Exotic Electricity Options and the Valuation. Assets. April 6, Abstract

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1 Exotic Elctricity Options and th Valuation of Elctricity Gnration and Transmission Assts Shiji Dn Blak Johnson y Aram Soomonian z April 6, 1998 Abstract This papr prsnts and applis a mthodoloy for valuin lctricity drivativs by constructin rplicatin portfolios from lctricity futurs and th risk fr asst. Futurs basd rplication is arud to b mad ncssary by th non-storabl natur of lctricity, which ruls out th traditional spot markt, stora-basd mthod of valuin commodity drivativs. Usin th futurs basd approach, valuation formula ar drivd for both spark and locational sprad options for both omtric Brownian motion and man rvrtin pric procsss. Ths valuation rsults ar in turn usd to construct ral options basd valuation formula for nration and transmission assts. Finally, th valuation formula drivd for nration assts is usd to valu a sampl of Ph.D candidat, IE & OR Dpt., Univrsity of California at Brkly, Brkly, CA dn@ior.brkly.du yassistant Profssor, EES & OR Dpt., Stanford Univrsity, Stanford, CA zvic Prsidnt, Risk Manamnt, Edison Entrpriss. 1

2 assts that hav bn rcntly sold, and th thortical valus calculatd ar compard to th obsrvd sals prics of th assts. 2

3 1 Introduction With drulation swpin throuh th US lctric powr industry and a fully comptitiv marktplac for lctricity takin shap, lctric utilitis and thir customrs accustomd to a cost-rcovry pricin structur for lctricity must adapt to markt basd pricin. Th risk manamnt nds this transition will nrat ar likly to mak lctricity drivativs on of th fastst rowin drivativs markts in th yars to com, as nancial institutions and othr nry markt participants work to provid th tools ncssary to mana th pric and invstmnt risks associatd with comptitiv markts. Whil many of th risk manamnt tools and mthods now wll stablishd in othr markts can b radily transfrrd to th lctricity markts, th uniqu charactristics of lctricity and lctricity markts also prsnt nw challns to th risk manamnt disciplin. Th most important of ths ar th challns that th non-storabl natur of lctricity prsnts to th traditional mthods of modlin pric procsss and valuin drivativs. Spcically, du to th non-storabl natur of lctricity, th traditional stora basd, no-arbitra mthods of valuin commodity drivativs ar unavailabl. As a consqunc, an ntirly nw mthodoloy is rquird to valu vn th simplst lctricity drivativs, such as lctricity futurs contracts. Also as a consqunc of non-storability, lctricity prics can (and do) dmonstrat proprtis such as stron man rvrsion ovr short tim horizons that would b inconsistnt with an cint markt for a storabl ood. A scond risk manamnt challn that lctricity markts prsnt is th nd to valu a ran of cross commodity transactions, such as spark and locational sprads. In this papr w prsnt tools for addrssin both of ths uniqu proprtis of lctricity and lctricity drivativs. W rst prsnt a mthod for valuin lctricity drivativs by rplicatin thm with futurs contracts rathr than by attmptin to stor or borrow lctricity in th spot markt. Th mthod thus allows traditional no- 3

4 arbitra basd mthods of drivativs valuation to procd without th implausibl assumption that lctricity can b stord. W thn prsnt closd form xprssions for th valu of a ran of cross-commodity drivativs, includin spark and locational sprad options. Th valuation formula ar providd both for th cas in which th undrlyin pric procsss ar omtric Brownian motion procsss, and for th mor plausibl cas in which thy ar man rvrtin. Finally, w dmonstrat how ths rsults can b usd to valu both nration and transmission assts, and prsnt a prliminary comparison btwn th valus ths modls nrat and th actual prics at which assts of this kind hav bn rcntly sold. Th rmaindr of th papr is oranizd as follows. In sction 2 w introduc th st of cross commodity drivativs w will considr in th papr, and idntify som of thir basic charactristics. In sction 3 w dscrib how ths drivativs can b rplicatd (and thus valud by arbitra) usin futurs contracts, and prsnt th principal valuation rsults of th papr. In sction 4 w us ths rsults to dvlop a ral options basd mthodoloy for valuin nration and transmission assts, and prsnt th rsults of our prliminary mpirical valuation of th ctivnss of th mthodoloy. 2 Cross Commodity Elctricity Drivativs Thr ar two principal catoris of cross-commodity lctricity drivativs; spark sprad, or hat rat linkd drivativs, and locational sprad drivativs. W considr ach blow. 4

5 2.1 Spark Sprad, or Hat Rat Linkd Drivativs Th primary cross-commodity transaction in lctricity markts is th spark sprad, which is basd on th dirnc btwn th pric of lctricity and th pric of a particular ful usd to nrat it. Th sprad btwn th pric of lctricity and a ful that can b usd to nrat it is of intrst sinc it is this sprad that dtrmins th conomic valu of nration assts that can b usd to transform th ful into lctricity 1. Th amount of ful that a particular nration asst rquirs to nrat a ivn amount of lctricity will of cours dpnd on th asst's cincy. This cincy is summarizd by th asst's hat rat, which is dnd as th numbr of British thrmal units (Btus) of th input ful (masurd in millions) rquird to nrat on mawatt hour (MWh) of lctricity. Thus th lowr th hat rat, th mor cint th facility. Th spark sprad associatd with a particular hat rat is dnd as th currnt pric of lctricity lss th product of th hat rat and th currnt ful pric. Thus th lowr th hat rat, th lowr th ful pric, and th hihr th lctricity pric, th larr th spark sprad. In a drulatd markt, prsumably only assts that hav a positiv spark sprads undr prvailin markt conditions will b opratd. dnition of th prvailin markt implid hat rat H as: This lads naturally to th H = numbr of 10,3 MMBtu ndd for a marinal nratin plant to nrat on MWh of lctricity = S E S G 10,3 MMBtu=MWh (2:1) whr S E is th spot pric of lctricity pr Mawatt hour (MWh) and S G is th spot pric of th nratin ful pr MMBtu. 1 This ida is analoous to th concpt of th \crack sprad" usd in th oil/rnin industry. In that xampl, w look at th sprad btwn crud oil and rnd products lik disl or asolin (s [8]). 5

6 With th notion of a hat rat stablishd, w dn Europan spark sprad basd put and call options. Dnition 1 (Spark Sprad): An Europan spark sprad call option writtn on ful G at a xd (or \strik") hat rat K H ivs th option holdr th riht but not th obliation to pay K H tims th unit pric of ful G at th option's maturity T and rciv th pric of on unit of lctricity. Lt S T E and S T G b th unit spot prics of lctricity and ful at tim T,rspctivly. Dnot th valu of th option at tim t by C 1 (SE;S t G;t). t Thn th payo of th option at maturity tim T is: C 1 (S T E;S T G;T) = max(s T E, K H S T G; 0) Dnition 2 An Europan spark sprad put option writtn on ful G at a xd hat rat K H ivs th option holdr th riht but not th obliation to pay th pric of on unit of lctricity and rciv K H tims th unit pric of ful G at maturity tim T. Dnot th valu of th option at tim t by P 1 (SE;S t G;t). t Thn th payo of th option at tim T is: P 1 (S T E ;ST ;T) = max(k G HS T, G ST; 0) E Th followin xampl provids a simpl illustration of how spark sprad options can b usd to mana lctricity pric risk. Apowr marktr in a rion whr th marinal nratin ful is natural as would lik to buy powr at tim T at a markt implid hat rat not to xcd K H. An armnt providin such a hat rat cap would nsur th marktr powr at tim T at a pric ivn by S G min(h; K H ). Assumin th marktr slls th powr into th spot markt at tim T, his payo will b (S T, E ST min(h T G ;K H )), which is qual to S T max(h T G, K H ; 0). Brinin 6

7 S T G insid th brackts, th payo is max(s T E, K H SG; T 0), which is xactly th sam as that of a Europan spark sprad call option with strik hat rat K H. Th powr marktr can thrfor achiv his oal by purchasin this spark sprad call. Throuhout th rmaindr of th articl, w mak th followin assumptions: Assumption 1 A complt st of futurs contracts for lctricity and for th rlvant nratin fuls ar tradd. Assumption 2 Th risk-fr intrst rat r is constant. Nxt w provid a put-call parity rlationship btwn th spark sprad put and call options, as wll as uppr and lowr bounds on thir valus. W dlay makin spcic assumptions about th pric procsss that lctricity and th nratin fuls follow until sction 3. Proposition 1 (Put-Call Parity) Lt F t E and F t G dnot th futurs prics of lctricity and nratin ful, rspctivly. Th followin parity rlationship holds for an Europan spark sprad put and call options with th sam xd hat rat K H and xpiration dat t. C 1 = P 1 +,rt (F t E, K H FG) t (2:2) Proof. At tim t, th payo of a lon position in on unit of spark sprad call option C 1 is max(s t E,K H SG; t 0); th payo of a short position in on unit of spark sprad put option P 1 is max(k H S t, G St ; 0). Consquntly, th payo of (C E 1, P 1 ) at maturity timtis(k H S t G, SE). t Th prsnt valu of (K H S t G, SE)is t,rt (F t E, K H FG). t Thrfor, C 1, P 1 =,rt (F t, K E HF t ) G 7

8 Proposition 2 (No-arbitra lowr/uppr bounds) Lt F t E and F t G dnot th futurs prics of lctricity and a nratin ful, rspctivly. Thn th valu of a spark sprad call option, C 1, has both a lowr bound and a uppr bound, i..,rt max(f t E, K H FG; t 0) C 1,rt F t (2:3) E Proof. Th rst inquality isby th Put-Call parity (2.2) and th fact P 1 0. Th scond inquality is du to th fact max(s t E, K H S t G; 0) S t E at tim t and th prsnt valus of max(s t E, K HS t G ; 0) and St E ar C 1 and,rt F t E, rspctivly. 2.2 Locational Sprad Options Du to transmission costs and constraints 2, substantial dirncs frquntly xist btwn th pric of lctricity at dirnt locations. W rfr to ths dirncs as locational sprads, and dn call options on thm as follows: Dnition 3 (Locational Sprad): An Europan call option on th locational sprad btwn location on and location two with maturity T ivs its holdr th riht but not th obliation to pay th pric of on unit of lctricity at location on at tim T and rciv th pric of on unit of lctricity at location two. Lt S T i b th unit pric of lctricity at location i (i =1; 2) at tim T. Dnot th valu of th option at tim t by C 2 (S1;S t 2;t). t Thn th payo of th option at tim T is: C 2 (S T 1 ;S T 2 ;T) = max(s T 1, S T 2 ; 0) A Europan locational sprad put option can b dnd in a similar way. 2 Not that th impact of transmission constraints is compoundd by th non-storability of lctricity, which forcs ral tim dlivry. 8

9 3 Valuation of Elctricity Drivativs In this sction w prsnt a futurs basd mthod of rplicatin lctricity drivativs, and illustrat th mthod by usin it to driv xplicit xprssions for th valu of th spark sprad and locational sprad options dnd abov. Valuation quations ar providd for ths instrumnts for both omtric Brownian motion pric procsss and man-rvrtin pric procsss. In both cass w xplicitly driv only th valu of th call options. Th valu of put options can thn b drivd usin th put-call parity rlationship prsntd in sction Futurs Basd Rplication of Elctricity Drivativs As notd abov, bcaus lctricity is non-storabl, th traditional stora-basd mthods of constructin rplicatin portfolios for commodity drivativs cannot b usd to valu lctricity drivativs. In plac of th stora basd mthods, w prsnt a mthod for rplicatin lctricity drivativs by dynamically tradin futurs contracts of th appropriat maturity. Sinc at maturity th pric of a futurs contract must convr to th thn currnt spot pric, th mthodoloy prmits xact rplication. Sinc th prcis natur of th rplicatin straty will naturally dpnd on th spcic drivativ bin rplicatd, to illustrat th mthod w us it to driv th rplicatin straty for spark and locational sprad options. W do so rst undr th assumption that th rlvant pric procsss follow omtric Brownian motion procsss, and thn undr th mor rasonabl assumption that thy follow man rvrtin procsss. 9

10 3.2 Gomtric Brownian motion Pric Procsss W rst considr cas in which th futurs pric procsss of lctricity and th appropriat nratin ful, F and F, follow omtric Brownian motion procsss df =F = dt + db 1 (3:1) df =F = dt + db 2 whr B 1 and B 2 ar two Winr procsss with instantanous corrlation.,,, and ar assumd to b constants for th momnt. Th mor nral cas whr thy can b functions of tim is considrd in th man-rvrtin modl Valuation of Spark Sprad Options Dnot th tim-t valu of a spark sprad call option which maturs at tim T by V (x; y; t) C 1 (F t;t ;F t;t ;T, t). F t;t is th commodity futurs pric at tim t with maturity dat T. By constructin an instantanous risk fr portfolio usin th lctricity and nratin ful futurs contracts and th risklss asst, it follows that C 1 must satisfy th partial dirntial quation (PDE):, V t [x2 V xx 2 x +2 x y xyv xy + y 2 V yy 2 y]=0 (3:2) with boundary conditions V (x; y; 0) = max(x,y; 0), V (x; 0; t) = x, and V (0; y; t) = 0. Proposition 3 (Valu of a spark sprad call option) Th closd form solution for C 1 is: whr C 1 (F t;t ;F t;t ;T, t) =,r(t,t) [F t;t d 1 = d 2 = v 2 = ln(f t;t =(K H F t;t )) + v 2 (T, t)=2 v p T, t d 1, v p T, t 2, N(d 1 ), K H F t;t N(d 2 )] (3:3) 10

11 Proof. Vrify V = C 1 (F t;t boundary conditions. ;F t;t ;T, t) solvs PDE (3.2) with th corrspondin Valuation of Locational Sprad Options Th valu of th locational sprad call option can b drivd in xactly th sam way th valu of th spark sprad call option was drivd abov. Spcically, dnin F ;1 and F ;2 to b th omtric Brownian motion pric procsss that th futurs prics of lctricity at locations 1 and 2 follow, w hav: Proposition 4 (Valu of a locational sprad call option) Th valu of C 2 is ivn by C 2 (F t;t ;1 ;F t;t ;2 ;T, t) =,r(t,t) [F t;t ;1 N(d 1 ), F t;t ;2 N(d 2 )] (3:4) whr d 1 = d 2 = t;t ln(f;1 =F t;t ;2 )) + v 2 (T, t)=2 v p T, t d 1, v p T, t v 2 = 2 ;1, 2 ;1 ;2 + 2 ;2 3.3 Man-rvrtin pric procsss In this sction w assum that th futurs pric procsss of lctricity F and of th rlvant nratin ful F follow th man-rvrtin procsss df = (, ln F )F dt + (t)f db 1 (3:5) df = (, ln F )F dt + (t)f db 2 whr (t) and (t) ar functions of tim t, and ar th lon-trm mans, and ar th man-rvrtin cocints, and B 1 and B 2 ar, as abov, two Winr procsss with instantanous corrlation. 11

12 3.3.1 Valuation of Spark Sprad options Dnot th tim-t valu of a spark sprad call option which maturs at tim T by V (x; y; t) C 1 (F t;t ;F t;t ;T, t). Applyin th sam rplication arumnts applid abov, it follows that C 1 must satisfy th partial dirntial quation:, V t [x2 V xx 2 x(t)+2 x (t) y (t)xyv xy + y 2 V yy 2 y(t)]=0 (3:6) with boundary conditions V (x; y; 0) = max(x,y; 0), V (x; 0; t) = x, and V (0; y; t) = 0. Proposition 5 (Valu of a spark sprad call option) Th clos-form solution for C 1 is: C 1 (F t;t ;F t;t ;T, t) =,r(t,t) [F t;t N(d 1 ), K H F t;t N(d 2 )] (3:7) whr d 1 = d 2 = v 2 = ln(f t;t R T t =(K H F t;t )) + v 2 (T, t)=2 v p T, t d 1, v p T, t [2 (s), 2 (s) (s)+ 2 (s)]ds T, t Proof. Vrify V = C 1 (F t;t boundary conditions. ;F t;t ;T, t) solvs PDE (3.6) with th corrspondin Svral comparativ static proprtis of th spark sprad call option valu can b drivd by invstiatin th sin of th partial drivativs of C 1 with rspct to its paramtrs. Proposition 6 As F t;t % (incrass), or F t;t & (dcrass) ) C 1 % (incrass) F t;t % and F t;t F t;t % ) C 1 % & or r & ) C 1 % 12

13 3.3.2 Valuation of locational sprad options Dnin F ;1 and F ;2 to b th man-rvrtin pric procsss that ovrn th futurs prics of lctricity at locations 1 and 2 and followin th drivation abov, w hav: Proposition 7 (Valu of a locational sprad option) Th valu of C 2 is ivn by C 2 (F t;t ;1 ;F t;t ;2 ;T, t) =,r(t,t) [F t;t ;1 N(d 1 ), F t;t ;2 N(d 2 )] (3:8) whr d 1 = d 2 = v 2 = R T t ln(f t;t ;1 =F t;t ;2 )) + v 2 (T, t)=2 v p T, t d 1, v p T, t [2 ;1(s), 2 ;1 (s) ;2 (s)+ 2 ;2(s)]ds T, t 4 Ral Options Valuation of Gnration and Transmission Assts Th riht to oprat a nration asst with hat rat H that uss nratin ful is clarly ivn by th valu of a spark sprad option with \strik" hat rat H writtn on nratin ful. Similarly, th valu of a transmission asst that conncts location 1 to location 2 is qual to th sum of th valu of th locational sprad option to buy lctricity at location 1 and sll it at location 2 and th valu of th option to buy lctricity at location 2 and sll it a location 1 (in both cass, lss th appropriat transmission cost). This quivalnc btwn th valu of appropriatly dnd spark and locational sprad options and th riht to oprat a nration or a transmission asst can b asily usd to valu such assts. In this sction w illustrat this approach by dvlopin a simpl spark sprad basd modl of th valu of a as-rd nration asst. Onc stablishd, w t th modl and us it to nrat stimats of th valu 13

14 of svral as-rd plants that hav rcntly bn sold. Th accuracy of th modl is thn valuatd by comparin th stimats constructd to th prics at which th assts wr actually sold. In th analysis w mak th followin simplifyin assumptions about th opratin charactristics of th nration assts undr considration: Assumption 3 Ramp-ups and ramp-downs of th facility can b don with vry littl advanc notic. Assumption 4 Th facility's opration (.. start-up/shutdown costs) and maintnanc costs ar constant. Ths assumptions ar rasonabl, sinc for a typical as turbin combind cycl conration plant th rspons tim (ramp up/down) is svral hours and th variabl costs (.. opration and maintnanc) ar nrally stabl ovr tim. To construct a spark sprad basd stimat of th valu of a nration asst, w stimat th valu of th riht to oprat th asst ovr its rmainin usful lif. This valu can b found by intratin th valu of th spark sprad option to oprat th facility ovr its rmainin usful lif. Spcically, Dnition 4 Lt on unit of th tim-t capacity riht of a natural as rd lctric powr plant rprsnt th riht to convrt K H units of natural as into on unit of lctricity by usin th plant at tim t, whr K H is th plant s hat rat. Th payo of on unit of tim-t capacity riht is max(s t E, K H S t G; 0), whr S t E and S t G ar th spot prics of lctricity and natural as at tim t, rspctivly. Dnot th valu of on unit of th tim-t capacity riht by u(t). 14

15 Dnition 5 Dnot th virtual valu of on unit of capacity of a as rd powr plant by V. Thn V is qual to on unit of th plant s tim-t capacity riht ovr th rmainin lif [0;T] of th powr plant, i.. V = Z T 0 u(t)dt. Without makin any distributional assumptions about th pric procsss that S t E and S t G follow, w hav th followin proposition: Proposition 8 Th valu of on unit of capacity V of a plant that has a usful lif Z T of T has both a lowr bound and an uppr bound: Z T 0,rt max(f t, K H F t ; 0)dt V 0,rt F t dt Proof. By dnition and Proposition 2. If w furthr assum that th pric procsss of lctricity and natural as spot and futurs prics follow th man-rvrtin procsss as ivn by (3.5), thn w hav u(t) =C 1 (t) whr C 1 (t) is ivn by Proposition 5. Th valu of a as-rd powr plant with lif tim T is thrfor V n = Z T 0 C 1 (t)dt (4:1) Similarly, ifw assum that th pric procsss of th lctricity futurs prics at two dirnt locations follow man-rvrtin procsss, th valu of a transmission lin connctin th two locations in a radial ntwork is V tran = Z T 0 C 2 (t)dt (4:2) Equations (4.1) and (4.2) ar two fundamntal valuation formula w propos for th valuation of nration and transmission assts in a comptitiv lctricity markt. 15

16 4.1 Application of th Modl to Rcnt Gnration Asst Sals To valuat th accuracy of (4.1), w t th modl and us it to construct stimats of th valu of svral nration assts that hav bn rcntly sold. For purposs of comparison, w also stimat th valu of ach asst usin a standard discountd cash ow calculation. In ordr to t th modl, w rst stimat th volatilitis of th pric procsss of th rlvant futurs contracts. Lt f t n maturs in n months, and assum that f t n kind considrd abov. Lt R t n ln f t n, thn b th pric of th futurs contract that follows a man-rvrtin procss of th dr n = 1 (b n, R n )dt + n db W stimat n usin th Nw York Mrcantil Exchan (NYMEX) lctricity futurs historical pric data. Th natural as volatility trm structur and th as-tolctricity pric corrlation ar also stimatd usin historical data on th NYMEX natural as (Hnry Hub) futurs contracts. Onc stimatd, ths paramtrs ar insrtd in th valuation formula drivd in th prvious sction to construct ral options basd stimats of th valu of th assts in qustion. To calculat th discountd cash owvalu of th assts w us th xpctd futur lctricity and natural as prics implid by th rowth-adjustd mans of th rlvant forward curvs. Th sampl of nration assts considrd consists of four as-rd powr plants which Southrn California Edison rcntly sold to Houston Industris. At prsnt, not all of th individual plant dollar invstmnts hav bn mad public. As a proxy w us th total invstmnt mad by Houston Industris ($237 million to purchas four plants - Coolwatr, Ellwood, Etiwanda and Mandalay), dividd by th total numbr of mawatts (MW) of capacity (2172MW) to t approximatly $110,000 16

17 Elctricity and Natural Gas Forward Curvs Futurs Pric $40.00 $35.00 PV PV COB NG NG $5.00 $4.50 $30.00 $25.00 $20.00 $15.00 $10.00 $4.00 $3.50 $3.00 $2.50 $5.00 $2.00 $0.00 $ Tim to Maturity (Month) Fiur 1: Elctricity and Natural Gas Futurs Pric Curv pr MW (or $110/kW) of capacity for th ntir packa of plants. Howvr, th Coolwatr Plant 3, in Datt, California, is th most cint (with an avra hat rat of 9,500) of th four plants in th packa and thus should hav a hihr valu pr MW. W thrfor assum that th implid markt valu for Coolwatr could 3 Th Coolwatr Plant is mad up of four units. Two 256MW Combind Cycl Gas Turbins plus a stam turbin; and two convntional turbins with capacity 65MW and 81MW ach. Som r-powr work has bn don on th larr units. 17

18 ran from $110,000 to $220,000 pr MW, or quivalntly, $110/kW to $220/kW. Usin th NYMEX lctricity and natural as futurs pric data on 02/11/97 and 11/10/97, (s Fiur 1), w comput both th option valu and DCF valu (usin a risk-adjustd discount rat of 10%) of a as-rd plant with various possibl hat rats assumin a rmainin opratin lif of ftn yars for th plant and an availability factor of 90% for pak hours. Fiur 2 shows th plot of th option valus and th DCF valus of a plant ofvarious possibl hat rats usin forward curvs at dirnt tims and at dirnt tradin hubs. Capacity Valu ($/kw) Gas Fird Powr Plant Capacity Valu ($/kw) (Plant lif tim: 15 yars; r_dist =10%; r_f =6%; rho =0.4) OPT Valu (PV11/10/97) DCF Valu (PV11/10/97) OPT Valu (COB11/10/97) DCF Valu (COB11/10/97) OPT Valu (PV02/11/97) DCF Valu (PV02/11/97) Hat Rat Fiur 2: Capacity Valu of a Gas-Fird Plant W can s that th option valus of capacity ar sinicantly hihr than th DCF valus. For hat rat hihr than 8500 Btu/kWh, th DCF valus of capacity 18

19 ar clos to zro. At th hat rat lvl of 9500 and usin th lctricity forward curvs at Palo Vrd, th thortical option-basd capacity valu of a plant comparabl to th Coolwatr Plant 4 rans from $188/kW to $321/kW. 5 Conclusions This articl has prsntd a mthodoloy for valuin lctricity drivativs by constructin rplicatin portfolios with futurs contracts and th risk fr asst. Futurs basd rplication is mad ncssary by th non-storabl natur of lctricity, which ruls out th traditional spot markt, stora-basd mthod of valuin commodity drivativs. Onc dvlopd, th mthodoloy was usd to drivvaluation formula for both spark and locational sprad options whn th prics of th undrlyin assts follow ithr omtric Brownian motion or man rvrtin procsss. Ths valuation rsults wr in turn usd to construct ral options basd valuation formula for nration and transmission assts. Application of th nration asst valuation formula to a sampl of rcnt asst sals susts that th spark sprad analysis nrats rasonabl stimats of th actual markt valu of th assts, and crtainly much mor accurat stimats than thos which traditional discountd cash ow mthods provid. In addition, th stimats nratd could almost crtainly b improvd by incorporatin a ratr lvl dtail about th plants and thir sits in th analysis. Doin so prsnts a natur avnu for futur rsarch. 4 Th natural tradin hub for Coolwatr to sll into is th Mad hub. Howvr, du th liquidity of th Palo Vrd nancial futurs contracts w us th PV futurs contracts as a proxy for th lctricity pric information for Coolwatr. In addition, th basis dirntial associatd with PV/Mad is typically not lar. 19

20 Rfrncs [1] Black, F. \Th Pricin of Commodity Contracts," Journal of Financial Economics, 3 (Spt. 1976), [2] Black, F., and M. Schols,\Th Pricin of Options and Corporat Liabilitis," Journal of Political Economy, 81 (May-Jun 1973), [3] Dn, Shiji (1998), \Financial Mthods in Comptitiv Elctricity Markts," Ph.D. Dissrtation (forthcomin), Univrsity of California at Brkly, CA. [4] Dixit, A.K., and R.S. Pindyck (1994), Invstmnt undr Uncrtainty, Princton Univrsity Prss, NJ. [5] Du, Darrll (1996), Dynamic Asst Pricin Thory, 2nd d., Princton Univrsity Prss, NJ. [6] Hsu, Michal, \Spark Sprad Options ar Hot!" Th Elctricity Journal, Elsvir Scincs, Vol. 11 No. 2 (March 1998), [7] Mrton, R.C., \Thory of Rational Option Pricin," Bll Journal of Economics and Manamnt Scinc, 4 (Sprin 1973), [8] Shimko, David (1994) \Options on Futurs Sprads: Hdin, Spculation and Valuation," Journal of Futurs Markts, v.14, no.2. [9] Smith, J.E., and R.F. Nau (1995), \Valuin Risky Projcts: Option Pricin Thory and Dcision Analysis," Manamnt Scinc, 41, no.5, [10] Schwartz, Eduardo S., \Th Stochastic Bhavior of Commodity Prics: Implications for Valuation and Hdin", Journal of Financ, (July 1997),