ineress include he applicaion of muliperiod asse pricing mehods o he valuaion of real asses eg, businesses, echnologies, large projecs, real esae as opposed o financial asses eg, socks, bonds, derivaives Dr Johnson has a PhD in Engineering Economic Sysems from Sanford Uni- Dr Aram Sogomonian was recenly named Chief Risk Officer of PacifiCorp e is currenly implemening his organizaion which will have primary responsibiliy for undersanding he firms risk exposures and how o manage hem Prior o his curren posiion, Dr Sogomonian was vice presiden a Pacifi- Corp Power Markeing wih primary responsibiliy for heading up he middle once and analyical funcions Prior o PacifiCorp, Dr Sogomonian was vice presiden of Risk Managemen a Edison Source Before coming o Edison Source, Dr Sogomonian was direcor of he risk analyics and asse pricing group a ouson-based Enron Capial and rade resources, which does projec evaluaion for Enron Dr Sogomonian holds a PhD in Managemen Science from he Anderson Graduae School of Managemen a UCLA, an MS degree in Operaions Research and BA degrees in Applied Mahemaics and Economics from he Universiy of California a Berkeley Dr Shi-Jie Deng is an Assisan Professor of Indusrial and Sysems Engineering a Georgia Insiue of echnology is research ineress include financial asse pricing and real opions valuaion, financial engineering applicaions in energy commodiy markes, ransmission pricing in elecric power sysems, sochasic modeling and simulaion Dr Deng has served as a consulan o several privae and public organizaions on issues of risk managemen and asse valuaion in he deregulaed elecriciy indusry Dr Deng holds a BS degree in Applied Mahemaics from Beijing Universiy in China, an MS degree in Mahemaics from he Universiy of Minnesoa a win Ciies, as well as MS and PhD degrees in Indusrial Engineering and Operaions Research from he Universiy of California a Berkeley