Understanding price volatility in electricity markets

Transcription

1 Proceedngs of the 33rd Hawa Internatonal Conference on System Scences - 2 Understandng prce volatlty n electrcty markets Fernando L. Alvarado, The Unversty of Wsconsn Rajesh Rajaraman, Chrstensen Assocates Abstract Ths paper llustrates notons of volatlty assocated wth power systems spot prces for electrcty. The paper demonstrates a frequency-doman method useful to separate out perodc prce varatons from random varatons. It then uses actual observed prce data to estmate parameters such as volatlty and the coeffcent of mean reverson assocated wth the random varaton of prces. It also examnes spatal volatlty of prces. prces can be qualtatvely observed. In the frst phase of our analyss, we establsh the perodcty characterstcs of ths sgnal. To do ths, we perform a Fourer Transform of the sgnal n queston. The absolute value of the sgnal under consderaton s gven n fgure 2. Keywords: electrcty spot prcng, rsk management. Introducton Prces n power markets vary both as a functon of tme of day (and week) and locaton. Prces also vary as a result of random and less predctable varatons n demand, temperature and varous market and system condtons. The two key questons of nterest n ths paper are: Can the perodc components of prces be dentfed n a smple and systematc manner? Can the random varatons be characterzed s some manner nvolvng just a few parameters? Ths paper develops procedures to estmate prce volatlty n an actual market characterzed by effcent spot prcng []. The assumpton s that the prces are avalable (retrospectvely) for every locaton n the system on an hourly bass for a one-month perod. The system prces observed over a one-month perod are decomposed nto a perodc and a random component usng frequency doman analyss. The random component s then further characterzed as a specfc type of random process borrowng from technques n wdespread use for optons prcng calculatons. Separatng out perodc components We begn the analyss by a characterzaton of the observed prces wthn the market of nterest. The perod analyzed nvolves one 3-day month (744 hours). The system margnal prce for the perod of study s llustrated n Fgure. A certan perodcty of Fgure : Actual prce varablty of the system prce for a one month perod Fgure 2: Spectral content of the prce varaton. At least fve perodc harmoncs are evdent as ndcated. Also, an exponental ft to the spectral magntude ampltudes s llustrated / $. (c) 2 IEEE

2 Proceedngs of the 33rd Hawa Internatonal Conference on System Scences - 2 The perodc harmoncs dentfed from the graph correspond to perods of 3 (daly varaton), 62 (every other day), 24 (every fourth day), 55 (every ffth day) and 27 (weekly varaton). The next step n our analyss s to separate the random component from these perodc varaton components. In order to do so, we perform a curve ft n the frequency doman spectrum, and adjust the heght of these components to the value of the ft curve. The ft process proceeds as follows: consder a small nterval of tme and let S be a varable functon of tme. Assume that the change n S over one tme nterval s z. For a Wener process, S = σ N(,) t, where N(,) s a zero mean unty standard devaton random dstrbuton. That s, S = S + σ N(,) A Wener process s used k+ k to descrbe random walks and Brownan partcle moton. Determne all the frequences for whch a perodc component s suspected. In our example, ths nvolves harmoncs 3, 62, 24, 55 and 27. At each of these frequences, determne a scalng factor as the rato of the ampltude of the exponental to the ampltude of the correpondng harmonc. Ths rato should always be less than one, otherwse the harmonc cannot possbly be consdered a perodc component, rather t must be consdered part of the nose. Scale the complex frequency doman ampltudes of each of these harmoncs (ncludng scalng ther alased counterparts n the upper porton of the spectrum) so that ther ampltude s precsely equal to the exponental ampltude. Ths s llustrated n Fgure 3(a). The remander after scalng conssts of a smple spectrum of a few solated frequences (5 n our example) that can be lustrated n the tme doman. Ths s llustrated n Fgure 3(b). After constructng the perodc sgnal n the frequency doman, restore t to the tme doman by nverse Fourer transformaton. The results of restorng both the perodc and the random components of the sgnal to the tme doman are llustrated n Fgure 3(c). Volatlty determnaton Havng removed the perodc components, t s now of nterest to determne the characterstcs of the random component, prmarly the prce. The determnaton of volatlty s a key element of the theory of optons prcng [2]. Volatlty plays many other mportant roles n the assessment of rsk n markets. We begn our dscusson of volatlty wth a characterzaton of a Wener process. To understand an ordnary Wener process, Fgure 3: Frequency spectrum after separaton between perodc and random sgnal components. System prce System prce Perodc component of prces Hour (expanded scale) Random component of prces Hour Fgure 4: Tme doman sgnal reconstructon for the tme doman and the random components of prce. A Wener process tends to result n varances that drft from a known mean value. It also exhbts / $. (c) 2 IEEE 2

3 Proceedngs of the 33rd Hawa Internatonal Conference on System Scences - 2 unnatural behavor near zero for tems that are never supposed to reach zero or negatve value (such as stock prces). A Wener process wll often lead to negatve values. An alternatve assumpton s to assume that t s the logarthm of the varable that drfts accordng to a Wener process. Ths s sometmes called an Ito process. In ths process: S = σ t k+ ln N(,) Sk That s, the logarthms of the prces follow a Wener process. Ths s called a lognormal process. Let there be n+ prces observed (744 n our example), where the ntervals are numbers from to n=743. Let the observed prce of nterest for nterval be S. For the usual case of estmatng prces of a stock, the usual defnton of u s: u S = ln S Ths defnton fals n stuatons where zero or negatve prces are expected. Negatve prces durng any tme perod can and do occur, sometmes as the result of low demand and nonzero startup and shutdown costs, and other tmes as a result of transmsson congeston. In fact, our example contans zero prces durng some hours. Thus, rather than usng the lognormal assumpton for observed prces, t s perhaps better to assume an ordnary Wener process wth a normal dstrbuton of prce dfferences. In ths case, we defne u as follows: u = S S The estmate of the standard devaton s of ths process s: s = u u n n = ( ) 2 If the prce of a stock ever reaches zero, t can smply be dscarded. Thus, negatve stock prces make no sense. where u represents the mean value of the u s. The volatlty σ for the random component of the process (expressed n yearly terms) s: σ = s τ where τ s the length of the nterval n years (or /2, n our example). Usng ths methodology for estmatng the volatlty, for the example at hand we obtan an annualzed volatlty of 2.8. Ιn order to better understand the meanng of ths volatlty, we now generate a completely artfcal prce process, usng ths mpled volatlty. We assume a zero-mean value for the prce. The tme step s one day, or / 36 years. Assumng that there s no drft (or rather, that any drft or perodc components s accounted as a determnstc addtve term), we then obtan that random prces can be generated as follows: S = σ N(,) Usng a startng prce of zero, a sample sequence generated by ths random process s llustrated n fgure 5(a). Observe the sgnfcant qualtatve dfferences between ths sequence and the actual observed prces. In partcular, observe the tendency of ths prce to drft. It s clear that the assumpton of a smple random walk process does not ft the emprcal evdence well. The reason for the dscrepancy s that (a) there appears to be a strong mean reverson component to the random varaton, and (b) there seems to be a jump behavor to the prces [3,4,5]. We now repeat the same experments but we use the followng expresson nstead: S = as + σ N(,) Here a represents a coeffcent of mean reverson (a<). The mean s zero n ths example. Fgure 5(b) llustrates the same smulaton as before, but ths tme wth a mean reverson coeffcent. Mean reverson s not suffcent to replcate realstc prce data. There seem to be jump processes assocated wth ths data. Jump processes have also been thoroughly studed n the lterature [Merton] / $. (c) 2 IEEE 3

4 Proceedngs of the 33rd Hawa Internatonal Conference on System Scences - 2 Fgure 5(c) llustrates the same smulaton but ncludng jump processes. The jump processes have a separate mean-reverson coeffcent and the probablty of upward jumps s dfferent from the probablty of downward jumps. Ths models seems to come the closest to replcatng actual prce behavor. analyss are summarzed n the hstogram llustrated n Fgure 6. It s clear from ths hstogram that most of the locatons have more or less the same volatlty n prces, but there are a few locatons (83 to be exact) that exhbt sgnfcantly hgher volatlty n ther prces. No locaton exhbted sgnfcantly lower volatlty than the system volatlty. 4 (a) Random component as a pure random walk (b) Random walk wth mean reverson (c) Random walk wth unsymmetrc jump processes and mean reverson Fgure 6: Hstogram of observed volatltes at all locatons. The frst column contans almost all of the 2 values. Thus, only a few locatons exhbt hgh prce volatlty. Fgure 5: Synthetc models for prce behavor. Volatlty by locaton The data we have used also ncludes data for the prce at ndvdual buses n the system. A total of more than 2 locatons are consdered n the study. For each prce at each locaton the same analyss as has been done for the system prces can be repeated. The result of ths analyss for the perod of nterest (one month) s a dstrbuton of average prces and a dstrbuton of locatonal volatltes. These are summarzed n ths secton. The objectve here was to study all 2-plus locatons and to determne the volatlty coeffcent for the prce at each locaton, to establsh whether volatlty s farly constant across the system or whether t vares sgnfcantly by locaton. The results of ths volatlty Conclusons An emprcal study of the prce volatlty n actual power markets has been conducted. Ths study suggests that frequency-doman methods are an outstandng tool to dentfy perodc components of prce varablty. The study also llustrates the features of a model that better represents volatlty of random components as a combnaton of random walk and jump processes wth mean reverson. References. F. Schweppe, M. Caramans, R. Tabors and R. Bohn, Spot prcng of electrcty (Kluwer Academc) J. Hull, Optons, futures and other dervatves (Prentce-Hall) / $. (c) 2 IEEE 4

5 Proceedngs of the 33rd Hawa Internatonal Conference on System Scences S. Deng, Stochastc Models of energy commodty prces and ther applcaton: mean-reverson wth jumps and spkes, PSERC report 98-28, 998 (avalable from 4. R. Merton, An Inter-temporal Captal Asset Prcng Model, Econometrca, vol 4, pp , V. Kamnsk, The challenge of prcng and rsk managng electrcty dervatves, n The US Power Market (Rsk Publcatons, Fnancal Engneerng Ltd.); / $. (c) 2 IEEE 5