The last two Learning Curve articles described a multi-factor model for energy prices based on the observed forward curve and showed how energy derivative prices can be calculated. This third instalment will describe a model which is a simplified version of our preferred multi-factor model, which is intuitive, simple to implement, analytically tractable, and also allows for the easy pricing of American-style options. The model can be viewed as a single factor spot energy model with mean reversion;
where S(t) represents the spot energy price at time t. In this model, energy price returns "mean revert" towards a long term, time dependent, level µ at a rate of *. The time-dependent long-term mean allows the model to fit exactly the observed market forward curve. For many market participants the issue of 'fitting' models for energy to the forward curve is an important one, as it is this which allows the model to incorporate seasonality in prices and ensures that options are correctly priced relative to the market. The model given by the above equation is a special case of the general model described in the first article of this series and is obtained by setting n = 1, and with the single volatility function given by *(tT)=*e*(T-t).
For this model, standard European options (i.e. caps, floor, collars and swaptions) can be priced analytically. Furthermore, the future forward curve is an analytical function of the future spot price, initial forward curve and model parameters. This result is important when performing Monte Carlo simulation or when building trees for the spot price. For example, when pricing options whose payoff depends on a particular forward contract we can terminate the simulation, or tree building procedure, at the maturity of the option instead of the maturity of the forward contract.
In order to value options with early exercise features and path dependent options, such as Asians, lookbacks, barriers, and swing options, it is necessary to implement numerical techniques such as trees. For the model of this article, trinomial trees for the spot energy price have been built, which are consistent with both the observed energy forward curve and term structure of interest rates, and price such options.
Figure 1 shows two market curves, with monthly maturities out to two years, that are representative of a downward term structure of forward prices (NYMEX Light, Sweet Crude Oil Futures observed on 1 October 1997), and a term structure that exhibits seasonality (NYMEX Henry Hub Natural Gas Futures observed on 17 December, 1997).
Figures 2 and 3 present diagrammatically the trees for the spot energy price resulting from our procedure with time steps every two months. The volatility parameters used in the tree construction were chosen by fitting the volatility function implied by the model to one year of observations of the futures prices.
Derivative pricing for American options is performed by the usual method of backwards induction--just like in a binomial tree. This starts at the end of the tree where the maturity condition for the option can be evaluated and worked backwards. At every time step the value of the option is the maximum of the value 'alive' (evaluated as a discounted expectation) and the value if the option is 'killed', via exercising, (evaluated as the intrinsic value).
Once the tree has been constructed it is possible to price exotic, or path dependent, options (such as Asian, barriers, lookback, and swing options). This technique involves tracking a subset of the path dependent variable (i.e. the average for an Asian option) through the tree and careful evaluation of the derivative price from this subset.
Using the methods described in this article it is possible to price many kinds of European or American exotic energy derivatives both accurately and efficiently. However, for certain types of derivatives, such as options which depend on a range of forward maturities, the multi-factor model described in our first Learning Curve article is more appropriate.
This week's Learning Curve was written byLes Clewlowand Chris Strickland. Both hold positions at theSchool of Finance and Economics, University of Technology, Sydney, Australia, and the Financial Options Research Centre, Warwick Business School, UK. They are also directors of Lacima Consultants.
