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Following the financial crisis of 2008, the need for more robust techniques to quantify the capital charge for risk management has become a pressing problem. Under Basel II/III, banks are allowed to calculate the capital charge using internally developed models subject to regulatory approval. An interesting problem for the regulator is to compare the resulting figures against the required capital under worst case scenarios. The existing literature on the latter problem, which is based on the marginal problem, assumes that no a-priori information is known about the dependencies of contributing risks. These problems are linear optimisation problems over a constrained set of probability measures, discretisation of which leads to large scale LPs. But this approach is very conservative and cannot be implemented robustly in practice, due to the scarcity of historical data. In our approach, we take a less conservative strategy by incorporating dependence information contained in the data in a form that still leads to LPs, an important feature of such problems due to their high dimensionality. Conceptually, our model is the discretisation of an infinite dimensional linear optimisation problem over a set of probability measures. For some specific cases we can prove strong duality, opening up the approach of discretising the dual instead of the primal. This approach is preferable, as it yields better numerical results. In this work we also apply our model to model-free path-dependent option pricing. Use of delayed column generation techniques allows us to solve problems several orders of magnitude larger than via the standard simplex algorithm. For high-dimensional LPs we also implement Nesterov's smoothing technique to solve the problems.

This work is devoted to portfolio optimisation problem arising in the context of constrained optimisation. Despite the classical convex constraints imposed on proportion of wealth invested in the stock this work deals with the pathwise constraints. The drawdown constraint requires an investor's wealth process to dominate a given function of its up-to-date maximum. Typically, fund managers are required to post information about their maximum portfolio drawdowns as a part of the risk management procedure. One of the results of this work connects the drawdown constrained and the unconstrained asymptotic portfolio optimisation problems in an explicit manner. The main tools for achieving the connection are Azema-Yor processes which by their nature satisfy the drawdown condition. The other result deals with the constraint given as a floor process which the wealth process is required to dominate. The motivation arises from the financial market where the class of products serve as a protection from a downfall, e.g. out of the money put options. The main result provides the wealth process which dominates any fraction of a given floor and preserves the optimality. In the second part of this work we consider a problem of a lifetime utility of consumption maximisation subject to a drawdown constraint. One contribution to the existing literature consists of extending the results to incorporate a general drawdown constraint for a case of a zero interest rate market. The second result provides the first heuristic results for a problem in a presence of interest rates which differs qualitatively from a zero interest rate case. Also the last chapter concludes with the conjecture for the general case of the problem.

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Backward stochastic differential equations appear in many areas of research including mathematical finance, nonlinear partial differential equations, financial economics and stochastic control. The first existence and uniqueness result for nonlinear backward stochastic differential equations was given by Pardoux and Peng (Adapted solution of a backward stochastic differential equation. System and Control Letters, 1990). They looked for an adapted pair of processes {x(t) / y(t)} / t is in [0 / 1]} with values in Rd and Rd&times / k respectively, which solves an equation of the form: x(t) + int_t^1 f(s,x(s),y(s))ds + int_t^1 [g(s,x(s)) + y(s)]dWs = X. This dissertation studies this paper in detail and provides all the steps of the proofs that appear in this seminal paper. In addition, we review (Cvitanic and Karatzas, Hedging contingent claims with constrained portfolios. The annals of applied probability, 1993). In this paper, Cvitanic and Karatzas studied the following problem: the hedging of contingent claims with portfolios constrained to take values in a given closed, convex set K. Processes intimately linked to BSDEs naturally appear in the formulation of the constrained hedging problem. The analysis of Cvitanic and Karatzas is based on a dual control problem. One of the contributions of this thesis is an algorithm that numerically solves this control problem in the case of constant volatility. The algorithm is based on discretization of time. The convergence proof is also provided.

In this thesis, we introduce a non-probabilistic model for the short-term interest rate. The key concepts involved in this new approach are the non-diffusive nature of the short rate process and the uncertainty in the model parameters. The model assumes the worst possible outcome for the short rate path when pricing a fixed-income product (from the point of view of the holder) and differs in many important ways from the traditional approaches of fully deterministic or stochastic rates. In this new model, delta hedging and unique pricing play no role, nor does any market price of risk term appear. We present the model and explore the analytical and numerical solutions of the associated partial differential equation. We show how to optimally hedge the interest rate risk of a fixed-income portfolio and price and hedge common and exotic fixed-income products. Finally, we consider extensions to the model and present conclusions and areas for further research.

Probability theory is the study of random phenomena. Many dynamical systems with random influence, in nature or artificial complex systems, are better modeled by equations incorporating the intrinsic stochasticity involved. In probability theory, stochastic partial differential equations (SPDEs) generalize partial differential equations through random force terms and coefficients, while stochastic differential equations (SDEs) generalize ordinary differential equations. They are both abound in models involving Brownian motion throughout science, engineering and economics. However, Brownian motion is just one example of a random noisy input. The goal of this thesis is to make contributions in the study and applications of stochastic dynamical systems involving a wider variety of stochastic processes and noises. This is achieved by considering different models arising out of applications in thermal engineering, population dynamics and mathematical finance.<br><div><br></div><div>1. Power-type non-linearities in SDEs with rough noise: We consider a noisy differential equation driven by a rough noise that could be a fractional Brownian motion, a generalization of Brownian motion, while the equation's coefficient behaves like a power function. These coefficients are interesting because of their relation to classical population dynamics models, while their analysis is particularly challenging because of the intrinsic singularities. Two different methods are used to construct solutions: (i) In the one-dimensional case, a well-known transformation is used; (ii) For multidimensional situations, we find and quantify an improved regularity structure of the solution as it approaches the origin. Our research is the first successful analysis of the system described under a truly rough noise context. We find that the system is well-defined and yields non-unique solutions. In addition, the solutions possess the same roughness as that of the noise.<br></div><div><br></div><div>2. Parabolic Anderson model in rough environment: The parabolic Anderson model is one of the most interesting and challenging SPDEs used to model varied physical phenomena. Its original motivation involved bound states for electrons in crystals with impurities. It also provides a model for the growth of magnetic field in young stars and has an interpretation as a population growth model. The model can be expressed as a stochastic heat equation with additional multiplicative noise. This noise is traditionally a generalized derivative of Brownian motion. Here we consider a one dimensional parabolic Anderson model which is continuous in space and includes a more general rough noise. We first show that the equation admits a solution and that it is unique under some regularity assumptions on the initial condition. In addition, we show that it can be represented using the Feynman-Kac formula, thus providing a connection with the SPDE and a stochastic process, in this case a Brownian motion. The bulk of our study is devoted to explore the large time behavior of the solution, and we provide an explicit formula for the asymptotic behavior of the logarithm of the solution.<br></div><div><br></div><div>3. Heat conduction in semiconductors: Standard heat flow, at a macroscopic level, is modeled by the random erratic movements of Brownian motions starting at the source of heat. However, this diffusive nature of heat flow predicted by Brownian motion is not observed in certain materials (semiconductors, dielectric solids) over short length and time scales. The thermal transport in these materials is more akin to a super-diffusive heat flow, and necessitates the need for processes beyond Brownian motion to capture this heavy tailed behavior. In this context, we propose the use of a well-defined Lévy process, the so-called relativistic stable process to better model the observed phenomenon. This process captures the observed heat dynamics at short length-time scales and is also closely related to the relativistic Schrödinger operator. In addition, it serves as a good candidate for explaining the usual diffusive nature of heat flow under large length-time regimes. The goal is to verify our model against experimental data, retrieve the best parameters of the process and discuss their connections to material thermal properties.<br></div><div><br></div><div>4. Bond-pricing under partial information: We study an information asymmetry problem in a bond market. Especially we derive bond price dynamics of traders with different levels of information. We allow all information processes as well as the short rate to have jumps in their sample paths, thus representing more dramatic movements. In addition we allow the short rate to be modulated by all information processes in addition to having instantaneous feedbacks from the current levels of itself. A fully informed trader observes all information which affects the bond price while a partially informed trader observes only a part of it. We first obtain the bond price dynamic under the full information, and also derive the bond price of the partially informed trader using Bayesian filtering method. The key step is to perform a change of measure so that the dynamic under the new measure becomes computationally efficient.</div>

Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2021 / Vovk and Shafer, [41], introduced game-theoretic framework for probability in mathematical finance. This is a new trend in financial mathematics in which no probabilistic assumptions on the space of price paths are made. The only assumption considered is the no-arbitrage opportunity widely accepted by the financial mathematics community. This approach rests on game theory rather than measure theory. We deal with various properties and constructions of quadratic variation for model-free càdlàg price paths and integrals driven by such paths. Quadratic variation plays an important role in the analysis of price paths of financial securities which are modelled by Brownian motion and it is sometimes used as the measure of volatility (i.e. risk). This work considers mainly càdlàg price paths rather than just continuous paths. It turns out that this is a natural settings for processes with jumps. We prove the existence of partition independent quadratic variation. In addition, following assumptions as in Revuz and Yor’s book, the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by model-free price paths is proven. / National Research Foundation (NRF)

In this manuscript, we develop a nite liquidity framework for two-asset markets. In contrast to the standard multi-asset Black-Scholes framework, trading in our market model has a direct impact on the asset's price. The price impact is incorporated into the dynamics of the first asset through a specific trading strategy, as in large trader liquidity models. We adopt Euler- Maruyama and Milstein scheme in the simulation of asset prices. Exchange and Spread option values are numerically estimated by Monte Carlo with the Margrabe option as a controlled variate. The time complexity of these numerical schemes is included. Finally, we provide some deep learning frameworks to implement these pricing models effectively. / Thesis / Master of Science (MSc)

Energy markets feature a wide range of unusual price behaviour along with a complicated dependence structure between electricity, natural gas, coal and carbon, as well as other variables. We approach this broad modelling challenge by firstly developing a structural framework to modelling spot electricity prices, through an analysis of the underlying supply and demand factors which drive power prices, and the relationship between them. We propose a stochastic model for fuel prices, power demand and generation capacity availability, as well as a parametric form for the bid stack function which maps these price drivers to the spot electricity price. Based on the intuition of cost-related bids from generators, the model describes mathematically how different fuel prices drive different portions of the bid stack (i.e., the merit order) and hence influence power prices at varying levels of demand. Using actual bid data, we find high correlations between the movements of bids and the corresponding fuel prices (coal and gas). We fit the model to the PJM and New England markets in the US, and assess the performance of the model, in terms of capturing key properties of simulated price trajectories, as well as comparing the model’s forward prices with observed data. We then discuss various mathematical techniques (explicit solutions, approximations, simulations and other numerical techniques) for calibrating to observed fuel and electricity forward curves, as well as for pricing of various single and multi-commodity options. The model reveals that natural gas prices are historically the primary driver of power prices over long horizons in both markets, with shorter term dynamics driven also by fluctuations in demand and reserve margin. However, the framework developed in this thesis is very flexible and able to adapt to different markets or changing conditions, as well as capturing automatically the possibility of changes in the merit order of fuels. In particular, it allows us to begin to understand price movements in the recently-formed carbon emissions markets, which add a new level of complexity to energy price modelling. Thus, the bid stack model can be viewed as more than just an original and elegant new approach to spot electricity prices, but also a convenient and intuitive tool for understanding risks and pricing contracts in the global energy markets, an important, rapidly-growing and fascinating area of research.