Pricing of gas swing contracts with indexed strike: a viscosity solution approach with applications

Swing contracts are structured products mostly traded on energy and gas markets, tailor-made to handle simultaneously price and volume risk arising from the modern liberalized markets. This thesis deals with some specific swing contracts relevant for the gas market. In gas market, swing contracts are also known as take or pay. They are long-term supply contracts which allow flexibility of delivery: the holder of such a contract has multiple exercise rights and can decide the amount exercised as well, hedging the volume risk caused by a frequent demand fluctuation which in practice is impossible to foresee in the long period. Moreover, such type of contracts can be also seen as a strip of spread options on gas market spot price and the contractual price (called strike): in this view, they can be used to hedge the price fluctuation risk. On the other hand, even if the holder can exercise the option with a volume control, such control has however to satisfy some upper and lower limits at all times as well as a total volume, so the given flexibility need to be optimised, i.e. one must know the optimal execution of this flexibility. Today, the correct valuation of these type of contracts is important both for trading purposes as well as for portfolio optimization. In fact, after the recent liberalization, the price of such contracts is negotiated between agents and no more set by regulators. On the other hand, the embedded flexibilities may be used not only to manage demand fluctuation, but also to make profit against local market price. In this thesis we model, in a continuous time framework, a gas swing contract in the spirit of [6], with one additional state variable corresponding to a stochastic strike price. Since, in real contracts, the strike is a market index which is updated monthly, this results in a mixed discrete/continuous stochastic control problem that we reduce to the usual continuous time situation by adding another state variable, corresponding to an index rolled-over in continuous time. The price of a swing contract is then equal to the value function of one sequence of Markov control problems, each one corresponding to a period between two consecutive changing dates of the index. After that, we prove that the value function of the corresponding control problem is the unique viscosity solution of the resulting Hamilton-Jacobi-Bellman (HJB) equation, and that the value function is smooth enough to ensure the existence of an optimal strategy, that we find out. Briefly spoken, this is the content of the first two chapters of this thesis where, after having framed the valuation problem as a stochastic optimal control problem, an introductory part on viscosity solutions is then applied to the particular problem at hand. This entails in some new results in the theory of viscosity solutions for parabolic nonlinear equations stemming from these swing options. After having found the HJB equations, which become nonlinear partial differential equations, the problem is then to solve them. To do this, in the third chapter we present a finite difference (FD) method for solving the HJB equations numerically. We derive explicitly the boundary conditions for a particular model, when both gas and strike price are supposed to be in the class of the one-factor Schwartz-Smith dynamics. More in detail, we suppose that the log-prices follow Ornstein-Uhlenbeck processes driven by two correlated Brownian motions. Also, the third chapter presents the popular Least Square Monte Carlo (LSMC) algorithm, originally developed by Longstaff-Schwartz for valuing American options and here extended to the present problem. This algorithm is based on the backward solution of the discrete-time version of the control problem. It regresses the continuation value in the Bellman equation to the current available information, obtained using Monte Carlo simulations. Two critical steps using the LSMC algorithm are the choice of both type and number of the basis functions used in the regression; we approach this problem adapting to our case the radial basis function approximation introduced in [11] for storage structured products, which seems very appropriate and interesting for such applications. Several comparison examples are then numerically analysed, in particular the effect that the number of basis functions and the number of simulated paths have on the solution of LSMC as well as the efficiency of the FD method. Some conclusion on the comparison between the two algorithms ends the chapter. The last chapter is from the published paper [17] and deals with the so-called make-up clauses, which extend the swing option previously studied by allowing to the holder of the contract more flexibility among years. From a technical point of view, a swing contract with an embedded make-up clause can not be any more split into yearly contracts, but one must consider the whole contract at once, typically lasting over several years (usually from 3 to 5). To approach the complexity of such problem, another numerical method popular among practitioners is introduced for the purpose of pricing, namely lattice of trees. After having presented the algorithm and analysed its computational cost, the fourth chapter ends with many numerical examples testing for the swing option price's dependency on various crucial parameter