Constructing forward price curves in electricity markets Stein-Erik Fleten Norwegian University of Science and Technology Jacob Lemming Risø National Laboratory, Denmark Motivation Risk management Production planning Capital budgeting Market data Bottom-up models Time series models Price models Trondheim, 1 1 Trondheim, 1 3 Presentation outline Forward/futures prices theory and illustrations Optimization model for building forward curves problems with modeling electricity prices smooth curve consistent with observed prices seasonality from forecast of bottom-up model Market data 19996 Kursliste Ukemarkedet Produkttype Produkt Sluttpris Kjøperkurs Selgerkurs Omsatt MW Levering fra Levering til T imer FUT GU5-99 142.88 142.5 143.25 115. 19991213 19991219 168 FUT GU51-99 143.25 143. 143.75. 1999122 19991226 168 FUT GU52-99 143.25 143. 143.5. 19991227 12 168 FUT GU1-147. 147. 148. 5. 13 19 168 FUT GU2-147. 147. 148. 5. 11 116 168 FUT GU3-147.5 146.5 148.5. 117 123 168 FUT GU4-147.38 146.25 148.5. 124 13 168 FUT GB2-147.5 147.25 147.75. 131 227 672 FUT GB3-131.13 129.75 132.5. 228 326 671 FUT GB4-125. 123.5 125. 5. 327 423 672 FUT GB5-123.5 122.25 124.75. 424 521 672 FUT GB6-115. 114.25 115.75. 522 618 672 FUT GB7-14.63 13.75 15.5. 619 716 672 FUT GB8-14. 13. 15.5 26. 717 813 672 FUT GB9-124.63 124.25 125.. 814 91 672 FUT GB1-134.25 133.5 135.. 911 8 672 FUT GS3-147.25 145.5 149.. 9 1231 217 FWD FWV1-136.4 136.3 136.5 27. 11 43 293 FWD FWV1-1 151.43 15.75 152.1. 111 143 2879 FWD FWSO- 116.48 116. 116.95. 51 93 3672 FWD FWSO-1 123.63 122.25 125.. 151 193 3672 FWD FWV2-146.63 146.25 147.. 1 1231 229 FWD FWV2-1 153.75 153. 154.5. 11 11231 229 FWD FWYR- 13.48 13.25 13.7 11. 11 1231 8784 FWD FWYR-1.25 139.5 141.. 111 11231 876 FWD FWYR-2 147.38 146.25 148.5. 211 21231 876 Trondheim, 1 2 Trondheim, 1 4 1
Term structure of futures and forward prices at Nord Pool 6-12-1999 NOK/MWh Futures 6 Sesong-forwards 4 År-forwards 2 1-1-1999 1-1- 1-1-1 1-1-2 1-1-3 Bottom-up models Matching supply and demand to find prices MPS (Samkjøringsmodellen), BALMOREL, MARKAL etc. Details of supply and demand, short and long term Expected spot prices <> value of future delivery Trondheim, 1 5 Trondheim, 1 7 Market data Gives directly the current market value of future exchange of electricity Limited number of products (delivery/maturity dates) traded Low liquidity of some products Little long-term information Trondheim, 1 6 NOK/MWh Samkjøringsmodellen 4 Terminstruktur 35 Gj. sn. 3 25 15 5 26 52 78 14 13 156 Trondheim, 1 8 Ukenummer 2
Forwards/futures Contracts without flexibility Fixed load profile: Constant power level Varying time intervals Next week... A full year three years from now Means of payment forwards: during delivery futures: mark to market Theory of derivative pricing Cost of carry forward price: F = Se rt Buy underlying, sell forward Cash flow: S At time T, sell underlying Cash flow: F S + S ( ) T T Trondheim, 1 9 Trondheim, 1 11 The development of the futures and forwards markets Liquidity increasing rapidly Growing interest from international players wanting to enter the Nordic market 3 Theory of derivative pricing NPV = S + Fe rt = TWh/year Forecast Brokered Nord Pool Trondheim, 1 1994 1995 1996 1997 1998 1999 1 2 1 However, electricity can not be stored cannot price forwards using physical arbitrage Forward prices determined by supply and demand for hedging and speculation Trondheim, 1 12 3
Hedgers: Producers, distributors, industry Speculators: Traders; producers, distributors, industry, financial institutions Main price component: Expected future spot prices Payoff of forward at maturity: ST F Expected discounted payoff: rt * e E ( ST F ) Forward/futures price = expected future spot price under risk neutral measure * F = E S ( ) T Trondheim, 1 13 Trondheim, 1 15 Theory of derivative pricing No arbitrage exists a risk neutral probability measure that makes pricing very easy: the price of any derivative equals their expected payoff discounted at the riskless rate Of limited use for pricing power forwards except when there are overlapping products Trondheim, 1 14 Electricity forwards Electricity is a flow commodity and must be delivered over a period of time Receive 1 MW (MWh/h) in time interval [T 2, T 1 ] Assume forward prices for any maturity date given by f(t,s) forward price at time t of a hypothetical contract with delivery at date s with an infinitesimal delivery period Trondheim, 1 16 4
Market value of receiving 1 MW from date T 1 to date T 2 : T2 T2 ( ) (,) r s t t = (,), T T 1 2 V f s s ds e f t s ds t T T 1 1 Forward price F(t,T 1,T 2 ) paid at a constant rate during the delivery period Market value of entering contract is zero: T2 r( s t) r( s t) = (, ) (, 1, 2) T1 e f t s e F t T T ds T2 (,, ) ω(; ) (, ) (,, ) 1 2 bid T 1 2 ask 1 FtTT sr f ts ds FtTT This relationship is a constraint in our optimization model for generating the theoretical forward prices f(t,s) Many products are overlapping, and for 43 of 1128 trading dates (3,8%) the constraints are violated arbitrage opportunities Trondheim, 1 17 Trondheim, 1 19 T2 (, 1, 2) = ω(; ) (, ) T1 F tt T rs e ω(;) sr = = e sr f ts ds rs re e T2 rt1 rt rs 2 e ds T1 The forward price is a weighted average of hypothetical forward prices over the delivery period Trondheim, 1 18 The model With P products and N number of maturity dates (we use daily and weekly resolution): 2P constraints and N variables Data from 1.4.1996, each trading day Objective: smoothness of generated curve capture seasonality by penalizing deviation from bottom-up model forecast Trondheim, 1 2 5
The model Minimize I I 1 2 2 lsq ( i i) + smo ( i 1 2 i + i+ 1) i= 1 i= 2 W x B W x x x Subject to T2 FtTT (,, ) ω(; srx ) FtTT (,, ) 1 2 bid s 1 2 ask s= T1 For all products, i.e. for all delivery periods T 1 and T 2 365 73 195 MPS 7-5-1999 Trondheim, 1 21 1-1-1999 1-1- 1-1-1 1-1-2 1-1-3 1 Trondheim, 23 Empirical illustrations 24 22 365 73 195 MPS 365 73 195 22 Trondheim, 1 1-1-1 1-1-2 1-1-3 1-1-4 1-1-5 24 22 4-1-1 MPS 1-1-1 1-1-2 1-1-3 1-1-4 1-1-5 Trondheim, 1 24 6
Test When seasonal blocks are split into monthly blocks or when monthly blocks are split into seasons, some of the true (market) seasonality is revealed Compare the curves generated before and after the split calculate contract prices of splitted products on the day before the split compare with market prices on the day of split Trondheim, 1 25 Test Price (NOK/MWh) 15 13 11 165 215 265 315 365 Days from today Market 8/1-99 Sine function TSG model Market 11/1-99 Smoothing Trondheim, 1 27 Test Average error % Our model Smoothness Sine func. 1997-4-18 4 1.4% 1.8% 5.4% 1997-1-6 4 4.3% 2.4% 8.4% 1997-12-23 3.6% 1.6% 5.% 1998-4-17 4.6% 15.7% 2.9% 1998-1-2 6 6.2% 5.% 14.8% 1998-12-3 3 3.1% 2.3% 8.1% 1999-4-23 4 2.9% 2.9% 2.6% 1999-1-8 6 4.3% 1.5% 1.3% Total 34 3.3% 4.2% 7.9% Total averages are over all 34 monthly products Risk premia A model sometimes used: all contracts priced at their expected payoff The price of risk is not zero Trondheim, 1 26 Trondheim, 1 28 7
Expected spot prices vs. term structure 7-5-1999 Forward_ Forward_ Forecast Risk premium Risk p. detailed NOK/MWh 6 4 2 1-1-1999 1-1- 1-1-1 1-1-2 Trondheim, 1 29 Rounding up A nonlinear program for generating forward curve and detecting arbitrage opportunities among forwards/futures Experimental results favourable in a comparison between the model and curves fitted to market data either by smoothing or by a sine function Use information in market prices Trondheim, 1 31 Future work Further tests Speed up solution times Get better historical forecasts Use as input to real option models Trondheim, 1 3 8