A BLACK-SCHOLES APPROACH FOR THE PRICING OF ELECTRIC POWER OPTIONS IN TURKISH POWER MARKET

A BLACK-SCHOLES APPROACH FOR THE PRICING OF ELECTRIC POWER OPTIONS IN TURKISH POWER MARKET AHMET YUCEKAYA Deparmen of Indusrial Engineering, Kadir Has Universiy, Faih, Isanbul, Turkey E-mail: ahmey@khas.edu.r Absrac The elecriciy price is a sochasic decision variable which depends on he load, emperaure, uni breakdowns, seasonal affecs, and workdays ec. Wholesale elecriciy cusomers aim o minimize heir cos hrough long erm bilaeral conracs. One way o deal wih he problem is o ge elecriciy opions in Turkish derivaive markes for fuure periods which need o be exercised before he expiraion dae. An opion gives he righ o consume 0.1 MWH of energy wih he srike price for each hour of he monh ha is he opion is exercised. We assume ha a wholesale power cosumer would like o have opions from he marke in an effor o ge physical energy a he opion expiraion. The opion is exercised only if he srike price is less han he average of he hourly day ahead power prices. This imposes a limi on he srike price ha is hen used in he Black-Scholes model. Using cyclic behavior of daily power prices and hisorical price daa provided by he marke auhoriy, a simple model is developed o forecas he hourly power prices. Then he inpu is used in Black-Scholes model o esimae he value of he opion. A numerical case sudy is developed and he resuls are presened. Index Terms Opion Pricing, Black-Scholes Model, Elecriciy Price Forecasing, Power Trading I. INTRODUCTION In deregulaed power markes he supply and demand of elecriciy deermine he marke prices. The supply and demand are wo differen and independen eniies of he sysem. Each consumer wih an oversized power consumpion rae is free o selec is own way of power supply. If he consumer has is own generaion unis, he power can be supplied by generaing elecriciy from he unis. The bilaeral conrac is common way ha is used for power supply. Two companies can have a conrac of which hey define he price, capaciy and delivery condiions. The risk is low and hence many large consumers prefer his opion. In deregulaed markes i is also possible for a company o purchase low-cos power and sell i on he spo marke in an effor o make profi. For such siuaions, he derivaive markes are used o purchase and sell he opions. An energy opion is an agreemen which is purchased from he derivaive markes o exercise i according o predefined condiions. The opion gives is owner he righ of using he energy for a defined period of ime and paying he corresponding srike price for he power. The European call opions are common conracs ha are used for such purposes. The opion can be exercised on he mauriy dae if i is economic. If i is no exercised, he opion expires a he end of he mauriy period. The value of he opion needs o be evaluaed as i will be an imporan inpu for he decision making. The risk of loss and probabiliy of profi should also have o be included o his process. There is pleny of research in he lieraure ha develops pricing mehods on he socks which mainly focusing on he well-known black and scholes model [1]. However, he pricing of energy derivaives require modified approaches as he elecriciy is a non-sorable, special and real-ime commodiy. The researches on energy derivaive pricing are limied. The auhor develops a mone carlo mehod o value a European call opion in elecriciy purchase agreemens in [2]. They firs explain he process ha large power consumers follow o purchase energy opions and heir moivaion and hen hey develop a load consumpion model o invesigae he role of uncerainy in he opion value. Auhors explain he common mehods used in pricing of energy derivaives in [3]. Then hey use geomeric Brownian moion and mone carlo simulaion o esimae he forward prices and esimae he value of an opion using black-scholes model. In [4], auhors explains basic conceps and formulaions for pricing elecriciy derivaives in compeiive markes. Auhors invesigae he mean revering models for energy commodiies especially for naural gas and crude oil in [5]. Numerical analyses are developed using binomial rees, finie difference mehod and mone carlo mehod. In [6], auhor presens he sochasic models for energy prices ha also include he jumps and spikes. Auhor focuses on he pricing of swing opions in energy markes in [7]. He presens pricing mehods for call opions, forwards, and opions wih payoff, hedging and risk. In [8], he auhor develop mehods o esimae he value of he forward opions. He use daa from he Nord Pool and shows ha black scholes mehod can be used for opion pricing. The price is volaile in developing markes like Turkey as he availabiliy and price of resources show significan changes over ime. The derivaive marke in Turkey has sared is operaions in 2005. The operaions for elecriciy opions sared in November 1

2011 which is relaively new. On he oher hand, he deregulaion of Turkish power marke sared in 2003 and he resrucuring process coninues. The process of privaizaion of plans ha belong o he sae has no been compleed ye. However, he independen sysem operaor mainains an open marke medium for supply side and demand side. The large power supplier, wholesale power consumer, power disribuion companies and ransmission organizaion are some of he naural member of he sysem. The energy opions ha are provided in he derivaive marke help he marke paricipans o price he elecriciy of fuure periods. The opion seller (wrier) sells an agreemen o he opion buyer in which a fixed amoun of power is guaraneed for a so-called srike price ha will be used during he defined ime inerval. An opion seller and buyer have o pay a fixed amoun of money as a deposi. The changes on asse prices are wihdrawn from one accoun and ransferred o oher accoun. If he remaining balance drops below a susainable level, he accoun holder is called for exra paymen o coninue he agreemen. The opion allows he buyer o use O MWh of energy for each hour of he monh. Financial ransacions are allowed and physical usage is no required. A power consumer can purchase he power from he spo marke in which he acceps all high price risks. On he oher hand, i is possible o forecas he forward elecriciy prices considering he supply and demand of he paricipans. I is naural ha he forecased prices have deviaions from he acual prices bu he forecased prices provide a confidence inerval and base ground for decision maker. In his research, we develop a price forecasing mehod and hen using he forecased prices we show ha i is possible o find a value for he energy opion. II. PROBLEM FORMULATION opion is: The payou equaion imposes a limi on he exercise price of he opion in such a way ha he exercise price is limied wih he average of marke prices. I can be expressed as follows: and hence using basic calculus we ge: If he opion is purchased a he curren ime hen he value of he opion according o Black-Scholes equaion: rt V S0N( d1) e XN( d2 ) (6) If he value of he opion is aracive, hen he opion can be used o ge power for he monh i is exercised. Ofen imes he opion buyer wais unil 1 o purchase he opion. Then he value of he opion a ime is: To find an accepable opion value, one can have an upper bound for an opion value by plugging in (5) o (7) and hence ge: and subjec o: As menioned earlier, he energy purchase agreemen esablished a ime is an opion ha requires buyer and seller o commi a required price. The opion gives he righ o use a O MWh of energy for each hour of he corresponding monh wih he cos of he srike price X. The appendix provides a lis of noaions. Assuming an M-days monh and h-hours days, he oal cos of exercising he opion becomes As an example le s consider a monh wih 30 days and 24 hours in each day. The cos of opion for he monh C opion = 720OX. Assume ha he energy is purchased from he spo marke if he opion is no exercised. Then oal cos of he energy supply is The opion is a European call opion and he opion is exercised if i is in he money. Hence he payou of he An opion buyer evaluaes he value of he offer and if V he value is larger han, here is a risk ha loss can occur. Noe ha a proper elecriciy price forecasing mehod should be employed o esimae he average of day-ahead hourly prices. The power price is a sochasic decision variable which depends on he load, emperaure, uni breakdowns, workdays ec. The hourly price in a day has a cyclic behavior wih random deviaions which need o be esimaed. Time series mehods, sochasic price generaion models, arima models are some mehods used in he lieraure o forecas he power prices[9]. In hese models, hisorical daa is used as a reference o esimae he fuure prices. Hisorical load daa, emperaure and hourly prices are some inpus ha are commonly used. Turkish power marke is dominaed by bilaeral conracs as he deregulaion and privaizaion are sill 2

incomplee. Governmen based resources sill have he imporan share. Besides he load and price daa along wih he generaion informaion are no available for marke paricipans. The compeiion is sill no widely effecive on he prices as he resources are limied. Therefore, we use a hisorical price based regression mehod o esimae he hourly prices. Suppose ha HP is he hisorical power price a ime for he pas year. Then we assume ha i is possible o have an equaion: wih furher assuming as he noise wih N(0,σ p ). The accurae calculaion of and leads o an approximaed esimae of real power price for hour. Noe ha and can be esimaed only if here is real daa and i is in usable forma. A. Soluion approach We consider a company ha purchases energy opions from he Turkish derivaive marke in an effor o consume he low-cos power or sale his power o marke for profi. The main objecive of he problem is o find a decision framework o value he opion. Black-scholes model is seleced as a ool o evaluae he value of he opion. In order o have an upper bound on he opion value, we developed a price forecasing mehod based on he hisorical price daa. We firs use he laes available daa ha is closer o he opion exercise dae o calculae he parameers and. Then use he parameers o calculae he hourly prices of he monh of ineres. The pseudo code of he soluion approach is given in Fig. 1. 1. START 2. Se = 1, mauriy ime =T, ineres = r 3. Ge hisorical energy opion daa for las 4 monhs 4. Esimae σ 5. Ge hisorical power price daa for each hour of las 6. year 7. Esimae and parameers for regression equaion 8. Esimae σ p for power prices 9. Generae hourly price scenarios for monh of ineres 10. based on and 11. Calculae P based on he prices 12. Calculae d1, d2, N(d1) and N(d2) 13. Esimae V ha will be an upper bound for an opion value 14. End Fig.1. Pseudo code of he muli-period opimizaion and simulaion Noe he parameers ha affec he elecriciy price have he same effec on he boh opion monh and he previous monhs. Hence, we assume ha i is no a srong assumpion o use he same regression formulae o calculae hourly elecriciy prices of exercise monh. Anoher imporan parameer o calculae is he volailiy of he opion price hrough ime. To esimae he volailiy of he opion price, we use he mehod employed by Blanco and Soronow, (2001). They use logarihmic price changes o calculae he volailiy of he opion. The naural log of price changes for each period are assumed as coninuous reurns and heir sandard deviaion is compued. Then he volailiy is esimaed based on he number of periods ha he price daa is available. Once he average power price is found and volailiy of opion price is esimaed, d1 and d2 which will be used o in cumulaive normal disribuion are compued. The black and scholes model provides an opion value for he opion buyer. This value hen can be used o make a decision for power supply. III. NUMERICAL EXAMPLE We assume ha a wholesale power supplier needs o have 10 MWh of power for he April, 2012. The value of O, amoun of power per opion is deermined as 0.1 MWH in Turkish Power marke. They can procure he power from he marke wih he cos of spo marke price which is open o risk of higher prices. They would like o purchase opion conracs from he derivaive marke a he beginning of March 2012. A feasibiliy analysis is needed before he acual decision is made. Since each opion provides 0.1 MWh of power for each hour of he April (which is 30x24 =720 hours), hey need o purchase 100 opions o saisfy he demand. Each opion requires a paymen of 1200 TL o iniiae he conrac and in oal a cos of 120000 TL o he company. The analysis presened below is only for one opion. I is assumed ha =0, T =1 monh (1/12), r = 10% compounded coninuously. The derivae marke in Turkey sared o do energy sales operaions on November 2011. Therefore, no annual daa is available for he opion prices. However, he opion prices of pas 5 weeks are colleced as shown in Table I. Table I. Opion prices in Turkish derivaive marke 3

The reurns of he opions are found using logarihmic scale of he change beween wo consecuive periods. Then he sandard deviaion of he reurns is used o find he annual volailiy of he opion. The nex sep is o esimae he parameers ha will be used o esimae he hourly elecriciy prices. The laes available daa ha can be used is he prices of February 2011 and 2012. Using he regression analysis, he parameers are esimaed as 0. 63827 and 97. 12. To show he relaionship beween acual and forecased prices, we plo he forecased and acual prices of February 2012 in Figure 2. 270 250 Acual prices Forecased prices value is an upper bound for he value of he opion. In order words, if he calculaed opion value of is less han 1,004 TL, hen he opion can be purchased. We perform a sensiiviy analysis o show he effec of he average power prices on he opion value. Fig. 4 shows he resuls. O p i o n V a l u e ( T L ) 3000 2500 2000 1500 1000 500 y = -31.411x + 5817.3 R 2 = 0.9924 P o w e r p r ic e (T L /M W h ) 230 210 190 170 150 130 110 90 70 0 100 200 300 400 500 600 700 hours Fig. 2. Acual and forecased power prices of February 2012 Using he relaionship P=97.12+0.63827HP, we esimae he power prices for April 2012. Fig. 3 shows he esimaed prices for April. The average of hourly prices plays an imporan role in Black-scholes model. P is esimaed as 152.11 TL/MWh and used as an inpu for he model. P o w e r p r i c e ( T L / M W h ) 230 210 190 170 150 130 110 90 0 100 200 300 400 500 600 700 800 Hours Fig. 3. Esimaed hourly day-ahead power prices for April 2012 The parameers for he cumulaive normal disribuion are esimaed as N(0.11)= 0.54 and N(-0.22)= 0.41. The Black scholes model is solved and he upper bound on he opion value V is esimaed as 1,004 TL. This 0 100 110 120 130 140 150 160 170 180 Average price (TL/MWh) Fig. 4. Acual and forecased power prices of February 2012 An inerval of 5 TL/MWh is seleced o deermine he average prices ha will be used in he sensiiviy analysis. The relaionship beween average power price and he opion value is close o linear. Noe ha as he esimaed average power price decreases, he value of opion increases. I is worh menioning ha he figure is only valid for he exercise monh for given parameers and should be esimaed for any new parameer se. CONCLUSION The value of opion plays an imporan role in derivaive markes no maer he opion is purchased for physical power supply or for financial ransacion purposes. The esimaed forward prices are commonly used o make a decision in he derivaive markes. In his research, we have developed a model ha combines esimaed power prices wih he black-scholes model o provide an opion value for he opion buyers. A sensiiviy analysis is provided ha shows he value of he opion for differen average prices. The model can be exended by including sochasic feaures of he opion and power prices. The price forecasing mehod can furher be improved if he load daa is employed. However, he model wih is curren form can be used in Turkish derivaive marke if he price forecasing mehod is esed and verified wih real daa. REFERENCES [1] S. Chen, B. Mulgrew, and P. M. Gran, A clusering echnique for digial communicaions channel equalizaion using radial basis funcion neworks, IEEE Trans. on Neural Neworks, vol. 4, pp. 570-578, July 1993. 4

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