Department of Mathematical Sciences

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Browsing Department of Mathematical Sciences by browse.metadata.advisor "Becker, Ronald I."

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    Building Interest Rate Curves and SABR Model Calibration
    (Stellenbosch : Stellenbosch University, 2015-03) Mbongo Nkounga, Jeffrey Ted Johnattan; Becker, Ronald I.; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences.
    ENGLISH ABSTRACT : In this thesis, we first review the traditional pre-credit crunch approach that considers a single curve to consistently price all instruments. We review the theoretical pricing framework and introduce pricing formulas for plain vanilla interest rate derivatives. We then review the curve construction methodologies (bootstrapping and global methods) to build an interest rate curve using the instruments described previously as inputs. Second, we extend this work in the modern post-credit framework. Third, we review the calibration of the SABR model. Finally we present applications that use interest rate curves and SABR model: stripping implied volatilities, transforming the market observed smile (given quotes for standard tenors) to non-standard tenors (or inversely) and calibrating the market volatility smile coherently with the new market evidences.
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    An error correction neural network for stock market prediction
    (Stellenbosch : Stellenbosch University, 2019-04) Mvubu, Mhlasakululeka; Sanders, J. W.; Becker, Ronald I.; Bah, Bubacarr; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
    ENGLISH ABSTRACT : Predicting stock market has long been an intriguing topic for research in different fields. Numerous techniques have been conducted to forecast stock market movement. This study begins with a review of the theoretical background of neural networks. Subsequently an Error Correction Neural Network (ECNN), Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) are defined and implemented for an empirical study. This research offers evidence on the predictive accuracy and profitability performance of returns of the proposed forecasting models on futures contracts of Hong Kong’s Hang Seng futures, Japan’s NIKKEI 225 futures, and the United State of America S&P 500 and DJIA futures from 2010 to 2016. Technical as well as fundamental data are used as input to the network. Results show that the ECNN model outperforms other proposed models in both predictive accuracy and profitability performance. These results indicate that ECNN shows promise as a reliable deep learning method to predict stock price.
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    Forecasting stock returns: A comparison of five models
    (Stellenbosch : Stellenbosch University, 2018-12) Ramuada, Vhahangwele Cedrick; Sanders, J. W.; Becker, Ronald I.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
    ENGLISH ABSTRACT : Forecasting the movement of stock returns prices has been of interest to researches for many decades. Due to the complex and chaotic nature of the stock market, it has been difficult for researches to find a model which can be used to accurately predict the movement of stock returns prices. Many statistical models have been proposed for forecasting the direction of movement of stock returns prices. The objective of this study was to use ARMA type models and an Artificial Intelligence Neural Network model to predict the direction of movement of stock returns prices of four JSE listed companies, namely, Netcare Group Ltd, Santam Ltd, Sanlam Group Ltd, and Nedbank Group. The models were assessed in terms of their ability to predict whether the next day’s returns price will go down or up. Four ARMA-type models, namely, ARMA-Maximum Likelihood, ARMAState Space, ARMA-Metropolis Hastings, AR(3)-AVGARCH(1,1)-Student-t model and an Artificial Neural Network (ANN) model were implemented to try to predict the direction of movement of stock returns prices. Historical (past) stock returns prices were used to make inference about future directional movement of stock returns prices. Empirical results show that the ARMA-Maximum Likelihood, ARMA-State Space, AR(3)-AVGARCH(1,1)- Student-t model, and Artificial Neural Network (ANN) models have a strong ability to predict whether the next day’s returns price will go down or up with acceptable accuracy. However, the ARMA-Metropolis Hastings model performed very poorly, its highest accuracy was a mere 68%. Overall, empirical results show that the Artificial Neural Network model was superior or outperformed all the ARMA-type models, the highest accuracy achieved by the model was 89%. The results of the Superior Ability Test also showed that the ANN model was indeed superior to the Box-Jenkins ARMA type models in at least 5 cases.
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    The Levy-LIBOR model with default risk
    (Stellenbosch : Stellenbosch University, 2015-03) Walljee, Raabia; Becker, Ronald I.; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences.
    ENGLISH ABSTRACT : In recent years, the use of Lévy processes as a modelling tool has come to be viewed more favourably than the use of the classical Brownian motion setup. The reason for this is that these processes provide more flexibility and also capture more of the ’real world’ dynamics of the model. Hence the use of Lévy processes for financial modelling is a motivating factor behind this research presentation. As a starting point a framework for the LIBOR market model with dynamics driven by a Lévy process instead of the classical Brownian motion setup is presented. When modelling LIBOR rates the use of a more realistic driving process is important since these rates are the most realistic interest rates used in the market of financial trading on a daily basis. Since the financial crisis there has been an increasing demand and need for efficient modelling and management of risk within the market. This has further led to the motivation of the use of Lévy based models for the modelling of credit risky financial instruments. The motivation stems from the basic properties of stationary and independent increments of Lévy processes. With these properties, the model is able to better account for any unexpected behaviour within the market, usually referred to as "jumps". Taking both of these factors into account, there is much motivation for the construction of a model driven by Lévy processes which is able to model credit risk and credit risky instruments. The model for LIBOR rates driven by these processes was first introduced by Eberlein and Özkan (2005) and is known as the Lévy-LIBOR model. In order to account for the credit risk in the market, the Lévy-LIBOR model with default risk was constructed. This was initially done by Kluge (2005) and then formally introduced in the paper by Eberlein et al. (2006). This thesis aims to present the theoretical construction of the model as done in the above mentioned references. The construction includes the consideration of recovery rates associated to the default event as well as a pricing formula for some popular credit derivatives.