The natural-gas and oil price run-up since hurricanes Katrina and Rita has subsided somewhat following a warmer than usual winter, record natural-gas storage levels, and successful conservation...
The Art of Gas Storage Valuation
Benefits and drawbacks of the most popular estimation methods or modeling techniques.
is a well studied mathematical process, modeling the return or the natural logarithm of the price process that is assumed to follow a Brownian motion. If the logarithm of the price process is a brownian motion, then the price process itself is said to follow a geometric brownian motion.
A simple way to view Brownian motion is to think of a random walk in continuous time. If one were to make up a list of properties that a probability distribution for price returns should have, they may include:
- Additivity: The distribution should be stable, i.e., the sum of the returns has the same distribution as the individual returns if the returns are independent. This property makes life a lot easier because the only thing that changes are the parameters of the distribution.
- Heavy Tails: The distribution should have a kurtosis ( i.e., fourth moment) greater than that of the normal distribution. It has been observed that extreme events in commodities markets occur more frequently than that predicted by a normal distribution.
- Finite Variance: The variance of the distribution should be finite. If the variance is not finite, then it would become harder to use the information about implied volatilities that is generated by the options market. The option values implied by the distribution still could be finite, but one would need to create a measure of dispersion (other than the standard deviation) to track the volatility of prices.
If the returns follow a Brownian motion, it can be shown that they are normally distributed. The most common complaint about the normal distribution is that it does not exhibit heavy tails. Before one tries to come up with an alternative, it would be helpful to keep the following facts in mind.
- The normal distribution is the only stable distribution where the mean, variance, and all higher moments are finite. For all other stable distributions that exhibit heavy tails, these tails are so heavy that they have an infinite variance.
- There does not exist a distribution that has all three properties: additivity, heavy tails, and finite variance. The normal distribution does not include the heavy tailed assumption. If one wants to include the heavy tailed property, one will have to sacrifice one of the other two properties. If one wants to model heavy tails using the volatility smile, then the additivity assumption will have to be abandoned.
Brownian motion is not perfect for modeling commodity prices, but it is pretty good when you look at the alternatives. Naturally, even when using the probability tree method, it is still difficult to model natural-gas storage correctly. Simply building the tree so that the prices are consistent with option and forward prices does not uniquely define the tree. For a given set of parameters, one can fit the tree to the forward price and volatility curves, and still get different values for a storage facility. In the financial markets, one can use additional information available from the captions and swaptions markets to obtain more clues about the tree parameters. It is more difficult to obtain