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This dissertation details the performance of two specific trading strategies which are based on the Johansen-Ledoit-Sornette (JLS) model. Both positive and negative bubbles are modelled as a log-periodic power law (LPPL) ending in a finite time singularity. The stock prices of the constituents of the FTSE/JSE Top40 index are taken as inputs to the JLS model from 3 June 2003 to 31 August 2015. It is shown that for certain time horizons into the past, the JLS based trading strategies significantly outperform random trading strategies. However this result is highly dependent on how far the model looks into the past, and if the model is calibrating to positive or negative bubbles. The lack of research with regards to the "stylized facts" of the JLS model, specifically relating to the time horizon and type of bubble, poses a significant hurdle in correctly identifying a LPPL structure in stock prices. These core features of the JLS model were developed from a number of positive bubbles that built up over many years. The results suggest that these features may not apply over all time horizons, and for both types of bubbles.

Traditional option pricing methods like Monte Carlo simulation can be time consuming when pricing and hedging exotic options under stochastic volatility models like the Heston model. The purpose of this research is to apply the Gaussian Process Regression (GPR) method to the pricing and hedging of exotic options under the Black-Scholes and Heston model. GPR is a supervised machine learning technique which makes use of a training set to train an algorithm so that it makes predictions. The training set is composed of the input vector X which is a n × p matrix and Y an n×1 vector of targets, where n is the number of training input vectors and p is the number of inputs. Using a GPR with a squared-exponential kernel tuned by maximising the log-likelihood, we established that this GPR works reasonably for pricing Barrier options and Asian options under the Heston model. As compared to the traditional method of Monte Carlo simulation, GPR technique is 2 000 times faster when pricing barrier option portfolios of 100 assets and 1 000 times faster computing a portfolio of Asian options. However, the squared-exponential GPR does not compute reliable hedging ratios under Heston model, the delta is reasonably accurate, but the vega is off.

The recent explosion of work on rough volatility and fractional Brownian motion has led to the development of a new generation of stochastic volatility models. Such models are able to capture a wide range of stylised facts that classical models simply do not. While these models have sound mathematical underpinnings, they are difficult to implement, largely due to the fact that fractional Brownian motion is neither Markovian nor a semimartingale. One idea is to investigate the behaviour of these models as maturities become very small (or very large) and consider asymptotic estimates for quantities of interest. Here we investigate the performance of small-time asymptotic formulae for the cumulant generating function of the Fractional Heston model as presented in Guennoun et al. (2018). These formulae and their effectiveness for small-time pricing are interrogated and compared against the Rough Heston model proposed in El Euch and Rosenbaum (2019).

The concurrence between the displaced lognormal forward-Libor model (DLFM), Gaussian Heath-Jarrow-Morton (GHJM) model and Hull-White (HW) model is explored. We briefly present the theory underpinning these models, specifically focusing on single factors. A useful volatility relation result adapted from Andersen and Piterbarg (2010) is derived. It relates the instantaneous volatility functions of the GHJM model and the DLFM model. The volatility relation allows us to state a specific GHJM model and derive a corresponding DLFM model that it is concurrent with. We take the Hull-White model and derive its corresponding GHJM model, the volatility of the GHJM model is then fed into the volatility relation in order to derive the corresponding DLFM model. This was sufficient mathematical proof of the concurrence, but numerical confirmation is also essential. The HW, GHJM and DLFM models were implemented, with applications to pricing European swaptions. Numerical results show that swaption prices are consistent across the three models. This provides good numerical evidence to support the concurrence between the DLFM and HW models.

Includes bibliographical references. / This dissertation is an investigation into realised volatility (RV) estimators. Here, RV is defined as the sum-of-squared-returns (SSR) and is a proxy for integrated volatility (IV), which is unobservable. The study focuses on a subset of the universe of RV estimators. We examine three categories of estimators: historical, high-frequency (HF) and implied. The need to estimate RV is predominantly in the hedging of options and is not concerned with speculation or forecasting. The main research questions are; (1) what is the best RV estimator in a historical study of S&P 500 data? (2) What is the best RV estimator in a Monte Carlo simulation when delta hedging synthetic options? (3) Do our findings support the stylized fact of `Asymmetry in time scales' (Cont, 2001)? In the answering of these questions, further avenues of investigation are explored. Firstly, the VIX is used as the implied volatility. Secondly, the Monte Carlo simulation generates stock price paths with random components in the stock price and the volatility at each time point. The distribution of the input volatility is varied. The question of asymmetry in time scales is addressed by varying the term and frequency of historical data. The results of the historical and Monte Carlo simulation are compared. The SSR and two of the HF estimators perform best in both cases. Accuracy of estimators using long term data is shown to perform very poorly.

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 thesis focuses on modelling the distributions of loss in consumer credit arrangements, both at an individual level and at a portfolio level, and how these might be influenced by loan-specific factors and economic factors. The thesis primarily aims to examine how these factors can be incorporated into a credit risk model through logistic regression models and threshold regression models. Considering the fact that the specification of a credit risk model is influenced by its purpose, the thesis considers the IFRS 7 and IFRS 9 accounting requirements for impairment disclosure as well as Basel II regulatory prescriptions for capital requirements. The thesis presents a critique of the unexpected loss calculation under Basel II by considering the different ways in which loans can correlate within a portfolio. Two distributions of portfolio losses are derived. The Vašíček distribution, which is the assumed in Basel II requirements, was originally derived for corporate loans and was never adapted for application in consumer credit. This makes it difficult to interpret and validate the correlation parameters prescribed under Basel II. The thesis re-derives the Vašíček distribution under a threshold regression model that is specific to consumer credit risk, thus providing a way to estimate the model parameters from observed experience. The thesis also discusses how, if the probability of default is modelled through logistic regression, the portfolio loss distribution can be modelled as a log-log-normal distribution.