AuthorFanelli, Viviana
  1. Introduction

    For many years in the natural gas industry the purchase and sale of gas has occurred through contracts which provide some flexibility for the timing of delivery.

    Sales companies or municipal utilities enter a contract in order to hedge against gas price volatility, while producers or wholesalers protect themselves against volumetric risk by putting a limit to the buyer's flexibility. A supplier of gas has to be able to supply the quantities needed by its client in every moment. Civilian gas demand depends very much on weather conditions. A very cold winter drives up demand for heating, and a very hot summer drives demand for cooling. As unusual weather variations are difficult to predict, the suppliers often need to supply more or less than the forecasted quantities. The extra quantity has to be produced or bought on the market and has a price associated with it. This association between price and volume can be managed through the stipulation of a gas sales agreement (GSA) (see Asche et al. (2002) and Creti and Villeneuve (2003)).

    Since energy markets are very volatile, contracts have to be optimized as frequently as the contract allows, in order to preserve its value or simply to contain losses. The optimization of the remaining contract depends not only on forecasted future values of the gas products, but also on past withdrawn volumes.

    Most of European natural gas contracts are long-term contracts including a so-called "Take-or-Pay" clause, that obliges the gas buyer to pay for the minimum contract quantity whether or not the gas is actually withdrawn or not. However, the minimum contract quantity represents only a certain part of the contractual quantity. The remaining part, defined as the downward quantity tolerance, can be o taken through a swing option.

    A gas swing option gives the holder the right, not the obligation, to adjust the volume of received gas at his discretion. The buyer receives a fixed baseload amount of gas at predetermined dates paying a strike price. In addition, over the life of the contract, the buyer can exercises a certain number n of swing rights to temporarily vary these deliveries, instead requesting to receive a different amount (swing amount) of gas. This request is usually made on a 1-period ahead basis, so that the supplier has time to adjust his delivery.

    The swing amount is subject to some constraints. The timing of these exercises is at the discretion of the buyer. Thus, depending on the needs of the buyer, the amount received can be swung up or down up to n times. So a swing option can be described as a set of American options, calls or puts, on a spread between the prices of natural gas and one or more different energy commodities, expiring on the last days of the delivery period defined by the gas sales agreement.

    Literature on commodity markets contains many papers on the analysis of swing options. Some of them are: Barbieri and Garman (1996); Garman and Barbieri (1997); Pilipovic and Wengler (1998); Clewlow et al. (2001); Jaillet et al. (2004); Wahab and Lee (2011); Edoli et al. (2013).

    In this paper we will consider swing call options under some non-trivial constrains. The most basic contract guarantees the option buyer the right to withdraw, that is to buy, extra quantities, up to the global limit of the contract at a contract price, calculated according to standardized oil-indexed formulae or increasingly often, indexed on a liquid enough gas price like the TTF. This contract poses challenges in its evaluation, since a number of factors have to be taken into account:

    * fixed or indexed contract price;

    * maximum and minimum levels of global volume;

    * maximum or minimum levels of any higher-granularity volumes;

    * impact of penalties and rights;

    * the market price of the reference product.

    In this paper we aim at finding the fair value of a real world gas swing option. In Section 2 we describe the considered gas swing contract. Section 3 deals with the swing option model development, including the time series statistical analysis and mathematical model representation. In Section 4 we perform the numerical implementation of the model and we show the obtained results. Section 5 concludes.

  2. Gas Swing Contracts

    Common swing contracts refer to a period of supply [0; T]. The period of supply is setup to contain D yearly delivery periods, so that a 5-year contract will have D = 5. The gas year is not equal to the calendar year, but starts on October 1st and ends on September 30th, so that gas year 2011/12 would comprise quarters Q42011-Q32012. Any delivery year is divided into N subperiods periods [t.sub.j] with j = 1,...,D, and i = 1,..,N, so that

    0 = [t.sub.1,1]

    In addition, any delivery year contains a certain number of swing rights. The swing rights are set to zero at the beginning of a new gas year. The right will typically be communicated in the sub-period that precedes the delivery period. We assume that the total number of swing rights n is equal to the number of sub-periods N and will be set to 365 days. These capacity constraints usually reflect effective transportation capacity limitations. It is not possible to communicate a swing right at the end of the last sub-period [t.sub.D,N]. The exercise decisions are taken daily, with n = 365.

    The swing contract can contain volumetric limits on several nested levels: hourly, daily, weekly, monthly and yearly. The higher-granularity maximum and minimum levels do not correspond to the lower-granularity or end-of-contract maximum and minimum levels. The different quantities are settled on different levels, where for example the daily maximum can be withdrawn on 250 days, and not on all the 365 days of the year before exceeding the contractual annual maximum quantity. These non-trivial volume constraints between daily and annual quantities can be written as:

    AQm [greater than or equal to] n * DQm; AQM [less than or equal to] n * DQM (2)

    where AQm is the minimum annual quantity that needs to be reached. DQm is the daily minimum quantity. AQM denotes the annual contract quantity, or the maximum amount of gas that can be withdrawn in a year, and DQM is the daily maximum quantity.

    The quantity of gas that is withdrawn on any day [t.sub.j,i] is denoted by [q.sub.j,i]. This quantity is limited by the daily constraints:

    DQm [less than or equal to] [q.sub.j,i] [less than or equal to] DQM for all j = 1,...D; i = 1,...,N (3)

    and by the annual constraints:

    AQm [less than or equal to] [[summation].sub.j,i] [q.sub.j,i] [less than or equal to] AQM for all j = 1, ... D; i = 1, ..., N (4)

    Where [[summation].sub.j,i] [q.sub.j,i] is the total annual quantity withdrawn. The maximum quantity of gas that a buyer could physically withdraw during one year, [Q.sub.max], is the number of swing rights times the DQM less the AQM:

    [Q.sub.max] = n * DQM - AQM (5)

    The swing contract has [Q.sub.max] = 0, that is the AQM coincides with the quantity n * DQM.

    The minimum amount of gas that the buyer has to take, [Q.sub.min], is the annual minimum quantity AQm less the number of swing rights times the daily minimum quantity:

    [Q.sub.min] = AQm - n * DQm (6)

    The above equations show that the volumetric constraints are non-trivial: the holder of the contract not only must consider the quantity [q.sub.j,i] for the current period, but also its effect on subsequent periods. In other words, even if the spread between the market price for the gas and the contract price is "wide" enough to withdraw maximum quantities...

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