It is widely believed among South African agricultural market participants that the United States' corn price, as represented by the Chicago Board of Trade-listed corn contract, is causal to the price of white and yellow maize traded on the South African Futures Exchange. Although a strong correlation exists between these markets, the corn contract is far from causal to the South African maize price, as indicated by Auret and Schmitt (2008). Similarly, South African market participants believe that volatility generated in the United States corn market spills over to the South African market. Given the perceived volatility spill-over from the corn market to the maize market, market participants might inadvertently include a higher volatility component in an option price in the South African maize market than is necessary. This study sought to quantify the amount of volatility spill-over to the South African white and yellow maize market from the United States corn contract. This task was accomplished by applying an Exponential Generalised Auto Regressive Conditional Heteroscedasticity model, within an aggregate shock framework, to the data. The findings indicated that the volatility spill-over from the United States corn market to the South African maize market is not statistically significant. This result suggests that volatility in the South African market is locally driven; hence, it should not be necessary for a South African listed option contract to carry an international volatility component in its price. It was also found that the returns data of the South African maize market is asymmetrically skewed, indicating that bad news will have a greater effect on the price of maize compared with good news. / Thesis (MCom (Risk Management))--North-West University, Potchefstroom Campus, 2013.

The modelling of volatility has long been seminal to finance and risk management in general, as it provides information on the spread of portfolio returns. In order to reduce the overall volatility of a stock portfolio, modern portfolio theory (MPT), within an efficient market hypothesis (EMH) framework, dictates that a well-diversified portfolio should have a market beta of one (thereafter adjusted for risk preference), and thus move in sync with a benchmark market portfolio. Such a stock portfolio is highly correlated with the market, and considered to be entirely hedged against unsystematic risk. However, the risks within and between stocks present in a portfolio still impact on each other. In particular, risk present in a particular stock may spill over and affect the risk profile of another stock included within a portfolio - a phenomenon known as volatility spill-over effects. In developing economies such as South Africa, portfolio managers are limited in their choices of stocks. This increases the difficulty of fully diversifying a stock portfolio given the volatility spill-over effects that may be present between stocks listed on the same exchange. In addition, stock portfolios are not static, and therefore require constant rebalancing according to the mandate of the managing fund. The process of constant rebalancing of a stock portfolio (for instance, to follow the market) becomes more complex and difficult during times of financial distress. Considering all these conditions, portfolio managers need all the relevant information (more than MPT would provide) available to them in order to select and rebalance a portfolio of stocks that are as mean-variance efficient as possible. This study provides an additional measure to market beta in order to construct a more efficient portfolio. The additional measure analyse the volatility spill-over effects between stocks within the same portfolio. Using intraday stock returns and a residual based test (aggregate shock [AS] model), volatility spill-over effects are estimated between stocks. It is shown that when a particular stock attracts fewer spill-over effects from the other stocks in the portfolio, the overall portfolio volatility would decrease as well. In most cases market beta showcased similar results; this change is however not linear in the case of market beta. Therefore, in order to construct a more efficient portfolio, one requires both a portfolio that has a unit correlation with the market, but also includes stocks with the least amount of volatility spill-over effects among each other. / MCom (Risk Management), North-West University, Potchefstroom Campus, 2013

It is widely believed among South African agricultural market participants that the United States' corn price, as represented by the Chicago Board of Trade-listed corn contract, is causal to the price of white and yellow maize traded on the South African Futures Exchange. Although a strong correlation exists between these markets, the corn contract is far from causal to the South African maize price, as indicated by Auret and Schmitt (2008). Similarly, South African market participants believe that volatility generated in the United States corn market spills over to the South African market. Given the perceived volatility spill-over from the corn market to the maize market, market participants might inadvertently include a higher volatility component in an option price in the South African maize market than is necessary. This study sought to quantify the amount of volatility spill-over to the South African white and yellow maize market from the United States corn contract. This task was accomplished by applying an Exponential Generalised Auto Regressive Conditional Heteroscedasticity model, within an aggregate shock framework, to the data. The findings indicated that the volatility spill-over from the United States corn market to the South African maize market is not statistically significant. This result suggests that volatility in the South African market is locally driven; hence, it should not be necessary for a South African listed option contract to carry an international volatility component in its price. It was also found that the returns data of the South African maize market is asymmetrically skewed, indicating that bad news will have a greater effect on the price of maize compared with good news. / Thesis (MCom (Risk Management))--North-West University, Potchefstroom Campus, 2013.

The modelling of volatility has long been seminal to finance and risk management in general, as it provides information on the spread of portfolio returns. In order to reduce the overall volatility of a stock portfolio, modern portfolio theory (MPT), within an efficient market hypothesis (EMH) framework, dictates that a well-diversified portfolio should have a market beta of one (thereafter adjusted for risk preference), and thus move in sync with a benchmark market portfolio. Such a stock portfolio is highly correlated with the market, and considered to be entirely hedged against unsystematic risk. However, the risks within and between stocks present in a portfolio still impact on each other. In particular, risk present in a particular stock may spill over and affect the risk profile of another stock included within a portfolio - a phenomenon known as volatility spill-over effects. In developing economies such as South Africa, portfolio managers are limited in their choices of stocks. This increases the difficulty of fully diversifying a stock portfolio given the volatility spill-over effects that may be present between stocks listed on the same exchange. In addition, stock portfolios are not static, and therefore require constant rebalancing according to the mandate of the managing fund. The process of constant rebalancing of a stock portfolio (for instance, to follow the market) becomes more complex and difficult during times of financial distress. Considering all these conditions, portfolio managers need all the relevant information (more than MPT would provide) available to them in order to select and rebalance a portfolio of stocks that are as mean-variance efficient as possible. This study provides an additional measure to market beta in order to construct a more efficient portfolio. The additional measure analyse the volatility spill-over effects between stocks within the same portfolio. Using intraday stock returns and a residual based test (aggregate shock [AS] model), volatility spill-over effects are estimated between stocks. It is shown that when a particular stock attracts fewer spill-over effects from the other stocks in the portfolio, the overall portfolio volatility would decrease as well. In most cases market beta showcased similar results; this change is however not linear in the case of market beta. Therefore, in order to construct a more efficient portfolio, one requires both a portfolio that has a unit correlation with the market, but also includes stocks with the least amount of volatility spill-over effects among each other. / MCom (Risk Management), North-West University, Potchefstroom Campus, 2013

The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013