Remark 3.10 As discussed in Section 3.2.3, the holder’s problem and by extension AHBN might have multiple optimal solutions. In accordance with our observations in Section 3.2.3, we henceforth select the w.l.o.g. unique strategy optimal in AHBN which maximizes the expected exercise profits of the swing option, see Remark 2.2.
3.5
Other Applications
While our approach has been developed for the purpose of hedging and valuing swing options, numerous other contracts come with contractual or physical constraints similar to those inherent in a swing contract. Our approximation scheme thus has applications for a broader class of hedging and valuation problems. In what follows, we discuss a few of these alternative applications.
3.5.1
Valuation of Mines and Oil Fields
An immediate application of our approach is in the valuation of mines and oil fields (see e.g., Brennan and Schwartz [1985], Ludkovski and Carmona [2010]). Prices of commodities such as oil, gold or copper are highly volatile and exhibit frequent spikes. Mines and oil fields can act as storage facilities which enable the transfer of the commodity from periods of low demand (low prices) to periods of high consumption. For example, the owner of a mine (or the agent holding the rights to manage the mine) may decide to withhold extraction until the price of the commodity increases further. In what follows, we discuss the valuation of oil fields, but direct parallels can be made with the case of mines.
We discuss the problem of valuing a contract which gives its holder the right to manage an oil field over a planning horizon T = {1, . . . , T }. At the beginning of each period t ∈ T (e.g., each day or hour), the holder of the contract may select the amount et of oil she wishes to extract
extracted. Thus, the payoff from extracting et(ξt) units of oil at t is given by
et(ξt)(St(ξt) − K).
The amount extracted during any one time-period t ∈ T cannot exceed the extraction rate et
for that period, while we assume that no oil may be injected into the field, that is,
0 ≤ et(ξt) ≤ et.
The cumulative amount of oil extracted from the field can never exceed the capacity c of the field, i.e.,
0 ≤X
t∈T
et(ξt) ≤ c.
Thus, the problem of valuing mines and oil fields can immediately be cast into our framework.
3.5.2
Valuation of Power Plants and Refineries
Power plants and refineries enable the conversion of a fuel commodity into another fuel com- modity. A power plant converts for example gas, oil or coal into electricity, while a refinery converts e.g., crude oil into gasoline and natural gas into commercial or industrial fuel gas. From a financial point of view, they thus enable capitalizing on the price differential between two commodities.
Power plants and refineries are costly to build and maintain while participants in energy mar- kets require access to the physical equipment. Indeed, numerous commodity contracts require delivery of the underlying. For this purpose, lease contracts (sometimes termed tolling agree- ments) were introduced, see e.g., Deng et al. [2001], Ludkovski [2005], Deng and Xia [2005]. These give their holder (i.e., the agent renting the facility) the right to plan the power plant dispatching policy, i.e., the level of production, over a finite planning horizon T.
Our method is applicable to the valuation of such contracts which are subject to the physical characteristics of the plant or refinery. We now focus on the valuation of power plants. At the
3.5. Other Applications 97
beginning of each time-period, the contract holder may select the amount et of electricity to
produce. This quantity may never exceed the production rate et of the plant,
0 ≤ et(ξt) ≤ et.
In order to produce an amount et of electricity, the contract holder must purchase het units
of the input commodity at price Gt(ξt), where h denotes the heat rate of the plant, which we
assume to be independent of the output level et. The heat rate thus corresponds to the number
units of the input commodity needed to produce 1 MWh of electricity (the higher the heat rate, the less efficient the plant). Production costs K are incurred for each unit of electricity produced. The payoff from generating et units of energy at time t is thus
et(ξt)(St(ξt) − hGt(ξt) − K).
Some lease contracts require the cumulative production of the plant over the lease period to lie below a certain level c, that is,
0 ≤X
t∈T
et(ξt) ≤ c,
so as to moderate plant wear.
Remark 3.11 Certain power plants (e.g., thermal power stations) impose restrictions on the rate of change of et: ρ−t ≤ et−1− et ≤ ρ+t. These are known as ramping constraints and can be
accommodated by our framework for example by constructing candidate strategies which satisfy these constraints and by keeping a sufficient number of micro-periods in between two consecutive macro-periods.
Remark 3.12 Certain least contract do not incorporate limitations on the cumulative amount of electricity generated. In this case, there always exists an optimal operation strategy for the power plant which is of bang-bang type with exercise threshold given by K.