Constructing forward price curves in electricity markets. Motivation

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1 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 Kursliste Ukemarkedet Produkttype Produkt Sluttpris Kjøperkurs Selgerkurs Omsatt MW Levering fra Levering til T imer FUT GU FUT GU FUT GU FUT GU FUT GU FUT GU FUT GU FUT GB FUT GB FUT GB FUT GB FUT GB FUT GB FUT GB FUT GB FUT GB FUT GS FWD FWV FWD FWV FWD FWSO FWD FWSO FWD FWV FWD FWV FWD FWYR FWD FWYR FWD FWYR Trondheim, 1 2 Trondheim, 1 4 1

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

2 Term structure of futures and forward prices at Nord Pool NOK/MWh Futures 6 Sesong-forwards 4 År-forwards 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 Trondheim, 1 8 Ukenummer 2

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

3 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, However, electricity can not be stored cannot price forwards using physical arbitrage Forward prices determined by supply and demand for hedging and speculation Trondheim,

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

4 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,

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

5 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 , each trading day Objective: smoothness of generated curve capture seasonality by penalizing deviation from bottom-up model forecast Trondheim, 1 2 5

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

6 The model Minimize I I 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 MPS Trondheim, Trondheim, 23 Empirical illustrations MPS Trondheim, MPS Trondheim,

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

7 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) 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 % 1.8% 5.4% % 2.4% 8.4% % 1.6% 5.% % 15.7% 2.9% % 5.% 14.8% % 2.3% 8.1% % 2.9% 2.6% % 1.5% 1.3% Total % 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,

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.

8 Expected spot prices vs. term structure Forward_ Forward_ Forecast Risk premium Risk p. detailed NOK/MWh 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

opportunities among forwards/futures Experimental results favourable in a comparison between the model and curves fitted to market data either by

Notes for Lecture 2 (February 7)

Notes for Lecture 2 (February 7) CONTINUOUS COMPOUNDING Invest $1 for one year at interest rate r. Annual compounding: you get $(1+r). Semi-annual compounding: you get $(1 + (r/2)) 2. Continuous compounding: you get $e r. Invest $1 for