Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems. One strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation theorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained. In recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

The Single Factor market Model is an important tool in contemporary research in finance. The assumption of parametric stability is usually invoked to facilitate statistical estimation of the model. Much of the importance of the model follows from its "beta" parameter which, ideally, measures the sensitivity of returns on a security to changes in a market factor. While stability of this parameter is desirable, this property is neither required nor expected from financial theory. Instances of nonstationarity are commonly observed. The objective of this thesis is to address, theoretically and empirically, the nature and causes of potential instability. To this end, analytical models are derived for the beta parameter. These models predict that the intertemporal behaviour of parametric estimates is characterized by information-related structural shifts and seemingly random perturbations. These predictions are given empirical support. Structural shifts are observed in conjunction with pre-selected information flows; a random component of the beta parameter is identified. The implications of these findings are considered.

This thesis examines the predictive ability of models for forecasting inflation and financial market volatility. Emphasis is put on evaluation of forecasts and the usage of large data sets. Variety of models are used to forecast inflation, including diffusion indices, artificial neural networks, and traditional linear regressions. Financial market volatility is forecast using various GARCH-type and high-frequency based models. High-frequency data are also used to obtain ex-post estimates of volatility, which is then used to evaluate forecasts. All forecast are evaluated using recently proposed techniques that can account for data snooping bias, nested, and nonlinear models.

Although options have been traded for many centuries, it has remained a relatively thinly traded financial instrument. Paradoxically, the theory of option pricing has been studied extensively. This is due to the fact that many of the financial instruments that are traded in the market place have an option-like structure, and thus the development of a methodology for option-pricing may lead to a general methodology for the pricing of these derivative-assets. This thesis will focus on the development of the theory of option pricing. Initially, a fundamental principle that underlies the theory of option valuation will be given. This will be followed by a discussion of the different types of option pricing models that are prevalent in the literature. Special attention will then be given to a detailed derivation of both the Black-Scholes and the Binomial Option pricing models, which will be followed by a proof of the convergence of the Binomial pricing model to the Black-Scholes model. The Black-Scholes model will be adapted to take into account the payment of dividends, the possibility of a changing inter est rate and the possibility of a stochastic variance for the rate of return on the underlying as set. Several applications of the Black-Scholes model will finally be presented. / Thesis (M.Sc.)-University of Natal, 1992.

When high-frequency data is available, realised variance and realised absolute variation can be calculated from intra-day prices. In the context of a stochastic volatility model, realised variance and realised absolute variation can estimate the integrated variance and the integrated spot volatility respectively. A central limit theory enables us to do filtering and smoothing using model-based and model-free approaches in order to improve the precision of these estimators. When the log-price process involves a finite activity jump process, realised variance estimates the quadratic variation of both continuous and jump components. Other consistent estimators of integrated variance can be constructed on the basis of realised multipower variation, i.e., realised bipower, tripower and quadpower variation. These objects are robust to jumps in the log-price process. Therefore, given adequate asymptotic assumptions, the difference between realised multipower variation and realised variance can provide a tool to test for jumps in the process. Realised variance becomes biased in the presence of market microstructure effect, meanwhile realised bipower, tripower and quadpower variation are more robust in such a situation. Nevertheless there is always a trade-off between bias and variance; bias is due to market microstructure noise when sampling at high frequencies and variance is due to the asymptotic assumptions when sampling at low frequencies. By subsampling and averaging realised multipower variation this effect can be reduced, thereby allowing for calculations with higher frequencies.

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A disappointingly large proportion of the citizens of developing countries remain excluded from services provided by formal financial enterprises. This segment of the market, often referred to as the bottom of the economic pyramid (BOP), is subjected to a host of barriers which prevent it from accessing traditional financial services. In society, money does not only represent material value, but it is also a social construct through which all members of societies express themselves. Hence these, often poor, members of developing societies consequently attain access to credit through alternative modes of financing – one of which is the financial cooperative. Financial cooperatives play a dual role in society; they provide an economic vehicle for individuals and also a platform for social interaction for its members, thereby building social capital within the society. The existence of social capital within a society increases the members’ ability to solve communal problems. Unfortunately, an inadequate amount of research has been done to aid our understanding of this, and indeed other, alternative financing models. This paper seeks to begin to address this by investigating the competitive advantage and critical success factors of financial cooperatives in Swaziland. To this end, a qualitative research with an exploratory approach was conducted, as this approach was found most appropriate to provide new insights on the key phenomena that was being investigated. Consequently, 24 in-depth interviews at five different levels of involvement with cooperatives in Swaziland were conducted. The interviews provided rich data which was analysed using content and frequency analysis. This study found that successful financial service providers serving the market at the BOP provide a solution that has an appropriate balance between functional and social features. Financial cooperatives as a means of financial service was found to be suitably designed to meet the needs of the underserved financial service market. The Financial Services Acquisition Journey model was developed from the research findings. In graphic form, this model offers an integrated representation on how to ensure enhanced customer experience whilst providing financial services for the underserved financial services market. / Dissertation (MBA)--University of Pretoria, 2014. / zkgibs2015 / Gordon Institute of Business Science (GIBS) / Unrestricted

Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 / In this study,we consider two problems.The first one is the problem of computing hedging portfolios for options that may have discontinuous payoff functions.For this problem we use the Malliavin property called the Clark-Ocone formula and give some examples for diferent types of pay of functions of the options of European type.The second problem is based on the computation of price sensitivities (derivatives of the probabilistic representation of the pay off functions with respect to the underlying parameters of the model) also known as`Greeks' of discontinuous payoff functions and also give some examples.We restrict ourselves to the computation of Delta, Gamma and Vega.For this problem we make use of the properties of the Malliavin calculus like the integration by parts formula and the chain rule.We find the representations of the price sensitivities in terms of the expected value of the random variables that do not involve the actual direct differentiation of the payout function,that is, E[g(XT ) ] where g is a payoff function which depend on the stochasticdic differential equation XT at maturity time T and is the Malliavin weigh tfunction. In general, we study the regularity of the solutions of the stochastic differentia lequations in the sense of Malliavin calculus and explore its applications to Mathematical finance. / Services SETA and National Research Foundation (NRF)

The dissertation deals with the allocation of resources, among profit producing elements within a company, to achieve satisfactory results over a planning horizon. The company is viewed as a confederation of profit making elements called Strategic Business Units (SBU) which are independent of each other and held together by a central authority. The dissertation uses a computerized simulation model of the deterministic type in a timesharing mode. The model is deterministic with some limited probabilistic The model is deterministic with some limited probabilistic effects. It is used to develop projected balance sheets and profit and loss statements for each period in a predetermined planning horizon and to evaluate the success of a set of alternative futures of the SBU. The set of SBU alternatives to be evaluated as the "company" may be arbitrarily chosen by the operator or chosen through the use of a near-optimal, integer programming algorithm for a variety of measurement criteria and subject to various restraints on the balance sheet. The research uses data generated by the long-range planning process at an intermediate sized, multinational corporation listed on the New York Stock Exchange. The data consist of assets, liabilities, and profit and loss statement items for each period in a planning horizon and for each of three alternatives of each SBU in the study. In addition, a beginning corporate balance sheet is required as are planned corporate expense items and the specification of operating restrictions. Research into the effect of several strategic policies, including dividend rate and debt to equity ratio, on the future prospects of the company in accordance with the optimal value of five different measurement criteria, is reported. The five measurement criteria are present value of shareholder equity, growth in earnings per share, growth in total assets, total assets and growth in sales. The appendix material contains listings of computer programs used in the model (written in the Basic computer language), the research data used, numerous computer printouts, and technical discussions on the model. Several tentative conclusions are listed, many areas for further research are suggested, and strengths and constraints of the model are discussed. It is concluded that the techniques developed have good potential for increasing cash generation and the efficiency of the investment process in a company; the dividend rate has a significant effect on how fast a company can grow; and the model is flexible and can be used for a number of investigative purposes to support company decision-makers. An interesting area for further research is the tentative conclusion that return on assets, when used as an optimization criterion, produces a significantly different set of SBU alternatives from the one which results from using the other measurement criteria.

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