This report is mainly based on the paper "Optimal multiple stopping and valuation of swing options" by R. Carmona and N. Touzi (1). Here the authors model and solve optimal stopping problems with more than one exercise time. The existence of optimal stopping times is firstly proved and they then construct the value function of American put options with multiple exercises in the case of the Black-Scholes model, characterizing the exercise boundaries of the perpetual case. Finally, they extend the analysis to the swing contracts with infinitely many exercise rights. In this report, we concentrate on explaining their rigorous mathematical analysis in detail, especially for the valuation of the perpetual American put options with single exercise and two exercise rights, and the characteristics of the exercise boundaries of the multiple stopping case. These results are presented as theorems in Chapter 2 and Chapter 3. / text

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Since the liberalisation of the energy market in Europe in the early 1990s, much opportunity to trade electricity as a commodity has arisen. One significant consequence of this movement is that market prices have become more volatile instead of its tradition constant rate of supply. Spot price markets have also been introduced, affecting the demand of electricity as companies now have the option to not only produce their own supply but also purchase this commodity from the market. Following the liberalisation of the energy market, hence creating a greater demand for trading of electricity and other types of energy, various types of options related to the sales, storage and transmission of electricity have consequently been introduced. Particularly, swing options are popular in the electricity market. As we know, swing-type derivatives are given in various forms and are mainly traded as over-the-counter (OTC) contracts at energy exchanges. These options offer flexibility with respect to timing and quantity. Traditionally, the Geometric Brownian Motion (GBM) model is a very popular and standard approach for modelling the risk neutral price dynamics of underlyings. However, a limitation of this model is that it has very few degrees of freedom, as it does not capture the complex behaviour of electricity prices. In short the GBM model is inefficient in the pricing of options involving electricity. Other models have subsequently been used to bridge this inadequacy, e.g. spot price models, futures price models, etc. To model risk-neutral commodity prices, there are basically two different methodologies, namely spot and futures or so-called term structure models. As swing options are usually written on spot prices, by which we mean the current price at which a particular commodity can be bought or sold at a specified time and place, it is important for us to examine these models in order to more accurately inculcate their effect on the pricing of swing options. Monte Carlo simulation is also a widely used approach for the pricing of swing options in the electricity market. Theoretically, Monte Carlo valuation relies on risk neutral valuation and the technique used is to simulate as many (random) price paths of the underlying(s) as possible, and then to average the calculated payoff for each path, discounted to today's prices, giving the value of the desired derivative. Monte Carlo methods are particularly useful in the valuation of derivatives with multiple sources of uncertainty or complicated features, like our electricity swing options in question. However, they are generally too slow to be considered a competitive form of valuation, if any analytical techniques of valuation exist. In other words, the Monte Carlo approach is, in a sense, a method of last resort. In this thesis, we aim to examine a numerical method involved in the pricing of swing options in the electricity market. We will consider an existing and widely accepted electricity price process model, use the finite volume method to formulate a numerical scheme in order to calibrate the prices of swing options and make a comparison with numerical solutions obtained using the theta-scheme. Further contributions of this thesis include a comparison of results and also a brief discussion of other possible methods.

The Swing options are the natural outcomes of the increasing uncertainty in the power markets, which came along with the deregulation process triggered by the UK government&rsquo / s action in 1990 to privatize the national electricity supply industry. Since then, the ways of handling the risks in the price generation process have been explored extensively. Producer-consumers of the power market felt confident as they were naturally hedged against the price fluctuations surrounding the large consumers. Companies with high power consumption liabilities on their books demanded tailored financial products that would shelter them from the upside risks while not preventing them from benefiting the low prices. Furthermore, more effective risk management practices are strongly dependent upon the successful parameterization of the underlying stochastic processes, which is also key to the effective pricing of derivatives traded in the market. In this thesis, we refer to the electricity spot price model developed jointly by Hambly, Howison and Kluge ([13]), which incorporates jumps and still maintains the analytical tractability. We also derive the forward curve dynamics implied by the spot price model and explore the effects on the forward curve dynamics of the spikes in spot price. As the main discussion of this thesis, the Grid Approach, which is a generalization of the Trinomial Forest Method, is applied to the electricity Swing options. We investigate the effects of spikes on the per right values of the Swing options with various number of exercise rights, as well as the sensitivities of the model-implied prices to several parameters.

Swing options had been part of natural gas market before its embedded option feature was appreciated. The flexibility of delivery is valuable because of characteristic features of energy commodities as non-storability, high frequency of events and seasonality. Swing options enable this flexibility. Holder of the option is allowed to react to market situation in flexible way and change the amount of delivery up or down in some known intervals. Total deviation from negotiated amount can't exceed some boundaries in case of "take-or-pay" condition. It is not unique general valuation form of such flexible contracts as swing options. General definition of Longstaff Schwartz Least Square approximation method (LSM) is provided at first. Then it is shown other standard valuation concept as finite difference method. It is also mentioned tree method and more complex dynamic stochastic programming method. Analysis of energy commodities time series of central Europe is done and it is shown example of LSM approach use in valuing swing option with underlying asset of base load electricity in Czech Republic.

The presented dissertation is focused on the applications of statistical methods and ap-proaches applied in the energy business. The need for the modeling of energy risks arose only recently when the energy business was opened to competition. Therefore, the prima-ry aim of the dissertation is to clarify the main principles of the energy business which are necessary for understanding both risk principles and motivation of the proposed models. I am largely focused on retail risks, i.e., the risks associated with delivery to end-consumers. In particular, I deal with energy contracts providing volume flexibility, recalled as swing options in the literature. Therefore, the second issue on which I am focusing is a group of demand-driven swing options whose more systematic analysis in the portfolio context has not been published so far. Examining the risk, I apply the deductive (probabil-istic) analysis which reveals interesting relations between correlations. The practical ap-plications also require inductive considerations resulting in the construction of statistical estimators relying on historical data. I propose an estimator of the volumetric correlation based on a classical theory whose bias is investigated via MC simulation. To analyze a par-ticular volume-price correlation, I introduced the notion of robust dependency. Applying bootstrap procedures, robust dependency can be used both for testing purposes and for sensitivity analysis of the sample correlation. There are many works available devoted to energy price models which are different from the price models applied on financial markets. Therefore, the third target of the dis-sertation is an empirical statistical analysis of both power and natural gas Czech spot pric-es which can serve as a basis for the development of price models adapted to the Czech market environment. Finally, the fourth aim is the evaluation of power contracts which is very specific. The outputs of the model are both a synthetic market price and a hedging strategy. The model is designed to provide flexibility in practical applications.