### Browsing by Subject "Financial Mathematics"

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- ItemOpen Access3-month bond option strategies: an analysis of performance from 1998 to 2010 in the South African market(2011) Ndebele, Ndumiso; Jones, Samantha; Touna-Mama, AlbertDue to the 2008 financial crisis, investors have become more risk averse in investing in equities and have increased their holdings in bonds as they are believed to be less risky. However, South African interest rates have been volatile over the past decade due to changes in the inflation rate. This has caused the returns of bond portfolios to be uncertain since bond prices are inversely related to interest rates. It is thus imperative to manage the interest rate risk inherent in bond portfolios so that institutional investors can achieve their mandates and targeted returns.
- ItemOpen AccessAn alternative model for multivariate stable distributions(2009) Jama, Siphamandla; Guo, RenkuanAs the title, "An Alternative Model for Multivariate Stable Distributions", depicts, this thesis draws from the methodology of [J36] and derives an alternative to the sub-Gaussian alpha-stable distribution as another model for multivariate stable data without using the spectral measure as a dependence structure. From our investigation, firstly, we echo that the assumption of "Gaussianity" must be rejected, as a model for, particularly, high frequency financial data based on evidence from the Johannesburg Stock Exchange (JSE). Secondly, the introduced technique adequately models bivariate return data far better than the Gaussian model. We argue that unlike the sub-Gaussian stable and the model involving a spectral measure this technique is not subject to estimation of a joint index of stability, as such it may remain a superior alternative in empirical stable distribution theory. Thirdly, we confirm that the Gaussian Value-at-Risk and Conditional Value-at-Risk measures are more optimistic and misleading while their stable counterparts are more informative and reasonable. Fourthly, our results confirm that stable distributions are more appropriate for portfolio optimization than the Gaussian framework.
- ItemOpen AccessThe Bates model : Fourier Transform for option pricing under jump-diffusions in the South African market(2011) Munhumwe, Blessing; Becker, RonaldThe purpose of this study is to price options under jump diffusions using Fourier Transforms and obtain the implied volatility surface from these option prices.
- ItemOpen AccessBenefits of a Tree-Based model for stock selection in a South African context(2014) Giuricich, Mario Nicolo; Bosman, PetrusQuantitative investment practitioners typically model the performance of a stock relative to its benchmark and the stock's fundamental factors in a classical linear framework. However, these models have empirically been found to be unsuitable for capturing higher-order relationships between a stock's return relative to a benchmark and its fundamental factors. This dissertation studies the use of Classification and Regression Tree (CART) models for stock selection within the South African context, with the focus being on the period from when the Global Financial Crisis began in early 2007 until December 2012. By utilising four types of portfolios, a CART model is directly compared against two traditional linear models. It is seen that during the period focused upon, the portfolios based on the CART model deliver the best excess return and risk-adjusted return, albeit in most cases modestly above the returns delivered by the portfolios based upon the linear models. This is observed in the hedge-fund style and long-only portfolios constructed. Moreover, it is observed that the CART-based portfolios' returns are not correlated with those from the linear-model-based portfolios. This observation suggests that CART models offer an attractive option to diversify model risk within the South African context.
- ItemOpen AccessBounds on baskets option prices(2005) De Swardt, N C; Polakow, DanielThe celebrated Black-Scholes option pricing model is unable to produce closed-form solutions for arithmetic basket options. This problem stems from the lack of an analitical form for the distribution of a sum of lognormal random variables. lVlarket participants commonly price basket options by assuming the basket follows lognormal dynamics, although it is known that this approximation performs poorly in some cicumstances. The problem of finding an analytical approximation to the sum of lognormally distributed random variables has been widely studied. In this dissertation we seek to draw these studies together and apply them in an option pricing setting. We propose some new option pricing formulae based on these approximations. In order to examine the utility of these new formulae and compare them to commonly used market approximations we present rigorous analytical bounds for the price of arithmetic basket options using the theory of comonotonicity. In this we follow the ideas in Deelstra et al. [7]. Additionally we provide an interval of hedge parameters (the Greeks). We carry out a numerical sensitivity analysis and identify circumstances under which the market approximation misprices basket options.
- ItemOpen AccessA comparison of three analytical approximations for basket option valuation(2013) Hagspihl, Christoph; Floor, JustinThree prominent analytical approximations for pricing basket options,by Levy (1992), Ju (2002) and Deelstra et aI. (2004), are tested for performance and accuracy. Sensitivity analysis shows that all three have greater errors in high volatility and long maturity environments, while Deelstra has weaknesses with small correlation and baskets with few stocks. Deelstra and Levy show tendencies to underprice and overprice respectively, while Ju's errors are more consistently around the true price. A mathematical understanding of the three techniques is also developed.
- ItemOpen AccessAn examination of liquidity risk and liquidity risk measures(2010) Bhyat, Aneez; Becker, RonaldLiquidity risk represents a vacuum of rigour in the otherwise well-researched area of risk management. In both practice and theory most of finance is silent regarding its scope and effect. This is principally due to a lack of consensus regarding its definition and measurement. Current liquidity risk measures differ fairly widely in both respects. This thesis attempts at addressing this by consolidating and examining the principle liquidity risk measures used in financial literature.
- ItemOpen AccessGeometric Asian option: Geometric Ornstein-Uhlenbeck process(2013) Zhou, Sen Lin; Mataramvura, SureAsian options, also known as average value options, are exotic options whose payoffs are dependent on the average prices of the underlying assets over the life of the options. The Asian options are very popular among the market participants when dealing with thinly traded commodities because the average property of the Asian options makes it very difficult to manipulate the payoffs of the options. Another reason for the popularity of Asian options is that they are cheaper than the corresponding portfolio of standard options to hedge the same exposure. The pricing of Asian options has been the subject of continuous studies. In previous studies, Asian options have been priced based on the assumption that the underlying asset follows a geometric Brownian motion. This dissertation, however, assumes that the underlying asset follows a geometric Ornstein-Uhlenbeck process and provides an explicit formula for the geometric Asian options. The geometric Ornstein-Uhlenbeck process is more economically appropriate than the geometric Brownian motion for modelling commodity prices, exchange rates and interest rates due to its mean-reverting property.
- ItemOpen AccessGrey diffenrential equation modeling on stock prices(2005) Xue, Qifeng; Guo, RenkuanIncludes bibliographical references (leaves 110-111).
- ItemOpen AccessInvestigation on the efficient frontier based on CVaR under copula dependence structure with applications to South African JSE stocks(2005) Damaseb, W B; Ouwehand, P; Demchuk, AWe study the feasihility of using a coherent monetary risk measure, Conditional Value at Risk (CVaR) also known as Expected Shortfall (ES), to optimise a portfolio of South African stocks. Value at Risk (VaR) is not a sub-additive risk measure and therefore does not possess one of the four properties that all coherent risk measures must satisfy. Using copula to describe the dependence structure between the instruments in our portfolio, we implement and backtest a CVaR optimization algorithm and compare the backtested results to those obtained using parametric and non-parametric/Monte Carlo VaR. Finally we optimise the portfolio of stocks and generate an efficient frontier specifying CVaR as the risk measure instead of the portfolio variance traditionally used in Markowitz and CAPM models.
- ItemOpen AccessModelling dependance in collateralied debt obligations with copulas(2010) Linley, Christopher; Becker, RonaldIn this paper we provide a review of credit derivatives, and some of the tools used to model them. We give a basic introduction to copulas and how they are used to model the depedence between single name credit derivatives. We then investigate various features of Gaussian and t copula dependence using numerical results obtained from Monte-Carlo simulation.
- ItemOpen AccessModelling of volatility of stock prices using GARCH models & its importance in portfolio construction(2009) Mtemeri, Tinotenda; Guo, RenkuanThis thesis is aimed at investigating the possibility to model the risk of stocks in financial markets and evaluating the adequacy and effectiveness of univariate GARCH models such as the symmetric GARCH and a few other variations such as the EGARCH, TARCH and PARCH in modelling volatility in monthly returns of stocks traded on the Johannesburg Stock Exchange. This is further used to investigate the importance of GARCH modelling in portfolio construction using Improved Sharpe Single Index Models. The data used for model estimation has been randomly selected from different sectors of the South African economy. GARCH models are estimated and validated for the data series of the randomly selected 15 JSE stocks. Conclusions are drawn regarding the different GARCH models, best lag structure and best error distributions for modelling. The GARCH (1,1) model demonstrates a relatively good forecasting performance as far as the short term forecasting horizon is concerned. However, the use of alternatives to the more common GARCH (1,1) and use of non-normal distributions is not clearly supported. Also, the use of higher order GARCH models such as the GARCH (1,2), GARCH (2,1) and GARCH (2,2) is not clearly supported and the GARCH (1, 1) remains superior overall to these models. The results obtained from this thesis are of paramount importance in portfolio construction, option pricing and formulating hedging strategies. An illustration of the importance of the G ARCH (1,1) model in portfolio construction is given and conclusions are drawn regarding its usefulness in improving our volatility estimations for purposes of portfolio construction.
- ItemOpen AccessModelling seasonality in South African agricultural futures(2007) Kirk, Richard; Wilcox, DianeThis study investigates the seasonality in agricultural commodity futures prices. Futures prices are modelled using the model developed by SÃ¸rensen (2002). The model defines the commodity spot price as the sum of a nonstationary state variable, a stationary state variable and a deterministic seasonal component. Standard no-arbitrage arguments are applied in order to derive futures and option prices. Model parameters are estimated using Kalman filter methodology and maximum likelihood estimation. Model parameters are estimated for white maize, yellow maize and wheat futures traded on the South African Futures Exchange (SAFEX). Furthermore, this research considers other models for commodity derivatives as well as pricing futures contracts in the presence of price limits.
- ItemOpen AccessModern portfolio optimization using robust estimation techniques(2005) Van Straaten, Conrad; Troskie, Casper GRather than following a normal distribution, share returns and market proxies have been shown to follow skewed distributions, with long tails in some cases. In this dissertation various robust estimation techniques are investigated in an attempt to minimise the influence that outliers may have on the estimation and to better estimate the input parameters for the Markowitz and Sharpe portfolio models. The main goal is to ascertain whether or not the input parameters determined, using the robust procedures, yield better results than the Ordinary Least Squares (OLS) procedure.
- ItemOpen AccessNonparametric smoothing in extreme value theory(2010) Clur, John-Craig; Haines, LindaThis work investigates the modelling of non-stationary sample extremes using a roughness penalty approach, in which smoothed natural cubic splines are fitted to the location and scale parameters of the generalized extreme value distribution and the distribution of the r largest order statistics. Estimation is performed by implementing a Fisher scoring algorithm to maximize the penalized log-likelihood function. The approach provides a flexible framework for exploring smooth trends in sample extremes, with the benefit of balancing the trade-off between 'smoothness' and adherence to the underlying data by simply changing the smoothing parameter. To evaluate the overall performance of the extreme value theory methodology in smoothing extremes a simulation study was performed.
- ItemOpen AccessPortfolio construction using index regression models(2008) Steyn, Dirk; Troskie, Casper GIn this dissertation we review the Sharpe Index Model and an innovation on this model introduced by Hossain, Troskie and Guo (2005b). These models are extended to the multi index framework. We then empirically investigate the impact of the models on portfolio creation over an extensive data set. Next we extend these models by modelling the regression residuals as ARMA and GARCH(l, 1) processes and investigate the effect on the resulting portfolios. We then introduce the topic of bounded influence regression and apply it to financial data by down weighting extreme returns prior to regression. A new weighting function is introduced in this dissertation and the effects on the efficient frontiers and resulting market portfolios for the chosen set of shares are investigated.
- ItemOpen AccessPricing 2-colour rainbows : nonparametric methods using copulae(2005) Knox, Sean DThis paper investigates the use of copulae for non parametric pricing of multivariate contingent claims. Price estimates and no-arbitrage bounds for various types of two-colour rainbow options on the South African equity and bond markets were calculated. Implied marginal risk-neutral distributions were derived nonparametrically from each assets option price spread. This was achieved in a very simple manner by assuming that, for each of the underlying assets in question, a continuum of option prices exist. Cubic splines were used to fit this continuum to the implied volatilities of the actual options available. Two nonparametric copulae were considered: an empirical copula based directly upon the data and a kernel copula derived from a smooth two-dimensional kernel approximation of the historic density function. In addition, various parametric copulae were considered for comparison purposes. The differences between each of these approaches was found to vary from one type of rainbow to another.
- ItemOpen AccessPricing of credit risk and credit risk derivatives : from theory to implementation(2008) Sewnath, Neville; Abraham, HaimIncludes abstract. Includes bibliographical references (leaves 223-230).
- ItemOpen AccessPricing options in a fuzzy environment(2008) Ramsden, Bevan; Guo, RenkuanAlthough Fuzzy Logic is not new, it is however only since 2004 that an axiomatic theory has been created that has all the desirable effects of Fuzzy Logic. This theory, named Credibility theory was proposed by Dr. Liu. Within this thesis we aim to utilize credibility theory to model the psychological impacts of market participants on European options. Specifically this is done by modifying the approach that was originally taken by Black and Scholes. The Hew model, which is known as the fuzzy drift parameter model, begins by replacing the deterministic drift within Brownian motion with a fuzzy parameter. This fuzzy parameter models the psychological impacts of market participants. Naturally as we are dealing in Chance theory 1 the risk neutral dynamics change from that of Black and Scholes and thus so does the price of European call options.
- ItemOpen AccessStatistical arbitrage in South African financial markets(2011) Govender, KieranEngle and Grangerâ€™s (1987) co-integrating framework provides a useful method of analyzing the dynamics of non-stationary data in both the short and long run. However, despite its popularity in various areas of research, the application of co-integration to financial data has been limited. This paper provides an example of the application of co-integration in a pairs trading strategy to identify mean reverting spreads. The strategy is implemented with an algorithmic trading setup that models the spread in a state-space form...