A Jump Diffusion Model for Spot Electricity Prices

In this paper we demonstrate the efficacy of a stochastic modelling approach involving both diffusion and a jump processes to describe the evolution of spot electricity prices in New South Wales. The model allows a deterministic time trend component and an unobserved process driven by both time varying volatility and occasional jumps. The structure allows us to cast the problem in a state space form and suitable modification of the Kalman filter enables us to infer the unobserved driving process. The one-step ahead predicted price based on this component structure performs reasonably well in capturing the patterns in the daily average spot prices. The work in this article may be viewed as an initial attempt and much more work in the area needs to be undertaken. This version: 23 October 2007 1 Integral Energy project at the MISG 2007 meeting aims to calibrate a jump diffusion model for NSW spot electricity price series. Introduction At the outset, to appreciate the background of this project, we quote Geman and Roncoroni (2006): “Over the last 10 years, major countries have been experiencing deregulation in generation and supply activities. One of the important consequences of this restructuring is that prices now determined according to the fundamental rule of supply and demand: there is a ‘market pool’ in which bids are placed by generators to sell electricity for the next day are compared to purchase orders.” Prior to that, the regulators used to set the price based on the cost of generation, transmission and distribution and the price to the consumer was essentially fixed for long period of time. A large fraction of the literature on electricity today belongs to the economics of deregulated electricity market from the perspective of the regulators (see Joskow and Kahn (2001)). In the market mechanism now operating today, the price will be determined by the interaction of the purchase orders placed by the retailers against the pool prices. The deregulation of the electricity market has also led to increased trading activities in both spot and related derivatives like forwards and options. The risk of spot-price has forced retailers to manage the risk of the spot price through various hedging mechanisms. Many retailers provide incentives to the consumers to enter into long term contract with predetermined price structure, but that still leaves the risk of buying price. It is in this context modelling the stochastic behaviour of the spot price of electricity has become important. One feature of the electricity market that is unique to this commodity is that electricity is not storable, although, it may be argued that the concept of storability applies to hydro electricity generation. Since, in general, it cannot be stored the spot price is likely to be determined by the spot concerns, e.g., spot demand and supply constraints. The ability to store any commodity has the effect of smoothing the evolution of the spot price to some extent. As a result of its absence, price spikes are a regular feature of the electricity spot prices in most countries that have deregulated this market. Price spikes are possibly due to disruption in transmission, unscheduled outages, extreme weather changes or a combination of all these events. Additional details about the characteristics of this market may be found in Geman and Roncoroni (2006). 2 We will now review some of the salient characteristics of the electricity prices in the deregulated market. In standard commodity-futures markets the concept of convenience yield plays a key role in the relationship between the spot and the forward prices. The convenience yield is a way of expressing the fact that an investor is sure of available supply when the demand for using that commodity arises at a future date. The non-storability of electricity makes the concept of convenience yield difficult to apply. This implies that the spot price itself should contain all the characteristics of the price process that would be necessary to impute prices of derivatives contracts written on electricity prices. Next, we outline the important temporal characteristics of spot electricity prices observed in most markets. A detailed description of these characteristics may be found in Geman and Roncoroni (2006). Mean reversion is an important feature of spot electricity prices. The prices tend to fluctuate around values determined by cost of production and the level of demand. The mean reversion level may be constant or periodic with a trend. Seasonality is another obvious characteristic. The prices change by time of day, week, month and the year in response to cyclical fluctuations in demand. Another feature already mentioned before is that price jumps or spikes. A point to note is that technically price does not jump to a new level (to stay there) but spikes and quickly reverts to their previous levels. This price spike has been the most difficult aspect from modelling purposes. It is, therefore, clear from the above discussions that a pure diffusion process would not adequately capture the characteristics for electricity price series. A pure diffusion process approach has worked well in stock price modelling. For the electricity market, however, we need to incorporate a jump component with an appropriate intensity function to capture the spikes. Many of the traditional modelling approaches applied to financial market data e.g. equity, foreign exchange, and interest rates etc do not work well with spot electricity prices. This has been the experience for most researchers in this area as discussed in Geman and Roncoroni (2006). With respect to the equity market though, the work by Kim, Oh and Brooks (1994) is an important contribution to detect jumps (as opposed to spike). Their focus has been whether jump risks in stock returns are diversifiable. In this paper we attempt to combine the ideas expressed in the cited literatures and explore a jump diffusion model for spot electricity prices in NSW. We allow both a deterministic time 3 dependent factor as well as a latent factor combined with Poisson jumps to capture the observed characteristics of the spot electricity price series. We show how to calibrate such a model to the market data and describe the appropriate algorithm for that. The algorithm we employ generates, in a natural way, one period ahead forecast of spot electricity price. This, in turn, helps us determine the “goodness of fit” of such a model. A Model for Spot Electricity Prices In modelling commodity prices the approach of Schwartz and Smith (2000) has become quite popular. Their analyses depend upon both short dated and long dated futures contracts of the commodity. It also relate to the convenience yield as normally applied to futures contracts. Since the electricity, as a commodity, is different in this respect due to non-storability of the commodity for possible future consumption, the short-term, long-term concept introduced by Schwartz and Smith may not strictly apply to this market. Nevertheless, the ideas contained in Schwartz and Smith have important bearing in dealing with the electricity market. It is clear from the earlier discussions that price jumps or price spikes are a natural characteristic of the electricity market and have to be built into the model. It is also useful as we may be able to adopt the models we develop here for pricing derivatives contracts on the electricity spot prices. To reliably model contingent claims prices we have to incorporate jumps in addition to the usual diffusion assumptions in the price process, which makes it far more complex compared to pricing derivatives on equities. In this context we need to be mindful of the theorem by Duffie, Pan and Singleton (2000) that leads to closed form solution, in most cases, of the contingent claims when the underlying security follows an affine jump-diffusion process (AJD). Although, we are not strictly focussing on electricity derivatives contract in this paper, we will strive to stay close to the AJD process so that our approach can be easily adapted for contingent claims pricing later. Many researchers traditionally model log of the spot price of the commodity as in Schwartz and Smith (2000). The existence of a significant jump component in the electricity prices it is worthwhile to re-consider whether a logarithmic transformation is useful. The logarithmic transformation affects the estimation of the jump component due its effect on the skewness of 2 Affine structure implies linear dependence on state variables. 4 the distribution of the series. Since the derivatives contracts are written on spot price level and not on its log transformation, developing models of log transformation of spot price will not be useful. Lucia and Schwartz (2002) find that the model of price level fits the forward contract prices better than the log-price level. In this paper we will, therefore, model the price level and not its log transformation. That way the models we develop will be better suited to pricing derivatives contracts on spot electricity prices. In the original approach of Schwartz and Smith (2000) the log of the commodity price is modelled via two factors, both unobserved. The first factor captures the short-term variations and is modelled by an Ornstein-Uhlenbeck (OU) process whereas the second factor (the longterm variations) is modelled by an Arithmetic Brownian process (ABM). The commodity examined in Schwartz and Smith is crude oil and it display non-stationarity. Hence the inclusion of the ABM process in their analysis is not only meaningful but is also a necessity since the OU process alone would not be able to capture the dynamics. Since our spot electricity price series is stationary (found by Augmented Dickey-Fuller tests) we need only include the OU process to capture the dynamics without the jumps. To capture the jump characteristic we include a jump component in the OU process. Another difference from the structure in Schwartz and Smith (2000) for electricity spot prices is the inclusion of a time-dependent, deterministic function to capture the observed seasonality in the series. This arises mainly due to the nature of household consumptions of electricity depending on the season we are in. This also indicates that the intensity process for the Poisson component capturing the jumps in the series may not be constant, and is more likely to depend on seasonal factors. With this background we are in a position to specify the spot price process ( ) mathematically in terms of a deterministic, time-dependent function , and a state variable . Although many researchers specify their models in continuous time setting and for implementation purposes use Euler discretisation, we prefer to stay in the discrete framework from the start. We set daily average electricity spot price, measured in dollars per megawatt-hour, t P ( ) f t