The popularity of the LIBOR Market Model (LMM) in interest rate modelling is a result of its consistency with market practice of pricing interest rate derivatives. In the context of a life insurance company, the LMM is calibrated to swaptions as they are actively traded for a wide variety of maturities and they serve as the natural hedge instruments for many of the long dated maturity products with embedded options. Before calibrating the model we extend the calibration process to address the issue of illiquidity in the South African swaption market. The swaption surface used in calibrating the model is generated with market implied quotes for the hedgeable component and thereafter using historical volatilities for the unhedgeable or illiquid component. Rebonato's 3 parameter correlation function proposed by Rebonato (2005) provides the best fit to historical data. We assume a general piecewise constant parameterisation for the instantaneous forward rate volatilities. These volatilities are then determined analytically using the Rectangular Cascade Calibration Algorithm from Brigo and Morini (2006). The calibration generates a stable volatility term structure with the instantaneous forward rate volatilities being positive and real. Through an extension of the calibration we are able to capture the benefits of a pure replication component and accommodate a large unhedgeable component in the price faced by life insurance companies in South Africa.

Includes bibliographical references. / In this thesis we replace the normal distribution assumption in the calculation of the prior equilibrium returns used in the model with a more general distribution which captures the skewness and fat tails exhibited by stock data. We consider the á stable distributions as an alternative distribution to the normal distribution. Consequently we also consider alternative measures of risk, the Value at Risk and the Conditional Value at Risk other than the variance used in the normal case.

This dissertation is a hedging back-study which assesses the effectiveness of interest- rate modelling and the hedging of interest-rate derivatives. Caps that trade in the Johannesburg swap market are hedged using two short-rate models, namely the Hull and White (1990) one-factor model and the subsequent Hull and White (1994) two-factor extension. This is achieved by using the equivalent Gaussian additive-factor models (G1++ and G2++) outlined by Brigo and Mercurio (2007). The hedges are constructed using different combinations of theoretical zero-coupon bonds. A flexible factor hedging method is proposed by the author and the bucket hedging technique detailed by Driessen, Klaasen and Melenberg (2003) is tested. The results obtained support the claims made by Gupta and Subrahmanyam (2005), Fan, Gupta and Ritchken (2007) and others in the literature that multi-factor models outperform one-factor models in hedging interest-rate derivatives. It is also shown that the choice of hedge instruments can significantly influence hedge performance. Notably, a larger set of hedge instruments and the use of hedge instruments with the same maturity as the derivative improve hedging accuracy. However, no evidence to support the finding of Driessen et al. (2003) that a larger set of hedge instruments can remove the need for a multi-factor model is found.

This dissertation investigates the cost of using single-factor models to exercise and hedge American options on South African coupon bearing bonds, when the simulated market term structure is driven by a two-factor model. Even if the single factor models are re-calibrated on a daily basis to the term structure, we find that the exercise and hedge strategies can be suboptimal and incur large losses. There is a vast body of research suggesting that real market term structures are in actual fact driven by multiple factors, so suboptimal losses can be largely reduced by simply employing a well-specified multi-factor model.

The credit risk of a portfolio is often managed via measures of counter-party exposure, such as potential future exposure (PFE) and expected exposure (EE), with these measures playing an important role in setting economic and regulatory capital levels. For the sake of risk measurement and risk management these exposure measures should be computed under the real-world probability measure. However, due to the similarity of these exposure calculations to those used in calculating credit valuation adjustments, some have begun to compute them under the risk-neutral measure instead. This is problematic, as the magnitudes of PFEs and EEs differ under different equivalent martingale measures and their associated numéraires. Working with the Hull-White (HW) model of the short rate, the effect of a change of measure on the PFE and EE profiles of vanilla interest rate swaps and European swaptions is shown under three common measures: the money-market account measure, the T-forward measure and the Linear Gaussian Markovian (LGM) measure. A modified Least Squares Monte Carlo (LSM) algorithm, which allows for substantial computational savings, is then introduced in order to approximate contract level exposures under each of the aforementioned probability measures. Finally, a change of measure is implemented within the modified LSM algorithm in order to approximate exposure profiles under the real-world measure. The modified LSM algorithm is particularly useful for computing exposure profiles of contracts without closed-form valuation formulae, which would otherwise take significantly longer to compute via a standard Monte Carlo approach.

The notion of portfolio tilting towards fundamental factors has been the subject of many empirical studies over the last few decades. With this being said, there is limited literature on the interaction effects between these individual factors. This dissertation focuses specifically on quality, value, low volatility and momentum and determines which factors have the largest impact on portfolio return. In addition to testing these single factor portfolios, the various interaction effects between the individual factors are investigated. This framework is divided into two parts. The first, is an empirical study on the JSE Top 100 over the 15 year period beginning September 2001 and ending September 2016. Quarterly and monthly rebalancing as well as transaction costs of 100 basis points (per trade) have been employed to mimic realistic investment management. Much of the framework used to incorporate these factors is adapted from Bender and Wang (2016) who tested these interactions on the S&P 500. The second, involves the construction of a controlled market model in an attempt to provide mathematical justification to the framework. The controlled model simulates stock price paths, in a Mil'shtein (1974) fashion, using Geometric Brownian Motion with a stochastic alpha component added to the drift. Factors are simulated randomly using correlated uniform distributions. The controlled model uses realistic market parameters and constructs the factor portfolio in the same manner as the empirical study.

Includes bibliographical references. / Markov-Functional models are a very powerful class of market models which calibrate and compute prices and Greeks quickly. This dissertation explains, in detail, how Markov-Functional models work as well as discussing all of the specific models developed in the literature. It contains the key points that can be found in the present literature. We explain, in detail, all of the concepts, from the theoretical framework down to the numerical implementation of the specific models. This involves explaining the framework for Markov-Functional models, describing specific models, obtaining a deeper understanding of how the model parameters affect the results, discussing the issues involved in the implementation, implementing various models and investigating the effect of numerical and market parameters on the outcome. Various concepts, not discussed in the present literature, such as considerations for selecting a discretization grid for the numerical implementation, are developed. The practical application of Markov-Functional models is considered as well as alternative fields, such as Actuarial science, where the model can be applied. In summary, this dissertation embodies a complete discussion of the current class of Markov-Functional models.

This paper implements four calibration methods on stochastic volatility models. We estimate the latent state and parameters of the models using three non-linear filtering methods, namely the extended Kalman filter (EKF), iterated extended Kalman filter (IEKF) and the unscented Kalman filter (UKF). A simulation study is performed and the non-linear filtering methods are compared to the standard least square method (LSQ). The results show that both methods are capable of tracking the hidden state and time varying parameters with varying success. The non-linear filtering methods are faster and generally perform better on validation. To test the stability of the parameters, we carry out a delta hedging study. This exercise is not only of interest to academics, but also to traders who have to hedge their positions. Our results do not show any significant benefits resulting from performing delta hedging using parameter estimates obtained from non-linear filtering methods as compared to least square parameter estimates.

This dissertation follows, scrupulously, the probability of default model used by the National University of Singapore Risk Management Institute (NUS-RMI). Any deviations or omissions are noted with reasons related to the scope of this study on modelling probabilities of corporate default of South African firms. Using our model, we simulate defaults and subsequently, infer parameters using classical statistical frequentist likelihood estimation and one-world-view pseudo-likelihood estimation. We improve the initial estimates from our pseudo-likelihood estimation by using Sequential Monte Carlo techniques and pseudo-Bayesian inference. With these techniques, we significantly improve upon our original parameter estimates. The increase in accuracy is most significant when using few samples which mimics real world data availability

Includes bibliographical references. / This research project examines stock price fragility, a measure developed by Greenwood and Thesmar (2011), which serves as a proxy for non-fundamental risk i.e. it aims to isolate the drivers of stock price volatility beyond traditional fundamental drivers, in particular examining the impact of concentrated stock ownership and correlated liquidity shocks on price volatility. Here, the measure is applied to the South African financial market. Subject to data complications, it is nevertheless shown that stock price fragility is a significant predictor of total return volatility owing to the ownership structure of South African funds, even when controlling for endogeneity, autocorrelation and heteroskedasticity in the model.