1. MODELLING SPOT PRICE OF ELECTRICITY IN NSW 2005 - 2008 Hilary Green with Nino Kordzakhia and Ruben Thoplan 15th International Conference Computing in Economics and Finance University of Technology, Sydney, Australia University of Technology, Sydney, Australia Wednesday 15 July, 2009

2. Data NEMCCO (managed the Australian National Electricity Market until July 1 2009)  AEMO http://www.aemo.com.au/ http://www.aemo.com.au/  Half Hourly data for year 2005 to 2008 (NSW only)  Daily averages for year 2005 to 2008

3. Volatility Mean Reversion SeasonalitySpike

4. Characteristics of Spot Price 1. Seasonality 2. Mean Reversion 3. Volatility 4. Spikes 5. Residuals Statistical Model

5. Previous work

6. Spot Price differs by Month, Day and Time of day Lower Prices on public holidays and weekends. Highest prices in June Highest prices between 5:30 pm and 7:30 pm

7. Highest Mean Prices Lowest Volatility

12. Methods 1. Obtain a seasonal model 2. Identify spikes in de-seasonalised data 3. Fit an exponential decay function to spikes 4. Time series analysis on residuals from step 3 5. Distribution of final residuals

13. Seasonal Analysis (Mon – Thurs) Table 1: Periodogram Results arranged by Intensity IntensityFrequencyPeriodYearsCycles 27.4920.0012805.99664.000.250 10.2630.0037268.66551.330.750 6.10140.0050201.49921.001.000 4.54720.0025402.99832.000.500 3.02590.0062161.19930.801.250 2.50660.022344.77680.224.500 2.05710.0099100.74960.502.000 1.73560.024840.29980.205.000 1.54260.011289.55760.442.250 1.51230.12537.98020.0425.250 1.49230.0074134.33280.671.500 1.17590.042223.70570.128.500 1.12970.045921.78360.119.250 1.10610.039725.18760.138.000 1.10250.021147.41130.244.250 4 year cycle 1⅓ year cycle 1, 2 year cycle

14. Seasonal Analysis r 2 = 0.46

15. De-seasonalised Data Identify spike threshold

16. Fitting spikes Cluster 1 Cluster k 11 kk 11 kk

17. Fit the spikes is the time of occurrence of max spike in cluster k is the size of the maximum spike in cluster k, generalised pareto fits

20. Residuals remain autocorrelated

21. Time series analysis An autoregressive moving model fitted to the twice differenced Post Poisson residuals. An autoregressive moving model fitted to the twice differenced Post Poisson residuals.  ARIMA(2,2,6) Ljung Box test results confirms appropriate model Ljung Box test results confirms appropriate model

23. ARIMA(2,2,6) Discrete-time IDPOLY model: A(q)y(t) = C(q)e(t) A(q) = 1 + 0.7815 q -1 - 0.1924 q -2 C(q) = 1 - 0.6694 q -1 - 1.118 q -2 + 0.5794 q -3 + 0.0661q -4 + 0.07229 q -5 + 0.07 q -6 + 0.07229 q -5 + 0.07 q -6 Estimated using ARMAX from twice differenced residuals (Sampling interval: 1) Loss function 0.0555158 and FPE 0.0566445

24. Ljung Box test results of residuals from ARIMA(2,2,6) Ljung Box test results of residuals from ARIMA(2,2,6) LagFitp-valQ_StatCV 1No0.0195.4773.842 2y0.0615.6055.992 3y0.1125.9877.815 4y0.1966.0439.488 5y0.2296.88911.071 6y0.3316.89712.592 7y0.4257.03514.067 8y0.5337.03815.507 9y0.5038.31816.919 10y0.5678.63918.307 11y0.6308.91719.675 12y0.55610.69221.026 13y0.54211.82222.362 14y0.51913.09523.685 15y0.45514.95324.996 16y0.52814.95326.296 17y0.54015.76927.587 18y0.52017.05028.869 19y0.56017.43930.144 20y0.60217.77531.410 21y0.48420.59632.671 22y0.54420.62033.924 23y0.58920.87135.173 24y0.16030.77936.415 25y0.18631.09037.653 26y0.21831.29438.885 27y0.25731.35440.113 28y0.24132.86941.337 29y0.27832.99242.557 30y0.31233.24343.773

25. Residual Analysis after ARIMA

26. Final residuals are not normally distributed not normally distributed long tailed long tailed General hyperbolic distributions invariant to summation invariant to summation invariant to rescaling invariant to rescaling bell shaped bell shaped long tailed long tailed allow skewness allow skewness popular in financial modelling popular in financial modelling -> Normal Inverse Gaussian Distribution (NIG)

27. Normal Inverse Gaussian Distribution (NIG) The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the Inverse Gaussian (IG) as the mixing distribution. X ~ NIG(α,β,μ,δ) if it has the following pdf: X ~ NIG(α,β,μ,δ) if it has the following pdf: μ μ : centre β, α β, α : skewness, kurtosis δ δ : scale K 1 is the modified Bessel function of third order and index 1.

28. Examples: Normal Inverse Gaussian Distribution

29. Distribution of Final Residuals (Mon-Thurs fit)

30. The rest of the time

31. Fridays, Weekends and Public Holidays Cycles 4 1.333 1 2 0.222 0.8 0.667 0.444 r 2 = 0.44

32. De-seasonalised Holiday Data

33. From holiday data

34. Final residuals are long tailed again

35. NIG fits the residuals from the ARMA model

36. Summary Used a complex process to model the various components of the Spot Price of electricity for two different weekday behaviour patterns

37. Future Research Forecasting Forecasting Constant lambda smoothing down parameter Constant lambda smoothing down parameter Poisson intensity  time dependent Poisson intensity  time dependent Include load into the model Include load into the model Pricing of Futures Pricing of Futures Descriptive models not useful Descriptive models not useful Aim to use reduced exp(AR(1)) model with filtered Poisson component to fit to Futures prices. Aim to use reduced exp(AR(1)) model with filtered Poisson component to fit to Futures prices.  futures pricing tool

38. References [1] Julio J.Lucia and Eduardo S.Schwartz, Electricity Prices and Power Derivatives: Evidence from the Nordic Power Exchange, (2002) [2] Peter Brockwell and Richard Davis, Introduction to Time Series and Forecasting, (Springer, 2002) [3] M.Burger, B.Klar, A.Muller and G. Schindlmayr, A spot market model for pricing derivatives in electricity markets, (2003) [4] H.Geman and A.Roncoroni, Understanding the Fine Structure of ElectricityPrices, (2006) [5] Jan Seifert and Marliese Uhrig-Homburg, Modelling jumps in electricity prices:theory and empirical evidence, (2007) [6] T.Meyer-Brandis and P.Tankov, Multi-factor jump-diusion models of electricity prices, (2008) [7] Thorsten Schmidt, Modelling Energy Markets with Extreme Values in Mathematical Control Theory and Finance, ed Sarychev A. et al., (Springer,2008)