The Joint Dynamics of Electricity Spot and Forward Markets: Implications on Formulating Dynamic Hedging Strategies

Transcription

1 Energy Laboratory MI EL Massachusetts Institute of echnology he Joint Dynamics of Electricity Spot an Forwar Markets: Implications on Formulating Dynamic Heging Strategies ovember 2000

2 he Joint Dynamics of Electricity Spot an Forwar Markets: Implications on Formulating Dynamic Heging Strategies Petter Skantze an Marija Ilic Energy Laboratory Massachusetts Institute of echnology Cambrige, Massachusetts Energy Laboratory Publication # MI_EL ovember 2000 For further information please contact Marija Ilic at or via at ilic@mit.eu

3 Abstract he eregulation of the electric utility inustry has brought with it a great eal of financial uncertainty for market participants. In this report we aress the question of how participants can use available markets in orer to mitigate this risk. In orer to evelop effective strategies for traing, one must first have a goo unerstaning of the ynamics of prices on all available markets. We therefore begin by aressing the relationship between financial an physical, spot an forwar markets. In oing so we examine the arbitrage pricing theory approach to moeling forwar prices, an evaluate its relevance for non-storable commoities. From the basic relationships between the markets, we arrive at stochastic moels which quantify future uncertainty in the marketplace. ext we consier the case of a loa serving entity serving loa uner a stanar offer contract. We show how the stochastic moels for loa an spot prices allows us to quantify the risk exposure of the LSE. ext we formulate the problem of how an LSE can optimally manage its risk using a perioically rebalance forwar portfolio, base on a mean variance objective function. We show that by using the propose price moels, we can convert the problem into a ynamic programming formulation, which can be solve with a number of computationally efficient tools.

4 Introuction Competitive power markets exhibit a level of price volatility unparallele in traitional commoity markets. he reason for this behavior lies in the nature of how electricity is prouce an consume, incluing lack of storage, inelastic loa, an strong seasonal effects on multiple time scales. hese characteristics of supply an eman are reflecte in the ynamics of market prices, an specifically in the joint ynamics of spot an forwar prices. his interaction is of tremenous interest to market participants who wish to use the forwar markets to manage their financial risk. In this report we aress the relationship between financial an physical, spot an forwar markets. In oing so we examine the arbitrage pricing theory approach to moeling forwar prices, an evaluate its relevance for non-storable commoities. From the basic relationships between the markets, we arrive at stochastic moels which quantify future uncertainty in the marketplace. hese moels are then be applie to the problem of ynamic heging of physical an financial obligations in electricity markets. 2 Power Markets here are three funamental markets available for traing electricity. he Spot Market (ay ahea), the physical forwar or bilateral marke an the financial forwar or futures market. While there is no exact mapping between prices across these markets, there is a strong interepenence. We here examine the interplay between the markets, an attempt to efine creible moels for the joint evolution of prices. 2. he Spot Market he spot market is conucte by either a power exchange or an ISO. Participants submit bis, generally on a ay ahea basis, an the market maker clears the market an announces an hourly locational system price. rae on this market is physical, meaning that physical eliver is always expecte. Participants who efault on a physical contract will be charge a penalty which is normally epenent on the price of real-time or balancing power in that region. 2.2 he Physical Forwar Market Physical forwars can be trae on an exchange or in a bilateral manner through over the counter (OC) transactions. Exchange trae forwars use stanarize contracts, with power being trae in monthly on- an off-peak blocks (see CBO efinition of power contract). he contract specifies a single MW quantity (q) an a single $/MWh price (F). he short position (seller of the forwar contract) is obligate to physically eliver power at a constant rate q to a location specifie in the contract (the HUB). he contract oes not specify the location at which the power is prouce or consume, but states that the short party is responsible for elivering the power from the generator location to the HUB, an the long position is responsible to eliver the power from the HUB to the loa location. For both sies this may involve purchasing aitional transmission contracts, or purchasing/selling power through the spot market. Such

5 provisions are not aresse in the contrac an the relative prices of the spot an transmission market will not affect the price of the forwar contract. he price of exchange trae physical forwars is quote aily by the exchange. he information provie inclues the high an low prices as well as the volume trae an the volume of open interest. he exchange quotes prices for every elivery month up to 5 months into the future. his vector of prices G(t), which constantly evolves new traes become public, make up the forwar curve for electricity. g jan00( t) g feb00( t) G( t) = : gmar0( t) Physical forwar contracts trae continuously while the exchange is open, until the fourth business ay prior to the first elivery ay of the contract. At this point traing terminates, an any party left with a short position is require to eliver power accoring to the provisions in the contract. A traer can avoi this by booking out his position, purchasing a long position which exactly offsets his short position for the same elivery month. 2.3 he Financial Futures Market Financial futures contracts for electricity are trae on exchanges such as YMEX an CBO. Financial contracts are similar to exchange trae physical contract in structure. he main ifference is that the parties entering into the contract have no intention of physically proucing or consuming the power, but rather use it as a financial hege against other positions in the market. he financial futures contracts are therefore settle trough the exchange of cash rather than power. In general the payoff function for a party holing the long position in a forwar contract is given by: payoff ( long) = S F( ), where S is the spot price at the maturity, an F() is the price of the futures contract at the time t it was entere into. he problem which occurs with electricity is that the elivery perio for the futures contract is one month, while the unerlying spot process is upate on a ay ahea basis. As a resul when the futures contract matures on the 4 th business ay prior to the first ay of the elivery perio, the sot prices for hours in the elivery month are not yet known. Hence the contract cannot be settle financially at this time. o circumvent this problem, exchanges have taken on two ifferent approaches, ex-post settling an ex-ante settling. Ex-post settling: In this approach, the futures contract is settle graually settle uring the elivery month. If two parties have entere into a futures contract for q MWs of on-peak power at a price F, then for every ay for the uration of the elivery perio, the following process etermines the cash flow:

6 . he on-peak price of power for the ay is calculate by averaging the hourly price of the 6 on-peak hours from the ay-ahea spot market. We enote this prices P peak. 2. he long position (buyer of contract) will pay the ifference between the P peak an F times the quantity of the contrac times the number of on-peak hours (6). If this quantity is negative then the cash flow will be from the short position to the long position. he total cash flow for the long position over the uration of the elivery month is given by: n 6 q peak ( P i F ) where n is the total number of ays in the elivery perio. Ex-ante settling: In this case, the futures contract is settle financially at its expiration ate, ie. on the 4 th ay prior to the beginning of the elivery perio. Since the ay-ahea spot price is not yet known for the elivery month, the price of a physical forwar contract for the same elivery perio an location is use in place of the ay-ahea spot. his effectively is a change in the unerlying commoity from which the futures contract is erive from a erivative the spot market to a erivative on the physical forwar market. he payoff function for the long position at maturity is given by: n q ( G(, ) F ) where q is the quantity of the contract in MWs, G(,) is the price of a physical forwar on the last ay of traing, an F is the price at which the futures contract was purchase. Both the ay-ahea spot an physical forwar are base on the same commoity, electricity elivere at a specific gri location. However there is no simple mapping between the ex-post average spot price an the ex-ante physical forwar price. his is a very crucial point to unerstan in electricity markets. While the settling proceure iffers form market to marke the ominant tren seems to be in the irection of ex-post settling, as seen in California an orpool. Unless otherwise specifie we will from here on assume that financial forwars settle ex-post Arbitrage Pricing an Price Convergence Arbitrage pricing theory (AP) [6] is base on the belief that pure arbitrage opportunities cannot survive in competitive markets. his assumption imposes constraints on the manner in which prices coevolve in the market. We consier this approach as it relates to three types of assets: stocks, storable commoities, an electricity. We aapt the following efinition of arbitrage. Consier a market with n traable assets, each with price X i (t). A portfolio is buil by purchasing an selling these contracts. he value of the portfolio is given by:

7 Π( t) = n w ( t) x ( t) where w i represents the quantity of asset i in the portfolio. W s can be negative if the market allows short-selling. Since future asset values are uncertain, the value of the portfolio at t>t 0 is a ranom variable. We efine an arbitrage opportunity as follows. Arbitrage exists if at time t 0 we can construct a portfolio with the following properties: Π( t 0 ) = 0 an for some t > t 0 Prob Prob i ( Π(t) < 0) = 0 ( Π(t) > 0) > 0 his means that we can construct a portfolio with zero cos which has zero probability of ecreasing in value an a strictly positive probability of increasing in value. Since the portfolio has zero initial cos any market participant can purchase an unlimite amount of the portfolio, an enjoy a risk free guarantee profit. he theory is that as arbiters start to take avantage of this opportunity, they will create an upwar price pressure on assets with positive weights in the arbitrage portfolio, an ownwar price pressure on assets with negative weights. Prices will then reach a new equilibrium where the arbitrage opportunity no longer exists Application of Arbitrage Pricing heory (AP) in Electricity Forwar an Futures Markets We now aress the relative prices of physical forwar an financial futures contracts with ex-post settling, in the framework of arbitrage pricing as efine in the previous section. We allow for the contract to be trae on ifferent exchanges, but assume that there is reasonable price transparency an liquiity in the market. he valiity of these aresse at the en of the section. Recall that the notation for the price of a physical forwar contract signe at time t for elivery at time, is enote by G(). he equivalent notation for a financial futures contract is F(). We now consier possible relative price levels of the physical an financial markets, an test their consistency with the absence of arbitrage assumption. First consier the event where at a time we observe a set of contracts for elivery month satisfying the relationship, F ( ) > G( ) A traer can then implement the following strategy. At time t:. Purchase q MW of physical forwar contracts. 2. Sell q MW of financial futures contracts. At time : i

8 . For each hour in the elivery perio, submit a sell bi of q MW of power into the ay ahea spot market at zero price. he power neee to eliver from the spot market is receive from the physical forwar contract. he cash flow from this strategy in each time perio is shown in the table. ote that all cash flows from forwar contracts are realize at the en of the contract. t buy physical 0 qf ( ) sell financial 0 qf( ) qs i sell spot 0 otal 0 q ( F( ) G( ) ) > 0 his strategy provies a guarantee profit with zero investmen an therefore it is an arbitrage opportunity which cannot be sustaine. ow consier the case, F ( ) < G( ) he traer aopts the following strategy. At time t:. Purchase q MW of financial futures contracts. 2. Sell q MW of physical forwar contracts. At time :. For every hour in the elivery perio, submit a buy bi for q MW to the ayahea spot market at the market maximum price (we later iscuss what happens if the spot market fails to clear). he electricity purchase in the spot market is use to eliver against the obligation from the physical forwar contract. he cash flows in each time perio is given by: t sell physical 0 - qf ( ) buy financial 0 qs qf( ) sell spot 0 otal 0 q ( G( ) F( ) ) > 0 his strategy provies a guarantee profit with zero investmen an therefore it is an arbitrage opportunity which cannot be sustaine. i qs i qs i

9 he strategies presente above show that in a market free of arbitrage opportunities, the price of a financial forwar cannot eviate from the price of a physical forwar, in ether a positive or negative irection. his conition must hol true not just at maturity, but uring the entire lifetime of the contracts. We thus arrive at the first constraint for electricity erivatives in an arbitrage free marketplace: G ( ) = F( ) Limits to Arbitrage Pricing Arguments While AP provies a convincing argument why physical an financial forwar prices must be equal at all times, actual observations in the market place show that the two market can iverge at times. he reasons for this inconsistency can be foun in the assumptions unerlying the arbitrage argument. he following points illustrates how market realities eviate from the theory:. Moving Equilibrium: Arbitrage pricing theory is base on an equilibrium argument. It states that in a market with active arbitruers, a set of prices which allow for risk-free profit with zero investment is not sustainable. As traers execute the arbitrage, they graually alter the relative prices until the system settles into an arbitrage free state. Markets in general, an electricity markets in particular, are continuously evolving ynamic systems. he effect is similar to that of a feeback control system riving the states of a system towars a continuously changing control input. Unless the input signal evolves at a significantly slower rate, the states will never settle to their equilibrium value. In the case of electricity markets, the valiity of the equilibrium argument will epen on two factors:. he rate at which new information about the future expecte value of the spot price enters into the market. Changes in traers perception of the future is the riving input into the futures market. Information which woul cause traers to change their perception woul inclue upates on future weather/loa conitions, or news of a generator or transmission line outage. 2. he volume an rate at which contracts are traing in the market. his represents the magnitue an spee of the feeback response, or the rate at which the market can react the new information. his is also known as a market s liquiity. We aress this issue in more etail as we introuce our ynamic moel for the evolution of the spot price. 2. Uniqueness of Prices: Unlike the spot marke the forwar markets o not have a unique clearing price. he price quote by the exchange is a weighte average of all traes in the last ay. However there is no guarantee that the traer can fin a counterparty willing to trae at exactly this average price at the time the arbitrage is execute. here generally is a gap between the highest price the market is willing to buy, an the

10 lowest price the market is willing to sell at. his is known as the bi-ask sprea. he magnitue of the bi-ask sprea is epenant on the liquiity of the market. 3. ransaction Costs: Exchanges are generally for profit enterprises. hey make a profit by charging a small fee for every contract which is execute on the exchange. he loss incurre by the traer ue to such fees is known as a transaction cost. In electricity markets, exchanges generally charge a fixe fee per MWh of power covere in the contract. In orer to execute an arbitrage, the guarantee profit must be greater than the total transaction cost incurre. he magnitue of the transaction cost is relatively minor. orpool for example charges approximately one cent per MW trae in a futures contract. 4. Market Failure: In esigning the arbitrage strategies we assume that a zero sell bi an max buy bi into the spot market is always accepte. here are situations where the spot market woul be unable to eliver aitional power at any price ue to shortage of generation assets or system security constraints. In such a case the spot market woul fail to clear as the aggregate eman an supply curves o now intersect. Uner such circumstances there are efault conitions specifying the charge/payment to be mae to each market participant. In the context of forwar markets, the contracts often have a clause for liquiate amages in the case of market failure. he party failing to eliver on a physical obligation must pay whatever financial amage is incurre by the opposing party to replace the power, or any penalty incurre by the opposing party for failing to eliver on its subsequent obligations. Similar clauses for liquiate amages can be inclue in financial forwars, thus effectively heging the traer against market failure. 2.4 Relationship of Spot an Forwar Markets So far we have consiere the relationship between physical an financial forwar contracts. Uner the arbitrage free assumption it coul be shown that prices in the two markets have to be equal at all times. ow we consier the relationship between the forwar price an the spot price. We apply arbitrage pricing theory to three markets, equity, storable commoities an electricity, to illustrate how the characteristics of the unerlying asset changes the pricing moel he price of a forwar contract on a stock Assume the current price of the stock, which pays no iviens, is S t an the risk free interest rate is r, continuously compoune. he price of a forwar contract on the stock (F()) with elivery ate must then be e r(-t) S t. o see why this is true consier the following cases:. If F()> e r(-t) S t, the investor shoul sell one forwar contrac borrow S t ollars at the risk free rate (assuming this is possible), an buy one unit of stock. he net cash flow at time t is zero. At time, the investor elivers the stock against the forwar

11 contrac receives F() ollars as payment for the forwar, an e r(-t) S t ollars to pay off his ebt. he net cash flow at time is F()-e r(-t) S t >0. his is a pure arbitrage opportunity, which cannot be sustaine in an efficient marke an therefore sets the upper limit to the forwar price. 2. If F()< e r(-t) S t, the investor shoul buy on forwar contrac short-sell one stock, an len S t at the risk free rate. he net cash flow at time t is zero. At time, the investor pays F() an receives eliver of the stock from the forwar contract. He uses this stock to repay his short-selling obligation. He also recovers e r(-t) S t from the money len. he net cash flow is e r(-t) S t -F()>0. his is again a pure arbitrage opportunity, setting the lower limit for the forwar price. In this case the upper an lower limits on the forwar price are ientical, an therefore, in an efficient market where participants can borrow an len at the risk free rate, the forwar price must be given by: F()=e r(-t) S t. his illustrates two important points. Firs uner no-arbitrage conitions, the forwar price of a stock is a eterministic function of the spot price an the time to maturity (-t). Secon, there is a smooth convergence of the spot an forwar prices at maturity he price of a forwar contract on a storable commoity Assume the current unit price of the commoity is S t, the present value of the total cost of storage incurre uring the length of the futures contract is U, an the risk free interest rate is r. he lower boun on the futures price for elivery at time is F()>(S t +U)e r(-t). If this oes not hol, an investor can receive a risk-free profit by borrowing S t +U at the risk free rate, purchase the commoity an pay off the storage cos an short a forwar contract in the commoity. he cash-flow at time t is zero, an the cash-flow at time is F()-(S t +U)e r(-t) >0. his is known as cash an carry arbitrage. Payoff at each time step from cash an carry arbitrage: t Buy commoity to be elivere -S t 0 against forwar contract. Sell forwar contract 0 F() Pay storage cost -U 0 Borrow now, repay at maturity S t +U -(S t +U)e r(-t) otal Cash Flow 0 F() -(S t +U)e r(-t) >0 Cash an carry arbitrage establishes an upper lower on the forwar price of the commoity. he boun converges to the spot price as we reach maturity (=t), an hence if the forwar price is consistently lower than the spot price then the two prices must converge.

12 he effects of cash an carry arbitrage can also be interprete as a ynamic relationship between spot an forwar prices. Assume that at time t we observe a forwar price F(), which violates the upper boun impose by AP. We woul expect the following behavior in the market.. In the spot marke eman will increase, as arbitreurs rush to buy the commoity in orer to store it. his put upwar pressure on the spot market price. 2. On the forwar marke the same arbitreurs sell forwar contracts in orer to execute the arbitrage, creating ownwar pressure on the forwar price. ow consier the reverse conition, when forwar prices rop below spot market levels. In this case, no pure arbitrage strategy is presen since it may not be possible to short sell a physical commoity on the spot market. However, consier the position of a market participant who is currently holing an inventory of the commoity. For this person, the optimal strategy will be to sell the inventory toay, an purchase cheap forwar contracts which can be use to restore the inventory at a later ate. If there is significant inventory in the marke this will put ownwar pressure on the spot price, an upwar pressure on the forwar price. One can question weather the bouns set by AP uner realistic market conitions. his is especially true for commoities with thin forwar markets an high transaction cost. However, weather or not the bouns are quantitatively accurate, the qualitative interaction between spot an forwar prices can certainly be observe.. An increase/ecrease in the forwar price will put upwar/ownwar pressure on the spot price. 2. A spike/rop in the spot price will put upwar/ownwar pressure on the forwar price. Consier the following scenario. omorrow OPEC announces that it will reuce its annual prouction of oil by 50%. Base on these news, the forwar price of oil increases sharply. ex arbitreurs recognize the isparity between spot an forwar prices, igniting a buying spree on the spot market. his causes an immeiate spike in the spot price of oil. he above scenario illustrates an interesting characteristic of storable commoities markets. he relationship between the state of physical prouction an consumption on one han, an the spot market price on the other, is non-causal. In other wors, a future rop in prouction leas to a spike in toay s spot price. ote however that the relationship between the spot price an the information flow is still causal. hat is, the spot market will only react when the rop in future prouction becomes known to market participants. he nee to moel the ynamic relationship between the spot an forwar prices in storable commoities has le to the notion of convenience yiel (y) which is efine as: ( r y)( t ) F( ) = ( S + U ) e t

13 he convenience yiels represents the premium the market is willing to pay in orer to physically hol the commoity toay, rather than a promise for elivery at time. We can moel y as a eterministic parameter, or a stochastic state of the system epening on the market he price of a forwar contract for electricity It is easy to see that cash an carry arbitrage is not possible for electricity. o execute the arbitrage one woul nee to purchase electricity at time store it somewhere, an eliver it against a forwar contract at time. Since electricity is not storable one cannot execute this type of arbitrage. As a resul the ynamic relationship between the spot an forwar price escribe above oes not hol for electricity. A goo example is the case of scheule unit outages. If it were announce toay that a major nuclear plant in ew Englan woul be out of commission for the month of July, this woul cause an immeiate increase in toay s price of a forwar contract with elivery in July. However it woul have no effect on the current spot price. We can therefore state that electricity spot prices are causal in the state of prouction an consumption of electricity. his will have a tremenous impact on how we moel electricity spot an forwar prices. Without the ability to execute an arbitrage between the spot an forwar markets, AP is useless in preicting the relationship between the two markets. Instea we have to aress the forces unerlying the eman an supply in forwar markets. One approach is to assume that the market as a whole is liqui enough that every participant hols a small fraction of the total risk. As a result the market effectively behaves in a risk neutral manner, even if the iniviual participants are risk averse, allowing us to pose the relationship, F ( ) = E t { S }. he risk neutral formulation is the basis for most risk management an option pricing theories in commoities markets. he problem with this assumption is that electricity markets are relatively illiqui, with a small number of participants. In light of this we here propose a more general moel allowing for the existence of a risk premium in the market. We moel the forwar price as a function of the spot price, the variance of the spot price, an a ranom isturbance (z F ), F F( ) = Φ( E ( S ),var ( S ), z ) he exact structure of the forwar risk premium is likely to vary from market to market. 3 Dynamic Heging t 3. Motivation We consier the situation where a loa serving entity (LSE) has obligate itself to serve a group of customers at a fixe rate. he contract is set up in such a manner that the customers may consume may consume as much or as little power as they want at any time without aitional penalties. his setup is similar to the current stanar offer t

14 contracts being offere to retail electricity customers. Furthermore we impose the constraint that the LSE owns no generating assets but purchases all power from the spot market. his exposure to the spot market leaves the LSE with significant financial risk. o mitigate this risk it can purchase financial futures contracts on electricity through the commoities exchange. We will here aress the problem of how to generate an optimal traing strategy for the LSE in the futures market. 3.2 Problem Formulation We will now generate mathematical moels for the financial risk face by the loa serving entity. his will inclue moeling the stochastic behavior of loas, spot prices, an futures prices. he notation use is summarize in the table below. R Fixe rate for customer (stanar offer) ($/MW) S k Spot price in ay. l k otal amount consume in ay (MW) F ti, Price of a forwar contract for elivery in month starting at, as seen at the time of purchase t i ($/MWh) q ti, otal quantity of forwar contracts purchase for elivery month starting at (MW/h) at time of purchase t i. M total number of ays in a month number of months in the heging perio Starting ay of heging Perio We begin by moeling the cash flow for the LSE before any purchases in the forwar market. his is the unhege cash flow (CF U ). CF + = M ( m ) U l m= k = Mm+ ( R S ) For simplicity we will here consier the case where the heging perio is a single month. he cash flow function then becomes: U + M l = CF = ( R S ) ext we consier the cash flow incurre from a portfolio of forwar contracts q i,m. his is the cash flow of the hege CF H. + M H H CF = qt S F j, ( i, ) = j= 0 he inex t j represents points in time when we allow the LSE to purchase forwar contracts. Since a forwar contract F cannot be purchase after the starting ate of the elivery month (), we inclue the constraint t i. he timeline of the heging process is shown in figure.

15 t 0 t t 2 t H- +M heging perio Figure elivery perio he heging perio [t 0,] is ivie into H heging intervals of equal length. he number of heging intervals use will generally epen on the transaction cost an liquiity in the forwar market. Finally we consier the total cash flow for the hege LSE (CF). + M H CF = l ( R S ) qt j, ( S Ft j, ) = j= 0 ot that the forwar contracts have no cash flow prior to the elivery perio, therefore CF is equal to the total profit receive by the LSE. As seen from the start of the heging perio, l, S, an F tj,, are all ranom variables. We therefore efine a value function V ti enoting the expecte profit seen by the heger at each step t i in the heging perio. V t = i E ti { CF} or in expane form, + M i Vt = Et l ( R S ) + qt t i i j, ( S F j, ) = j= 0 V t0 represents the initial expecte return of the unhege portfolio. he objective of the heging strategy is to maximize the expecte value of the portfolio while minimizing the risk (or variance) of the return. We efine a mean variance objective function [8]as, max J = E V V rvar V V. q ti, { } { } t 0 t Unerlying Stochastic Moels o solve the stochastic optimization problem formulate above, we require specific information about the joint probability istributions of loa, spot prices, an forwar prices. o arrive at these istributions we postulate stochastic moels for the time evolutions of the ranom variables. he moels escribe below are simplifie versions of the full blown bi base price moel escribe in [], an generates aily spot prices. he moel mimics the supply an eman bis into the spot market. Deman bis are assume to be inelastic, while the aggregate supply bi curve is moele as a time varying exponential function. he stochastic processes escribing the time evolution of the loa an supply states are all Markov, reflecting the temporal relationship between

16 spot market price an supply/eman states iscusse in earlier sections. Another key aspect of the moel is the link between loa an price ynamics. his accounts for the correlation between loa an price uncertainty when formulating the heging strategy. Spot price Moel: al + b S = e Loa Moel: l b l l = α ( µ l + σ z + ) + ) l l Supply Moel: b b b b = α ( µ b + σ z Forwar Price Moel: F = E S + p var, where, + = S = S. b b F F { } { S } + σ z ote that var t (S ) is a eterministic function of (-t), an can therefore be thought of as a parameter of the moel. Specifically we write: η = [vart ( S t+ ), vart ( St + 2),..., vart ( St + )] We can express this function in terms of the state vector x, x = l b, [ ] the output vector y, = l S F V, [ ] y, the control variable corresponing to forwar market purchases, = u q, an stochastic inputs, l b F z = z z z. [ ] We also efine a vector of parameters θ,,,,,, θ l b l b l b F = [ µ α, σ, p, η] We can now write the ynamic constraints, x = Ax + Bu + Cz + y = f ( x, u, z ) ote that while the state ynamics is linear, the output variables are a nonlinear function of the states. 3.4 Properties of the Optimization Problem Base on the stochastic moels escribe in the previous section, we can escribe the properties of the moel an the value function V. he moel is Markov, meaning that all information of future outputs is containe in the current values of the state. As a result the value function V is also Markov. he changes in value of V over time is ue to changes

17 in the unerlying state vector x. Furthermore x changes as a function of the noise vector z. Since z is a stochastic process with inepenent increments, V will also be an inepenent increment process. We formally write this as: V V ) is inepenen t of (V + V ), for 0 ( + + τ + + τ τ Furthermore V is a martingale process, E ( V ) + τ = V. he proof for this is a simple application of iterate expectations. Using these characteristics of the value function, we can now rewrite the objective function of the heging problem, J = E{ V V 0 } rvar{ V V 0 }. First we write the total change in the value function over the heging perio as a sum of incremental changes, V V 0 = Vi + Vi. Since V is an inepenent increment process, the variance becomes a linear function of the increments, an we can write the objective function as, J = { V } r Var{ V V } E V i+ i i+ i. Furthermore, since V is martingale, the increments are zero mean, an we can write, J = 0 2 { } { V } r E ( V V ) E V i i i i+ ext we join the summation an arrive at the new objective function, J = 2 { } { V } re ( V V ) E V Which can be written as: g x, w, u ) V V i i i i. ( ) r( V V ) 2 ( = i where g i is a function of the current state an the isturbance vector. he objective function now becomes, J = E g ( x, w, u ) his problem can be solve recursively by starting at the en of the heging perio, J = E { } g ( x, w, u ) + g ( x, w., u which conforms to the stanar ynamic programming formulation [5]. For this type of formulation there is a wie range of literature regaring efficient solution techniques. ), 4 Conclusion an Future Work In this report we show how the unique properties of electricity prouction an consumption influence the ynamics between electricity spot an forwar markets.

18 Specifically, the price level on the spot market epens only on the current state of eman an supply, an is inepenent on forwar market prices. We apply this unerstaning of market ynamics to the problem of ynamic heging of the obligation to serve loa uner stanar offer contracts. A price process is presente which mimics the behavior of loa an supply bis into the spot market. It emphasizes the temporal relationship between loa an supply states an spot prices. Applying this type of moel we show how the ynamic heging approach can transforme into a stanar ynamic programming optimization problem. his result will allow us to apply efficient DP solution techniques to the heging problem. Future work will focus on solving the ynamic programming problem as pose, an also exten it to the full blown bi base price moel as escribe in []. We also want to exten the heging approach to inclue risk management by power proucers. Specifically, by applying principal component analysis, we can transform the cash flow from a generator with unit commitment constraints, into a Markov type process which lens itself to the techniques outline in this report [2].

19 References [] Skantze, P., Gubina, A., Ilic, M., Bi-base Stochastic Moel for Electricity Prices: he Impact of Funamental Drivers on Market Dynamics, MI Energy Laboratory echnical Report EL , ovember [2] Skantze, P., Visuhipan, P., Ilic, M., Valuation of Generation Assets with Unit Commitment Constraints uner Uncertain Fuel Prices, MI Energy Laboratory echnical Repor ovember [3] Skantze, P., Chapman, J., Stochastic Moeling of Electric Power Prices in a Multi-Market Environment, ransactions of IEEE PES Winter Meeting, Singapore, January [4] Ilic, M., Skantze, P., Electric Power Systems Operation by Decision an Control: he Case Revisite, IEEE Control Systems Magazine, Vol. 20, o. 4, August [5] Bertsekas, D. P., Dynamic Programming: Deterministic an Stochastic Moels, Prentice-Hall 987. [6] Hull, J., Options, Futures an other Derivatives, 4 th e., Prentice-Hall, 999. [7] Bertsimas, D., Kogan, L., Lo, A. W., Pricing an Heging of Derivative Securities in Incomplete Markets: An ε-arbitrage Approach, BER Working Paper Series, Cambrige, MA, ovember 997. [8] g, K., Sheble, B., Exploring Risk Management ools, ransactions of the IEEE/IAFE/IFORMS 2000 Conference on Computational Intelligence for Financial Engineering, ew York, Y, March [9] Kawai, M., Spot an Futures Prices of onstorable Commoities Uner Rational Expectations, Quarterly Journal of Economics, Vol. 98, Issue 2, May 983. [0] Hirshleifer, D., Resiual Risk, raing Costs, an Commoity Futures Risk Premia, Review of Financial Stuies, Vol., Issue 2, July 988. [] Ho,.S., Intertemporal Commoity Futures Heging an the Prouction Decision, Journal of Finance, volume 39, Issue 2, June 984.