Last week's Learning Curve proposed a very general model for the evolution of energy prices that has been found to provide a good representation of reality. This week's will tackle the pricing of energy derivatives. The model considers the stochastic behaviour of the whole energy forward curve simultaneously and can be represented by the following equation;
F(t,T) represents the forward curve and is the price at time t for future delivery at time T. *(t,T) is the volatility function at time t of the forward/futures. Forward curve models easily handle the seasonality (including the intra-day and intra-week effects found in the electricity markets) in both price and volatility experienced in energy markets.
One of the most widely-traded energy derivatives is a cap agreement. An energy price cap limits the floating price of energy the holder will pay on a predetermined set of dates to a fixed cap level. It is well known that the cap can be interpreted as a portfolio of standard European call options with its price given by summing the price of the individual caplets. Similarly, an energy price floor limits the minimum price the holder will pay and is therefore a portfolio of standard European put options, with a collar simply a portfolio of a long position in a cap and a short position in a floor.
From the general energy model, analytical formulae for standard options (and hence for caps and floors) can be obtained. For example the price of a standard European call at time t with strike price K which matures at time T is given by
P(t,T) represents the discount factor until the maturity of the option. European puts can be valued via put-call parity. The above formula will appear very familiar to many market participants, as it is similar to the well-known Black formula even though here it is a very general, and multiple, stochastic factors affect the evolution of the forward curve. The only difference is in the calculation of the volatility term, w. Depending on the specification of the volatility functions this term can either be evaluated analytically or numerically, but either way prices can be obtained very efficiently.
Figure 1 shows the prices of monthly caps on natural gas. The prices are determined consistent with market futures data on 12/17/97 and for cap rate ranging from USD2.0 to USD2.4. The volatility functions used are taken from last week's Learning Curve.
Another product currently very popular is an option on an energy swap, or energy swaption. These options can be seen as options on portfolios of forward contracts. The value at time t of an energy swaption, with maturity date T, to swap a series of floating spot price payments on dates si = T +iT ,, i=1,..., m for a fixed strike price K can be shown to be
where the operator Et [.] represents the expected value at time t. In order to price instruments of this type Monte Carlo simulation is used and so the expectation is evaluated as the average of the simulated values. This equation shows the need to model the whole forward curve simultaneously, rather than just the spot price, and our modelling framework is ideally suited to performing these simulations starting with modelling the entire forward curve.
Table 1 shows the results of pricing at-the-money natural gas swaptions using the same input data as before. The maturity of the options range from one month to one year, with swaption tenors ranging between one and six months. All resets were assumed to be monthly.
The modelling framework of this article easily extends to pricing path-dependent options such as Asians, lookbacks, and barriers. Next weeks Learning Curve will look at a single factor restriction of the general model that results in a trinomial tree for the spot energy price which allows pricing of American style options.
This week's Learning Curve was written by Les Clewlowand Chris Strickland. Both hold positions at the School of Finance and Economics, University of Technology, Sydney, Australia, and the Financial Options Research Centre, Warwick Business School, U.K. They are also directors of Lacima Consultants.
