Includes bibliographical references (leaves 29-31). / This paper investigates the European-style arithmetic Asian option pricing solution of Vyncke, Dhaene, and Goovaerts (2004) who apply the concept of comonotonicity to obtain upper and lower bounds for the true option price. A moment-matching formula is used to and a weighted average solution of the two bounds, thereby obtaining a fast approximation to the true price. This method is implemented and tested against an accurate Monte Carlo benchmark and compared with some other well-known closed-form approximations. Although a summary of some of the theoretical aspects underpinning the solution is provided to build intuitive understanding, the focus of the paper lies instead in the empirical analysis. The Vyncke et al. solution is found to be very accurate across a range of input parameters and out-performs competing solutions in some important cases, most notably high volatility and long maturities.

Industry and academia have thus far focussed on three classes of volatility models, namely, constant volatility, local volatility and stochastic volatility. Pathdependent volatility models are a lesser known class of models which possess the key characteristic of completeness together with the ability to generate a wide range of volatility dynamics with respect to the underlying asset (Guyon, 2014). This dissertation highlights the usefulness and practicality of these models for application in the South African market, while drawing comparisons with other widely used models. The tests cover both pricing and hedging of vanilla European options on the FTSE JSE Top 40. The Black-Scholes, Heston and CEV models are used as comparative benchmarks for each of the other classes of models.

It is often a goal of the risk management of a portfolio of interest rate sensitive instruments to minimize the impact of movements in market rates on the value of the portfolio. This can be done by considering the sensitivity of the portfolio to each of the market rates that are used to bootstrap a yield curve. However, this is likely to lead to an excessive amount of trading due to an investment in a large number of hedging securities. As an alternative, we consider using principal components analysis (PCA) to condense most of the variability in the market rates into a much smaller number of risk factors, called the principal components. One can then construct a hedging portfolio so as to make the portfolio immune to shocks in these principal components, and hence to the most common movements in the yield curve. We compare the effectiveness of these two hedging strategies for hedging a portfolio of interest-rate options, both in the absence and presence of transaction costs. We also consider the additional feature of being able to update each hedging methodology on a daily basis and rebalance the hedge portfolios accordingly.

One of the central debates in finance concerns the Efficient Market Hypothesis (EMH)—wherein markets are assumed to be efficient in the absolute sense. However, the possibility of time-varying weak-form market efficiency has received increasing attention in recent years. Under the Adaptive Market Hypothesis (AMH) it is postulated that market efficiency is dynamic, which advocates using models with non-constant coefficients. The concept of evolving efficiency has yielded a Test for Evolving Efficiency (TEE) and following that, a Generalised Test for Evolving Efficiency (GTEE) – both with an associated Kalman filtering (KF) technique. Unfortunately, these methods assume that the inherent stochastic processes are Gaussian despite widespread evidence that many real financial time series are nonGaussian. Unlike the classical KF, modern filters such as the maximum correntropy Kalman filters (MCC-KF) have been shown to be less sensitive to non-Gaussian uncertainties. These filters utilise a similarity measure known as correntropy– which incorporates higher order information than the mean square criterion that is utilised in the classical KF. As a result, they have been shown to improve filter robustness against outliers or impulsive noises. In this paper, the South African and American stock markets are tested for adaptive market efficiency using both the standard KF and the MCC-KF. A simulation study shows that the MCC-KF is a more robust estimator of adaptive efficiency but it less accurately estimates unknown system parameters. The South African stock market is found to be inefficient prior to August 2004 but achieves efficiency thereafter. Testing the S&P500 does not provide evidence of inefficiency in the American stock markets. The GTEE, implemented with the MCC-KF, is selected as the bestperforming test for the S&P500.

Includes bibliographical references. / A popular approach to formulating the initial-boundary value problem for fibre suspension flows is that in which fibre orientation is accounted for in an averaged sense, through the introduction of a second-order orientation tensor A. This variable, together with the velocity and pressure, then constitutes the set of unknown variables for the problem. The governing equations are balance of linear momentum, the incompressibility condition, an evolution equation for A, and a constitutive equation for the stress. The evolution equation contains a fourth-order orientation tensor A, and it is necessary to approximate A as a function of A, through a closure relation. The purpose of this these is to examine the well-posedness of the equations governing fibre fibre suspension flows, for various closure relations. It has previously been shown by GP Galdi and BD Reddy that, for the linear closure, the problem is wellposed provided that the particle number, a material constant, is less than a critical value. The work by Galdi and Reddy made of a model in which rotary diffusivity is a function of the flow. This thesis re-examines these issues in two different ways. First, the second law of thermodynamics is used to establish the constraints that the constitutive equations have to satisfy in order to be compatible with this law. This investigation is carried out for a variety of closure rules. The second contribution of the thesis concerns the existence and uniqueness of solutions to the governing equations, for the linear and quadratic closures; for a model in which the rotary diffusivity is treated as a constant, local and global existence of solutions are established, for sufficiently small data, and in the case of the linear closure, for admissible values of the particle number. The existence theory uses a Schauder fixed point approach.

No description available.

Volatility modelling is used predominantly in order to explain the volatility smile observed in the market. Stochastic volatility models are mainly used to capture the curvature of a volatility smile while local volatility models generally model the skew. Jackel and Kahl ¨ (2008) present a hyperbolic-local hyperbolic-stochastic volatility (Hyp-Hyp) model which aims to improve upon existing local and stochastic volatility models such as the stochastic alpha, beta, rho (SABR) and constant elasticity of variance (CEV) models. The advantageous features of the Hyp-Hyp model are corroborated by implementing the model. Jackel and Kahl ¨ (2008) investigate the accuracy of a scaled analytical approximation for implied volatility, based on approximations presented by Watanabe (1987) and Fouque et al. (2000), for the Hyp-Hyp model. They use the approximation to derive an expression for the delta of an option. This dissertation analyses the Hyp-Hyp model, as well as the approximation, by deriving expressions for other sensitivities and by investigating the effect of the Hyp-Hyp model parameters on the volatility smile. The accuracy of the analytical approximation for functional forms other than those defined by the Hyp-Hyp model is explored. A derivation of the approximation is undertaken, presenting corrections to the expressions introduced by Kahl (2007) and used by Jackel and Kahl ¨ (2008).

This study investigates extreme market events which occur in the tails of a distribution. The extreme events occur with a very low probability, but with significant consequences, which is what makes them of interest. In this study 20 years of data from both the S&P 500 and the JSE All Share index have been used. An extreme value approach has been taken to quantify the risks associated with extreme market events. To achieve this a two phased process is used to calculated the Value at Risk and Expected Shortfall. The first phase involved running the daily returns through the GARCH model, and then extracting the residuals. The second phase involves using the Block Maxima Method, or Peaks over Threshold method to fit the residuals to the Generalized Extreme Value Distribution or the Generalized Pareto Distribution. Finally, the impact of estimation frequency is considered for each of the models. In conclusion, taking an extreme value approach to provide a statistically sound method to calculate risk, even when the parameters of the model are updated less frequently, this is preferable to simpler models where the parameter estimates are updated daily.

No description available.

Since its development in 1997, the LIBOR market model has gained widespread use in interest rate modelling, largely owing to its consistency with the Black futures formula for pricing interest rate caps and floors. From its original construction(s), the LIBOR market model specifies a discrete set of forward rates that correspond to a fixed tenor structure, e.g. market tenors. This implies the pricing of interest rate contingent claims is restricted to claims with cashflow dates that coincide with the fixed tenor structure. In this light, several interpolation schemes have been suggested to handle the pricing restrictions, however at the cost of introducing possible arbitrage opportunities. The present dissertation studies four such interpolation schemes, paying particular attention to arbitrage-free interpolation schemes: Piterbarg deterministic interpolation, Schlogl deterministic interpolation, Schlogl stochastic interpolation, and Beveridge-Joshi stochastic interpolation.