Adjoint Differentiation's (AD) ability to calculate Greeks efficiently and to machine precision while scaling in constant time to the number of input variables is attractive for calibration and hedging where frequent calculations are required. Algorithmic adjoint differentiation tools automatically generates derivative code and provide interesting challenges in both Computer Science and Mathematics. In this dissertation we focus on a manual implementation with particular emphasis on parallel processing using Graphics Processing Units (GPUs) to accelerate run times. Adjoint differentiation is applied to a Call on Max rainbow option with 3 underlying assets in a Monte Carlo environment. Assets are driven by the Heston stochastic volatility model and implemented using the Milstein discretisation scheme with truncation. The price is calculated along with Deltas and Vegas for each asset, at a total of 6 sensitivities. The application achieves favourable levels of parallelism on all three dimensions implemented by the GPU: Instruction Level Parallelism (ILP), Thread level parallelism (TLP), and Single Instruction Multiple Data (SIMD). We estimate the forward pass of the Milstein discretisation contains an ILP of 3.57 which is between the average range of 2-4. Monte Carlo simulations are embarrassingly parallel and are capable of achieving a high level of concurrency. However, in this context a single kernel running at low occupancy can perform better with a combination of Shared memory, vectorized data structures and a high register count per thread. Run time on the Intel Xeon CPU with 501 760 paths and 360 time steps takes 48.801 seconds. The GT950 Maxwell GPU completed in 0.115 seconds, achieving an 422⇥ speedup and a throughput of 13 million paths per second. The K40 is capable of achieving better performance.

This dissertation explores the adaptive market hypothesis (AMH) first proposed by Lo (2004) which incorporates the efficient market hypothesis (EMH) of Malkiel and Fama (1970) and its behavioural exceptions. The AMH differs from the EMH, in that it assumes that the efficiency level of a market can fluctuate over time, whereas the EMH does not. The original test of evolving efficiency (TEE) was developed by Emerson et al. (1997) and Zalewska-Mitura and Hall (1999) and has an underlying GARCH-M model. Later, the generalised test of evolving efficiency (GTEE) was developed by Kulikova and Talyor (in progress), which has an underlying stochastic GARCH-M model proposed by Hall (1991). In this dissertation, the stochastic volatility test of evolving efficiency (SV-TEE) is developed using an underlying Stochastic Volatility-in-Mean (SVM) model introduced by Koopman and Uspensky (2002). The QMLE technique introduced by Harvey (1989) and the classical and Extended Kalman Filter techniques are described so that the TEE, the GTEE and the SV-TEE can be calibrated together with the hidden volatility process estimation. The empirical study tests the adaptive efficiency of four markets - two developed (London Stock Exchange and New York Stock Exchange), a mature developing (Johannesburg Stock Exchange) and an immature developing (Nairobi Stock Exchange). The best-performing tests were selected for each market and it was observed that there were constant and adaptive efficiencies in the developed and mature developing markets, and constant inefficiency in the immature developing market. The SV-TEE was not selected as the best-performing test for any of the markets - possibly because the time period considered for each market was too short.

The risk neutral density function is the distribution implied by the market price of derivative securities, namely options. It encloses the assumption that arbi-trage free conditions persist in the market. Given the historical evolution of stock prices, an investor will form some belief about the future progression of the stock price.

Includes bibliographical references. / In this dissertation we set to find the dual optimal policy of a dividend payout scheme for shareholders with a risk-averse utility function and the retention level of received premiums for an insurance company with the option of reinsurance. We set the problem as a stochastic control problem. We then solve the resulting second-order partial differential equation known as Hamilton-Jacobi-Bellman equation. We find out that the optimal retention level is linear with the current reserve up to a point whereupon it is optimal for the insurance company to retain all business. As for the optimal dividend payout scheme, we find out that it is optimal for the company not to declare dividends and we make further explorations of this result.

Stock-based loans are an increasingly popular form of loan that are collateralised using stocks. Since these loans are often non-recourse loans, the lenders are subject to the risk that the collateral is worth less than the loan, and the borrower defaults. This dissertation will consider the credit risk faced by lenders when issuing these loans. To achieve this, this dissertation will propose different models to quantify this risk using various credit measures. A sensitivity analysis to key model parameters is then conducted. Some brief comments about capital requirements will also be made.

Includes bibliographical references. / Corporate bonds are an attractive form of investment as they provide higher returns than government bonds. This increase in returns is usually associated with an increase in risk. These risks include liquidity, market and credit risk. This dissertation will focus on the modelling of a corporate bond's credit risk by considering how to estimate the credit default swap (CDS) spread of a firm's bond. A structural credit model will be used to do this. In this dissertation, we implement an extension of Merton's model by Hull, Nelken and White (2004), which is based on the use of the implied volatilities of options on the company's stock to estimate model parameters. Such an approach provides an insight into the relationship between credit markets and options markets.

Particle filtering in stochastic volatility/jump models has gained significant attention in the last decade, with many distinguished researchers adding their contributions to this new field. Golightly (2009), Carvalho et al. (2010), Johannes et al. (2009) and Aihara et al. (2008) all attempt to extend the work of Pitt and Shephard (1999) and Liu and Chen (1998) to adapt particle filtering to latent state and parameter estimation in stochastic volatility/jump models. This dissertation will review their extensions and compare their accuracy at filtering the Bates stochastic volatility model. Additionally, this dissertation will provide an overview of particle filtering and the various contributions over the last three decades. Finally, recommendations will be made as to how to improve the results of this paper and explore further research opportunities.

Includes bibliographical references / In this dissertation, a methodology is developed for constructing a volatility surface for a managed fund by extending the work of Bakshi et al. (2003) and Taylor (2014). The power utility assumption (with constant relative risk aversion for a specific maturity) and historical returns series data are used for the identified factors in influencing the return of the fund and the fund itself. The coefficient of relative risk aversion for a specific maturity and market is estimated from quoted option prices on a market index. This is used in combination with the identified factors and fund return series to estimate the risk-neutral skewness of the fund. An optimisation procedure is then used to determine the volatility smile of the fund for a specific maturity. Thereafter, the volatility surface of the fund is constructed by repeating each step for different maturities. Although this methodology produces sensible results, the optimisation routine used is sensitive to initial values and constraints.

The aim of this study was to calculate a classification function for discriminating between five grape cultivars with a view to determine the cultivar of an unknown grape juice. In order to discriminate between the five grape cultivars various multivariate statistical techniques, such as principal component analysis, cluster analysis, correspondence analysis and discriminant analysis were applied. Discriminant analysis resulted in the most appropriate technique for the problem at hand and therefore an in depth study of this technique was undertaken. Discriminant analysis was the most appropriate technique for classifying these grape samples into distinct cultivars because this technique utilized prior information of population membership. This thesis is divided into two main sections. The first section (chapters 1 to 5) is a review on discriminant analysis, describing various aspects of this technique and matters related thereto. In the second section (chapter 6) the theories discussed in the first section are applied to the problem at hand. The results obtained when discriminating between the different grape cultivars are given. Chapter 1 gives a general introduction to the subject of discriminant analysis, including certain basic derivations used in this study. Two approaches to discriminant analysis are discussed in Chapter 2, namely the parametrical and non-parametrical approaches. In this review the emphasis is placed on the classical approach to discriminant analysis. Non-parametrical approaches such as the K-nearest neighbour technique, the kernel method and ranking are briefly discussed. Chapter 3 deals with estimating the probability of misclassification. In Chapter 4 variable selection techniques are discussed. Chapter 5 briefly deals with sequential and logistical discrimination techniques. The estimation of missing values is also discussed in this chapter. A final summary and conclusion is given in Chapter 7. Appendices A to D illustrate some of the obtained results from the practical analyses.

Consider a market consisting of two correlated assets: one liquidly traded asset and one illiquid asset that can only be traded at time 0. For a European derivative written on the illiquid asset, we find a hedging strategy consisting of a constant (time 0) holding in the illiquid asset and dynamic trading strategies in the liquid asset and a riskless bank account that minimizes the expected square replication error at maturity. This mean-variance optimal strategy is first found when the liquidly traded asset is a local martingale under the real world probability measure through an application of the Kunita-Watanabe projection onto the space of attainable claims. The result is then extended to the case where the liquidly traded asset is a continuous square integrable semimartingale, and we again use the Kunita-Watanabe decomposition, now under the variance optimal martingale measure, to find the mean-variance optimal strategy in feedback form. In an example, we consider the case where the two assets are driven by correlated Brownian motions and the derivative is a call option on the illiquid asset. We use this example to compare the terminal hedging profit and loss of the optimal strategy to a corresponding strategy that does not use the static hedge in the illiquid asset and conclude that the use of the static hedge reduces the expected square replication error significantly (by up to 90% in some cases). We also give closed form expressions for the expected square replication error in terms of integrals of well-known special functions.