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The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

In this dissertation, the price of variance swaps under stochastic volatility models based on the work done by Barndorff-Nielsen and Shepard (2001) and Heston (1993) is discussed. The choice of these models is as a result of properties they possess which position them as an improvement to the traditional Black-Scholes (1973) model. Furthermore, the popularity of these models in literature makes them particularly attractive. A lot of work has been done in the area of pricing variance swaps since their inception in the late 1990’s. The growth in the number of variance contracts written came as a result of investors’ increasing need to be hedged against exposure to future variance fluctuations. The task at the core of this dissertation is to derive closed or semi-closed form expressions of the fair price of variance swaps under the two stochastic models. Although various researchers have shown that stochastic models produce close to market results, it is more desirable to obtain the fair price of variance derivatives using models under which no assumptions about the dynamics of the underlying asset are made. This is the work of a useful analytical formula derived by Demeterfi, Derman, Kamal and Zou (1999) in which the price of variance swaps is hedged through a finite portfolio of European call and put options of different strike prices. This scheme is practically explored in an example. Lastly, conclusions on pricing using each of the methodologies are given. / Dissertation (MSc)--University of Pretoria, 2019. / Mathematics and Applied Mathematics / MSc (Financial Engineering) / Unrestricted

In this project, we are aiming to solve option pricing and hedging problems numerically via Backward Stochastic Differential Equations (BSDEs). We use Markovian BSDEs to formulate nonlinear pricing and hedging problems of both European and American option types. This method of formulation is crucial for pricing financial instruments since it enables consideration of market imperfections and computations in high dimensions. We conduct numerical experiments of the pricing and hedging problems, where there is a higher interest rate for borrowing than lending, using the least squares Monte Carlo and deep neural network methods. Moreover, based on the experiment results, we point out which method to chooseover the other depending on the the problem at hand.

The robust pricing and hedging approach in Mathematical Finance, pioneered by Hobson (1998), makes statements about non-traded derivative contracts by imposing very little assumptions about the underlying financial model but directly using information contained in traded options, typically call or put option prices. These prices are informative about marginal distributions of the asset. Mathematically, the theory of Skorokhod embeddings provides one possibility to approach robust problems. In this thesis we consider mostly robust pricing and hedging problems of Lookback options (options written on the terminal maximum of an asset) and Convex Vanilla Options (options written on the terminal value of an asset) and extend the analysis which is predominately found in the literature on robust problems by two features: Firstly, options with multiple maturities are available for trading (mathematically this corresponds to multiple marginal constraints) and secondly, restrictions on the total realized variance of asset trajectories are imposed. Probabilistically, in both cases, we develop new optimal solutions to the Skorokhod embedding problem. More precisely, in Part I we start by constructing an iterated Azema-Yor type embedding (a solution to the n-marginal Skorokhod embedding problem, see Chapter 2). Subsequently, its implications are presented in Chapter 3. From a Mathematical Finance perspective we obtain explicitly the optimal superhedging strategy for Barrier/Lookback options. From a probability theory perspective, we find the maximum maximum of a martingale which is constrained by finitely many intermediate marginal laws. Further, as a by-product, we discover a new class of martingale inequalities for the terminal maximum of a cadlag submartingale, see Chapter 4. These inequalities enable us to re-derive the sharp versions of Doob's inequalities. In Chapter 5 a different problem is solved. Motivated by the fact that in some markets both Vanilla and Barrier options with multiple maturities are traded, we characterize the set of market models in this case. In Part II we incorporate the restriction that the total realized variance of every asset trajectory is bounded by a constant. This has been previously suggested by Mykland (2000). We further assume that finitely many put options with one fixed maturity are traded. After introducing the general framework in Chapter 6, we analyse the associated robust pricing and hedging problem for convex Vanilla and Lookback options in Chapters 7 and 8. Robust pricing is achieved through construction of appropriate Root solutions to the Skorokhod embedding problem. Robust hedging and pathwise duality are obtained by a careful development of dynamic pathwise superhedging strategies. Further, we characterize existence of market models with a suitable notion of arbitrage.

MASTER THESIS ABSTRACT TITLE: Trading Strategy in Incomplete Market AUTHOR: Tomáš Bunčák DEPARTMENT: Department of Probability and Mathematical Statistics, Charles University in Prague SUPERVISOR: Andrea Karlová We focus on the problem of finding optimal trading strategies (in a meaning corresponding to hedging of a contingent claim) in the realm of incomplete markets mainly. Although various ways of hedging and pricing of contingent claims are outlined, main subject of our study is the so-called mean-variance hedging (MVH). Sundry techniques used to treat this problem can be categorized into two approaches, namely a projection approach (PA) and a stochastic control approach (SCA). We review the methodologies used within PA in diversely general market models. In our research concerning SCA, we examine the possibility of using the methods of optimal stochastic control in MVH, and we study the problem of our interest in several settings of market models; involving cases of pure diffusion models and a jump- diffusion case. In order to reach an exemplary comparison, we provide solutions of the MVH problem in the setting of the Heston model via techniques of both of the approaches. Some parts of the thesis are accompanied with numerical illustrations.

Master-Arbeit Univ. St. Gallen, 2006.