Energy markets around the world are rapidly being deregulated, exposing participants to potentially enormous risks. In response to this there has been an explosion in the use of derivatives for managing these risks. Although the Black-Scholes model is readily applied to the pricing of options in many markets, it is easy to see that the use of the formula for pricing and hedging the majority of energy options can lead to major errors. Recently, considerable attention therefore has been paid to the modeling of energy prices, especially electricity.
When deciding on a modelling approach for energies it is necessary to bear the following in mind: most energies have a forward curve; energy prices typically experience mean reversion; there are multiple sources of uncertainty affecting the forward curve; there are jumps clearly observed in most data sets; and most traders want the ability to price and hedge energy derivatives quickly. The key to modeling energy prices is to consider the behaviour of the whole forward curve, rather than just the spot energy price. A general model for energy forward curves, with multiple sources of uncertainty is given by the following:
F(t,T) represents the forward curve and is the price at time t for future delivery at time T. *i(t,T) is the i'th independent volatility function (of which there are n) at time t of the forward/futures return. These volatility functions are just like Black-Scholes volatilities except that they are volatility curves which apply to the forward curve as a whole. An important property of this model is that the shape of the volatility functions *(.) is directly related to the behaviour of the spot price.
In energy markets with liquid forward markets the volatility functions can be estimated by principal components analysis of the historical behaviour of the forward curve. Figure 1 illustrates some typical results for natural gas futures contracts from NYMEX over the period November 1995 to December 1997.
The curves labelled "fn 1", "fn 2" and "fn 3" represent the volatility functions *i(t,T) for i = 1, 2 and 3 in the equation for the evolution of the forward curve. Because of their shapes these risk factors are often interpreted as a 'shift', 'tilt', and 'bend' to the forward curve respectively and combine to give the 'overall' volatility of the forward curve.
The mean reversion of the spot price is reflected in the fact that the volatility of forward prices declines with increasing time to maturity. An important property of the overall volatility is that it does not fall off exponentially, as is the case with simple mean reversion models, but flattens out at a non-zero level. This is a generic property of energy markets and together with the significance of the second and third volatility functions means that simple mean reversion models are not sufficient and that more sophisticated multi-factor models such as the one proposed here must be used.
Additional complications arise in the new electricity markets as reliable forward curves are not yet available, but as volatility functions are directly related to the behaviour of the spot price and they can be estimated from this set of data alone. Also, even a cursory examination of the behaviour of electricity prices shows intra-day and intra-week seasonality in both price and volatility. Our own analysis indicates that each day of the week can have its own distinct pattern. The forward curve based model of this article easily copes with these seasonality patterns via the initial forward curve and time dependent volatility functions.
The model described so far will not, however, capture the sudden extreme peak behaviour caused by, for example, unexpected loss of generation capacity. This behaviour can be modeled by adding a jump process. The strong mean reversion in the prices naturally causes the jumps to rapidly disappear. Therefore a model of the following form is proposed;
The jump information is represented by the independent Poisson processes dq(t). Associated with the jump process are parameters describing the frequency of jumps (average number of jumps per year), the mean jump size, and a jump volatility.
The model described by the above equation works well for electricity--Figure 2 shows simulated half hourly spot electricity prices under this model which can be qualitatively compared with a typical three month period of the Australian New South Wales electricity pool prices in Figure 3.
Notice how the combination of high time varying volatility, jumps and high mean reversion creates a range of sharp peaks in price very similar to the actual NSW spot prices.
The models described in this article have been found to model energy prices very well. The next two Learning Curves will show that they are also able to both price and hedge a wide range of energy contracts.
This week's Learning Curve was written by Les Clewlowand Chris Strickland. Both hold research positions at the Financial Options Research Center Warwick Business School, U.K., and the School of Finance and Economics University of Technology, Sydney, Australia. They are also directors of Lacima Consultants.
