When one tries to model a system, several variables can affect the system’s behavior. The modeler tries to extract a minimal subset of variables that can satisfactorily explain most of the effects under study: in this case, a natural-gas storage facility (NGSF). The question we would like to answer is: What is its value over a fixed time period?
To determine this, we must identify the attributes, or inputs, that are the key determinants of this value, and then determine how the value changes as a function of the inputs.
Before we go about trying to value natural-gas storage, we should try and come up with a list of important considerations in any valuation process. The valuation method should use:
The method should be able to capture the intricacies of most NGSFs, including:
Also, the data requirements should be modest, and the calculation time should be reasonable.
The five main methods for valuing storage are:
But not all gas storage valuations are created equal, as will be explained below.
This method provides a range of values to indicate what storage could or should be worth. The logic behind this method is the following: How can one replicate the services that one currently has without storage, and how much would it cost to provide such services?
For example, instead of storage, one could use long-haul transportation, coupled with an LNG peaking contract to achieve the deliverability requirements. Then one could compare the cost of these services versus the cost of purchasing storage.
The drawbacks of this method are:
Nevertheless, this method is useful because it does provide a window as to what the storage asset class should be worth relative to the alternative services available in the marketplace.
The intrinsic value method is the most common method that is used to value storage and it is the benchmark upon which all other valuation is based. The idea behind the intrinsic method is simple.
Suppose we assume that the forward curve is fixed or static and we wish to determine the value of storage at the current time instant. The optimal value obtained using the static curve is the intrinsic value of storage. Another way of looking at the intrinsic value of storage is as follows: If one had to choose the storage hedges today, and cannot alter them in the future, the maximum value that can be locked in with certainty is the intrinsic value.
The intrinsic method has the following advantages. It is:
The biggest drawback of the intrinsic value method is that it fails to assign any value to storage due to price volatility. As a result, the values generated by this method are not realistic and therefore of limited significance. Despite its shortcomings, the intrinsic value is calculated as a part of any valuation process.
The spread option method is a natural extension to the intrinsic value method described earlier. One of the drawbacks of the intrinsic method is that it fails to assign any value to the optionality of storage. The spread option method attempts to overcome this limitation by looking at the option value of each spread rather than the static value.
In the following discussion, we illustrate some of the issues that we have to deal with when using the spread option method. The spread option matrix should be arbitrage free. It is difficult to recognize a priori whether the spread values are consistent or arbitrage-free. In other words, the spread option matrix should contain values such that the value of any basket of spreads should be equal to the value of the net spread position. When one uses a correlation matrix that is obtained using historical data, there is no guarantee that the spread option matrix being generated is arbitrage free. One needs to be very careful when using a correlation matrix for the spreads because even if it works for the volatilities obtained from the implied volatility strip for one day, there is no guarantee that it will work for another set of volatility values.
The spread option method requires a correlation matrix that is difficult to generate. What is the correlation number? Suppose the current date is June 1, 2005, and we are looking at the August 2005 to January 2006 spread. The correct number is the average price correlation between the August 2005 forward and January 2006 forward for the time period starting June 1, 2005, and ending Aug. 1, 2005. It is important to take this into consideration when determining this matrix, or ask how this matrix was generated.
In this method, the correlation does not have a term structure property. A good question to ask is the following: How are the correlation numbers generated? Most users use a 40-day price history, evaluate the logarithm of the price ratios between consecutive days to create a series of returns, and then use the pairs of time series to calculate the correlation.
But as we had described earlier, this is not what the correlation number represents. The correlation number should be the average correlation between the price returns for the two forwards that are being considered. As time elapses and one approaches the front 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.
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 a 10-year deal, the user has to make up the numbers while ensuring that the matrix remains positive semi-definite. In such a scenario, mean reversion becomes paramount. Otherwise the value of the option becomes very large. The user then has to tweak the correlation coefficient so that the spread is a reasonable value. At this point, the problem has moved out of the control of the user. Despite these limitations, the spread option method is widely used for a number of reasons. It can be built relatively quickly, the inputs can be seen, and the process is transparent. Users develop a certain degree of comfort with these models and continue to use them.
In a Monte Carlo simulation, one attempts to generate prices according to a distribution. We have the forward price and volatility curves. This translates into a distribution for a price vector that has a mean (forward price curve), a standard deviation (forward volatility curve), and a correlation matrix. Using a random number generator, one can generate a large number of sample vectors according to the distribution. For each of these price vectors, we have to value the storage deal. Taking the average over all these sample values provides us with the value of storage.
A large number of simulations are required to reduce sample error. As the number of simulation runs increases, the size of the sample error decreases by the square root of the number of runs. As a result, it takes a long time to reduce the size of the error. Techniques used in Monte Carlo simulation (e.g., Control Variate method) can be used to reduce the sample error, but it is hard to apply the control variate method to the case of natural-gas storage. Furthermore, calculating the moments requires special care. When one uses the Monte Carlo simulation to calculate the price deltas (or first moments), one needs to take special care.
To calculate the delta, the value of the deal usually is calculated, then the price is “tweaked” by a small amount and the new value is calculated. The difference yields the price delta. The problem that arises is that each of the calculations has a sample error. How much of the price difference is due to the change in prices and how much is due to the sample error?
Monte Carlo methods are inappropriate for valuing American options. A fundamental problem arises with the use of Monte Carlo simulation for valuing American options. The problem arises from the way the prices are generated in a Monte Carlo simulation. In a Monte Carlo simulation, the prices are generated ex post—in other words, the model assumes that the deal has been completed, it then looks at the settlement prices for each of the time periods, and it uses this price vector to determine the optimal injection and withdrawal schedule. In an American option—where an option can be exercised early for each given possible sample price path—the holder cannot “peek” along the path into the future to determine what the settlement price will be in order to determine whether or not to exercise. The user has to look at the information available only at the time of exercise. Hence, the value obtained using the Monte Carlo method will always be higher because of the advantage that this method has over the reality. The modeler now has to adjust the parameters so that the values are consistent with those being seen in practice. It should also be pointed out that the Monte Carlo method can produce an inaccurate representation of how the value of storage is monetized.
In the Monte Carlo method, the assumption is that the information is available to the model after the fact. So the algorithm goes through the following steps:
The algorithm looks at a price path, determines the optimal schedule, and obtains the value. This is not the case in practice. The problem here is that in reality, all the decisions are not made simultaneously. On this sample path, one cannot implement the rolling intrinsic hedge strategy. If you were required to be delta neutral, this would require that any gas injected into the ground has a corresponding sale out in the future, but one could change the timing of the sale anytime before the gas was actually withdrawn from the ground. In conclusion, the Monte Carlo simulation fails to capture the true dynamics of storage valuation.
The probability tree method is the most rigorous among all the methods, and also the most difficult to implement. The probability tree method uses the following inputs:
The first step in this process is construction of a probability tree consistent with the forward price curve and the forward volatility curve. If one were using this tree to price a forward contract, one should get the same price as the forward price used as an input. If one used this tree to value a simple option—i.e., call, put or straddle—one should get the same price as one would get pricing such an option using the implied volatility in the standard Black-Scholes formula.
At this point, the tree consists of a set of nodes (which represent prices) and arcs that represent the transition probabilities of moving from one node to another. The next step is to define at each node a set of inventory states that represent the level of inventory at that node, i.e., price/time period, to create a forest. A transition from one inventory state to another represents an injection or withdrawal depending on the states. One can then use value the problem on the tree using the stochastic dynamic programming approach.
The underlying mathematics in most probability tree models (and most spread option and Monte Carlo methods as well) assumes that the prices of the asset being studied follows a geometric Brownian motion. The Brownian motion 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:
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.
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 this information in the energy markets due to the lack of liquidity. Consequently the tree parameters have to be determined from historical data.
How many factors are appropriate? The theoretical answer is, the more factors, the better the fit. Increasing the number of factors can increase only the goodness of the fit. One needs to be careful about overfitting the data.
The practical answer is, as few factors as possible. A good model should be able to capture most of the desired effects with a few factors. What the appropriate number should be is open to debate.
How does the model implement daily volatilities? If one uses the implied volatility data from a futures exchange, e.g., NYMEX in the United States, one has to examine the underlying index to see whether it is consistent with the storage market. In the United States, the underlying natural-gas futures contract is a monthly contract. Taking physical delivery under this contract implies that the underlying physical gas is delivered prorata over the course of the month. Hence the price is the average of the daily spot prices that trade for that month.
If one uses these volatility numbers to value storage, the value most likely will be understated because the implied vol-atility on the monthly index will be lower than the implied volatility on the daily prices. One would have to determine the daily volatility from the monthly number using an appropriate model.