MODELLING SPOT PRICE OF ELECTRICITY IN NSW 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

Data NEMCCO (managed the Australian National Electricity Market until July )  AEMO  Half Hourly data for year 2005 to 2008 (NSW only)  Daily averages for year 2005 to 2008

Volatility Mean Reversion SeasonalitySpike

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

Previous work

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

Highest Mean Prices Lowest Volatility

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

Seasonal Analysis (Mon – Thurs) Table 1: Periodogram Results arranged by Intensity IntensityFrequencyPeriodYearsCycles year cycle 1⅓ year cycle 1, 2 year cycle

Seasonal Analysis r 2 = 0.46

De-seasonalised Data Identify spike threshold

Fitting spikes Cluster 1 Cluster k 11 kk 11 kk

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

Residuals remain autocorrelated

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

ARIMA(2,2,6) Discrete-time IDPOLY model: A(q)y(t) = C(q)e(t) A(q) = q q -2 C(q) = q q q q q q q q -6 Estimated using ARMAX from twice differenced residuals (Sampling interval: 1) Loss function and FPE

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 1No y y y y y y y y y y y y y y y y y y y y y y y y y y y y y

Residual Analysis after ARIMA

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)

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.

Examples: Normal Inverse Gaussian Distribution

Distribution of Final Residuals (Mon-Thurs fit)

The rest of the time

Fridays, Weekends and Public Holidays Cycles r 2 = 0.44

De-seasonalised Holiday Data

From holiday data

Final residuals are long tailed again

NIG fits the residuals from the ARMA model

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

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

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)