published_or_final_version / Mathematics / Master / Master of Philosophy

Counterparty credit risk (CCR) is the risk that a counterparty in a deal will not be able to meet their contractual obligations in the future. While CCR is an important task for any risk desk, it has often been underestimated due to the miss-conception that some counterparties were deemed to be either too big to fail or too big to be allowed to default. This was highlighted by the 2008 nancial crisis that saw respected banks, such as Lehman Brothers, and nancial service providers, such as AIG, default on their obligations. Since then there has been renewed interest in CCR, with the focus being on actively pricing and hedging it. In this work CCR is invistigated including its intersection with other forms of risk. CCR mitigation techniques are explored, followed by the formal quanti cation of CCR in the form of credit value adjustments (CVA). The analysis of CCR is then applied to interest rate derivatives, more speci cally forward rate agreements (FRAs) and interest rate swaps (IRSs). The e ect of correlation on unilateral and bilateral CVA between counterparties, including risk factors such as the interest rate, is investigated. This is invistigated under two credit risk modelling frameworks, the structural and intensity based frameworks. It is shown that correlation has a none-negligible e ect on both unilateral and bilateral CVA for FRAs and IRSs. Correlation structures, namely the Gaussian and the Student-t copula, are used to induce dependency in order to understand their e ect on both unilateral and bilateral CVA. It is shown that the choice of copula does not have signi cant e ect on either unilateral or bilateral CVA.

We present a tree algorithm, called the willow tree, for financial derivative pricing. The setup of the tree uses a fixed number of spatial nodes at each time step. The transition probabilities are determine by solving linear programming problems. The willow tree method is radically superior in numerical performance when compared to the binomial tree method.

We present a tree algorithm, called the willow tree, for financial derivative pricing. The setup of the tree uses a fixed number of spatial nodes at each time step. The transition probabilities are determine by solving linear programming problems. The willow tree method is radically superior in numerical performance when compared to the binomial tree method. / Science, Faculty of / Mathematics, Department of / Graduate

Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the multi-period binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option. We conclude with the pricing of exotic options with special emphasis on path dependent options. Monte Carlo simulation technique is applied as this method is very versatile in cases where there is no closed form analytical formula. The method is slow and time consuming but very flexible even for multi dimensional problems.

by Siu-ming Cha. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 123-132). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Objective --- p.1 / Chapter 1.2 --- Problem --- p.1 / Chapter 1.2.1 --- Motivation --- p.1 / Chapter 1.2.2 --- Approaches --- p.3 / Chapter 1.3 --- Contributions --- p.4 / Chapter 1.4 --- Organization of this Thesis --- p.5 / Chapter 2 --- Independent Component Analysis --- p.8 / Chapter 2.1 --- Overview --- p.8 / Chapter 2.2 --- The Blind Source Separation Problem --- p.8 / Chapter 2.3 --- Statistical Independence --- p.10 / Chapter 2.3.1 --- Definition --- p.10 / Chapter 2.3.2 --- Measuring Independence --- p.11 / Chapter 2.4 --- Developments of ICA Algorithms --- p.15 / Chapter 2.4.1 --- ICA Algorithm: Removal of Higher Order Dependence --- p.16 / Chapter 2.4.2 --- Assumptions in ICA Algorithms --- p.19 / Chapter 2.4.3 --- Joint Approximate Diagonalization of Eigenmatrices(JADE) --- p.20 / Chapter 2.4.4 --- Fast Fixed Point Algorithm for Independent Component Analysis(FastICA) --- p.21 / Chapter 2.5 --- Principal Component Analysis and Independent Component Anal- ysis --- p.23 / Chapter 2.5.1 --- Theoretical Comparisons between ICA and PCA --- p.23 / Chapter 2.5.2 --- Comparisons between ICA and PCA through a Simple Example --- p.24 / Chapter 2.6 --- Applications of ICA in Finance: A review --- p.27 / Chapter 2.6.1 --- Relationships between Cocktail-Party Problem and Fi- nance --- p.27 / Chapter 2.6.2 --- Security Structures Explorations --- p.28 / Chapter 2.6.3 --- Factors Interpretation and Visual Analysis --- p.29 / Chapter 2.6.4 --- Time Series Prediction by Factors --- p.29 / Chapter 2.7 --- Conclusions --- p.30 / Chapter 3 --- Factor Models in Finance --- p.31 / Chapter 3.1 --- Overview --- p.31 / Chapter 3.2 --- Factor Models and Return Generating Processes --- p.32 / Chapter 3.2.1 --- One-Factor Model --- p.33 / Chapter 3.2.2 --- Multiple-Factor Model --- p.34 / Chapter 3.3 --- Abstraction of Factor Models in Portfolio --- p.35 / Chapter 3.4 --- Typical Applications of Factor Models: Portfolio Mangement --- p.37 / Chapter 3.5 --- Different Approaches to Estimate Factor Model --- p.39 / Chapter 3.5.1 --- Time-Series Approach --- p.39 / Chapter 3.5.2 --- Cross-Section Approach --- p.40 / Chapter 3.5.3 --- Factor-Analytic Approach --- p.41 / Chapter 3.6 --- Conclusions --- p.42 / Chapter 4 --- ICA and Factor Models --- p.43 / Chapter 4.1 --- Overview --- p.43 / Chapter 4.2 --- Relationships between BSS and Factor Models --- p.43 / Chapter 4.2.1 --- Mathematical Deviation from Factor Models to Mixing Process --- p.45 / Chapter 4.3 --- Procedures of Factor Model Constructions by ICA --- p.47 / Chapter 4.4 --- Sorting Criteria for Factors --- p.48 / Chapter 4.4.1 --- Kurtosis --- p.50 / Chapter 4.4.2 --- Number of Runs --- p.52 / Chapter 4.5 --- Experiments and Results I: Factor Model Constructions --- p.53 / Chapter 4.5.1 --- Factors and their Sensitivities Extracted by ICA --- p.55 / Chapter 4.5.2 --- Factor Model Construction for a Stock --- p.60 / Chapter 4.6 --- Discussion --- p.62 / Chapter 4.6.1 --- Remarks on Applying ICA to Find Factors --- p.62 / Chapter 4.6.2 --- Independent Factors and Sparse Coding --- p.63 / Chapter 4.6.3 --- Selecting Securities for ICA --- p.63 / Chapter 4.6.4 --- Factors in Factor Models --- p.65 / Chapter 4.7 --- Conclusions --- p.66 / Chapter 5 --- Factor Model Evaluations and Selections --- p.67 / Chapter 5.1 --- Overview --- p.67 / Chapter 5.2 --- Random Residue: Requirement of Independent Factor Model --- p.68 / Chapter 5.2.1 --- Runs Test --- p.68 / Chapter 5.2.2 --- Interpretation of z-value --- p.70 / Chapter 5.3 --- Experiments and Results II: Factor Model Selections --- p.71 / Chapter 5.3.1 --- Randomness of Residues using Different Sorting Criteria --- p.71 / Chapter 5.3.2 --- Reverse Sortings of Kurtosis and Number of Runs --- p.76 / Chapter 5.4 --- Experiments and Results using FastICA --- p.80 / Chapter 5.5 --- Other Evaluation Criteria for Independent Factor Models --- p.85 / Chapter 5.5.1 --- Reconstruction Error --- p.86 / Chapter 5.5.2 --- Minimum Description Length --- p.89 / Chapter 5.6 --- Conclusions --- p.92 / Chapter 6 --- New Applications of Independent Factor Models --- p.93 / Chapter 6.1 --- Overview --- p.93 / Chapter 6.2 --- Applications to Financial Trading System --- p.93 / Chapter 6.2.1 --- Modifying Shocks in Stocks --- p.96 / Chapter 6.2.2 --- Modifying Sensitivity to Residue --- p.100 / Chapter 6.3 --- Maximization of Higher Moment Utility Function --- p.104 / Chapter 6.3.1 --- No Good Approximation to Utility Function --- p.107 / Chapter 6.3.2 --- Uncorrelated and Independent Factors in Utility Ma mizationxi- --- p.108 / Chapter 6.4 --- Conclusions --- p.110 / Chapter 7 --- Future Works --- p.111 / Chapter 8 --- Conclusion --- p.113 / Chapter A --- Stocks used in experiments --- p.116 / Chapter B --- Proof for independent factors outperform dependent factors in prediction --- p.117 / Chapter C --- Demixing Matrix and Mixing Matrix Found by JADE --- p.119 / Chapter D --- Moments and Cumulants --- p.120 / Chapter D.1 --- Moments --- p.120 / Chapter D.2 --- Cumulants --- p.121 / Chapter D.3 --- Cross-Cumulants --- p.121 / Bibliography --- p.123

No abstract available.

Wadley’s problem frequently emerges in dosage-mortality data and is one in which the number of surviving organisms is observed but the number initially treated is unknown. Data in this setting are also often overdispersed, that is the variability within the data exceeds that described by the distribution modelling it. The aim of this thesis is to explore distributions that can accommodate overdispersion in a Wadley’s problem setting. Two methods are essentially considered. The first considers adapting the beta-binomial and multiplicative binomial models that are frequently used for overdispersed binomial-type data to a Wadley’s problem setting. The second strategy entails modelling Wadley’s problem with a distribution that is suitable for modelling overdispersed count data. Some of the distributions introduced can be used for modelling overdispersed count data as well as overdispersed doseresponse data from a Wadley context. These models are compared using goodness of fit tests, deviance and Akaike’s Information Criterion and their properties are explored. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.

Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the multi-period binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option. We conclude with the pricing of exotic options with special emphasis on path dependent options. Monte Carlo simulation technique is applied as this method is very versatile in cases where there is no closed form analytical formula. The method is slow and time consuming but very flexible even for multi dimensional problems.

No abstract available.