Energy Spot Price Models and Spread Options Pricing

Transcription

1 Energy Spo Price Models and Spread Opions Pricing Samuel Hikspoors and Sebasian Jaimungal a a Deparmen of Saisics and Mahemaical Finance Program, Universiy of orono, 100 S. George Sree, orono, Canada M5S 3G3 o Appear in Inernaional Journal of heoreical and Applied Finance In his aricle, we consruc forward price curves and value a class of wo asse exchange opions for energy commodiies. We model he spo prices using an affine wo-facor meanrevering process wih and wihou jumps. Wihin his modeling framework, we obain closed form resuls for he forward prices in erms of elemenary funcions. hrough measure changes induced by he forward price process, we furher obain closed form pricing equaions for spread opions on he forward prices. For compleeness, we address boh an Acuarial and a risk-neural approach o he valuaion problem. Finally, we provide a calibraion procedure and calibrae our model o he NMEX Ligh Swee Crude Oil spo and fuures daa, allowing us o exrac he implied marke prices of risk for his commodiy. 1. Inroducion he energy commodiy markes are fundamenally differen from he radiional financial securiy markes in several ways: Firsly, hese markes lack he same level of liquidiy ha he majoriy of financial markes enjoy. Secondly, he sorage coss of mos commodiies ranslae ino peculiar price behavior; some commodiies are exremely difficul o sore or canno be sored a all elecriciy being a prime example. hirdly, parly due o he srucural issues surrounding energy price deerminaion, elecriciy prices are he Naural Sciences and Engineering Research Council of Canada helped suppor his work. 1

2 2 Samuel Hikspoors and Sebasian Jaimungal ypically exposed o very high volailiy and o large shocks. Finally, commodiy prices end o have srong mean revering rends. hese sylized empirical facs are well documened in, for example, Clewlow and Srickland 2000), Carmona and Durrleman 2003), Eydeland and Wolyniec 2003) and Hull 2005). he world wide energy commodiies markes have creaed a need for a deeper quaniaive undersanding of energy derivaives pricing and hedging. We conribue o his program firsly by proposing a wo-facor mean-revering spo price process, boh wih and wihou a jump componen, and secondly by carrying ou he explici valuaion of spread opions wrien on wo forward prices. he spo price model is similar in spiri o he wo-facor model proposed in Pilipovic 1997); however, in ha work he second facor follows a geomeric Brownian moion and, herefore, in he long run, he argeed mean blows up. Insead, we chose he mean-revering level of he firs facor o mean-rever o a second long-run mean. Our modeling framework can hen be viewed as a perurbaion on he sandard one-facor mean-revering approach. his is an appealing approach as he one-facor model has been exensively sudied and approximaely fis forward price curves. Adding a perurbaion on op of his firs order model allows us o correc some of he deficiencies of he one-facor model while mainaining racabiliy. In paricular, he second facor perurbaion does no change he saionary behavior ha he one-facor model enjoys. We delegae he deails of our purely diffusive model, and is relaion and differences o he classical Pilipovic 1997) model, o Secion 2.1 and of our jump-diffusion model o Secion 4.1. Much like he financial markes, energy markes are rife wih derivaive producs. However, one produc sands ou among he many ha are raded over he couner: spread opions. hese opions provide he owner wih he righ o exchange a prespecified quaniy of one asse for anoher, a a fixed cos. An even more popular opion is he spread opion on forward prices which allows he holder o exchange wo forward conracs, possibly wih differing mauriy daes, raher han he commodiy. he holder of such an opion receives a mauriy a payoff of ) ϕf 1), 1, F 2), 2 ) := max F 1), 1 α F 2), 2 K, 0. 1) I is well known ha even when he commodiy prices are modeled as geomeric Brownian

3 Energy Spo Price Models and Spread Opions Pricing 3 moions GBMs), no closed form soluion exiss for K 0. As such, we focus on he zero exchange cos case K = 0 which we succeed in valuing in closed form under our wo-facor mean-reversion modeling assumpions. Given our closed form soluions, he general case K 0 can be valued using eiher Mone Carlo or PDE mehods wih our K = 0 resul acing as a conrol variae. In a financial markes conex, before proceeding o he valuaion of derivaives, he objecive measure is ransformed o an equivalen risk-neural measure. However, in he conex of energy derivaives, due o he illiquidiy issue, such a measure change is by no means necessary. herefore, raher han immediaely moving o a risk-neural valuaion procedure, we firs presen a simple Acuarial valuaion approach for pricing exchange opions in Secion 2.2. his approach has been adoped by some indusry paricipans and is jusified by assuming ha he risks associaed wih he energy prices are nondiversifiable see for example Hull 2005)). Margrabe 1978) firs valued exchange opions assuming asse prices are GBMs under he risk-neural measure and by uilizing a measure change induced by reaing one of he asses as numeraire. However, since commodiies are no liquid, heir spo prices canno ac as a numeraire. Noneheless, we show ha here is a closely relaed measure change which renders he valuaion in closed form even under he Acuarial approach. Alhough some indusry paricipans adop an Acuarial valuaion procedure, riskneural approaches are sill very popular. In Secion 3, we specify a class of equivalen risk-neural measures which mainains he srucure of he real-world dynamics. his allows us o reuse much of he valuaion echnology developed in Secion 2.2. Once again, we show ha an equivalen measure provides closed form pricing equaions for spread opions. Mos energy commodiies are adequaely modeled by diffusive processes; however, elecriciy prices hemselves conain sever jumps. Secion 4 is devoed o a jump-diffusion generalizaion of our previous resuls appropriae for spark-spread opions exchange opions in which a fuel commodiy is exchanged for elecriciy. Using he affine srucure of our wo-facor model wih jumps, we obain he forward prices as a soluion o a sysem of coupled ODEs which we explicily solve. Furhermore, hrough measure changes and

4 4 Samuel Hikspoors and Sebasian Jaimungal Fourier ransform mehods, à la Duffie, Pan, and Singleon 2000) and Carr and Madan 1999), we obain closed form formula for he price of spark-spread opions on forwards. We complee he paper in Secion 5 wih a calibraion procedure ha fis he model o spo and forward prices simulaneously. Calibraing o boh spo and forward prices allows us o furher exrac he marke prices of risk implied by he daa. We apply he calibraion procedure o he NMEX Ligh Swee Crude Oil spo and fuures daa and repor on he sabiliy of he implied model parameers as well as on he implied marke prices of risk. Ineresingly, he real-world mean-reversion raes are found o be higher han he risk-neural mean-reversion raes. Furhermore, he rae of mean-reversion of he sochasic long-run mean level is lower han he mean-reversion rae of he log-spo price process. We find ha hese feaures are refleced in he marke prices of risk hemselves. 2. Real World Dynamics and Pricing 2.1. Model Specificaions A quick glance a hisorical spo prices for energy markes shows ha radiional geomeric Brownian moion models, even as a firs order model, are inadequae. A successful model mus include mean reversion as an essenial feaure. For early use of such models see he papers by Gibson and Schwarz 1990) and Corazar and Schwarz 1994). hese early one-facor models are a good firs order model; however, as energy derivaives will ofen have mauriies exending ino monhs, or even years, such firs order models require improvemen. hey invariably canno mach he erm srucure of forward prices for example. o his end, Pilipovic 1997) firs suggesed he wo-facor mean-revering model: ds = βθ S ) d + σ S S dw 1), 2) dθ = α θ d + σ θ θ dw 2), 3) where he wo Brownian risk facors are correlaed: d[w 1), W 2) ] = ρ d. In his model, θ represens he sochasic long-run mean ha spo prices S rever o. his addiional degree of sochasiciy helps o correc some of he biases ha a fixed long-run mean produces. In his paramerizaion, he mean revering level is a geomeric Brownian moion

5 Energy Spo Price Models and Spread Opions Pricing 5 and can herefore grow wihou bound leading o non-saionary spo price processes. o circumven his problem, and o assis wih obaining closed form formulae for spread opions, we propose o model he log spo-price wih an affine wo-facor mean-revering process. Much like Pilipovic s model, he firs facor mean-revers o sochasic level; however, we ensure ha he sochasic mean-revering level also mean-revers o a second long-run mean. Wih such a paramerizaion, he disribuion of spo-prices is saionary, prices do no grow wihou bound, and he model remains wihin he Affine modeling class. If he individual asses are driven by a wo-facor model, hen four driving facors are required o value spread opions wo for each asse. Le {W i) } 0 and {Z i) } 0, wih i = 1 or 2, denoe hese four Brownian risk facors and F = {F } 0 denoe he naural filraion generaed by hese processes. he measure P will denoe he real-world probabiliy disribuion and {Ω, F, P} is used o denoe he complee sochasic basis for he probabiliy space. he spo-prices {S i) } 0, wih i = 1 or 2, are obained via an exponeniaion of he driving risk-facors. More specifically, he spo-prices are defined as follows: S i) := exp g i) ) + X i), i = 1, 2. 4) Seasonaliy is an imporan feaure of some commodiy prices; we herefore include he seasonaliy erm g i) using he following popular ansäz: n ) g i) = A i) 0 + A i) k sin2π k ) + B i) k cos2π k ). 5) k=1 For calibraion sabiliy, n is ypically kep small: n = 1 or 2. In our subsequen calculaions we leave g i) general, assuming only smoohness and differeniabiliy. o complee he specificaion of he wo-facor model which drives he spo-prices, X i) is assumed o saisfy he following coupled SDEs: ) dx i) = β i i) X i) ) d i) = α i φ i i) d + σ i) i) X dw, 6) d + σ i) dzi). 7) Here, β i conrols he speed of mean-reversion of X i) α i conrols he speed of mean-reversion of he long-run level i) mean φ i ; σ i) X o he sochasic long-run level i) ; o he arge long-run and σi) conrol he size of he flucuaions around hese means.

6 6 Samuel Hikspoors and Sebasian Jaimungal o reduce he parameer space, he measure P is chosen such ha he following simple correlaion srucure is imposed on he Brownian moions: d[w 1), W 2) ] = ρ 12 d, 8) d[w i), Z i) ] = ρ i d, i = 1, 2, 9) and all oher cross correlaions are zero. his srucure allows he main driving facors X i) o be correlaed o one anoher and heir own idiosyncraic long-run mean-revering processes i), while his srucure forces he long-run revering facors 1) and 2) have an insananeous correlaion of zero. I is a sraighforward, albei edious, maer o generalize his correlaion srucure. he coupled SDEs 6)-7) can be solved by i) firs solving 7) for i) sandard mean-revering Ornsein-Uhlenbeck process o obain i) = φ i + i) s φ i ) e α i s) + σ i) and hen ii) subsiuing his resul ino 6) o solve for X i) correlaion and feedback of i) represened as X i) = G i) s, + e βi s) X s i) + M i) s, s i) + σ i) X where γ i := s o which is he e α i u) dz i) u ; 10) while accouning for he ino X i). Afer some edious calculaions X i) s e β i u) dw i) u + σ i) β i α i β i, and G i) s, and M i) s, are he deerminisic funcions s can be M i) u, dz i) u,11) M i) s, := γ i e β i s) e α i s) ), 12) G i) s, := φ i 1 e β i s) ) φ i M i) s,. 13) Armed wih our wo-facor model and he soluions 10)-11), we now focus our aenion on he valuaion of spo spread opions and defer he valuaion of spreads on forwards and model calibraion o secions 3.4 and 5 respecively Spo Spread Valuaion : An Acuarial Approach Much like he financial markes, energy markes are rife wih derivaive producs. However, one produc sands ou among he many ha are raded over he couner: spread opions. hese opions provide he owner wih he righ o exchange a specified quaniy

7 Energy Spo Price Models and Spread Opions Pricing 7 of one asse for anoher, a a fixed cos. he holder of such an opion receives a mauriy payoff of ) ϕs 1), S2) ) := max S 1) αs2) K, 0. 14) When he cos of exchanging K is se o zero, he opion is known as a Margrabe or exchange opion Margrabe, 1978). Various approximaions for he general spo) spread opion, under simple diffusion processes, have been sudied in he lieraure and he reader is referred o Carmona and Durrleman 2003) for an excellen overview and furher references. In he conex of elecriciy markes, his opion is known as he spark-spread opion and α represens he hea rae of a given plan. he hea rae encapsulaes he plan s profiabiliy by specifying he number of unis of he underlying commodiy such as oil or naural gas) which produces one uni of power his produc is sudied in Secion 4. If he exchange is beween crude oil and a refined produc such as gasoline) he opion is known as a crack-spread opion. Many oher specific examples of exchange opions exis in he energy secor. In all cases, he exchange opion can be used o hedge agains marke movemens of spo prices or, alernaively, o speculae on hose moves. In eiher case, a valuaion framework is required. I is difficul and someimes no viable o sore elecriciy and energy commodiies; his resuls in an illiquid spo marke. Harrison and Pliska 1981) demonsraed ha he absence of arbirage is equivalen o he exisence of a measure, no necessarily unique, under which he relaive price process of radable asses o he money marke accoun are maringales. Such measures are known as a risk-neural measures. However, his conclusion has one imporan assumpion unresriced and liquid rading of he underlying asse. In illiquid spo) energy markes, i may be dubious o adop a risk-neural pricing framework, and alhough we ulimaely proceed wih ha program, we firs ake an Acuarial approach. By assuming ha he risks associaed wih he energy prices are non-diversifiable, i is possible o jusify an Acuarial approach o pricing derivaives Hull, 2005) which values an opion as is discouned real-world expeced payoff. Definiion. he Acuarial valuaion principle assigns he following price o a -mauriy

8 8 Samuel Hikspoors and Sebasian Jaimungal coningen claim wih payoff ϕs 1), S2) ): Π, := P, ) E P [ ϕ S 1), S2) )]. 15) he noaion E [A] represens he expecaion of A condiional on he filraion F. hroughou he aricle we assume he possibly random) ineres raes o be independen from oher risk facors and wrie he price a ime of a zero-coupon bond mauring a as P, ). In he radiional valuaion procedure, expecaions are aken under he risk-neural measure Q; here, however, he relevan measure is he real-world one P. his complicaes he problem somewha. When he asse and he derivaive are radable, i is possible o use a numeraire change o value he Margrabe opion; in he presen conex he asse canno be used as numeraire and he relevan measure is no he risk-neural one. Noneheless, i is possible o adop a similar sraegy; o his end, define he auxiliary process [ H i), := EP S i) ]. If he expecaion in 16) is compued under a risk-neural measure, hen H i), represens he -mauriy forward price, which moivaes us o coin H i), he -mauriy pseudoforward price process. 16) A mauriy his price process coincides wih he spo-price H i), = Si) allowing he Acuarial value of he exchange opion o be wrien: [ ) ] Π, = P, ) E P H 1), α H2),. 17) + he pseudo-forward price process has wo oher noable properies: i) is expecaion [ ] is bounded a all finie imes E P H i), = H i) 0, < + for all <, and ii) i is a P-maringale. hese wo properies allow a normalized version of H i), ransforming he measure P ino a paricularly convenien measure for pricing. o assis in his measure change can, in some sense, be inerpreed as being induced by using he pseudoforward price process as a numeraire asse. he following heorem conains one of our main ools.

9 Energy Spo Price Models and Spread Opions Pricing 9 heorem. 2.1 Le {η } 0 denoe he Radon-Nikodym process ) d P η := := H2),. dp H 2) 0, 18) hen, for any A F PA) = E P [IA) η ], 19) and in paricular W i W 2) = W 2) Z 2) = Z 2) W 1) = W 1) Z 1) = Z 1) and Z i i = 1, 2) defined by [ σ 2) X e β 2 u) + ρ 2 σ 2) M 2) u, ] du 20) [ ] σ 2) M 2) u, + ρ 2σ 2) X e β 2 u) du 21) [ ] ρ 12 σ 2) X e β 2 u) du 22) [ ] ρ 1 ρ 12 σ 2) X e β 2 u) du 23) are P Wiener processes wih correlaion srucure d[ W 1), W 2) ] = ρ 12 d, 24) d[ W i), Z i) ] = ρ i d, i = 1, 2, 25) and all oher cross correlaions zero. Proof. Given properies i) and ii) above and η 0 = 1, i is clear ha η is a Radon- Nikodym derivaive process and equaion 19) immediaely follows. o demonsrae ha W i) and Z i) are P-Wiener processes subsiue 11) ino H i), expecaion explicily as follows: [ { H i), = E P exp g i) Here, +σ i) X + Gi), + e β i ) X i) e β i u) dw i) u + M i), i) + σ i) M i) u, dzi) u ) = exp g i) + Gi), + Ri), + e β i ) X i) + M i), i) R i), := h, ; 2β i) 2 h, ; α i + β i ) [ ) 2 σ i) X + γ i σ i) ) 2 + 2ρi γ i σ i) [ ) 2 γ i σ i) + ρi γ i σ i) X σi) X σi) ] ] }] + h, ; 2α i) 2 and hen compue he 26) ) 2 γ i σ i), 27)

10 10 Samuel Hikspoors and Sebasian Jaimungal wih h, ; a) := 1 e a ) )/a. 28) he Girsanov kernel can now be read off direcly from 26) and he soluion for X i) i) given in 10)-11). hrough Girsanov s heorem we find ha 20)-23) are P-Wiener processes. Corollary 2.2 he Acuarial valuaion formula 15) can be ransformed o ) [ )] Π, := P, ) E P ϕ H 1),, H2), = P, ) H 2), Ee P ϕ H 1),, H2), H 2), and. 29) Proof. Use η o change he measure. I now remains o compue he expecaion appearing in 29) under he ransformed measure P. Recall ha a process M is a P-maringale if and only if he process M d P/dP) is a P-maringale. Consequenly, boh H, := H 1), /H2), and dp/d P) are P-maringales. his, ogeher wih Corollary 2.2, reduces he Acuarial price of he Margrabe spread opion o Π, = P, ) H 2), Ee P [H, α) + ] 30) Since H, is a P-maringale, is drif under he P-measure is zero. Puing his ogeher wih equaion 26), we find ha H, saisfies he SDE: dh, H, = σ 1) X e β 1 ) d W 1) σ 2) X e β 2 ) d W 2) + σ 1) his expression clearly shows ha H, M 1), 1) d Z σ 2) M 2), d Z 2) 31) is a geomeric Brownian moion wih ime dependen bu deerminisic) volailiy; consequenly, is erminal value can be expressed in erms of is iniial value via H, = H, exp U, ), where U, is a normal random variable wih mean equal o 1σ 2, ) 2 and variance equal o σ, ) 2. Here, [ σ, ) 2 P = E e σ 1) X e β 1 s) d W s 1) σ 2) X e β 2 s) d W s 2) +σ 1) = 2 R 1), + 2 R2) M 1) 1) s, d Z s σ 2) M 2) 2) s, d Z s, 2 ρ 12 σ 1) X ) 2 ] σ2) X h, ; β 1 + β 2 ). 32)

11 Energy Spo Price Models and Spread Opions Pricing 11 he deerminisic funcions M i) s, and Ri), are as in 12) and 27) respecively. I is now a sraighforward maer o recover he final resul of his secion a Black-Scholes like expression for he Acuarial price of he exchange opion. Proposiion 2.3 he Acuarial value a ime of he -mauriy exchange opion is [ ] Π, = P, ) H 1), Φd + σ, ) α H 2), Φd) 33) wih σ, as in 32) and d defined as d := log H, α 1 2 σ, ) 2 σ,. 34) 3. Risk-Neural Dynamics and Pricing In complee marke seings, here exiss a unique equivalen measure which induces he relaive price process of radable asses o be maringales. his measure is known as he risk-neural measure Q. In he presen conex, he underlying asse is no radable in he usual sense due o he illiquidiy issue and poenially large sorage coss. In he previous secion we deal wih his issue by resoring o an Acuarial valuaion procedure and assigned a price equal o he discouned expecaion of he erminal payoff under he real-world measure. However, one can in principle sill uilize risk-neural mehodologies adjusing for he incompleeness of he marke seings. Wihin such incomplee markes here may exiss many equivalen risk-neural measures; i is he job of he marke as a whole, via rading of derivaives, o decide which measure prevails a any one given poin in ime. In his nex secion, we provide a class of equivalen maringale measures ha mainains he srucure of our real-world dynamics for asse prices. hese measures are hen used o obain forward prices and value spread opions Measure Change In his secion, we inroduce a class of risk-neural measure changes which mainains he real-world srucure of he asse dynamics. he following Lemma inroduces he new measure induced by a four dimensional marke price of risk vecor. Lemma 3.1 Le {ζ } 0 denoe he Radon-Nikodym process, ) dq { } ) ζ := = E λ 1) u dw u 1) + ψ u 1) dz u 1) + λ 2) u dw u 2) + ψ u 2) dz u 2), 35) dp 0

12 12 Samuel Hikspoors and Sebasian Jaimungal where EA ) is he Dolean-Dade s exponenial of he process A. hen for any A F we have, QA) = E P [A ζ ]. 36) In paricular he following are Q-Wiener processes: W 1) = W 1) Z 1) = Z 1) W 2) = W 2) Z 2) = Z 2) wih correlaion srucure, [ d W 1), W 2)] [ d W i), Z i)] 0 { } λ 1) u + ρ 12 λ 2) u + ρ 1 ψ u 1) du 37) { } ρ1 λ 1) u + ψ u 1) du 38) { } ρ12 λ 1) u + λ 2) u + ρ 2 ψ u 2) du 39) { } ρ2 λ 2) u + ψ u 2) du 40) = ρ 12 d, 41) = ρ i d, i = 1, 2, 42) and all oher cross-correlaions zero. Proof. Decompose he correlaed processes ino uncorrelaed processes and apply Girsanov s heorem. Noice ha here are no resricions on he form of he marke prices of risk oher han he usual inegrabiliy ones. In paricular, he drifs under he risk-neural measure Q are no consrained o he risk-free rae. his is precisely he effec of incompleeness in he presen conex. he following heorem applies consrains on he marke prices of risk such ha he risk-neural dynamics and he real-world one are of he same form. heorem. 3.2 If he marke price of risk processes are chosen as follows: λ i) = λ i) + λ i) X Xi) ψ i) = ψ i) + ψ i) X Xi) + λ i) i), 43) + ψ i) i), 44)

13 Energy Spo Price Models and Spread Opions Pricing 13 subjec o he consrains i, j) {1, 2), 2, 1)} ) ψ i) X = ρ i λ i) X, 45) λ i) + ρ 12 λ j) + ρ i ψ i) = 0, 46) λ i) X + ρ 12λ j) X + ρ iψ i) X = λ i) + ρ 12λ j) α i = α i σ i) α i φ i = α i φ i + σ i) β i = β i + σ i) X ρ i λ i) ) + ρ iψ i), 47) ) + ψi), 48) ρi λ i) + ψ i))), 49) λ i) + ρ 12λ j) ) + ρ iψ i), 50) hen he risk-neural dynamics of X i) In paricular, and i) remain wihin he same class as 6)-7). dx i) = β i i) d i) X i) ) d + σ i) = α i φ i i) ) d + σ i) X dw i) 51) dzi). 52) Proof. Inser he expressions for he Q-Wiener processes W i) Collec similar erms and equae coefficiens. and Z i) ino 6)-7). his ansäz may seem resricive; however, even hough he risk-neural dynamics remains wihin he same class as he real-world one, he coefficiens driving ha dynamics may be significanly differen. his flexibiliy is sufficien for he simulaneous calibraion of he risk-neural and real-world model parameers, while remaining parsimonious Forward Prices Since he risk-neural dynamics of he driving diffusion processes 51)-52) are of he same form as hey were under he objecive measure 6)-7), he forward price curves can easily be exraced from equaion 26). his is because, wihin a risk-neural framework, he forward prices are defined as [ ] F i), := EQ S i), 53) he precise risk-neural analog of he pseudo-forward price defined in 16). remains is o change he P-parameers for he Q-parameers. All ha

14 14 Samuel Hikspoors and Sebasian Jaimungal Proposiion 3.3 he forward prices associaed o commodiy i = 1, 2 are given by ) F i), ) = exp g i) + Ri), + G i), + e β i ) X i) + M i), i) 54) where he expressions for M i),, G i), and R i), are supplied in equaions 12)-13) and 27) respecively wih he appropriae change of parameers α i α i and so on...). hese resuls can be viewed as an exension of he one-facor model Carea and Figueroa 2005) sudy, albei wihou jumps. In secion 4, we address he wo facor model wih jumps Spo Spread Valuaion o value he opion under a risk-neural measure, we follow along he same lines as in Secion 2.2. In he presen conex, he measure change is he one induced by using he forward price o drive he measure change. o his end, define a new measure Q via he Radon-Nikodym derivaive process ) [ ] d Q E Q S 2) = [ ] = F 2), dq E Q 0 S 2) F 2) 0,. 55) All seps leading o Proposiion 2.3 remain valid in his new conex, and raher han repeaing hem, we simply quoe our final risk-neural pricing resul. Proposiion 3.4 he risk neural value a ime of he exchange opion wih mauriy is: Π, = P, ) where d defined as [ F 1), Φd + σ, ) α F 2), Φd) ] 56) d := log F, σ, ) 2 α 2 σ, 57) and σ, so on...). as in 32) wih all P-parameers replaced by Q-parameers i.e., α i α i and I is imporan o noe ha he marke provides quoes for he forward curve F i),, i = 1, 2 for a se of mauriies = { 1,..., n }. hese curves can be used o calibrae he riskneural parameers. Once he model has been calibraed o marke daa, he resuling

15 Energy Spo Price Models and Spread Opions Pricing 15 pricing rules are jus as simple o use as he Black-Scholes formula for a European opion on a single asse. Alhough he explici expressions for he forward prices F i), effecive volailiy σ, and he are somewha bulky, hey involve nohing more complex han exponeniaion and are herefore very efficien o calculae Forward Spread Valuaion In he previous secions we focused on valuing a spread opion based on he fuure spo prices; however, a more popular derivaive produc involves he spread beween he forward prices of he wo asses possibly wih differing mauriies). forwards pay ) ) ϕ F 1), 1, F 2), 2 = F 1), 1 α F 2), 2 + Such spreads on a he mauriy dae where i is implici ha 1, 2. We can once again use a measure change o simplify he calculaions, his ime i is convenien o use he 2 - mauriy forward price of asse 2, i.e. F 2), 2, o induce a measure change. In paricular we define he Radon-Nikodym derivaive process ) d Q := F 2), 2. dq F 2) 0, 2 he ime price of he forward spread opion is herefore [ ) ] Π F, := P, )E Q F 1), 1 α F 2), 2 = P, )F 2) Q, 2 E e [ ] F ;1, 2 α) +. 60) + Here, F ;1, 2 := F 1), 1 /F 2), 2 is he raio of he wo relevan forward prices. In analogy wih our earlier calculaions, he relaive process F ;1, 2 58) 59) is a Q-maringale and herefore is Q-dynamics is drifless. Following along he same argumens as in Secion 2.2, i is easy o show ha F ;1, 2 = F ;1, 2 exp{u ;, 1, 2 } where U ;, 1, 2 is a normal random variable wih mean equal o 1 2 σ ;, 1, 2 ) 2 and variance equal o σ ;, 1, 2 ) 2. he explici form for he variance is ) 2 σ;, 1, 2 ) 2 := γ 1 σ 1) [h, 1 ; 2α 1 ) h, 1 ; 2α 1 )] ) 2 + γ 2 σ 2) [h, 2 ; 2α 2 ) h, 2 ; 2α 2 )] { ) 2 ) } 2 [h, + σ 1) X + γ 1 σ 1) + 2ρ1 γ 1 σ 1) X σ1) 1 ; 2β 1 ) h, 1 ; 2β 1 ) ]

16 16 Samuel Hikspoors and Sebasian Jaimungal { ) 2 σ 2) X + + { 2 { 2 γ 1 σ 1) γ 2 σ 2) γ 2 σ 2) ) } 2 [h, + 2ρ2 γ 2 σ 2) X σ2) 2 ; 2β 2 ) h, 2 ; 2β 2 ) ] } [h, 1 ; α 1 + β 1 ) h, 1 ; α 1 + β 1 ) ] ) 2 + 2ρ1 γ 1 σ 1) X σ1) ) 2 + 2ρ2 γ 2 σ 2) X σ2) } [h, 2 ; α 2 + β 2 ) h, 2 ; α 2 + β 2 ) ] 2ρ 12 σ 1) X σ2) X exp { β 1 1 ) β 2 2 ) } h, ; β 1 + β 2 ) 61) Scholes like pricing resul. Proposiion 3.5 he risk neural value a ime of he forward spread opion 58) is Π F, = P, ) [ ] F 1), 1 Φd + σ, ) α F 2), 2 Φd ) wih σ, )2 as in 61) and d defined as d := log F ;1,2 1 α 2 σ, )2. 63) σ, No surprisingly, he pricing resul is very similar o he one in Proposiion 3.4 and reduces o i when = 1 = 2. 62) 4. Spo Prices wih Jumps he wo facor diffusion model capures he main characerisics of mos energy spo prices, however, i canno accoun for he possibiliy of sudden jumps in he price daa. Such behavior is paricularly imporan for modeling elecriciy prices and various spreads coningen on elecriciy and oher hedging asses. he mos imporan example of such ) opion is he so-called spark-spread opion which pays F 1), 1 αf 2), 2 K a he mauriy dae. Here F 1), 1 := E Q [S 1) 1 ] represens he elecriciy forward price, F 2), 2 represens he forward price of he commodiy used o generae elecriciy, and α represens he hea rae which encapsulaes he number of unis of energy ha he plan produces per uni of commodiy. Noice ha he srucure of his opion allows forward prices of differing mauriies o be used as he underlying. As commened earlier on, closed form soluions, even for he purely diffusion case, are no accessible for general srike levels; consequenly, we limi ourselves o he exchange opion wih a srike of zero. + where, h, ; a) is defined in 28). he pricing equaion 60) now reduces o a Black-

17 Energy Spo Price Models and Spread Opions Pricing Model Specificaion For breviy, we now focus solely on he risk-neural valuaion procedure, and provide model specificaions direcly under he risk-neural measure. ypically, when elecriciy prices jump hey rever back o normal levels very quickly. A widely used model specificaion incorporaes jumps and diffusions simulaneously as follows: d lns ) = αθ lns )) d + σ dw + dq. 64) Regardless of he specificaion of he jump process Q, such models suffer from unrealisically large diffusive volailiies and mean-reversion raes. his occurs because he process mus rever very quickly o normal levels afer a large jump, implying a high mean-reversion rae α. his in urn induces an arificially large diffusive volailiy, since oherwise all diffusive componens would rever o he mean exremely quickly and, excluding he jumps, he pahs would appear essenially deerminisic. We avoid hese problems by spliing he jump componen from he diffusion componen and modeling hem separaely. Specifically, define he power) spo price by S 1) { } := exp g 1) + X 1) + J where X i) and i) saisfy he usual wo-facor SDEs 51)-52), and he new jump componen J is defined via, 65) dj = κ J d + dq, 66) wih Q a compound Poisson process: Q := N 1 l i, where N is a ime inhomogeneous Poisson process wih aciviy rae λs), and {l i } he se of i.i.d. jump sizes wih disribuion funcion F l u). Furhermore, J denoes he value of J prior o any jump a ime. he jump componen J mean-revers o zero wih rae κ; ypically, κ will be quie large because elecriciy prices rever back o normal very quickly afer a jump. his empirical fac has no direc bearing on he valuaion procedure, however, i does aribue o he manner in which we have spli he jump componen from he diffusion componen. We allow he aciviy rae o vary wih ime o permi seasonaliy effecs in he rae of jump arrivals; however, we resric i o be deerminisic i is possible o generalize o sochasic aciviy raes; however, he addiional modeling flexibiliy renders

18 18 Samuel Hikspoors and Sebasian Jaimungal he calibraion process unsable. Finally, i is well known ha diffusions and jump processes canno have any insananeous correlaions, while his does no preclude he jump size from depending on he Brownian risk facors we make he naural assumpion ha N and {l i } are independen of all he Q-Brownian processes Forward Prices Equipped wih his jump-diffusion model, we now derive he forward price F 1), associaed wih he spo S 1). As usual, he forward price is he risk-neural expecaion of he asse price a he mauriy f, X 1), 1), J ) := F 1), := EQ [S 1) ] = EQ [ { }] exp g 1) + X1) + J. 67) Raher han compuing his expecaion direcly, we make use of he affine form of he processes along he lines of Duffie, Pan, and Singleon 2000). Since f is a Q-maringale, i saisfies he zero-drif condiion Af = 0 where A is he generaor of he process, X 1), 1), J ). he affine ansäz: { } f, X 1), 1), J ) = exp A, ) + B, ) X 1) + C, ) 1) + D, ) J, 68) wih erminal condiions A, ) = g 1), B, ) = 1, C, ) = 0 and D, ) = 1, reduces he PDE Af = 0 o he equivalen sysem of coupled ODEs: X )2 A + α 1 φ 1 C + σ1) 2 )2 B 2 + σ1) 2 B β 1 B = 0, 69) C + β 1 B α 1 C = 0, 70) D κd = 0, 71) C 2 + ρ 1 σ 1) X σ1) BC = λu) e D u 1 ) df l u).72) Alhough raher edious, sandard mehods can be used o solve his sysem and obain he forward price. Proposiion 4.1 he forward price for he wo-facor jump-diffusion spo process S 1) is { } F 1), = exp A 1), + e β 1 ) X 1) + M 1), 1) + e κ ) J, 73) where he deerminisic funcion M 1), is provided in equaion 12), A 1), = g 1) + λs) ϕ ) l e κ s) 1 ) ds

19 Energy Spo Price Models and Spread Opions Pricing 19 [ α 1 γ 1 φ 1 h, ; α1 ) h, ; β 1 ) ] + 1 ) 2 γ 2 1 σ 1) [ h, ; 2α1 ) + h, ; 2β 1 ) 2h, ; α 1 + β 1 ) ] + 1 ) 2 σ 1) X h, ; 2β1 ) + ρ 1 γ 2 1 σ 1) [ X σ1) h, ; 2β1 ) h, ; α 1 + β 1 ) ], 74) and ϕ l u) is he m.g.f. of he individual jump sizes l i, ϕ l u) := E Q [ e u l 1 ] = e u z df l z). 75) hese resuls can be viewed as an exension of he one-facor model Carea and Figueroa 2005) sudy Spark Spread Valuaion We now urn o he pricing of he exchange) spark spread opion wih -erminal ) payoff F 1), 1 αf 2), 2 and 1, 2. As usual, he forward prices are expressed as [ ] + { } F i), := EQ S i) where S 1) := exp g 1) + X 1) + J is he wo-facor jump-diffusion { } spo price presened in he previous secion and S 2) := exp g 2) + X 2) is he pure diffusion process of Secion 3. We begin our analysis by rewriing he risk-neural pricing formula in erms of an equivalen measure induced by he forward price process of he purely diffusive asse. In paricular, Π F, := P, ) E Q [ ) F 1), 1 α F 2), 2 where he measure Q is induced by ) d Q dq := E Q E Q 0 [ ] S 2) 2 [ S 2) 2 ] = + ] = P, ) F 2), 2 E e Q [F ;1, 2 α) + ], 76) F 2), 2 F 2) 0, 2, 77) and we inroduced he raio process F ;1, 2 := F 1), 1 /F 2), 2. he process F ;1, 2 is once again a Q-maringale; however, because of he presence of he jump componen, his fac alone does no allow us o exrac is disribuion. Insead, we make use of ransform mehods. Carr and Madan 1999) were among he firs o illusrae ha Fas Fourier ransform FF) mehods can be used o efficienly value European opions. he reader is referred o heir work for implemenaion deails and oher efficiency ricks.

20 20 Samuel Hikspoors and Sebasian Jaimungal he FF mehods require he m.g.f. of he logarihm of he effecive sochasic process in our case he process F ;1, 2. o his end, define Z := ln F ;1, 2 so ha Z = A 1), 1 A 2), 2 + e β 1 1 ) X 1) e β 2 2 ) X 2) + M 1), 1 1) M 2), 2 2) + e κ 1 ) J, 78) and define he corresponding m.g.f. process Ψ Z Q u) := E e [e u Z ]. 79) he process Ψ Z u) is clearly a Q-maringale; consequenly, i saisfies he zero drif condiion AΨ Z u) = 0 for every u where i is defined) where A is he generaor of he ) process, X 1), X 2), 1), 2), J under Q. Furhermore, since our modeling framework is affine, we employ he ansäz Ψ Z u) := exp { A, ) + B 1), ) X 1) + B 2), ) X 2) + C 1), ) 1) + C 2), ) 2) + D, ) J } 80) Ψ Z u) := exp {u Z } 81) Here, A, ), B 1), ), B 2), ), C 1), ), C 2), ), and D, ) are all deerminisic funcions of ime. Noe ha 1 and 2 have been removed from he argumens for easier readabiliy. Since he boundary condiion 81) mus hold for all erminal values of he auxiliary processes X 1), X 2), 1), 2), J ), he deerminisic funcions mus saisfy he induced boundary condiions ) A, ) = u A 1), 1 A 2), 2, B 1), 1 ) = ue β 1 1 ), B 2), 2 ) = ue β 2 2 ), C 1), 1 ) = um 1), 1, C 2), 2 ) = um 2), 2, D, 1 ) = ue κ 1 ). Expanding he PDE Af = 0, rewriing i in erms of an equivalen sysem of coupled ODEs and solving ha sysem similar o he analysis in Secion 4.2) provides he final resul. 82) Proposiion 4.2 he ransform Ψ Z Ψ Z u) = exp Q u) := E e [ ] e u Z is given by { A, + u e β 1 1 ) X 1) e β 2 2 ) X 2) + M 1), 1 1) M 2), 2 2) + e κ 1 ) J )} 83)

21 Energy Spo Price Models and Spread Opions Pricing 21 where M i), is defined in 12), ) A, = u A 1), 1 A 2), 2 + λs) ϕ l ue κ 1 s) ) 1 ) ds + u [ α 1 φ 1 γ 1 e α 1 1 ) h, ; α 1 ) + α 2 φ 2 γ 2 e α 2 2 ) h, ; α 2 ) ) 2 γ 2 σ 2) e 2α 2 2 ) h, ; 2α 2 ) + α 1 φ 1 γ 1 e β 1 1 ) h, ; β 1 ) { ) 2 ) } 2 σ 2) X + 2ρ2 γ 2 σ 2) X σ2) + γ 2 σ 2) e 2β 2 2 ) h, ; 2β 2 ) α 2 φ 2 γ 2 e β 2 2 ) h, ; β 2 ) ρ 1 ρ 12 γ 1 σ 2) X σ1) e α 1 1 ) β 2 2 ) h, ; α 1 + β 2 ) { ) } 2 + 2ρ 2 γ 2 σ 2) X σ2) + 2 γ 2 σ 2) e α 2+β 2 ) 2 ) h, ; α 2 + β 2 ) { } ] + ρ 12 σ 1) X σ2) X + ρ 1ρ 12 γ 1 σ 2) X σ1) e β 1 1 ) β 2 2 ) h, ; β 1 + β 2 ) [ 1 ) 2 + u 2 γ 2 1 σ 1) e 2α 1 1 ) h, ; 2α 1 ) + 1 ) 2 γ 2 2 σ 2) e 2α 2 2 ) h, ; 2α 2 ) { 1 ) 2 + γ 2 1 σ 1) 1 ) } 2 + σ 1) X + ρ1 γ 2 1 σ 1) X σ1) e 2β 1 1 ) h, ; 2β 1 ) { 1 ) 2 + γ 2 2 σ 2) 1 ) } 2 + σ 2) X + ρ2 γ 2 2 σ 2) X σ2) e 2β 2 2 ) h, ; 2β 2 ) { ) } 2 γ 1 σ 1) + ρ1 γ 1 σ 1) X σ1) e α 1+β 1 ) 1 ) h, ; α 1 + β 1 ) { ) } 2 ρ 2 γ 2 σ 2) X σ2) + γ 2 σ 2) e α 2+β 2 ) 2 ) h, ; α 2 + β 2 ) ] ρ 12 σ 1) X σ2) X e β 1 1 ) β 2 2 ) h, ; β 1 + β 2 ), 84) ϕ l u) is he MGF of he individual jump sizes see 75)), and he funcion h, ; ) is given in equaion 28). Now ha he ransform is explici, i is possible o use sandard Fourier analysis echniques o value he spread opion. Under some mild assumpions on he m.g.f. of jump sizes, i is possible o analyically coninue he m.g.f. o he enire complex plane. For compleeness in he exposiion, we remind he reader how he pricing equaion 76) appears in Fourier ransformed variables. Firsly, Π F, = P, ) F 2) Q, 2 E e [F ;1, 2 α) + ] = P, ) e α F 2) Q, 2 E e [ ] e Z α 1) +, 85) where α := lnα). By inroducing ηx) := e x 1) +, he expecaion in equaion 85)

22 22 Samuel Hikspoors and Sebasian Jaimungal reduces o he produc of Fourier ransforms Q E e [ ] e Z α 1 1) + = 2π η p) f Z αp) dp, 86) where ηp) and f Z αp) are he Fourier ransforms of ηx) and he probabiliy densiy of Z α, respecively. I is well known ha ηp) := e ipx ηx) dx = 1 pi p) 87) whenever Ip) > 1. A simple change of variables reveals ha f Z αp) := e ipx Q f Z αx) dx = E e [ ] e ipz α) = e iαp Ψ Z ip). 88) Puing hese resuls ogeher leads o our final pricing equaion up o a numerical inegraion. Proposiion 4.3 he price a ime of he exchange opion is Π, = P, ) e α F 2), 2 e iαp Ψ Z ip) pp + i) wih Ψ Z ) as in Proposiion 4.2. Some final remarks are crucial a his poin: dp 2π, 89) 1. In our framework, he price process of a spo exchange opion is simply given by seing = 1 = 2 in equaion 89). 2. he inegral par of equaion 89) seems formidable; however, he coefficiens are nohing more complicaed han exponenials and here exiss very efficien numerical mehods, such as FF, for performing he inegrals. herefore, we do no pursue his furher, and insead refer he reader o he monograph by Con and ankov 2004) for furher informaion and references on hese opics. 3. he marke reveals he enire forward curve and, of course, he risk-free zero coupon bond prices. Before using he valuaion formula, he model mus be calibraed o hese marke prices. Once he parameers are calibraed, hen he pricing equaion 89) will provide consisen no-arbirage prices o he various spread opions.

23 Energy Spo Price Models and Spread Opions Pricing I is possible o repea his analysis when boh asses conain jumps. Needless o say, he resuling equaions will be bulkier bu no fundamenally more complicaed), and alhough he change of measure will be more suble, i posses no real problems. However, in real applicaions, boh asses ypically do no conain sudden jumps, as one is usually he raw commodiy used o produce elecriciy. 5. Some care mus be aken o ensure he inegraion pah in 89) remains in he inersecion of he regions Ip) > 1 and where he complex coninuaion of he funcion Ψ Z z) is analyic in z. However, for ypical jump disribuions, such as double exponenial and normal, Ψ Z z) will be analyic in he region Ip) > 1, and any simple pah lying in Ip) > 1 will do. 5. Model Calibraion In his secion, we finally address he issue of parameer esimaion. We perform his las sep in wo sages. Firsly, in Secion 5.1 we provide a deailed review of an efficien mehod for calibraing he pure diffusion wo-facor model o marke fuures prices, resuling in he risk-neural model parameers. We also describe how jump parameers can be simulaneously esimaed from marke spo prices. Secondly, we describe how a mehod borrowed from ineres rae model calibraion can be used o esimae he realworld model parameers from a knowledge of spo and fuure prices. his simulaneous calibraion of fuures and spo prices o he risk-neural and real-world measures furher allows us o exrac he implied marke prices of risk. An alernaive approach o realworld calibraion is o use a well known Kalman Filer approach. Such approaches do no uilize fuures prices daa and can be quie useful. For more deails on he calibraion of various wo-facor models o spo daa and furher references on he opic we refer o he work of Barlow, Gusev, and Lai 2004). Secion 5.2 concludes wih concree applicaions of our saisical mehodology o he NMEX Ligh Swee Crude Oil daa and some furher commens.

24 24 Samuel Hikspoors and Sebasian Jaimungal 5.1. Mehodology Before proceeding o he calibraion process, recall ha he log of he forward price { } associaed wih he spo S i) := exp g i) + X i) is given by Secion 3.2): log F i), = g i) + Gi), + R i), + e β i ) X i) + M i), i) 90) = g i) + Gi), + R i), + e β i ) log S i) g i) ) + M i), i) 91) := U i), + M i), i). 92) Here, he funcion U i), is inroduced o simplify noaion. Given he spo price daa a ime, U i), is compleely deermined, while he las erm M i), i) depends on he he hidden long-run mean level. herefore, a sandard nonlinear leas-squares opimizaion canno be applied direcly. Insead, we will express he hidden facor in erms of he remaining model parameers and obain an opimal fi o he observed fuures curve a various ime poins. Le F i) p, p q denoe he observed fuures prices a p { 1,..., m } wih delivery ime p q { p 1,..., p n p } and denoe by Θ a poin in he risk-neural) parameer space Ω of our model. For each given quoed imes p, obain i) p Θ) as a funcion of he remaining parameers) such ha i minimizes he following sum of squares: n p Sum p, Θ) := q=1 he opimal i) p #i) p Θ) = np q=1 [ log F i) p, p q ] i) 2. log F p,q 93) p Θ) is easily found o be [ )] M i) p,q p log F i) U i) p,q p p,q p [ ] np q=1 M i) 2. 94) p,q p Subsiuing his opimal value ino he iniial sum of squares 93), summing over he range of iniial imes { p } and performing a nonlinear leas-squares opimizaion as follows: Θ := ArgMin Θ Ω m n m p=1 q=1 [ U i) p, p q + M i) p, p q ] 2 #i) p Θ) log F i) p,q, 95) p provides an opimal fi of he model o fuures prices, herefore obaining our riskneural model parameers β, α, φ, σ X, σ, ρ). An implemenaion of his mehodology naurally requires boh fuures prices and spo prices a he corresponding fuures quoe imes.

25 Energy Spo Price Models and Spread Opions Pricing 25 I is worh menioning ha his mehod does no direcly exend o jump-diffusion spo) models since he coefficiens of he X and J erms in he forward price 73) are unequal. his prevens a simple facorizaion ino funcions ha are known given he spo prices and he hidden process 1). o circumven ha problem a sandard alernaive mehodology is o exrac he jump parameers from he spo price daa only. Such a calibraion can be carried ou in wo ways: i) by cuing off all daa poins lying below a given level, so ha only spikes remains. From hese daa poins one can hen infer he value of he various jump parameers see for example he discussion in Clewlow and Srickland 2000)); or ii) by uilizing paricle-filer approaches which generalizes he Kalman filer o non-normal innovaions see for example Aiube, Baidya, and io 2005)). he sandard assumpion ha he calibraed jump parameers are unchanged when moving o he risk-neural world is hen invoked. Given, he jump parameers i is now possible o repea he previous fuures price calibraion process o obain he risk-neural diffusive componens. We now urn o he real-world P-parameers β, α, φ, σ X, σ, ρ) esimaion problem. Since under any diffusive model for spo prices, a change of measure from he real-world o risk-neural canno aler he volailiy srucure of he model, from equaion 95) we obain σ X, σ and ρ under P. he remaining se of P-parameers β, α, φ) are relaively sraighforward o obain. Firsly, we obain β and φ via linear regression on he spo price daa assuming a mean revering one-facor model for X as a proxy o our wo-facor model. he one-facor mean-reversion level φ becomes, in our model, he sochasic longrun mean level. Secondly, we perform a similar regression on he esimaed hidden process # which was obained by minimizing he error on an individual fuures curve basis see equaion 94)). Equaion 94) provides a daa se which we can use as an inpu in a regression o find α. We find his procedure o be very sable and, as shown in he nex secion, leads o reliable parameer esimaion Some Resuls: Crude Oil In his secion, we presen he calibraion resuls of our wo-facor pure diffusion model Secion 3) o he NMEX Ligh Swee Crude Oil spo and fuures daa for he period 1/10/2003 o 25/07/2006. In able 1, we repor he calibraion resuls for he real-world

26 26 Samuel Hikspoors and Sebasian Jaimungal α β φ α β φ σ X σ ρ able % 63% he calibraed real-world and risk-neural model parameers using he NMEX Ligh Swee Crude Oil spo and fuures daa for he period 1/10/ /07/2006. and risk-neural parameers. here are a few noable observaions: i) boh real-world mean-reversion raes α and β are significanly larger han he risk-neural mean-reversion raes α and β, ii) he real-world long-run mean-reversion level φ is larger han he risk-neural long-run mean φ, and iii) in boh he real-world and risk-neural measures, he mean-reversion raes α and α) of he long-run mean are smaller han he meanreversion raes β and β) of he log-spo X. $100 $80 Price $60 $40 $20 Marke Spo Simulaion Long Run Mean $0 08/17/03 03/04/04 09/20/04 04/08/05 10/25/05 05/13/06 Dae Figure 1. he NMEX Ligh Swee Crude Oil spo prices and simulaed spo prices based on he calibraion in able 1. In Figure 1, we plo he spo price daa ogeher wih he sochasic long-run mean level # implied by he fuures prices. For comparison, we also include one simulaed

27 Energy Spo Price Models and Spread Opions Pricing 27 sample pah based on a simulaion of he prices using he real-world model parameers in able 1. Figure 2 illusraes he relaive roo-mean squared-error RMSE) for each forward curve using he model parameers repored in able 1. he average RMSE per curve is 0.7% wih only a few daes having relaive errors larger han 1%. Recall ha he model parameers are fixed over all curves, and are no adjused on a curve by curve basis. Wih his in mind, we believe he fi is excellen. 1.8% 1.6% 1.4% Relaive RMSE 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% 08/17/03 03/04/04 09/20/04 04/08/05 10/25/05 05/13/06 Dae Figure 2. he relaive roo-mean squared-error of each forward curve based on he calibraion in able 1. We also invesigaed he sabiliy of our esimaion procedure hrough ime. We calibraed he model o he firs 1 calender days and hen o he firs 2 calender days and so on. he ime periods are approximaely equally spaced a 88 days from 1/10/2003 o 25/07/2006. We repor hese calibraion resuls in able 2. he mos sable parameers are he volailiy σ X of he X process, he volailiy σ of he sochasic long-run mean level, he mean-reversion level α of he sochasic long-run mean and he correlaion coefficien ρ. he remaining parameers, alhough no as unvarying as he previous four, are well behaved. None of he parameers suddenly explode or end o zero, and always remain realisic.

28 28 Samuel Hikspoors and Sebasian Jaimungal # Days β α φ σ X σ ρ % 19% % 54% % 56% % 60% % 58% % 52% % 58% % 63% Average: % 57% Sdev: % 4% 0.14 able 2 his able shows he evoluion of he esimaed risk-neural parameers hrough ime as more recen daa is added o he calibraion procedure. he average and sandard deviaion are repored using 176 days onwards. Finally, since we were successful in exracing he real-world and risk-neural parameers, we furher exrac he implied marke prices of risk hrough heorem 3.2. he evoluion of he implied marke prices of risk λ and Ψ are displayed in Figure 3. Ineresingly, hey are very srongly correlaed o one anoher, becoming almos indisinguishable afer one and a half years. his is due o he high correlaion coefficien of ρ = Also, boh marke prices of risk are negaive for essenially he enire ime period. his is a reflecion of he real-world mean-reversion raes α and β) and real-world long run mean-reversion level φ) being higher han he risk-neural ones α, β and φ)). he marke herefore aaches slower reversion raes and lower long run levels han he implied hisorical levels.

29 Energy Spo Price Models and Spread Opions Pricing Marke Price of Risk λ ψ -11 8/17/03 3/4/04 9/20/04 4/8/05 10/25/05 5/13/06 Dae Figure 3. his diagram depics he evoluion of he implied marke prices of risk using he calibraed real-world and risk-neural parameers. 6. Conclusions We inroduced a diffusive wo-facor mean-revering process for modeling spo prices of energy commodiies. he wo-facor diffusive model exends he one-facor meanrevering model by making he long-run mean a sochasic degree of freedom which iself mean-revers o a specified level. We also generalized he model o incorporae jumps in he price process such as hose observed in elecriciy prices. o mainain realisic meanreversion raes and diffusive volailiies we decoupled he jump and diffusive processes. Given our affine modeling framework, we were successful in obaining expressions for he forward price curves in erms of elemenary funcions. hrough a measure changed induced by he forward price process, our modeling framework allows us o obain closed form pricing equaions for various spread opions. We obained pricing equaions under boh an Acuarial and risk-neural valuaion procedures. Finally, we provided a mehod for calibraing boh he diffusion and jump-diffusion models o spo and forward prices simulaneously. his simulaneous calibraion procedure furher allowed us o exrac he implied marke prices of risk. Using he NMEX ligh swee crude oil daa se, we demonsraed ha he calibraion procedure produces

30 30 Samuel Hikspoors and Sebasian Jaimungal realisic and sable implied risk-neural and real-world model parameers. 7. Acknowledgemens he auhors would like o hank Bill Bobey for assisance wih acquiring he NMEX daa and Hans uener, OPG Energy Markes, for fruiful discussions on various aspecs of he energy markes.

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