Volumetric Hedging in Electricity Procurement

Transcription

1 Volumetric Hedging in Electricity Procurement Yumi Oum Deartment of Industrial Engineering and Oerations Research, University of California, Berkeley, CA, Shmuel Oren Deartment of Industrial Engineering and Oerations Research, University of California Berkeley, CA Shijie Deng School of Industrial and System Engineering Georgia Tech Atlanta, GA, Abstract Load serving entities (LSE) roviding electricity service at regulated rices in restructured electricity markets, face rice and quantity risk. We address the hedging roblem of such a risk averse LSE. Exloiting the correlation between consumtion quantities and sot rices, we develoed an otimal, zero-cost hedging function characterized by ayoff as function of sot rice. We then show how such a hedging strategy can be imlemented through a ortfolio of call and ut otions. I. INTRODUCTION The introduction of cometitive wholesale markets in the electricity industry has ut high rice risk on market articiants, articularly on load serving entities (LSEs). The unique non-storable nature of electricity as a commodity eliminates the buffering effect associated with holding inventory, and makes the ossibility of sudden large rice changes more likely. Significant market risks that are faced by LSEs are not related to rice alone. Volumetric risk (or quantity risk), caused by uncertainty in the electricity load is also an imortant exosure for LSEs since they are obligated to serve the varying demand of their customers at fixed regulated rices. Electricity volume directly affects the comany s net earnings and more imortantly the sot rice itself. Hence, hedging strategies that only concern rice risks for a fixed amount of volume cannot fully hedge market risks faced by LSEs. The rice and volumetric risks are esecially severe to LSEs because suly and demand conditions usually shift adversely together as demonstrated by the California electricity crisis in 2 and 2, which led three large LSEs in California to bankrutcy or near bankrutcy. As a way of mitigating rice risks in electricity markets, derivatives such as futures, forwards and otions have been used. An electricity forward contracts obligates a arty to buy and the other arty to sell a secified quantity on a given date in the future at a redetermined fixed rice. At the maturity date if the market rice is higher than the contracted forward rice, then the buyer would make a rofit, conversely, if the market rice is lower than the forward rice then the buyer will suffer a loss. The rofits and losses are aid when the delivery is comleted. Put or call otions are also used for different tyes of risk hedging: A call (ut) otion on electricity suly obligates the seller to reimburse the buyer for sot rices above (below) the strike rice. LSEs would also use call otions to avoid the risk of higher rices while still being able to forgo the contract and enjoy the benefit of lower sot rices. While it is relatively simle to hedge rice risks for a secific quantity, such hedging becomes difficult when the demand quantity is uncertain, i.e., volumetric risks are involved. When volumetric risks are involved a comany should be hedged against fluctuations in total cost, i.e., quantity times rice but unfortunately, there are no simle market instruments that would enable such hedging. Furthermore, the common aroach of dealing with demand fluctuations for commodities by means of inventories is not ossible in electricity markets where the underlying commodity is not storable. The non-storability of electricity combined with the steely rising suly function and long lead time for caacity exansion results in strong ositive correlation between demand and rice. When demand is high, for instance due to a heat wave, the sot rices will be high as well and vice versa. For examle, the correlation coefficient between hourly rice and load for two years from Aril 998 in California was.539. [2] also calculated the correlation coefficients between normalized average weekday rice and load for 3 markets: for examle,.7 for Sain,.58 for Britain, and.53 for Scandinavia. There are some markets where this rice and load relationshi is weak but in most markets load is the most imortant factor affecting rice of electricity. The correlation between load and rice amlifies the exosure of an LSE having to serve the varying demand at fixed regulated rices and accentuates the need for volumetric risk hedging. An LSE urchasing a forward contract for a fixed quantity at a fixed rice based on the forecasted demand quantity will find that when demand exceeds its forecast and it is underhedged the sot rice will be high and most likely will exceed its regulated sale rice, resulting in losses. Likewise, when demand is low below its forecast, the sot rice at which the LSE will have to settle its surlus will be low and most likely below its urchase rice, again resulting in losses. Because of the strong causal relationshi between electricity consumtion and temerature, weather derivatives have been considered to be an effective means of hedging volumetric risks in the electricity market. Weather derivatives, whose During this eriod, all the regulated utilities in the California market rocured electricity from the sot market at the Power Exchange (PX). They were deterred from entering into long term contracts through direct limitations on contract rices and disincentives due to ex ost rudence requirements.

2 ayoff is triggered by weather conditions, exloit the correlation between electricity demand and weather condition. For examle, if the ucoming winter is milder than usual, the electricity demand would be low leaving an LSE with low revenue. The LSE can rotect against such situation by buying a Heating Degree Days (HDD) ut otion, which gives a ositive ayoff if the winter was realized milder than the HDD strike value denotes and zero ayoff otherwise. However, the seculative image of such instruments makes them undesirable for a regulated utility having to justify its risk management ractices and the cost associated with such ractices to a regulator. In this aer we roose an alternative to weather derivatives which involves the use of standard forward electricity contracts and rice based ower derivatives. This new aroach to volumetric hedging exloits the aforementioned correlation between load and rice. Secifically, we address the roblem of develoing an otimal hedging ortfolio consisting of forward and otions contracts for a risk averse LSE when rice and volumetric risks are resent and correlated. Electricity markets are generally incomlete markets in the sense that not every risk factor can be erfectly hedged by market traded instruments. In articular, the volumetric risks are not traded in the electricity markets. Thus, we cannot naively adot the classical no-arbitrage aroach for hedging volumetric risks. Our roosed methodology is based on the alternative aroach offered by the economic literature for dealing with risks that are not riced in the market, by considering the utility of economic agents bearing such risks. Secifically, we maximize the exected utility over the LSE s rofit in order to investigate the otimal hedging strategies for rice and volumetric risks. Hedging roblems dealing with non-traded quantity risk has been analyzed in the agricultural literature. A ioneering article [] shows that the correlation between rice and quantity is a fundamental feature of this roblem and calculated the variance-otimizing hedge ratio of futures contracts. [3] shows that quantity uncertainty rovides a rationale for the use of otions. They derived exact solutions for hedging decision on futures and otions assuming a CARA utility function and multivariate normality for the distribution of rice and quantity. However, they assumed that only one strike rice of otions is available. In the electricity market literature, [7] directly deals with an LSE s roblem in a multi-eriod setting, but they don t consider otions as their hedging instruments. Our result shows that an otimal hedging ortfolio for the LSE includes otions with various strike rices. The idea of volumetric hedging using a sectrum of otions was also roosed in [6] from the ersective of a Public Utility Commissions who could imose such hedging on the LSE as a means of ensuring resource adequacy and market ower mitigation. Determining the otimal number of contracts from a set of available otions requires the solution of a difficult otimization roblem, even in a single-eriod setting since ayoffs of otions are non-linear. Instead, in this aer we tackle the roblem by first determining a continuous otimal ayoff function that reresents ayoff of a hedging ortfolio as a function of sot rice, and then develoing a relicating strategy based on a ortfolio of standard instruments. The idea of obtaining the otimal ayoff function is adoted from [4] which derives and analyzes otimal ayoff functions that should be acquired by a value-maximizing non-financial roduction firm facing multilicative risk of rice and quantity. Instead of assuming the existence of certain instruments, they derive the ayoff function that the otimal ortfolio will have. We extend their model to a more general setting and furthermore to the relication of the otimal ayoff function using available forwards and otions. We also derive an otimal ayoff function in a closed form for an LSE considering a constant absolute risk aversion (CARA) utility function under a bivariate normal assumtion on the distribution of quantity and logarithm of rice. The remainder of the aer is organized as follows. In section 2, we rovide a mathematical model and obtain the otimal ayoff function. In section 3, we exlore a way of relicating the otimal ayoff derived in section 2 using forwards and available call and ut otions. Section 3 shows an examle and section 4 concludes the aer. II. OBTAINING THE OPTIMAL PAYOFF FUNCTION A. Mathematical Formulation Consider an LSE who is obligated to serve an uncertain electricity demand q at the fixed rice r. 2 Assume that the LSE rocures electricity, that it needs in order to serve its customers, from the wholesale market at sot rice. We consider a single-eriod roblem where hedging instruments are urchased at time and all ayoffs are received at time. Hedging ortfolio has an overall ayoff structure x(), which deends on the realization of the sot rice at time. Note that our hedging ortfolio may include money market accounts, letting the LSE borrow money to finance hedging instruments. Let y(, q) be the LSE s rofit from serving the load at the fixed rate r at time. Then, the total rofit Y (, q, x()) after receiving ayoffs from the contracts in the hedging ortfolio is given by where Y (, q, x()) = y(, q)+x(). () y(, q) =(r )q. The LSE s reference is characterized by a concave utility function U defined over the total rofit Y ( ) at time. LSE s beliefs on the realization of sot rice and load q are characterized by a joint robability function f(, q) for ositive and q, which is defined on the robability measure P. On the other hand, let Q be a risk-neutral robability measure by which the hedging instruments are riced, and g() be the robability density function of under Q. Note that this 2 In fact, most of US states which oened their retail markets into cometition have frozen their retail electricity rices.

3 robability measure may not be unique since the electricity market is incomlete, however, in this aer we ignore this issue. We formulate the LSE s roblem as follows: E [ U[Y (, q, x())] ] max x() s.t. B EQ [x()] = (2) where E[ ] and E Q [ ] denote exectations under the robability measure P and Q, resectively. B is the time- rice of a risk-free asset aying $ at time. The constraint (2) means that the manufacturing cost of the ortfolio is zero, because a contract is riced as the exected value of discounted ayoff under the risk-neutral robability measure. This zerocost constraint imlies that urchasing derivative contracts may be financed from selling other derivative contracts or from the money market accounts. In other words, under the assumtion that there is no limits on the ossible amount of instruments to be urchased and money to be borrowed, our model finds a ortfolio from which the LSE obtains the maximum exected utility over total rofit. B. Otimality Conditions The Lagrangian function for the above constrained otimization roblem is given by L(x()) = E [ U(Y (, q, x())) ] λe Q [x()] = E [ U(Y ) ]f ()d λ x()g()d with a Lagrange multilier λ and the marginal density function f () of under P. Differentiating L(x()) with resect to x( ) results in L [ Y ] x() = E x U (Y ) f () λg() (3) by the Euler equation. Setting (3) to zero and substituting Y x =from () yields the first order condition for the otimal solution x () as follows: E [ U (Y (, q, x ())) ] = λ g() (4) f () Here, the value of λ should be the one that satisfies the zerocost constraint (2). If g() =f () for all, then (4) imlies that the otimal ayoff function makes an exected marginal utility from the variation in q to be the same for any. C. CARA utility A CARA utility function has the form: U(Y )= a e ay where a is the coefficient of absolute risk aversion. We see from the secial roerty U (Y ) = au(y ) of a CARA utility function that the following condition holds: E[U(Y ) ] = λ a g() f (), which imlies that the utility which is exected at any rice level is roortional to g() f (). It follows from U (Y )=e ay and (4) that the otimal condition is E [ e ] a(y(,q)+x ()) = λ g() f () for an LSE with a CARA utility function. Then, x ()= ( a ln f () λ g() E[ e ay(,q) ] ) = ( ln λ +ln f () a g() +lne[ e ay(,q) ] ) (5) The Lagrange multilier λ in the equation should satisfy the zero-cost constraint (2), which is x ()g()d =. That is, ( ln λ +ln f () g() +ln E[ e ay(,q) ] ) g()d = (6) Solving (6) for ln λ gives ( ln λ = ln f () g() +lne[ e ay(,q) ] ) g()d Substituting this into equation (5) gives the otimal solution: x () = ( ln f () a g() +lne[ e ay(,q) ] ) ( E Q[ ln f () ] + E Q [ ln E [ e ay(,q) ]] ) (7) a g() Note that if we can assume P Q in the electricity market, then the otimal ayoff function under CARA utility becomes x () = ( ln E [ e ay(,q) ] [ [ ]] E ln E e ay(,q) ) (8) a and thus the utility exected at after receiving the otimal ayoff is E[U(y + x ()) ] = ex(e[ln E[U(y) ]]) This imlies that the otimal ortfolio is such that the exected utility from the varying demand at given is the same for all. Bivariate lognormal-normal distribution under P Q: Suose for simlicity that the LSE assigns to each state the same robabilities as those given by the risk-neutral density function (i.e., P Q). We calculate the otimal ayoff function (8) assuming the distribution of (, q) to be bivariate lognormal-normal 3 : (log, q) N(u, q, v,σ 2 q,ρ). 2 We use the conditional distribution of q given, which is q N ( q + ρ σ q (log u ),σ v q( 2 ρ 2 ) ), to obtain ln E [ e ay(,q) ] [ ] =ln e a(r )q = qa(r ) ρ σ q (log u )a(r ) v + 2 σ2 q( ρ 2 )a 2 (r ) 2. 3 rice follows lognormal distribution and load follows normal distribution, but they are correlated each other.

4 Then, the otimal solution under P Q from (8) becomes x () = q( E[]) +ρ σ q ( log E[ log ] u ( E[]) ) v ρ σ q v r ( log E[log ] ) + 2 σ2 q( ρ 2 )a ( 2 E[ 2 ] 2r( E[]) ) For, a lognormal random variable with arameter (u,v 2 ), we have E[log ] =u, E[] =e u + 2 v2, E[ log ] =(u + v 2 )e u+ 2 v2, and E[ 2 ]=e 2u+2v2. By substituting these, we obtain the following otimal ayoff function: x ()=( q + σq( 2 ρ 2 )ar)( e u + 2 v2 ) (9) +ρ σ q ( r)(log u ) ρσ q v e u + 2 v 2 v + 2 σ2 q( ρ 2 )a ( 2 e 2u ) +2v 2 We note that scaling the quantity variable takes secial care. Consider scaling the quantity so that q = cq instead of q. Then q N(c q, (cσ q ) 2 ). One might be led to think that the otimal ayoff function would be just x () obtained using (, q ), multilied by c; however, that is not true. The only case where scaling the quantity by c results in an otimal ayoff function cx (), is when σ q = cσ q. III. REPLICATING THE OPTIMAL PAYOFF FUNCTION In the revious section, we ve obtained the ayoff function x () that the otimal ortfolio should have. In this section, we construct a ortfolio that relicates the ayoff x(). In [5], Carr and Madan showed that any twice continuously differentiable function x() can be written as in the following form: x() =[x(s) x (s)s]+x (s) + s x (K)(K ) + dk + s x (K)( K) + dk for an arbitrary ositive s. This formula suggests a way of relicating the ayoff x(). Let F be the forward rice for a delivery at time. Evaluating the equation at s = F and rearranging it gives x() =x(f ) +x (F )( F ) + F x (K)(K ) + dk + F x (K)( K) + dk. Note that, ( F ), (K ) + and ( K) + at each term are ayoffs at time of a bond, forward contract, ut otion, and call otion, resectively. x (K)dK units of call otions with strike K for every K>F gives the same ayoff as x(). The above imlies that unless the otimal ayoff function is linear, the otimal strategy involves urchasing (or selling short) a sectrum of both call and ut otions with continuum of strike rices. This result roves that LSEs should urchase a ortfolio of otions to hedge rice and quantity risk together. Even if rices go u with increasing loads, more call otions with higher strike rices are exercised, having an effect of utting rice cas on each incremental load. In ractice, electricity derivatives markets, as any derivatives markets, are incomlete. Consequently, the market does not offer otions for the full continuum of strike rices, but tyically only a small number of strike rices are offered. Our urose is to best-relicate the otimal ayoff function using existing otions only. Therefore, we need to decide what amount of otions to urchase for each available strike rice so that the total ayoff from those otions is equal or close to the ayoff rovided by the otimal ayoff function. Let K,,K n be available strike rices for ut otions, K,,K m be available strike rices for call otions where < K < < K n < F < K < < K m and let K =, K n+ = F = K, and K m+ =. Consider the following relicating strategy, which consists of x(f ) units of bonds, x (F ) units of forward contracts, 2 ((x (K i+ ) x (K i )) units of ut otions with strike rice K i, i =,,n, 2 (x (K i+ ) x (K i )) units of call otions for a strike rice K i, i =,,m. This strategy relicates a ayoff function x() with an error e() given below: e() = 2 {(x () x (K j ))(K j+ ) (x (F ) x (K n ))(F )}, if (K j,k j+ ) for any j =,,n, e() = 2 {(x () x (K n ))(F ) (x (F ) x ())(F )} if (K n,f), e() = 2 {(x (K ) x ())( F ) (x () x (F )}( F ) if (F, K ), and e() = 2 {(x (K j ) x ())( K j ) (x (K ) x (F ))( F )} if (K j,k j ) for any j =2,,m. We see that the error from the relicating strategy is very close to zero if there exist ut and call otions with strike rice F (i.e., K n F K ) and if is realized very close to one of the strike rices. The error will be smaller if strike rices are offered in smaller increments, esecially Therefore, x(f ) units of bonds, x (F ) units of forward contracts, x (K)dK units of ut otions with strike K for every K<F, and

5 Probability.8 x ρ = ρ =.3 ρ =.5 ρ =.7 ρ = Profit($/day): y = (r )q x 4 Fig.. Profit distribution for various correlation coefficients. Generated 5 airs of (, q) from a bivariate normal distribution of (log, q) with a various correlation ρ s, where log N(3.64,.35 2 ) and q N(3, 3 2 ), and lotted estimated robability density functions of the rofit using normal kernel (assuming r = $/MWh). for rice intervals with high robabilities of containing. 4 IV. AN EXAMPLE In this section, we illustrate the method that we derived in the revious sections. We consider the on-eak hours of a single summer day as time. Parameters were aroximately based on the California Power Exchange data of daily dayahead average on-eak rices and % of the total daily oneak loads from July to Setember, 999. Secific arameter values are imosed as follows: Price is distributed lognormally with arameters u = 3.64 and v =.35, in both the real-world and riskneutral world: log N(3.64,.35 2 ) in P and Q. Note that the exect value of the rice under this distribution is $4.5/MWh. Load that the LSE needs to serve is distributed normally with mean q = 3 and variance σ 2 q =3 2. Correlation coefficient between log and q is.7. The fixed rate r charged to the customers is. The LSE s risk reference is decided by CARA utility with the risk aversion a =.5. We would like to oint out a significant correlation-effect on rofit distributions. Figure shows that the rofit distributions become quite different as the correlation between load and logarithm of rice changes. Considering that the correlation coefficient of our data is.7, we observe that the correlation coefficient cannot be ignored in the analysis of rofit. The otimal ayoff functions (9) are drawn in Figure 2 for various correlation coefficients between log and q. Generally, 4 In fact, the NYMEX offers the following strike rices for PJM electricity otions: twenty strike rices in increments of $.5 er megawatt hour above and below the at-the-money strike rice, and the next strike rices in increments of $. above the highest and below the lowest existing strike rices for a total of at least 6 strike rices. The at-the-money strike rice is nearest to the revious day s close of the underlying futures contract. Strike rice boundaries are adjusted according to the futures rice movements. (source: low rofit from high loads for very high sot rices and from low load for very low sot rice is comensated with the cases where sot rices and loads are around the exected value. This can be seen from the grah where as the sot rice goes away from r, ositive ayoff is received from the otimal ortfolio while the ayoff is negative around r. We also note that larger ayoff can be received when the correlation is smaller. This is because the variance of rofit is bigger when the correlation is smaller as we can see from Figure. Therefore, even when the correlation is zero, the otimal ayoff function is nonlinear. Figure 3 illustrates the otimal numbers of contracts to be urchased in order to obtain the ayoff x () for an LSE with a CARA utility function. It indicates large variations in the number of contracts urchased under the otimal ortfoli as the correlation coefficient changes. We see that the numbers of otions contracts are very high relative to the mean volume. This is because we don t restrict the model with constraints such as credit limits. The zerocost constraint (2) that we included in our model allows borrowing as much money as needed to finance any number of derivative contracts. x*() 4 x ρ = ρ =.3 ρ =.5 ρ =.7 ρ = Sot rice Fig. 2. The otimal ayoff function under CARA utility when rice and load follow bivariate lognormal-normal distribution V. CONCLUSION Price risk and its management in the electricity market have been studied by many researchers and it is well understood. However, rice risk should bestudied in conjunction with volumetric risk (quantity risk), which is also significant. Volumetric risk has great imact on the rofit of load-serving entities; therefore, there is a great need for methodology addressing volumetric risk management. Weather derivatives are widely used to hedge volumetric risks since there is strong correlations between weather variables and ower loads. In contrast, we roose an alternative aroach that exloits the high correlation between sot rices and loads to construct a volumetric hedging strategy based on standard ower contracts. In a one-eriod setting, we obtain

6 Forwards Quantity (MWh) 2.5 x ρ = ρ =.3 ρ =.5 ρ =.7 ρ = Forward rice F ($/MWh) (a) x (F ) REFERENCES [] McKinnon, R. I., Futures markets, Buffer stocks, and income stability for rimary roducers, Journal of olitical economy, 75, , 967. [2] Li, Y. and Flynn, P., A Comarison of Price Patterns in Deregulated Power Markets, UCEI POWER Conference, Berkeley, March 24. [3] Moschini, G. and Laan, H., The hedging role of otions and futures under joint rice, basis, and roduction risk, International economic review, 36(4) Nov [4] Brown, G.W., and Toft, K.B., How firms should hedge, The review of financial studies, Fall 22, 5(4), , 22. [5] Carr, P., and Madan, D., Otimal ositioning in derivative securities, Quantitative finance Vol., 2,. 9-37, 2. [6] Chao, H. and Wilson, R., Resource adequacy and market ower mitigation via otions contracts, UCEI POWER conference, Berkeley, March 24 [7] Wagner, M., Skantze, P., Ilic, M., Hedging Otimization Algorithms for Deregulated electricity markets, Proceedings of the 2th Conference on Intelligent Systems Alication to Power Systems 23, Quantity on otions (MWh) ρ = ρ =.3 ρ =.5 ρ =.7 ρ =.8 ACKNOLEDGEMENTS This work was suorted by NSF Grants EEC 93, ECS 342 and by the Power System Engineering Research Center (PSERC) Strike rice K ($/MWh) (b) x (K) Fig. 3. The grahs show numbers on forward and otions contracts to be urchased to relicate the otimal ayoff x () that is obtained for the LSE with CARA utility. In this examle, the otimal ortfolio includes forward contracts for x (4.5) MWh, ut otions on x (K)dK MWh for K<4.5 and call otions on x (K)dK MWh for K>4.5. the otimal zero-cost ortfolio consisting of bonds, forwards and otions with a continuum of strike rices. Also the aer shows how to relicate the otimal ayoff using available Euroean ut and call otions. In a different aer we have obtained similar results for a mean-variance utility function and alternative joint robability distributions on quantity and rice The model and methodology are alicable to other commodity markets and with different rofit functions. There are more extensions which can be made to the current model. First, the zero-cost assumtion allows the LSE unlimited borrowing at time to buy the otions contracts. Imosing credit limits or Value-at-Risk limits on the hedging strategy would make the model more alicable. Second, the electricity market is incomlete, so the risk-neutral robability measure we choose would not be exactly the same as what the market uses for ricing. Therefore, a ricing error would exist, which can lead to inefficient hedging. A model that accounts for ossible errors in choosing the risk-neutral robability measure would be a good extension for alications in the actual electricity markets.