Valutazione di uno stoccaggio attraverso le opzioni swing

In trading companies, such as Enoi S.p.a. where I took my internship, it is necessary to evaluate the storage value and the profit we can get by making use of all the information available to date. For this purpose we have to consider both the financial and physical aspects of storage: we have to do a prevision of the price of natural gas, that depends on different factors and changes every day, we have to consider the volume constraints given by the capacity and by the quantity that we can inject or withdraw every day, that depends on the pressure inside the storage, we have to optimize the payoff. In this thesis we propose to extend the financial instrument of swing options, adapting them to the physical constraints of the storage. A swing option is an exotic derivative that gives the holder a number of exercise rights to buy a volume of a certain commodity between a lower and an upper bound at a fixed strike price at some predetermined set of exercise periods. In addition, the holder has the obligation to take a cumulated volume within a predetermined interval by the expiration date. This lower bound of volume is given by the Take or Pay clause and it obliges the buyer to take a minimum part of the total amount quantity established by the contract. The remaining part, usually defined as a downward quantity tolerance, can, but does not have to, be off-taken. Swings permit the holder to exercise the right to receive greater or smaller amounts of gas, subject to daily as well as periodic constraints, predetermining a fix purchase price according to some factors. This kind of contract allows more flexibility and consequently is more complex. Firstly, we extract information from market prices and volatilities and we build the behavior of an underlying spot price using a stochastic process, thus obtaining the forward curve. Here we choose firstly the Ornstein-Uhlenbeck process, also known as a one factor mean-reverting process, which explicitly incorporates seasonal effects; subsequently we suggest a two factor spot price model, also called Gabillon model, that considers a combination of a mean-reversion factor and two stochastic processes, a short term one and a long term one. Generalizing the previous findings, we can build a multifactor model of the forward curve, generated from the sum of Brownian motion terms weighted by a time dependent volatility function. All the evaluations are under the hypothesis that the market is complete and a risk-neutral measure Q exists, is unique and ensures that the market is arbitrage free. Once established all the constraints and generated spot prices paths we can evaluate the swing option value at the evaluation date, given by the expected value of the total payoff plus a penalty function. To solve the previous equation we employ two numerical methods: Monte Carlo simulations with least squares regression techniques and trinomial trees. Firstly we extend the Longstaff and Schwartz method to swing options, modifying the algorithm for N exercise rights, and later to the storage evaluation. To apply this method we have to work backwards in time and compute the continuation value. The second method implies the building of a trinomial tree that describes the underlying spot price at every step and, working backwards in time, we build a tridimensional trinomial tree with more levels that take into consideration both the number of rights and the volume quantities. We give a general overview of the dynamic hedging strategies, in particular we show the Delta algorithm used to hedge of a swing option and we give a numerical method for approximation. At the end we include the numerical results obtained with MATLAB using the least square Monte Carlo method and the Trinomial trees for swing options evaluation. Here we give a swing option numerical example. Then, we extend the program to obtain a gas storage evaluation and we make some comparisons.