EQUILIBRIUM FORWARD GAS PRICES WITH STORAGE
[Burcu Cigerli, Rice University, 7133209432, [email protected]]
Overview
We build on the analysis of forward markets in electricity undertaken by Bessembinder and Lemmon
(2002) to develop an equilibrium model for forward contracting in natural gas markets. Unlike electricity,
natural gas can be economically stored. This allows less variability in prices compared to electricity,
although the fact that storage is costly does not eliminate opportunities for arbitrage. We also build on a
paper by Routledge, Seppu and Spatt (1999), who consider equilibrium pricing of electricity contracts
when the option to store natural gas convert it into electricity links the natural gas and electricity markets.
Routledge et al conclude that electricity prices will be positively skewed, and, as in Bessembinder and
Lemmon (2002), we conclude that skewness will affect equilibrium forward premium and optimal forward
positions.
Methods
Rather than using traditional cost-of carry models for pricing forward contracts for storable commodities,
we use the equilibrium model of forward contracting developed by Bessembinder and Lemmon (2002).
Although, they develop their model based on the assumption that the underlying commodity, namely
electricity, is non-storable their set up can be modified to account for storage. The modified model yields
equilibrium forward prices and quantities for producers and retailers. In addition, we model the seasonal
demand for natural gas. Traditionally, natural gas has been a seasonal fuel that is in higher demand during
the winter, because of the need for heating. 1
We start with date t=1, the low demand season. At date t=1, producers, storage operators and retailers buy
and sell gas in the spot market. In addition, producers and retailers, but not the storage operators, take
forward positions to deliver at date t=2. Storage operators purchase gas from the spot market in the low
demand season, inject it into reservoirs and withdraw it during the high demand season. To solve the
model, we start from high demand season, optimize the agents' objective functions in the spot market while
taking into account previously selected forward positions as given. Then, we go backwards and find the
optimal level of forward quantity and prices.
At date t=2, producers maximize the following objective by winding down their forward positions and
simultaneously engaging in spot market trade:
where
and
are some constants and
increases with output and
is the fixed cost.
amount of gas sold by producer
, which implies that the marginal cost of production
denotes the wholesale spot price of gas, and
in the wholesale spot market.
is the
is the amount of gas contracted to
deliver (or purchase if negative) at date t=1 at the fixed forward price
Storage operators maximize:
1
This is changing to some extent in the southern United States as natural gas is used to provide more
intermediate and peak load electricity, and demand for the latter increases in the summer months due to air
conditioning. Nevertheless, for simplicity we assume just two periods of demand.
where
is the amount of gas withdrawn from the reservoir.
is a some constant where
Retailers maximize:
where
is the fixed cost which is less than
,
.
is the quantity sold (purchased if negative) forward by retailer
and
is the quantity sold in
the retailer market at fixed retail price,
.
Assuming that total physical production is equal to total retail demand and using the fact that the net
demand for total forward contracts is zero, we can solve for the market-clearing spot price.
Then we work backwards to get the optimal forward price and quantity sold at date t=1. We find an
equilibrium forward price in terms of the expected spot price, 2 the variance of the spot price and the
skewness of spot price at t=2.
At date t=1, a storage operator has to inject at least
to withdraw
at date t=2, where
is
the loss of gas from storage and extraction operations. We also assume that the storage operator discounts a
mean-variance period utility function at an interest rate . Hence, there is also an opportunity cost
associated with gas storage.
In the basic model, we assume that there is no capacity limit. We then modify the model to allow for a date
t=3 where there is a peak demand that exhausts the available supply and storage capacity. At date t=3 prices
then have to ration market demand to the storage and pipeline capacity constraints. These excess prices will
give and incentive to storage operators to invest in additional capacity. We will then examine how available
capacity will evolve over time as a function of the basic parameters in the model.
Conclusions
Our model with two dates concludes that forward contract price will be a biased forecast of future spot
price. Having storage option reduces the bias but not abandons.
Our model with lumpy investment with three dates is yet to be solved.
References
Bessembinder, H. and M.L. Lemmon (2002): “Equilibrium pricing and optimal hedging in electricity
forward markets,” Journal of Finance 57, 1347-1382.
Bar-Ilan, A. and W.C. Strange (1999): “The timing and intensity of investment,” Journal of
Macroeconomics 21, 55-77
Eijkel, R. and J.L. Moraga-Gonzalez (2010): “Do firms sell forward contracts for strategic reasons? An
application to the Dutch wholesale market for natural gas”, Working Paper, University of Groningen
Neuhoff, K. and C. von Hirschhausen (2005): “ Long-term vs. short-term contracts: A European
perspective on natural gas,” Cambridge, UK: University of Cambridge
Routledge, B. , D. Seppi, and C. Spatt (1999): "The spark spread: An equilibrium model of crosscommodity price relationships in electricity ," Working Paper, Carnegie Mellon University.
Routledge, B. , D. Seppi, and C. Spatt (2002): "Equilibrium forward contracts for commodities," Journal of
Finance 55, 1297-1338.
2
For simplicity we assume that all agents have the same information and expectations for the future spot
price.