A NON-GAUSSIAN ORNSTEIN-UHLENBECK PROCESS FOR ELECTRICITY SPOT PRICE MODELING AND DERIVATIVES PRICING

Transcription

1 A NON-GAUSSIAN ORNSTEIN-UHLENBECK PROCESS FOR ELECTRICITY SPOT PRICE MODELING AND DERIVATIVES PRICING FRED ESPEN BENTH, JAN KALLSEN, AND THILO MEYER-BRANDIS Absrac. We propose a mean-revering model for he spo price dynamics of elecriciy which includes seasonaliy of he prices and spikes. The dynamics is a sum of non-gaussian Ornsein-Uhlenbeck processes wih jump processes giving he normal variaions and spike behaviour of he prices. The ampliude and frequency of jumps may be seasonally dependen. The proposed dynamics ensures ha spo prices are posiive, and ha he dynamics is simple enough o allow for analyical pricing of elecriciy forward and fuures conracs. Elecriciy forward and fuures conracs have he disincive feaure of delivery over a period raher han a a fixed poin in ime, which leads o quie complicaed expressions when using he more radiional muliplicaive models for spo price dynamics. We demonsrae in a simulaion example ha he model seems o be sufficienly flexible o capure he observed dynamics of elecriciy spo prices. We also discuss he pricing of European call and pu opions wrien on elecriciy forward conracs.. Inroducion The aim of his paper is o propose a model for he elecriciy spo price dynamics. We sugges a model ha, while capuring he essenial characerisics of spo prices o a reasonable degree, is simple enough o yield closed form expressions for elecriciy forward and fuures conracs and oher derivaives. The sandard approach in he lieraure is o model he logarihmic elecriciy spo prices hrough a mean-revering process see e.g., Lucia and Schwarz [] and Geman and Roncoroni [9], such ha in he classical Gaussian seing he spo price dynamics becomes lognormal. For such models i is nooriously difficul o derive manageable analyical expressions for he corresponding forward and fuures conracs. We propose insead o model he spo price dynamics direcly by an Ornsein-Uhlenbeck process which, however, is non-gaussian. More precisely, we consider a sum of Ornsein-Uhlenbeck processes, each of which is revering o a mean a a differen speed and having pure jump processes wih only posiive jumps as sources of randomness. Our proposed spo price dynamics has hus an addiive srucure in conras o exponenial srucure and gives posiive prices. We remark ha our proposed model is moivaed from a sochasic volailiy model proposed by Barndorff-Nielsen and Shephard [] in a differen conex. The approaches o model prices on elecriciy markes can mainly be divided ino wo caegories: spo price models and models for fuures prices. A model in he laer caegory Dae: May 23, 25. Key words and phrases. Elecriciy markes, spo price modeling, forward and fuures pricing, addiive processes, Ornsein-Uhlenbeck processes.

2 2 BENTH, KALLSEN, AND MEYER-BRANDIS describes direcly fuures prices insead of modeling spo prices from which fuures prices hen are derived. The advanage of such models is ha he marke can be assumed o be complee and he usual risk neural pricing machinery may be employed. Recen approaches are o ransfer conceps from he Heah-Jarrow-Moron heory for ineres raes o elecriciy markes o model he complee fuures price curve see Bjerksund e al. [5] and Benh and Koekebakker [3]. The problem wih hese approaches is ha due o he nonsorabiliy of elecriciy one canno use arbirage argumens o derive informaion abou spo prices from he analysis of fuures price models. Also, since fuures and forwards ypically have delivery periods of fully monhs, quarers or years, fuures curve moves are much less volaile han changes in spo prices. However, many derivaives on elecriciy prices depend heavily on hourly or daily prices which makes he modeling of spo prices necessary. Models in his caegory ry o fi he dynamics of hourly mos ofen day ahead prices on he spo marke which hen can be used o derive prices of fuures and more complicaed derivaives. Bu since no-arbirage relaions beween spo and fuures prices do no exis, addiional assumpions have o be made in order o price derivaives based on spo price models. More specifically, one has o idenify he marke price of risk of he including risk facors which hen yield an equivalen maringale pricing measure. I seems o be reaonable o assume a leas wo risk facors in spo price models, one responsible for shor erm hourly behaviour wih srong volailiy and one for more long erm behaviour observed on fuures markes. There are several key characerisics of elecriciy spo prices ha can be observed more or less disincly in all elecriciy markes. We recall he hree mos imporan ones. Firs, elecriciy prices ypically show very sharp spikes. This is due o inelasic demand combined wih an exponenially increasing curve of marginal coss. In imes of sudden changes of demand or supply for example caused by weaher condiions his resuls in srong jumps of elecriciy prices. However, afer sudden changes anoher feaure of elecriciy prices is ha hey raher rapidly end o rever back o a mean level, which makes a mean revering process appropriae o model spo prices. Finally, all involved magniudes such as he mean level, jump inensiies and jump sizes exhibi seasonal behaviour over one day, week, and year. As an illusraive example we presen in Figure he developmen of he daily spo prices a Nordpool, he Nordic Elecriciy Exchange, spanning over he period of approximaely 3 years saring in April 997. The paper is organized as follows: In he nex secion we define he spo price model, and analyze some of is properies. Secion 3 calculaes elecriciy forward and fuures prices based on he proposed spo price dynamics. Expressions for he fair premium of call and pu opions wrien on hese elecriciy forwards and fuures are analyzed in Secion 4. Finally, in Secion 5 we conclude. 2. Elecriciy Spo Price Process Le Ω, P, F, F } [,T ] be a complee filered probabiliy space, wih T < a fixed ime horizon. If S denoes he spo price of elecriciy a ime, hen we se 2. S µ X.

3 ELECTRICITY SPOT PRICE MODELING 3 4 Nordpool 3 years spo days Figure. The daily spo price on Nordpool spanning from April, 997 unil July 4, 2 Here µ is a deerminisic, periodic funcion and he sochasic process X is described by he following dynamics: X Y i where 2.2 dy i λ i Y i d σ i dl i, Y i y i, i,..., n, and are posiive weigh funcions such ha n, λ i are posiive consans and σ i are posiive bounded funcions. The processes L i, i,..., n are assumed o be independen increasing càdlàg pure jump processes ha can be represened in erms of heir jump measures N i d, dz, i,..., n: L i z N i ds, dz. We suppose L i o be inegrable and ha he jump measures N i d, dz have deerminisic predicable compensaors ν i d, dz for all i,..., n. These processes are also referred o as addiive or Sao processes and hey have independen bu no necessarily saionary incremens. Seing Ñ i ds, dz N i ds, dz ν i ds, dz

4 4 BENTH, KALLSEN, AND MEYER-BRANDIS we denoe by L i he compensaed jump process L i z Ñid, dz. In our model we will only deal wih compensaors of he form ν i d, dz ρ i d ν i dz, where ρ i is a deerminisic funcion. The cumulan funcion is defined as } 2.3 ψ i θ e iθz ν i ds, dz, where θ R. An explici represenaion of he elecriciy spo price is 2.4 S µ y i e λi σ i ue λi u dl i u. We see ha S is mean revering o he periodic funcion µ o which each componen Y i conribues wih a speed given by and λ i. The processes L i conrol he price variaion including boh he daily volaile variaion and he price spikes where σ i conrols he seasonal variaion of he jump sizes while ρ i conrols he seasonal variaion of he jump inensiy. Posiiviy of spo prices is guaraneed because he processes L i are increasing. Noe he periodic funcion µ is no he mean level of he spo price bu is relaed o his in he following way. The mean level of he spo price, le us denoe i by θ, is usually idenified by calibraing a periodic funcion evenually including a rend line o spo prices afer having aken ou spikes of sudden bigger price variaions. Le us assume in our model, he processes Y i, i,.., l are responsible for modelling he daily volaile variaion while he processes Y i, i l,.., n are modelling he price spikes. Then i makes sense o require he average spo price exluding he spikes o be he mean level θ: [ ] l E µ Y i θ. Furher, in order o model he daily volailiy i may seem reasonable o assume he jump inensiy ρ i and jump size σ i o be consan over ime. In his case we can se ρ i and L i becomes a subordinaor i.e. increasing Lévy process for i,.., l. I is hen known ha Y i, i,.., l, become saionary processes given he righ saring poin Y i see e.g. []. Moreover, saring a a deerminisic value y i he processes Y i are converging o he corresponding saionary processes as ime goes by. If we denoe he firs momens of hese saionary processes by β i : σ i λ zνi i dz, i,.., l, we deermine he periodic funcion µ in his case by he relaion l µ θ β i. Finally, we assume ha he risk-free ineres rae in he marke is r >, which we will undersand as he reurn from a zero-coupon bond invesmen.

5 ELECTRICITY SPOT PRICE MODELING 5 Example 2.. I is no he purpose of his paper o do precise saisical analysis or calibraion. However, in order o make a firs assessmen of he model we simulae a pah of he spo dynamics using Ornsein-Uhlenbeck processes described by he following specificaions: according o he ime horizon considered in Fig. λ σ νdz ρ OU.6 Γ.7; 6 OU 2.5 Γ; 4 OU 3.7 Γ; sin π Here Γα; γ denoes he Gamma disribuion wih parameers α and γ. The firs wo Ornsein-Uhlenbeck processes are responsible for he volaile variaion around µ and he hird one models he spikes preferably occuring in winer. For µ we have aken a sinus funcion wih yearly period. The pah of he spo price is simulaed over a ime horizon equal o he one we considered in Fig. for he Nordpool spo prices. As we can see in Fig. 2, he simulaed pah seems o capure well, a leas visually, he essenial feaures exhibied by he Nordpool spo price sample pah, like e.g. seasonaliy and disinc price spikes in he winer. 4 Simulaed spo price spo days Figure 2. Simulaion of a spo price pah using he specificaions given in Example Pricing of Forwards and Fuures in he Elecriciy Marke In he Nordpool power marke and oher power exchanges around he world here is rade in forwards and fuures conracs based on elecriciy. The main disincion of such

6 6 BENTH, KALLSEN, AND MEYER-BRANDIS conracs compared o oher commodiy markes is ha elecriciy forwards and fuures delivers he underlying commodiy over a period, raher han a a fixed ime. In he financial power marke, hese producs are cash seled measured agains he spo price in he selemen period. Consider a forward conrac which delivers elecriciy over he period [, T 2 ], where < T 2 T. The elecriciy is delivered as a flow of rae S/T 2 in he selemen period, giving a oal delivery of Su du/t 2. The conracs a Nordpool is seled financially, in he sense ha he holder of he conrac receives he money equivalen of his delivery. Using he arbirage-free pricing mechanism and assuming ha selemen akes place a he end of he delivery period, he elecriciy forward has he price F,, T 2 a ime given as [ T2 ] 3. F ;, T 2 E Q Su du F, where Q is an equivalen maringale measure. Since here is no radable underlying, all equivalen measures Q P will become maringale measures. Thus, o derive elecriciy forward prices we need o idenify he marke price of he involved risk facors in order o choose he pricing measure Q. Noe in passing ha he forward price coincides wih he price of he correponding fuures conrac when he risk-free ineres rae r is consan, as we assume in his paper. Furhermore, afer inerchanging expecaion wih inegraion wih respec o ime, i holds ha T2 F ;, T 2 fu, τ dτ, T 2 where f, τ E Q [Sτ F ]. Noe ha f, τ is he price of a forward conrac a ime delivering elecriciy a he fixed ime τ. Thus, he elecriciy forwards can be considered as a coninuous sream of forward conracs wih fixed delivery imes over he delivery period. Due o he simple srucure of our model we may use his o derive a very racable dynamics of he elecriciy forwards, he ask we now urn our aenion o. To specify a class of pricing measures, we consider equivalen maringale measures Q which are characerized hrough Radon-Nikodym derivaives of he form where M i dq n dp EM i, R φ i z, Ñidz, d, T for some posiive deerminisic inegrand φ i z,. Then M i and he corresponding Doleans- Dade exponenial EM i are acually maringales see Lemmaa 4.2 and 4.4 in [] which

7 ELECTRICITY SPOT PRICE MODELING 7 also hold in his case, and he Girsanov heorem for random measures yields ha he jump measure N i dz, d has he compensaion φ i z, ν i dz, d under Q. Le We have he following resul: ˆν i dz, d : φ i z, ν i dz, d. Proposiion 3.. The price of an elecriciy forward F,, T 2 a ime and delivery period [, T 2 ],, is given as 3.2 F ;, T 2 F ;, T 2 where F ;, T 2 T 2 µu λ i T 2 y i e λiu and L i is he compensaed jump process under Q, i.e. L i z Ndz, ds ˆν i dz, ds}. R u σ i se λ i s e λ it 2 s d L i s, } σ i se λiu s zˆν i dz, ds du R Proof. The argumen goes by a sraighforward calculaion, using he independen incremen propery of L i under Q F ;, T 2 E Q [Su F ] du T 2 } T2 µu y i e λ iu du T 2 w T2 [ u ] i E Q σ i se λiu s dl i s F du T 2 T2 u µu y } i e λiu σ i se λiu s zˆν i dz, ds du T 2 R w T2 [ u ] i E Q σ i se λiu s d L i s F du T 2 T2 u µu y } i e λiu σ i se λiu s zˆν i dz, ds du T 2 R w T2 i σ i se λiu s d L i sdu T 2

8 8 BENTH, KALLSEN, AND MEYER-BRANDIS F ;, T 2 This concludes he proof. λ i T 2 σ i se λ it 2 s e λ i s d L i s In paricular, if he L i s were subordinaors i.e. increasing Lévy processes such ha under Q he processes L i were again subordinaors wih E Q [L i ] ˆπ i, we would ge ha L i is a compensaed subordinaor under Q and T2 u F ;, T 2 µu y i e λiu ˆπ i σ i se ds } λiu s du. T 2 I may be desirable o represen he elecriciy forward price in erms of he spo price. Since in our model we have represened he spo price essenially as he sum of Ornsein- Uhlenbeck processes, we are able o derive a represenaion of he forward price in erms of hese only. The nex proposiion saes he exac resul: Proposiion 3.2. The price of an elecriciy forward F,, T 2 a ime and delivery period [, T 2 ],, is given as T2 u F ;, T 2 µu σ i se λiu s zˆν i dz, ds du T 2 R Y i e λ i e λ it 2. λ i T 2 Proof. Observe ha Xu Y i e λiu u Thus, we can calculae as follows o derive he desired resul: F ;, T 2 T 2 E Q [Su F ] du σ i se λiu s dl i s. µu T 2 [ u ] Y i e λiu E Q σ i se λiu s dl i s F }du T 2 µu Y i e λiu [ u ] } E Q σ i se λiu s d L i s F du T 2 µu u R σ i se λ iu s zˆν i dz, ds

9 ELECTRICITY SPOT PRICE MODELING 9 u Y } i e λiu σ i se λiu s zˆν i dz, ds du R T2 u µu σ i se λiu s zˆν i dz, ds du T 2 R Y i e λ it 2 e λ i. λ i T 2 This concludes our proof. Remark ha he explici form of he forward dynamics can be used o calibrae he model o observaions. Raher han firs esimaing he parameers in he spo dynamics, and nex deriving forward prices, we can sar ou wih he forward dynamics in Prop. 3.2 as he model, and esimae he parameers using forward observaions. From his poin of view, one can say ha we use a Heah-Jarrow-Moron approach aken from ineres rae heory o model he forward dynamics, wih he advanage ha here is an underlying spo dynamics conneced o his. Noe ha since he daa is observed under he probabiliy P, one needs o consider he objecive dynamics, and no he risk-neural as saed in he proposiion. In he Nordpool marke here exis forward conracs where he delivery period is overlapping. One can for example rade in conracs wih yearly delivery, bu a he same ime he marke also offers conracs wih delivery in each quarers of he year. In heory, such conracs mus saisfy he no-arbirage condiion F ;, T 2 n τ i τ i T 2 F ; τ i, τ i where τ < τ 2 <... < τ n T 2. This is indeed he case wih our model. In order o illusrae he advanage of our addiive model compared o exponenial models, we calculae he dynamics of a swap in he corresponding exponenial model. For now, assume ha S is given by 3.3 S exp µ X where X is as before bu no necessary resriced o have only posiive jumps. Then he swap dynamic can be calculaed as follows. Proposiion 3.3. Suppose ha he spo dynamics is given by 3.3, and ha S is inegrable. Then he elecriciy forward price F ;, T 2 a ime wih delivery over he period [, T 2 ] is given by 3.4 F ;, T 2 n Gu M i ; udu If one sars ou wih a specificaion of he forward dynamics direcly, i is no always he case ha one can associae a spo dynamics. See Benh and Koekebakker [3] for more on his.

10 BENTH, KALLSEN, AND MEYER-BRANDIS where Gu exp µu and M i ; u is given by y i e λiu dm i ; u M i ; u u R } e zσ i re λ i u r ˆν i dz, dr e zσ i re λ i u r N i dz, d ˆν i dz, ds}. Proof. The argumen goes by a sraighforward calculaion. F ;, T 2 T 2 T 2 T 2 E Q [Su F ] du [ E Q exp exp µu y i e λiu µu [ u E Q exp T 2 exp y i e λiu u σ i se λ iu s dl i s u µu R y i e λiu exp σ i se λiu s dl i s T 2 Gu R n M i ; udu This concludes he calculaion. u R ]} σ i se λiu s dl i s F du e zσ i se λ i u s ˆν i dz, ds ] e zσ i se λ i u s ˆν i dz, ds } F du u R e zσ i se λ i u s ˆν i dz, ds } e zσ i se λ i u s ˆν i dz, ds du

11 ELECTRICITY SPOT PRICE MODELING Noe in 3.4 he inegral expression for which here is no closed form expression in general. This is he main drawback wih he exponenial models, which, furhermore, lead o quie complicaed expressions for he price of call and pu opions. The addiive model, on he oher hand, lends iself o an analysis of opion prices in a raher nea way. The reader should noice ha he proof in he proposiion above assumes ha he jump process L i is of finie variaion. We can easily exend he resul o general jump processes wih infinie variaion, however, he main message is no alered. The elecriciy forward prices do no allow for any explici pricing mechanism when he spo is based on an exponenial model. 4. Pricing of Opions on Elecriciy Forwards and Fuures The Nordpool elecriciy marke organizes sandardized rading in European call and pu opions wrien of elecriciy forwards and fuures. We analyze he pricing of hese based on he choice of risk-neural measure Q made in he secion above. By employing he mehod of Fourier ransform along wih he cumulan funcions of he jump processes involved, we can derive expression for he price ha a leas lend hemselves o numerical pricing by fas Fourier ransform echniques. Before seing off, le us inroduce he following noaion: Define Σ i,, T 2 as 4. Σ i,, T 2 σ i λ i T 2 Then he forward dynamics can be wrien F ;, T 2 F ;, T 2 e λ i e λ it 2. Σ i u;, T 2 d L i u. We also inroduce he noaion ψ i,t θ o denoe a sor of cumulan of L i wih respec o he measure Q, more specifically T ] T ψ i },T θ : ln E Q [expi θsdl i s e iθsz ν i dz, ds for deeminisic funcions θs. Le K be he srike price a ime T, where T, he sar of he delivery period of he underlying elecriciy forward. The price of a pu opion conrac a ime T wrien on a forward wih delivery period [, T 2 ] is given by he expecaion 4.2 p; T ;, T 2 e rt E Q [max K F T ;, T 2, F ]. Observe ha he payoff funcion for a pu opion gx maxk x, is in L [,. Moreover, by exending he funcion naurally o be zero on he negaive half of he real line, we have a payoff funcion which belongs o he space L R, and hus he Fourier ransform of i can be defined. Le us more generally consider payoff funcions g L R, and he corresponding opion price which becomes 4.3 p; T ;, T 2 e rt E Q [gf T ;, T 2 F ]. We have he following resul:

12 2 BENTH, KALLSEN, AND MEYER-BRANDIS Proposiion 4.. If gf T,, T 2 L Q, hen we have ha 4.4 p; T ;, T 2 e rt g Φ,T F ;, T 2 where he funcion Φ,T is defined via is Fourier ransform Φ,T y exp ψ i,t yσ i,, T 2, and is he convoluion produc. Proof. The proof goes via he use of he Fourier ransform. Recall firs ha gx ĝye iyx dy 2π where ĝ is he Fourier ransform of g. Thus, we have E Q [gf T ;, T 2 F ] ĝye Q [expiyf T ;, T 2 F ] dy 2π ĝy expiyf ;, T 2 E Q [expiy 2π ĝy expiyf ;, T 2 E Q [expiy 2π The independen incremen propery of L i yields T T Σ i s;, T 2 d L i s F ] Σ i s;, T 2 d L i s F ] E Q [gf T ;, T 2 F ] ] T ĝy expiyf ;, T 2 E Q [expiy Σ i s;, T 2 d L i s dy 2π ĝy exp iyf ;, T 2 ψ i,t yσ i ;, T 2 dy 2π This proves he resul. Of course, he payoff funcion of a call opion does no belong o he space L R, and a-priori we canno use he echnique above o calculae he price of a call opion. Neverheless, is price is easily obained from he call-pu pariy. Alernaively, one can inroduce an exponenial damping of he payoff as in Carr and Madan [6]. We refer he reader o Carr and Madan [6] for he deails on his. The reader should noe ha i is in general no possible o work ou such explici expressions when basing he elecriciy forward prices on an exponenial model. 5. Conclusion The mos common models used for spo price dynamics in elecriciy markes are of geomeric ype, and in general no feasible for calculaing expression for elecriciy forward and fuures prices. Alhough hey may describe well he sylized facs of elecriciy spo prices, hey become unfeasible for furher analysis of derivaives pricing. We have proposed dy dy.

13 References 3 an addiive model ha mees he feaures of spo prices like seasonaliy and price spikes. The model ensures posiiviy of prices since he sochasic price flucuaions are modeled by employing an increasing jump process. We have demonsraed ha he process is far beer manageable in order o represen and price derivaives, since forward and fuures prices can be calculaed analyically and plain vanilla opions wrien on hese can be analyzed by Fourier echniques. In his paper we only included a firs visual es of our model by simulaing a sample pah of he spo price. This es confirmed our hope ha he model is well suied o capure he sylized facs of spo prices a leas o an accepable degree. I is he purpose of fuure work o make a more precise saisical analysis of he qualiy of he model. Acknowledgemens We are graeful o Seen Koekebakker for inspiring discussions. References [] Barndorff-Nielsen, O. E., and Shephard, N. 2: Non-Gaussian Ornsein-Uhlenbeck-based models and some of heir uses in financial economics, J.R.Sais. Soc. B, 63 par2. [2] Benh, F.E., Ekeland, L., Hauge, R., and Nielsen, B.F. 23. On arbirage-free pricing of forward conracs in energy marke. Appl. Mah. Finance, 4, pp [3] Benh, F.E. and Koekebakker, S. 25. Sochasic modeling of forward and fuures conracs in elecriciy markes. Manuscrip [4] Black, F The pricing of commodiy conracs. J. Financial Econom., 3, pp [5] Bjerksund, P., Rasmussen, H. and Sensland, G. 2. Valuaion and risk managemen in he Nordic elecriciy marke. Working paper, Insiue of finance and managemen sciences, Norwegian School of Economics and Business Adminisraion. [6] Carr, M., Madan, D.B. 999, Opion valuaion using he fas Fourier ransform, J. Comp. Finance, 24, pp [7] Clewlow, L. and Srickland, C. 2. Energy Derivaives: Pricing and Risk Managemen. Lacima Publicaions. [8] Eydeland, A. and Wolyniec, K. 23. Energy and Power Risk Managemen. John Wiley & Sons. [9] Geman, H., Roncoroni, A.:Undersanding he fine srucure of elecriciy prices, working paper. [] Kallsen, J.: Opimal porfolios for exponenial Lévy processes, Mah Meh Oper Res 2 5, pp [] Lucia, J. and Schwarz, E. S. 22. Elecriciy Prices and Power Derivaives: Evidence from he Nordic Power Exchange. Rev. Derivaives Research, 5, pp [2] Pilipovic, D Energy Risk. McGraw Hill. [3] Schwarz, E. S The sochasic behaviour of commodiy prices: Implicaions for valuaion and hedging. J. Finance, LII3, pp Fred Espen Benh, Cenre of Mahemaics for Applicaions, Universiy of Oslo, P.O. Box 53, Blindern, N 36 Oslo, Norway, and, Agder Universiy College, School of Managemen, Serviceboks 422, N-464 Krisiansand, Norway address: fredb@mah.uio.no URL: hp:// Jan Kallsen, Deparmen of Mahemaics, Munich Universiy of Technology, D Garching Munich, Germany address: kallsen@ma.um.de Thilo Meyer-Brandis, Cener of Mahemaics for Applicaions, Universiy of Oslo, P.O. Box 53, Blindern, N 36 Oslo, Norway, address: meyerbr@mah.uio.no