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.

month of the spread option, the correlation will decrease. The correlation will tend to decrease over time as the nearer month spread reaches maturity.

What are the correct volatilities to use for the spreads? The implied volatility strip data is widely available. For a given option, the implied volatility is the average volatility over the life of the option. In practice, the volatility is low at first and tends to spike up as the option approaches maturity. Merely using the flat volatility may give rise to the wrong value of the spread option.

#### Other Spread Model Issues

The problems with the spread option model are more obvious when the curve is in backwardation. The problem with this approach is that the probability of exercise does not enter into the calculation. If one attempts to incorporate these probabilities into the model, the situation gets hopeless. To do this, one has to use a decision tree (this is different from the probability price tree that we will talk about later). Without getting into the details, the point to be taken away from the above discussion is that as the size of the problem gets larger, the problem grows so quickly that it is almost impossible to solve. Furthermore, option value is highly sensitive to the correlation coefficient. The option value changes substantially as a function of the correlation coefficient. This is especially true for spreads that are at-the-money.

Of course, using a Margrabe option pricing model does not assume mean reversion in the price processes for the two assets that are being considered. Remember that not only are the prices mean reverting, but the spreads are mean reverting as well. This implies that for longer-term options, the value of the spread option will be overstated. One has to reduce the correlation coefficient to get the value of the spread option down to a more realistic level.

The covariance matrix must be positive semi-definite. The definition for positive semi-definitiveness is a bit complicated. Intuitively, it means the following: if V is the variance-covariance matrix for a portfolio of asset returns, then the variance for any portfolio must be nonnegative. If this assumption were not satisfied, one could imagine a stock portfolio where the variance of the return were negative. This property is something we take for granted.

When a user is trying to price a multi-year deal, the correlation matrix needs to be expanded on the fly, and there is no guarantee that this condition will be met. The point to be made here is that the data requirements are substantial. By replacing the static value of the spread by the option value, the spread option method is trying to capture the dynamic behavior of storage. But it fails to capture the true dynamic behavior associated with storage—the changing nature of the correlation, the non-uniformity of the implied volatilities over time, early option exercise, and the complex interdependence between the spread options. The data requirements of the model are large, and it is difficult to have a good feel for the correlation numbers.

For modeling