In this paper, we investigate a numerical algorithm for the pricing of swing options, relying on the so-called optimal quantization method.The numerical procedure is described in details and numerous simulations are provided to assert its efficiency. In particular, we carry out a comparison with the Longstaff-Schwartz algorithm.Key words: Swing options, stochastic control, optimal quantization, energy.
IntroductionIn increasingly deregulated energy markets, swing options arise as powerful tools for modeling supply contracts [14]. In such an agreement between a buyer and a seller, the buyer always has to pay some amount even if the service or product is not delivered. Therefore, the buyer has to manage his contract by constantly swinging for one state to the other, requiring delivery or not. This is the kind of agreement that usually links an energy producer to a trader. Numerous other examples of energy contracts can be modeled as swing options. From storages [6,8] to electricity supply [17,7], this kind of financial device is now widely used. And it has to be noticed that its field of application has recently been extended to the IT domain [12].Nevertheless, the pricing of swings remains a real challenge. Closely related to a multiple stopping problem [10,9], swing options require the use of high level numerical schemes. Moreover, the high dimensionality of the underlying price processes and the various constraints to be integrated in the model of contracts based on physical assets such as storages or gas fired power plants increase the difficulty of the problem.Thus, the most recent technics of mathematical finance have been applied in this context; from trees to Least Squares Monte Carlo based methodology [25,16,18], finite elements [26] and duality approximation [20]. But none of these algorithms gives a totally satisfying solution to the valuation and sensitivity analysis of swing contracts.The aim of this paper is then to introduce and study a recent pricing method that seems very well suited to the question. Optimal Vector Quantization has yet been successfully applied to the valuation of multi-asset American Options [2, 1, 3]. It turns out that this numerical technique is also very efficient in taking into account the physical constraints of swing contracts. For sake of * Corresponding author. 1 simplicity we shall focus on gas supply contracts. After a brief presentation of such agreements and some background on Optimal Quantization methods [22], we show that a careful examination of the properties of the underlying price process can dramatically improve the efficiency of the procedure, as illustrated by several numerical examples.The paper is organized as follows: in the first section, we describe in details the technical features of the supply contracts (with firm or penalized constraints) with an emphasis on the features of interest in view of a numerical implementation: canonical decomposition and normal form, backward dynamic programming of the resulting stochastic control problem, existenc...