### Browsing by Subject "Mathematics of Finance"

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- ItemOpen AccessAccurate estimation of risk when constructing efficient portfolios for the capital asset pricing model(2010) Zwane, Samkelo Sifiso; Clark, Allan; Troskie, Casper GIn this paper, we investigate the behaviour of the efficient frontier and optimal portfolio of the Troskie-Hossain Capital Asset Pricing Model (TrosHos CAPM) and Sharpe Capital Asset Pricing Model (Sharpe CAPM) when the covariance structure of the residuals is correlated under the Markowitz formulation. By building in the dynamic time series models: AR, GARCH and AR/GARCH we were able to model the autocorrelation and heteroskedasticity of the residuals.
- ItemOpen AccessBayesian estimation of stochastic volatility models with fat tails and correlated errors applied to the South African financial market(2011) Savanhu, Richard; Becker, RonaldIn this study we apply Markov Chain Monte Carlo methods in the Bayesian framework to estimate Stochastic Volatility models using South African financial market data. A single move Gibbs sampler is used to sample parameters from the posterior distribution. Volatility is used as measure of an asset's risk. It is particularly important in risk management, derivatives pricing, and portfolio selection. When pricing derivatives it is important to quote the correct volatility trading in the market, hence there is need for good estimates of volatility. To capture the stylised facts about asset returns we used the model extended for fat tails and correlated errors. To support this model against the basic model of Taylor (1986), we computed Bayes Factors of Jacquier, Polson and Ross (2004). The extended model was found to be far superior to the basic model.
- ItemOpen AccessBuilding a statistical linear factor model and a global minimum variance portfolio using estimated covariance matrices(2009) Matoti, Lundi; Wilcox, Diane; Gebbie, TIncludes bibliographical references (leaves 73-75).
- ItemOpen AccessDynamic and robust estimation of risk and return in modern portfolio theory(2008) Mupambirei, Rodwel; Troskie, Casper GThe portfolio selection method developed by Markowitz gives a rational investor a way of evaluating different investment options in a portfolio using the expected return and variance of the returns. Sharpe uses the same optimization approach but estimates the mean and covariance in a regression framework using the index models. Sharpe makes a crucial assumption that the residuals from different assets are uncorrelated and that the beta estimates are constant. When the Sharpe model parameters are estimated using ordinary least squares, the regression assumptions are violated when there is significant autocorrelation and heteroskedasticity in the residuals. Furthermore, the presence of outlying observations in the data leads to unreliable estimates when the ordinary least squares method is used. We find significant correlation in the residuals from different shares and thus we use the Troskie-Hossain model which relaxes this assumption and ultimately produces an efficient frontier that is almost identical to the Markowitz model. The combination of the GARCH and AR models to remove both autocorrelation and heteroskedasticity is used on the single index model and it causes the efficient frontier to shift significantly to the left. Using dynamic estimation through the Kalman filter, it is noticed that the beta coefficients are not constant and that the resulting efficient frontiers significantly outperform the Sharpe model. In order to deal with the problem of outlying observations in the data, we propose using the Minimum Covariance Determinant, (MCD) estimator as a robust version of the Markowitz formulation. Robust alternatives to the ordinary lea.st squares estimator are also investigated and they all cause the efficient frontier to shift to the left. Finally, to solve the problem of collinearity in the multiple index framework, we construct orthogonal indices using principal components regression to estimate the efficient frontier.
- ItemOpen AccessEmpirical evidences of coherent market hypothesis(2002) Kao, Peter Ta-Chao; Guo, RenkuanIn this dissertation, empirical explorations of basic properties of the CMH-based returns distribution will be conducted on the Johannesburg Stock Exchange. This is followed by a the-oretical exploraion of the stochastic differential equations that governs the underlying market dynamics.
- ItemOpen AccessEmpirical evidences of stock split market effects(2011) Mhuru, Trust Taruona; Guo, RenkuanUnder normal financial market circumstances (i.e., not under the shadow of financial crisis) it iscommon to believe that buying shares from large institutions leads to high profit. This is becausethe shares are of high trading value due to the solid financial foundation and superiorperformances of large institutions or companies. In contrast to these traders' belief, largecompanies often exercise "stock split" to strengthen the confidence on the company and encourage more investments in the company. A "stock split" increases the number of shares outstanding without increasing the company's capital. A conjecture is that a "stock split" action will increase the market liquidity because of the price decrease of each share; consequently, market trading activities would be intensifying such that log-return will be higher and the volatility also higher accordingly. The financial market literature shows that the impacts of "stock split" were controversial. In other words, the influences on the market of "stock split" did not always behave as the management expected. In this thesis, we intend to use limited available stock split data from NASDAQ to explore some empirical evidences on the impacts of "stock split". We also propose a DEAR-based trend analysis in log-return and market volatility measured by daily trading range for technical analysis on "stock split" impacts.
- ItemOpen AccessEmpirical modelling of high-frequency foreign exchange rates(2004) Packirisamy, Someshini; Guo, RenkuanThere is a wealth of information available on modelling foreign exchange time series data, however, research studies on modelling and predicting high frequency foreign exchange data is less prominent. Furthermore, there does not appear to be much evidence supporting work on the modelling and prediction of high frequency South African Rand/United States Dollar (ZAR/USD) exchange rates. A fair amount of noise is embedded in high frequency time series data, especially the ZAR/USD exchange rates, and the modelling of these time series requires the use of specialized models. In addition, lengthy high frequency foreign exchange data is largely unavailable for the South African market. This dissertation undertakes empirical explorations to model high frequency foreign exchange time series (primarily the ZAR/USD time series), through the use of multi-agent neural networks, linear Kalman filters and fuzzy Markov chain theory.
- ItemOpen AccessAn examination and implementation of the libor market model(2006) Jardine, James; Becker, RonaldThe relatively young field of quantitative finance has grown over the past thirty years with the cherry-picking of a wide variety of techniques from the disciplines of finance, mathematics and computer science. The Libor Market Model, a model for pricing and risk-managing interest rate derivatives, is a prime example of this cherry-picking, requiring an understanding of the interest rate markets to understand the problem to be modelled, requiring some deep mathematics from probability theory and stochastic calculus to build the model, and requiring a level of computer expertise to efficiently implement the computationally demanding requirments of the model. This dissertation intends to draw from a wide literature to bring into one body of work a treatment of the Libor Market Model from start to finish.
- ItemOpen AccessAn investigation of short rate models and the pricing of contigent claims in a South African setting(2010) Jones, Chris; Becker, RonaldThis dissertation investigates the dynamics of interest rates through the modelling of the short rate { the spot interest rate that applies for an in-infinitesimally short period of time. By modelling such a rate via a diffusion process, one is able to characterize the entire yield curve and price plain vanilla options. The aim is to investigate which of the more popular short rate models is best suited for pricing such options, which are actively traded in the market. Thus one can then use such models to price more exotic options, as such options are typically less frequently traded in the market.
- ItemOpen AccessThe LIBOR market model in the South African setting(2009) Engelbrecht, Stephanus Francois; Bosman, Petrus
- ItemOpen AccessMarkov-Switching models and resultant equity implied volatility surfaces: a South African application(2012) Fairbrother, Mark; Becker, RonaldStandard Geometric Brownian Motion is the stock model underlying Black-Scholes famous option pricing formula. There are however numerous problems with this stock model as certain features do not follow some empirical stylised facts we see from the observation of actual asset prices. In particular, the constant parameter idea behind Geometric Brownian Motion is flawed. It is argued that information flow dictates stock price movements and information is a function macro-economic regimes shifts. As such, we propose an alternative model, one in which the parameters in the Standard Geometric Brownian Motion change according to an underlying Hidden Markov Process. This new model, termed a Markov-Switching model, is presented in extensive detail. Parameter Estimation methods, Simulation Methods and Option Pricing Theory are explored. Summary algorithms are presented so that this dissertation may be used as a good reference guide for those wishing to apply Markov-Switching Models. The model is tested by fitting the model on South African data and using the discussed option theory to create various implied volatility surfaces. The surfaces produced appear to obey some of the empirical observations and theoretical ideas around expected implied volatility surfaces, indicating that the Markov-Switching model has some value for option pricing.
- ItemOpen AccessThe mathematics of insider trading(2008) Chui, Chi Kin; Ouwehand, PeterOver the past decade the research into the topic of incorporating non-market information has accelerated. This dissertation aims to serve as a monograph of the contemporary body of research to the insider problem, under a Brownian setting in a complete market.
- ItemOpen AccessModelling conditional covariances with orthogonal factor models(2011) Jensen, Tracy; Haines, LindaThe recent sub prime crisis has resulted in an increased focus on risk management and monitoring in the financial industry. One of the essential components of risk management and monitoring is a reliable ex-ante covariance matrix of various financial time series. Therefore a reliable model which can handle a large number of time series is required to calculate an ex-ante or conditional covariance matrix.
- ItemOpen AccessOption pricing using hidden Markov models(2006) Anderson, Michael; Guo, RenkuanThis work will present an option pricing model that accommodates parameters that vary over time, whilst still retaining a closed-form expression for option prices: the Hidden Markov Option Pricing Model. This is possible due to the macro-structure of this model and provides the added advantage of ensuring efficient computation of option prices. This model turns out to be a very natural extension to the Black-Scholes model, allowing for time-varying input parameters.
- ItemOpen AccessParameter estimation of a bivariate diffusion process : the Heston model(2011) Nomoyi, Siyabulela; Varughese, MelvinThe main objective of the research is to estimate the parameters on the Heston (1993) model, which models the movement of asset prices assuming that the asset price volatility is stochastic. The paper concentrates on estimating these parameters by approximating the transitional probabilities of the diffusion process with a saddlepoint distribution. By solving a system of ordinary differential equations that are in terms of the systemâ€™s cumulants, and using these solutions to calculate the saddlepoint, the transitional probabilities of the diffusion process can be approximated.
- ItemOpen AccessPricing equity options on multiple underlyings in the South African context(2008) Preston, Bradley; Bosman, PetrusIt is well documented that financial asset prices returns are not normally distributed. Historical return distributions exhibit fatter tails and positive skewness that is not explained by a normal distribution. Moreover, the standard Black-Scholes option pricing framework that assumes that asset prices follow geometric Brownian Motion does not explain option prices observed in the market. In particular much work has been done trying to explain the volatility skew.
- ItemOpen AccessPricing path dependent options under variance gamma dynamics(2007) Anderson, Craig; Ouwehand, PeterWord processed copy. Includes bibliographical references (leaves 82-83).
- ItemOpen AccessRisk-return portfolio modelling(2007) Gilbert, Emmeleen Ulita; Troskie, Casper GMarkowitz introduced the concept of modelling the risk associated with a given security as the variance of the expected return and showed how under certain conditions an investors portfolio can be managed by balancing the expected return of the portfolio and its variance. Building on Markowitz original framework, William Sharpe, extended these ideas by connecting a portfolio to a risky asset. This extension became known as the Sharpe Index Model. There are number of assumptions governing the residuals of the Sharpe index model, one being that the error terms of the stocks are uncorrelated. The Troskie-Hossain innovation to the Sharpe Index model relaxes this assumption. We evaluate the Troskie-Hossain model relative to the Sharpe Index Model and Markowitz portfolio, and find that the Troskie-Hossain model approximates the Markowitz efficient frontier and optimal portfolio very closely. Further examining the residuals, we find evidence of autocorrelation and heteroskedasticity. Using ARMA to model the autocorrelation of the residuals has very little impact on the efficient frontier when working with log returns. However when working with simple returns the ARMA shifts the efficient frontier to the left. We find that GARCH(l , 1) models capture most of the autocorrelation in the squared residuals for both simple returns and log returns and shifts the efficient frontier to the left. Modelling a non-constant conditional mean and non-constant conditional variance (ARMA and GARCH) has proven difficult. The more complex a model becomes the more difficult the estimation. We investigate the effects of dividend yields on the efficient frontier, as well as using simple returns vs log returns in portfolio construction. Including dividend yields in our return data shifts the efficient frontier upwards. However only the a's are increased, and the f3's and f3 t-statistics of the shares remain the same. This shift effect of dividends has no impact on the time series or heteroskedastic models. The simple returns efficient frontier lies above that of the log returns efficient frontier. The a 's for simple returns are very different to those of log returns, however the f3's lie in a similar region to those of log returns.
- ItemOpen AccessSimulation of asset prices using LÃ©vy processes(2008) Riemer, Mark L; Ouwehand, PeterThis dissertation focuses on a LÃ©vy process driven framework for the pricing of financial instruments. The main focus of this dissertation is not, however, to price these instruments; the main focus is simulation based. Simulation is a key issue under Monte Carlo pricing and risk-neutral valuation- it is the first step towards pricing and therefore must be done accurately and with care. This dissertation looks at different kinds of LÃ©vy processes and the various approaches one can take when simulating them.
- ItemOpen AccessStrategic asset selection taxonomy : fund of hedge funds(2010) Mokoma, Kaibe; Becker, Ronald; Schlebusch, ThomasThis thesis develops a logical methodology to be used to assess the hedge fund managers' return time series in comparison with their peers. This enables Fund of Hedge Funds portfolio manager to identify those with required factors to be included in a portfolio. The models that had been used as the industry standard for some time are derived on the assumption of normal distribution. Hence they use only mean and standard deviation to explain all data phenomenal attributes of time series. This study project uses higher order moments and some performance measures to rank order feasible portfolios of different hedge fund strategies based on their calculated metrics. Then determine the significance of t-Statistics, thus to observe the likelihood of achieving a particular return level relative to the downside associated with that target return and also on the behavioral hypothesis that investors prefer more to less. The study proposes and examines an alternative performance measures to facilitate the investment decision making. An indication of how this may be applied across a broad range of problems in hedge funds analysis. Some performance measures capture the higher order moments of the return distributions. This method makes intuitive sense since one of the key mandates of the hedge funds is to seek to capture most upside while protecting against downside.