Using Affine Jump Diffusion Models for Modelling and Pricing Electricity Derivatives

Abstract

A seasonal affine jump diffusion spike model with regime switching in the long‐run equilibrium level is applied to model spot and forward prices in the Scandinavian power market. The spike part of the model incorporates different coefficients of mean reversion in the spike and normal variables and thus improves the spot–forward relationship, particularly at time periods when spikes occur. The regime switching part of the model contains two separate regimes to distinguish between periods of high and low water levels in the reservoirs, reflecting the availability of hydropower in the market. The performance of the models is compared with that of other models proposed in the literature in terms of fitting the observed term structure, as well as by generating simulated price paths that have the same statistical properties as the actual prices observed in the market. In particular, the model performs well in terms of capturing the spikes and explaining their fast mean reversion as well as in terms of reflecting the seasonal volatility observed in the market. These issues are very important for the pricing and hedging of derivative instruments.

Keywords:

• Regime‐switching spike model
• affine jump diffusion models
• electricity derivatives
• seasonal risk premium
• JEL Classification: G13
• G12 and G33

Acknowledgements

This paper has benefited from the comments of two anonymous referees and from discussions and comments of participants in the 2005 European Energy Conference in Bergen, Norway, the 2005 Cass Business School Energy Risk Management Seminar in London and the International Symposium in Advances in Financial Forecasting in Athens Greece. We are grateful to Viz Risk Management Services AS and Nord Pool for providing us with the data for this study.

Notes

1. See Geman (2005).

2. See Nord Pool report (www.nordpool.no) and Lucia and Schwartz (2002) for a description of the different instruments.

3. Annualised volatility = daily standard deviation *

4. Seasonality in electricity prices is also examined by Lucia and Schwartz (2002), who show that prices are lower during non‐peak hours and non‐working days. Thus for a more in depth analysis on these stylised facts we refer the interested reader to Lucia and Schwartz (2002).

5. The assumption of constant market price of risk is consistent with the changes in the state variables and aggregate wealth in the economy being constant and have constant covariance (Merton, 1973; Cox et al. 1985). Another method is to allow λ to be function of time, and thus calibrate perfectly the model to the forward curve. However, since the aim of this paper is to examine which model provides the best fit for derivatives pricing, the use of constant λ is recommended. See Lucia and Schwartz (2002).

6. Note that equation (6) contains an integral term. In our case we expand the integrand in a first‐order Taylor series and perform the integration analytically (Cartea and Figueroa, 2005).

7. Similar data have also been used by other studies such as Koekebakker and Ollmar (2001). The forward curve thus is computed by a program called ELVIZ, which uses a maximum smoothness function with a sinusoidal prior continuous forward price function, that prices all traded contracts within the bid/ask spread, using equation (8). For more details on the forward curve calculation see www.viz.no.

8. Lucia and Schwartz (2002), for example, use actual traded forward contracts with 28‐day frequency.

9. See Section 4 of the paper for further discussion on the difference in the speed of mean‐reversion between different power markets.

10. Alternatively we can also use a piecewise constant function of one year period, using monthly or seasonal dummies, as in Manoliu and Tompaidis (1999). Despite the fact that this is a flexible approach, it can be criticised on the grounds that dummy variables are very sensitive to anomalies in the sample such as jumps; hence the method does not provide a smooth function for the seasonal component, which may create discontinuities in the forward prices (Lucia and Schwartz, 2002).

11. Note that the probability of the Brownian motion scaled by the volatility parameter, in capturing these returns is almost zero.

12. Note that we found that especially for the upward spikes the difference in RMSE was almost 60 NOK/MWh.

13. The results for the Regime Switching Spike model are not reported, as they are identical to the Spike model, which is expected since the only difference between the two is in the long run level of the equilibrium to which prices revert. This, however, does not affect the daily returns levels, which should be similar for both models.

14. Note that we also ran simulations for log‐normally and exponentially distributed jump sizes. However the results showed that there was a high probability that prices would reach level of more than 1500 NOK/MWh, which has never been observed in the market. Hence care has to be taken when using such distributions, since they have to be truncated (see, for instance, Weron, 2005).

15. The generator is defined by the property that is a martingale for any f in its domain (Ethier and Kurtz, 1986).