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The objective of this dissertation is to model the volatility of financial time series data using ARCH, GARCH and stochastic volatility models. It is found that the ARCH and GARCH models are easy to fit compared to the stochastic volatility models which present problems with respect to the distributional assumptions that need to be made. For this reason the ARCH and GARCH models remain more widely used than the stochastic volatility models. The ARCH, GARCH and stochastic volatility models are fitted to four data sets consisting of daily closing prices of gold mining companies listed on the Johannesburg stock exchange. The companies are Anglo Gold Ashanti Ltd, DRD Gold Ltd, Gold Fields Ltd and Harmony Gold Mining Company Ltd. The best fitting ARCH and GARCH models are identified along with the best error distribution and then diagnostics are performed to ensure adequacy of the models. It was found throughout that the student-t distribution was the best error distribution to use for each data set. The results from the stochastic volatility models were in agreement with those obtained from the ARCH and GARCH models. The stochastic volatility models are, however, restricted to the form of an AR(1) process due to the complexities involved in fitting higher order models. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

Typescript. / Thesis (Ph. D.)--University of Hawaii at Manoa, 1977. / Bibliography: leaves 92-95. / Microfiche. / vii, 95 leaves ill

American options are the most commonly traded financial derivatives in the market. Pricing these options fairly, so as to avoid arbitrage, is of paramount importance. Closed form solutions for American put options cannot be utilised in practice and so numerical techniques are employed. This thesis looks at the work done by other researchers to find an analytic solution to the American put option pricing problem and suggests a practical method, that uses Monte Carlo simulation, to approximate the American put option price. The theory behind option pricing is first discussed using a discrete model. Once the concepts of arbitrage-free pricing and hedging have been dealt with, this model is extended to a continuous-time setting. Martingale theory is introduced to put the option pricing theory in a more formal framework. The construction of a hedging portfolio is discussed in detail and it is shown how financial derivatives are priced according to a unique riskneutral probability measure. Black-Scholes model is discussed and utilised to find closed form solutions to European style options. American options are discussed in detail and it is shown that under certain conditions, American style options can be solved according to closed form solutions. Various numerical techniques are presented to approximate the true American put option price. Chief among these methods is the Richardson extrapolation on a sequence of Bermudan options method that was developed by Geske and Johnson. This model is extended to a Repeated-Richardson extrapolation technique. Finally, a Monte Carlo simulation is used to approximate Bermudan put options. These values are then extrapolated to approximate the price of an American put option. The use of extrapolation techniques was hampered by the presence of non-uniform convergence of the Bermudan put option sequence. When convergence was uniform, the approximations were accurate up to a few cents difference.

Most of the notations and terminological conventions used in this thesis are Statistical. The aim in risk management is to describe the risk factors present in time series. In order to group these risk factors, one needs to distinguish between different stochastic processes and put them into different classes. The risk factors discussed in this thesis are fat tails and mean reversion. The presence of these risk factors fist need to be found in the historical dataset. I will refer to the historical dataset as the original dataset. The Ljung- Box-Pierce test will be used in this thesis to determine if the distribution of the original dataset has mean reversion or no mean reversion. / Mathematical Sciences / M.Sc. (Applied Mathematics)

This document demonstrates the use of the C++ programming language as a simulation tool in the efficient pricing of exotic European options. Extensions to the basic problem of simulation pricing are undertaken including variance reduction by conditional expectation, control and antithetic variates. Ultimately we were able to produce a modularized, easily extend-able program which effectively makes use of Monte Carlo simulation techniques to price lookback, Asian and barrier exotic options. Theories of variance reduction were validated except in cases where we used control variates in combination with the other variance reduction techniques in which case we observed increased variance. Again, the main aim of this half thesis was to produce a C++ program which would produce stable pricings of exotic options.

Firms can use financial derivatives to hedge risks and thereby decrease the probability of bankruptcy and increase total expected tax shields. Firms also can adjust their operational policies in response to fluctuations in prices, a strategy that is often referred to as "operational hedging". In this paper, I investigate the relationship between the optimal financial and operational hedging strategies for a firm, which are endogenously determined together with its capital structure. This allows me to examine how operational hedging affects debt capacity and total expected tax shields and to make quantitative predictions about the relationship between debt issues and hedging policies. I also model the effects of asymmetric information about firms' investment opportunities on their financing and hedging decisions. First, I examine the case in which both debt and hedging contracts are observable. Then, I study the case in which firms' hedging activities are not completely transparent. The models are tested using a data set compiled from the annual reports of North American gold mining companies. Supporting evidence is found for the key predictions of the model under asymmetric information. / Business, Sauder School of / Graduate

In this dissertation, we adopt the viability approach to mathematical finance developed in the book of Karatzas and Kardaras (2020), and extend it to settings where portfolio choice is constrained. We introduce in Chapter 2 the notions of supermartingale numeraire, supermartingale deflator, and viability. After that, we characterize all supermartingale deflators under conic constraints on portfolio choice. Most importantly, we prove a fundamental theorem for equity market structure and arbitrage theory under such conic constraints, to the effect that the existence of the supermartingale numeraire is equivalent to market viability. Further, and always under the assumption of viability, we establish some additional optimality properties of the supermartingale numeraire. In the end of Chapter 2, we pose and solve a problem of robust maximization of asymptotic growth, under some realistic assumptions. In Chapter 3, we state and prove the Optional Decomposition Theorem under conic constraints. Using this version of the Optional Decomposition Theorem, we deal with the problem, of superhedging contingent claims. In Chapter 4, we consider yet another portfolio optimization problem. Under simultaneous conic constraints on portfolio choice, and drawdown constraints on their generated wealth, we try to maximize the long-term growth rate from investment. Application of the Azema-Yor transform allows us to show that the optimal portfolio for this optimization problem is a simple path transformation of a supermartingale numeraire portfolio. Some asymptotic properties of this portfolio are also discussed in Chapter 4.

A thesis submitted to the School of Economic and Business Sciences, Faculty of Commerce, Law and Management, University of the Witwatersrand in fulfilment of the requirements for the degree of Master of Commerce (M.Com) in Finance, Johannesburg June 2017 / Traditional finance theory surrounding the risk-return relationship is underpinned by the CAPM which posits that a single risk factor, specifically market risk, is priced into asset returns. Even though it is a popular asset pricing model, the CAPM has been widely criticised due to its unrealistic assumptions and the APT was developed to address the CAPM’s weaknesses. The APT framework allows for a multitude of risk factors to be priced into asset returns; implying that it can be used to model returns using either macroeconomic or microeconomic factors. As such, the APT allows for non-traditional factors, such as investor sentiment, to be included. A macroeconomic APT framework was developed for nine countries using the variables outlined by Chen, Roll, and Ross (1986) and investor sentiment was measured by the FEARS index (Da, Engelberg, & Gao, 2015). Regression testing was used to determine whether FEARS is a statistically significant explanatory variable in the APT model for each country. The results show that investor sentiment is a statistically significant explanatory variable for market returns in five out of the nine countries examined. These results add to the existing APT literature as they show that investor sentiment has a significant explanatory role in explaining asset prices and their associated returns. The international nature of this study allows it to be extended by considering the role that volatility spill-over or the contagion effect would have on each model. / XL2018

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)