The deregulation of regional electricity markets has led to more competitive prices but also higher uncertainty in the future electricity price development. Most markets exhibit high volatilities and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. A good understanding of the stochastic price dynamics is required for the purposes of risk management and pricing derivatives. In this thesis we examine a simple spot price model which is the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of this spot price process at maturity T. With some restrictions on the set of possible martingale measures we show that the risk neutral dynamics remains within the class of considered models and hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulas for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process. Further contributions of this thesis include a short discussion of interpolation methods to generate a continuous forward curve based on the forward contracts with delivery periods observed in the market, and an investigation into optimal martingale measures in incomplete markets. In particular we present known results of q-optimal martingale measures in the setting of a stochastic volatility model and give a first indication of how to determine the q-optimal measure for q=0 in an exponential Ornstein-Uhlenbeck model consistent with a given forward curve.

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This thesis introduces a selection of models for optimal execution of financial assets at the tactical level. As opposed to optimal scheduling, which defines a trading schedule for the trader, this thesis investigates how the trader should interact with the order book. If a trader is aggressive he will execute his order using market orders, which will negatively feedback on his execution price through market impact. Alternatively, the models we focus on consider a passive trader who places limit orders into the limit-order book and waits for these orders to be filled by market orders from other traders. We assume these models do not exhibit market impact. However, given we await market orders from other participants to fill our limit orders a new risk is borne: execution risk. We begin with an extension of Guéant et al. (2012b) who through the use of an exponential utility, standard Brownian motion, and an absolute decay parameter were able to cleverly build symmetry into their model which significantly reduced the complexity. Our model consists of geometric Brownian motion (and mean-reverting processes) for the asset price, a proportional control parameter (the additional amount we ask for the asset), and a proportional decay parameter, implying that the symmetry found in Guéant et al. (2012b) no longer exists. This novel combination results in asset-dependent trading strategies, which to our knowledge is a unique concept in this framework of literature. Detailed asymptotic analyses, coupled with advanced numerical techniques (informing the asymptotics) are exploited to extract the relevant dynamics, before looking at further extensions using similar methods. We examine our above mentioned framework, as well as that of Guéant et al. (2012), for a trader who has a basket of correlated assets to liquidate. This leads to a higher-dimensional model which increases the complexity of both numerically solving the problem and asymptotically examining it. The solutions we present are of interest, and comparable with Markowitz portfolio theory. We return to our framework of a single underlying and consider four extensions: a stochastic volatility model which results in an added dimension to the problem, a constrained optimisation problem in which the control has an explicit lower bound, changing the exponential intensity to a power intensity which results in a reformulation as a singular stochastic control problem, and allowing the trader to trade using both market orders and limit orders resulting in a free-boundary problem. We complete the study with an empirical analysis using limit-order book data which contains multiple levels of the book. This involves a novel calibration of the intensity functions which represent the limit-order book, before backtesting and analysing the performance of the strategies.

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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

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.

The Vanna-Volga (VV) method has been in wide use as one of the major tools for several years among foreign exchange (FX) trading desks. Despite its popularity, the properties of the VV method are not well studied and understood. This thesis attempts to understand better why and when the VV method makes sense, and how to use it better. Often under practical circumstances the state of calibration can be described as being frequent but imperfect. To take advantage of this level of calibration, we studied the properties and benefits of the Karasinski method, and extended this method to a few useful applications. We have found that the Karasinski method, if used with a reasonably calibrated model, can provide significant performance improvement over the VV method.The VV and Karasinski chapters contain most of the original research in this thesis; there are a wealth of discoveries made in these chapters. Novel methods and applications related to the VV and Karasinski methods are proposed, and some of which can be readily applied to the practical trading environment. To make the VV and Karasinski methods work well in practice, the numerical issues for computing the price and Greeks have been carefully addressed with finite difference schemes that are second-order convergent and fast to compute. As an example of easy-to-compute but difficult-to-calibrate model candidates for the Karasinski method, the Multi-Heston model has been discussed too. A sound computational preparation enables the VV and in particular Karasinski methods to enjoy high viability as being fast, efficient and practical. This thesis is tailored to the purpose of making a detailed study on these useful methods whose great potential has not been adequately understood and fully realised.

In this thesis, we consider a class of stochastic dynamics running backwards, so called backward stochastic differential equations (BSDEs) in the literature. We demonstrate BSDEs can be reformulated as functional differential equations defined on path spaces, and therefore solving BSDEs is equivalent to solving the associated functional differential equations. With such observation we can solve BSDEs on general filtered probability space satisfying the usual conditions, and in particular without the requirement of the martingale representation. We further solve the above functional differential equations numerically, and propose a numerical scheme based on the time discretization and the Picard iteration. This in turn also helps us solve the associated BSDEs numerically. In the second part of the thesis, we consider a class of BSDEs with quadratic growth (QBSDEs). By using the functional differential equation approach introduced in this thesis and the idea of the Cole-Hopf transformation, we first solve the scalar case of such QBSDEs on general filtered probability space satisfying the usual conditions. For a special class of QBSDE systems (not necessarily scalar) in Brownian setting, we do not use such Cole-Hopf transformation at all, and instead introduce the weak solution method, which is to use the strong solutions of forward backward stochastic differential equations (FBSDEs) to construct the weak solutions of such QBSDE systems. Finally we apply the weak solution method to a specific financial problem in the credit risk setting, where we modify the Merton's structural model for credit risk by using the idea of indifference pricing. The valuation and the hedging strategy are characterized by a class of QBSDEs, which we solve by the weak solution method.

We propose two alternative approaches to evaluate numerically credit basket derivatives in a N-name structural model where the number of entities, N, is large, and where the names are independent and identically distributed random variables conditional on common random factors. In the first framework, we treat a N-name model as a set of N Bernoulli random variables indicating a default or a survival. We show that certain expected functionals of the proportion L_N of variables in a given state converge at rate 1/N as N [right arrow - infinity]. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of epsilon ² and computational complexity of order epsilon⁻², independent of N. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives. In the second framework, we extend the approximation of Bush et al. [13] to a structural jump-diffusion model with discretely monitored defaults. Under this approach, a N-name model is represented as a system of particles with an absorbing boundary that is active in a discrete time set, and the loss of a portfolio is given as the function of empirical measure of the system. We show that, for the infinite system, the empirical measure has a density with respect to the Lebesgue measure that satisfies a stochastic partial differential equation. Then, we develop an algorithm to efficiently estimate CDO index and tranche spreads consistent with underlying credit default swaps, using a finite difference simulation for the resulting SPDE. We verify the validity of this approximation numerically by comparison with results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the flexibility of the model and its extensions to match CDO spreads from precrisis and crisis periods.

In this thesis we address problems associated with financial modelling from a Bayesian point of view. Specifically, we look at the problem of calibrating financial models, measuring the model uncertainty of a claim and choosing an optimal hedging strategy. Throughout the study, the local volatility model is used as a working example to clarify the proposed methods. This thesis assumes a prior probability density for the unknown parameter in a model we try to calibrate. The prior probability density regularises the ill-posedness of the calibration problem. Further observations of market prices are used to update this prior, using Bayes law, and give a posterior probability density for the unknown model parameter. Resulting Bayes estimators are shown to be consistent for finite-dimensional model parameters. The posterior density is then used to compute the Bayesian model average price. In tests on local volatility models it is shown that this price is closer than the prices of comparable calibration methods to the price given by the true model. The second part of the thesis focuses on quantifying model uncertainty. Using the framework for market risk measures we propose axioms for new classes of model uncertainty measures. Similar to the market risk case, we prove representation theorems for coherent and convex model uncertainty measures. Example measures from the latter class are provided using the Bayesian posterior. These are used to value the model uncertainty for a range of financial contracts priced in the local volatility model. In the final part of the thesis we propose a method for selecting the model, from a set of candidate models, that optimises the hedging of a specified financial contract. In particular we choose the model whose corresponding price and hedge optimises some hedging performance indicator. The selection problem is solved using Bayesian loss functions to encapsulate the loss from using one model to price and hedge when the true model is a different model. Linkages are made with convex model uncertainty measures and traditional utility functions. Numerical experiments on a stochastic volatility model and the local volatility model show that the Bayesian strategy can outperform traditional strategies, especially for exotic options.