The credit and liquidity crisis of 2007 has triggered a number of inconsistencies in the interest rate market, questioning some of the standard methods and assumptions used to price and hedge interest rate derivatives. It has been shown that using a single risk-free curve (constructed from market instruments referencing underlying rates of varying tenors) to forecast and discount cash flows is not theoretically correct. Standard market practice has evolved to a multi-curve approach, using different curves to forecast and discount cash flows. The risk-free discount curve is proxied by the Overnight-Indexed Swap (OIS) curve. In South Africa there is no liquid market for OIS. In this dissertation a method is developed to estimate the ZAR OIS curve. A cointegration relationship between the SAFEX Overnight Rate, and the 3-month JIBAR rate is shown to exist. This relationship is used in a dual bootstrap algorithm, to simultaneously estimate the ZAR OIS curve and 3-month JIBAR tenor curve, while maintaining arbitrage relationships. The tractability of this method is shown, by pricing options written on ZAR OIS.

Includes bibliographical references. / This study investigates the performance of a statistical arbitrage portfolio in the South African equity markets. A portfolio of liquid stock pairs that exhibit cointegration is traded for a ten year period between the years 2003 and 2013. Without transaction costs, the portfolio has an encouraging Sharpe ratio of 2.1. When realistic transaction costs are factored in, the Sharpe ratio drops to 0.43.The results underline the theoretical profitability of statistical arbitrage as a trading strategy and highlight the importance of transaction costs in a real-world setting.

This dissertation investigates the computational efficiency and accuracy of three methodologies in the pricing of a Bermudan option, under the constant elasticity of variance (CEV) model. The pricing methods considered are the finite difference method, least squares Monte Carlo method and recursive marginal quantization (RMQ) method. Specific emphasis will be on RMQ, as it is the most recent method. A plain vanilla European option is initially priced using the above mentioned methods, and the results obtained are compared to the Black-Scholes option pricing formula to determine their viability as pricing methods. Once the methods have been validated for the European option, a Bermudan option is then priced for these methods. Instead of using the Black-Scholes option pricing formula for comparison of the prices obtained, a high-resolution finite difference scheme is used as a proxy in the absence of an analytical solution. One of the main advantages of the recursive marginal quantization (RMQ) method is that the continuation value of the option is computed at almost no additional computational cost, this with other contributing factors leads to a computationally efficient and accurate method for pricing.

When optimising the likelihood function one often encounters various stationary points and sometimes discontinuities in the parameter space (Gupta and Mehra, 1974). This is certainly true for a majority of multi-factor affine term structure models. Practitioners often recover different parameter optimisations depending on the initial parameters. If these parameters result in different option prices, the implications would be severe. This paper examines these implications through numerical experiments on the three-factor Vasicek and Arbitrage-free Nelson-Siegel (AFNS) models. The numerical experiments involve Kalman filtering as well as likelihood optimisation for parameter estimation. It was found that the parameter sets lead to the same short rate process and thus the same model. Moreover, likelihood optimisation in the AFNS does not result in different parameter sets irrespective of the starting point.

The traditional Black-Scholes (BS) model relies heavily on the assumption that underlying returns are normally distributed. In reality however there is a large amount of evidence to suggest that this assumption is weak and that actual return distributions are non-Gaussian. This dissertation looks at algorithmically generating a Volatility Targeting Strategy (VTS) which can be used as an underlying asset. The rationale here is that since the VTS has a constant prespecified level of volatility, its returns should be normally distributed, thus tending closer to an underlying that adheres to the assumptions of BS.

The Trolle and Schwartz (2009) interest rate model prices interest rate derivatives in a generalised stochastic volatility framework. It is a reformulation of the multifactor Heath, Jarrow and Morton (1992) framework with stochastic volatility terms presented in an analogous fashion to the seminal Heston (1993) model. The Trolle and Schwartz (2009) model provides semi-analytical pricing formulas for zerocoupon bonds and zero-coupon bond options. These formulas are extended to price interest rate caplets, and therefore caps, as well as swaptions. These formulas are described as semi-analytical because of the use of numerical methods as well as their dependency on unobserved state variables. These state variables are estimated by applying an extended Kalman filter on a dataset of interest rates and interest rate derivative prices. Although Trolle and Schwartz (2009) confirm the accuracy of their model when testing against empirical prices, they do not provide an analysis of the consistency between the semi-analytical formulas and Monte Carlo pricing. Presenting this test for consistency seeks to confirm the validity of these pricing formulas. The aim of this dissertation is to implement the Trolle and Schwartz (2009) model and discuss the performance of the semi-analytical pricing formulas against a Monte Carlo simulation. Emphasis will be placed firstly on reviewing the derivations outlined in Trolle and Schwartz (2009) and secondly, building a Monte Carlo framework capable of comparing prices with the semi-analytical pricing formulas. Simulated data will be considered for the purpose of confirming that the estimation of the state vector is sufficiently accurate. Thereafter, an analysis on an empirical dataset can determine whether the results hold across different sets of data.

The Linear-Rational Framework for the modelling of interest rates is a framework which allows for the addition of spanned and unspanned factors, while maintaining a lower bound on rates and tractable valuation of interest rate derivatives, particularly swaptions. The advantages of having all these properties are significant. This dissertation presents the Linear-Rational Framework, and specializes the factor process to a class of diffusion models which allows for the degree of state dependence of volatility to be estimated. This dissertation then finds that the estimated state dependent volatility structure is significantly different to that of typical models, where it is set it a priori. The effect the added degree of freedom has on the model implied swaption skew is then analysed.

It is common practice to use the return series from closing prices in order to estimate the values of variables to be used in Modern Portfolio Theory (MPT). In fact the closing price series is generally what is referred to when price data or a financial time series is mentioned. We know this series to be made up of discrete points recorded as the last traded price on a specific day. But we also know this gives no indication of where the price has moved during the day. It is also widely believed that the price breaking through a certain level can be an indication of future movements. The highs and the lows, regardless of what one may believe they represent exactly, do, together with the closing prices, give a more complete view of the behaviour of a moving price. Yet they are for the most part left unused. It is known that the series of highs and lows are published and are widely available, at least as available as closing prices. The question that needs to be answered is: are the highs and lows valuable enough to warrant their use in MPT, and if so, what would this entail? It will be attempted, through an empirical study, to determine whether or not it is worthwhile to incorporate the highs and lows into the existing framework of MPT, and how this might be accomplished. It must be discovered more clearly whether there is extra information in the highs and lows that is not in the closings. The high and low series must also be used in core procedures like the calculation of the efficient frontier or the determination of actual portfolios in order to see if there is an appreciable difference to just using closings. This should give is an indication of how one might use the highs and lows in MPT if they are indeed deemed valuable.

The stochastic differential equation (SDE) describing the spot FX rate is of central importance to modelling FX derivatives. A Monte Carlo estimate of the discounted individual payoffs of FX derivatives is taken to arrive at the price, provided there does not exist a closed form solution for the price. One propagates the FX spot rate through time under risk-neutral dynamics to realise the before-mentioned payoffs. A drawback to Monte Carlo becomes evident when the model dynamics become more complicated, such as when more dimensions are added to the dynamics of the model. These additional dimensions can be stochastic volatility and/or stochastic domestic and foreign short rates. This dissertation describes the calibration of such a model using mixed Monte Carlo, as described in Cozma and Reisinger (2015), to both model-generated and market data. Profit and loss analysis of hedging FX derivatives using the mixed Monte Carlo method is conducted when hedging against both model-generated and market data .

Includes abstract. Includes bibliographical references.