Thesis (MSc)--Stellenbosch University, 2015 / 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. / AFRIKAANSE OPSOMMING : In onlangse jare, is die gebruik van Lévy-prosesse as ’n modellerings instrument baie meer gunstig gevind as die gebruik van die klassieke Brownse bewegingsproses opstel. Die rede hiervoor is dat hierdie prosesse meer buigsaamheid verskaf en die dinamiek van die model wat die praktyk beskryf, beter hierin vervat word. Dus is die gebruik van Lévy-prosesse vir finansiële modellering ’n motiverende faktor vir hierdie navorsingsaanbieding. As beginput word ’n raamwerk vir die LIBOR mark model met dinamika, gedryf deur ’n Lévy-proses in plaas van die klassieke Brownse bewegings opstel, aangebied. Wanneer LIBOR-koerse gemodelleer word is die gebruik van ’n meer realistiese proses belangriker aangesien hierdie koerse die mees realistiese koerse is wat in die finansiële mark op ’n daaglikse basis gebruik word. Sedert die finansiële krisis was daar ’n toenemende aanvraag en behoefte aan doeltreffende modellering en die bestaan van risiko binne die mark. Dit het verder gelei tot die motivering van Lévy-gebaseerde modelle vir die modellering van finansiële instrumente wat in die besonder aan kridietrisiko onderhewig is. Die motivering spruit uit die basiese eienskappe van stasionêre en onafhanklike inkremente van Lévy-prosesse. Met hierdie eienskappe is die model in staat om enige onverwagte gedrag (bekend as spronge) vas te vang. Deur hierdie faktore in ag te neem, is daar genoeg motivering vir die bou van ’n model gedryf deur Lévy-prosesse wat in staat is om kredietrisiko en instrumente onderhewig hieraan te modelleer. Die model vir LIBOR-koerse gedryf deur hierdie prosesse was oorspronklik bekendgestel deur Eberlein and Özkan (2005) en staan beken as die Lévy-LIBOR model. Om die kredietrisiko in die mark te akkommodeer word die Lévy-LIBOR model met "default risk" gekonstrueer. Dit was aanvanklik deur Kluge (2005) gedoen en formeel in die artikel bekendgestel deur Eberlein et al. (2006). Die doel van hierdie tesis is om die teoretiese konstruksie van die model aan te bied soos gedoen in die bogenoemde verwysings. Die konstruksie sluit ondermeer in die terugkrygingskoers wat met die wanbetaling geassosieer word, sowel as ’n prysingsformule vir ’n paar bekende krediet afgeleide instrumente.

Thesis (MComm (Business Management))--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: his study aims to investigate whether the phenomena found by Shnoll et al. when applying histogram pattern analysis techniques to stochastic processes from chemistry and physics are also present in financial time series, particularly exchange rate and index data. The phenomena are related to fine structure of non-smoothed frequency distributions drawn from statistically insufficient samples of changes and their patterns in time. Shnoll et al. use the notion of macroscopic fluctuations to explain the behaviour of sequences of histograms. Histogram patterns in time adhere to several laws that could not be detected when using time series analysis methods. In this study general approaches are reviewed that may be used to model financial markets and the volatility of price processes in particular. Special emphasis is placed on the modelling of highfrequency data sets and exchange rate data. Following previous studies of the Shnoll phenomena from other fields, different steps of the histogram sequence analysis are carried out to determine whether the findings of Shnoll et al. could also be applied to financial market data. The findings of this thesis widen the understanding of time varying volatility and can aid in financial risk measurement and management. Outcomes of the study include an investigation of time series characteristics in terms of the formation of discrete states, the detection of the near zone effect as proclaimed by Shnoll et al., the periodic recurrence of histogram shapes as well as the synchronous variation in data sets measured in the same time intervals.

Copulas functions are a flexible tool for modelling the dependence structure between variables. The joint and marginal distributions of Copulas are not constrained by the assumptions of normality. This study examines the dependence structure between the gold, platinum prices and the ZAR/U.S.D exchange rate using Copulas. The study found that marginal distributions of Copulas follows the ARMA (1, 1)-EGARCH (1, 1) and ARMA(1, 1)-APARCH (1, 1) models under different error terms including the normal, the student-t and the skew student-t error terms. It used the Normal, the Student-t, the Gumbel, the rotated Gumbel, the Clayton, the rotated Clayton, the Plackett, the Joe Clayton and the Normal time varying Copulas to analyse the dependence structure between returns prices of gold, platinum and ZAR/U.S.D exchange rate. The results showed evidence of a positive strong dependence between the returns prices of gold, platinum and returns on the Rand/U.S.D exchange rate for constant and time varying Copulas. The result also showed a co-movement of exchange rates and gold and platinum prices during a rise or declining prices of gold and platinum. The results imply that fluctuations in gold and platinum prices generate Rand/U.S.D exchange rate volatility. / Statistics / M. Sc. (Statistics)

Lo, Kai Lisa. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 43-46). / Abstract also in Chinese. / Chapter 1. --- Introduction --- p.7 / Chapter 2. --- Literature Review --- p.10 / Chapter 3. --- Measuring the Labor Share with US Data --- p.14 / Chapter 3.1 --- Definition and Measurement --- p.14 / Chapter 3.2 --- Some Crude Evidence --- p.16 / Chapter 4. --- A Theoretical Relationship between Labor Share and Inflation in an Open Economy --- p.19 / Chapter 4.1 --- A Static Closed-economy Pricing Model --- p.20 / Chapter 4.2 --- Dynamic Model Based on Quadratic Adjustment Costs --- p.22 / Chapter 4.3 --- An Open-economy Dynamic Pricing Model --- p.30 / Chapter 5. --- An Empirical Investigation --- p.34 / Chapter 5.1 --- Data --- p.34 / Chapter 5.2 --- Estimation Results --- p.36 / Chapter 5.2.1 --- General Findings --- p.37 / Chapter 5.2.2 --- The Role of Adjustment Costs --- p.39 / Chapter 5.2.3 --- Predicting U.S. Inflation --- p.40 / Chapter 6. --- Conclusions --- p.42 / References --- p.43 / Figures and Tables --- p.47 / Data Appendix --- p.56

Wu, Zhouheng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 44-46). / Abstract also in Chinese. / Chapter 1. --- Introduction --- p.1 / Chapter 2. --- Literature Review --- p.3 / Chapter 3. --- Overview of Key Factors that affect inflation in China --- p.6 / Chapter 3.1 --- Output Growth --- p.6 / Chapter 3.2 --- Money Supply --- p.7 / Chapter 3.3 --- Exchange Rate --- p.7 / Chapter 3.4 --- Wage --- p.8 / Chapter 3.5 --- Other Exogenous Shocks --- p.10 / Chapter 4. --- The Model --- p.11 / Chapter 4.1 --- Households --- p.12 / Chapter 4.2 --- Production Firms --- p.16 / Chapter 4.2.1 --- Non-Traded Sector --- p.16 / Chapter 4.2.2 --- Traded Sector --- p.19 / Chapter 4.3 --- Import Prices --- p.20 / Chapter 4.4 --- Monetary Policy Rules --- p.21 / Chapter 4.5 --- Domestic and External Shocks --- p.23 / Chapter 4.6 --- Market Clearing Conditions --- p.24 / Chapter 5. --- Calibration --- p.26 / Chapter 5.1 --- Calibration of parameter values --- p.26 / Chapter 5.2 --- Theoretical Impulse Responses and Variance Decomposition --- p.28 / Chapter 6. --- Model Fitness --- p.33 / Chapter 7. --- Conclusion Remarks --- p.35 / References --- p.38 / Appendix / Appendix A Equilibrium Conditions --- p.41 / Appendix B Steady State --- p.43 / Appendix C Simulation --- p.45 / Appendix D Data Description and Empirical Results --- p.56

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

The aim of this thesis is to estimate the volatility function of continuoustime stochastic models. The estimation of the volatility of the following wellknown international stock market indexes is presented as an application: Dow Jones Industrial Average, Standard and Poor’s 500, NIKKEI 225, CAC 40, DAX 30, FTSE 100 and IBEX 35. This estimation is studied from two different perspectives: a) assuming that the volatility of the stock market indexes displays shortrange dependence (SRD), and b) extending the previous model for processes with longrange dependence (LRD), intermediaterange dependence (IRD) or SRD. Under the efficient market hypothesis (EMH), the compatibility of the Vasicek, the CIR, the Anh and Gao, and the CKLS models with the stock market indexes is being tested. Nonparametric techniques are presented to test the affinity of these parametric volatility functions with the volatility observed from the data. Under the assumption of possible statistical patterns in the volatility process, a new estimation procedure based on the Whittle estimation is proposed. This procedure is theoretically and empirically proven. In addition, its application to the stock market indexes provides interesting results.