Hoi Wai-leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 62-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preminilaries --- p.8 / Chapter 2.1 --- Gauss and Jacobi Sums --- p.8 / Chapter 2.2 --- Basic Properties of Characters and Jacobi Sums --- p.10 / Chapter 2.3 --- Characters on Ok/Q --- p.11 / Chapter 2.4 --- Bernoulli Numbers and Bernoulli Polynomials --- p.15 / Chapter 2.5 --- Evaluation of Bp-1(k)-Bp-1 (mod p) --- p.16 / Chapter 2.5.1 --- "k = 2,3,4,6" --- p.16 / Chapter 2.5.2 --- "k = 5,8,10,12" --- p.17 / Chapter 3 --- The Main Result --- p.19 / Chapter 3.1 --- p-adic Gamma Functions and Gross-Koblitz Formula --- p.19 / Chapter 3.2 --- The Sum Σ - j(modp) --- p.22 / Chapter 3.3 --- The Main Theorem --- p.23 / Chapter 4 --- Evaluation of J(xk-r： Xk-s) (mod Q2) --- p.29 / Chapter 4.1 --- Basic Result --- p.29 / Chapter 4.2 --- Results for k = 8 --- p.30 / Chapter 4.3 --- Results for k = 12 --- p.32 / Chapter 4.4 --- "Results for k = 10,5" --- p.35 / Chapter 5 --- Evaluation of Binomial Coefficients --- p.41 / Chapter 5.1 --- Results for k = 8 --- p.41 / Chapter 5.2 --- Results for k = 12 --- p.44 / Chapter 5.3 --- Results for k = 10 --- p.49 / Chapter 5.4 --- Numerical Examples --- p.54 / Chapter 5.4.1 --- k = 8 --- p.54 / Chapter 5.4.2 --- k = 10 --- p.56 / Chapter 5.4.3 --- k = 12 --- p.58 / Bibliography --- p.61

I have developed a tiling interpretation of the q-binomial coefficients. The aim of this thesis is to apply this combinatorial interpretation to a variety of q-identities to provide straightforward combinatorial proofs. The range of identities I present include q-multinomial identities, alternating sum identities and congruences.

One of the most successful and most controversial innovative financial products in recent years has been collateralised debt obligations (CDOs). The dimensionality of dependency embedded in a typical CDO structure poses great challenges for researchers - in both generating realistic default dynamics and correlation, and in the mean time achieving fast and accurate model calibration. The research presented in this thesis contributes to the class of bottom-up models, which, as opposed to top-down models, start by modelling the individual obligor default process and then moving them up through the dependency structures to build up the loss distributions at the portfolio level. The Gaussian model (Li 2000) is a static copula model. It has only on correlation parameter, which can be calibrated to one CDO tranche at a time. Its simplicity achieves wide spread industry application even though it suffers from the problem of ’correlation smile’. In other words, it cannot fit the market in an arbitrage-free manner in the capital-structure dimension. The first contribution of this thesis is the sensitivities analysis with regard to model parameters of expected losses of CDO tranches in the Gaussian and NIG copula models. The study provided substantial insight into the essence of the dependency structure. In addition, we apply the intensity approach to credit modelling in order to imply market distributions non-parametrically in the form of a binomial lattice. Under the same framework, we developed a series of three models. The static binomial model can be calibrated to the CDS index tranches exactly, with one set of parameters. The model can be seen as a non-parametric copula model that is arbitrage free in the capital-structure dimension. Static models are not suitable to price portfolio credit derivatives that are dynamic in nature. The static model can be naturally developed into a dynamic binomial model and satisfies no-arbitrage conditions in the time dimension. This setup, however, reduces model flexibility and calibration speed. The computational complexity comes from the non-Markovian character of the default process in the dynamic model. Inspired by Mortensen (2006), in which the author defines the intensity integral as a conditioning variable, we modify the dynamic model into a Markovian model by modelling the intensity integral directly, which greatly reduces the computational time and increases model fit in calibration. We also show that, when stochastic recovery rates are involved, there is a third no-arbitrage condition for the expected loss process that needs to be built into the Markovian model. For all binomial models, we adopt a unique optimisation algorithm for model calibration - the Cross Entropy method. It is particularly advantageous in solving large-scale non-linear optimsation problems with multiple local extrema, as encountered in our model.

One of the most successful and most controversial innovative financial products in recent years has been collateralised debt obligations (CDOs). The dimensionality of dependency embedded in a typical CDO structure poses great challenges for researchers - in both generating realistic default dynamics and correlation, and in the mean time achieving fast and accurate model calibration. The research presented in this thesis contributes to the class of bottom-up models, which, as opposed to top-down models, start by modelling the individual obligor default process and then moving them up through the dependency structures to build up the loss distributions at the portfolio level. The Gaussian model (Li 2000) is a static copula model. It has only on correlation parameter, which can be calibrated to one CDO tranche at a time. Its simplicity achieves wide spread industry application even though it suffers from the problem of ’correlation smile’. In other words, it cannot fit the market in an arbitrage-free manner in the capital-structure dimension. The first contribution of this thesis is the sensitivities analysis with regard to model parameters of expected losses of CDO tranches in the Gaussian and NIG copula models. The study provided substantial insight into the essence of the dependency structure. In addition, we apply the intensity approach to credit modelling in order to imply market distributions non-parametrically in the form of a binomial lattice. Under the same framework, we developed a series of three models. The static binomial model can be calibrated to the CDS index tranches exactly, with one set of parameters. The model can be seen as a non-parametric copula model that is arbitrage free in the capital-structure dimension. Static models are not suitable to price portfolio credit derivatives that are dynamic in nature. The static model can be naturally developed into a dynamic binomial model and satisfies no-arbitrage conditions in the time dimension. This setup, however, reduces model flexibility and calibration speed. The computational complexity comes from the non-Markovian character of the default process in the dynamic model. Inspired by Mortensen (2006), in which the author defines the intensity integral as a conditioning variable, we modify the dynamic model into a Markovian model by modelling the intensity integral directly, which greatly reduces the computational time and increases model fit in calibration. We also show that, when stochastic recovery rates are involved, there is a third no-arbitrage condition for the expected loss process that needs to be built into the Markovian model. For all binomial models, we adopt a unique optimisation algorithm for model calibration - the Cross Entropy method. It is particularly advantageous in solving large-scale non-linear optimsation problems with multiple local extrema, as encountered in our model.

One of the most successful and most controversial innovative financial products in recent years has been collateralised debt obligations (CDOs). The dimensionality of dependency embedded in a typical CDO structure poses great challenges for researchers - in both generating realistic default dynamics and correlation, and in the mean time achieving fast and accurate model calibration. The research presented in this thesis contributes to the class of bottom-up models, which, as opposed to top-down models, start by modelling the individual obligor default process and then moving them up through the dependency structures to build up the loss distributions at the portfolio level. The Gaussian model (Li 2000) is a static copula model. It has only on correlation parameter, which can be calibrated to one CDO tranche at a time. Its simplicity achieves wide spread industry application even though it suffers from the problem of ’correlation smile’. In other words, it cannot fit the market in an arbitrage-free manner in the capital-structure dimension. The first contribution of this thesis is the sensitivities analysis with regard to model parameters of expected losses of CDO tranches in the Gaussian and NIG copula models. The study provided substantial insight into the essence of the dependency structure. In addition, we apply the intensity approach to credit modelling in order to imply market distributions non-parametrically in the form of a binomial lattice. Under the same framework, we developed a series of three models. The static binomial model can be calibrated to the CDS index tranches exactly, with one set of parameters. The model can be seen as a non-parametric copula model that is arbitrage free in the capital-structure dimension. Static models are not suitable to price portfolio credit derivatives that are dynamic in nature. The static model can be naturally developed into a dynamic binomial model and satisfies no-arbitrage conditions in the time dimension. This setup, however, reduces model flexibility and calibration speed. The computational complexity comes from the non-Markovian character of the default process in the dynamic model. Inspired by Mortensen (2006), in which the author defines the intensity integral as a conditioning variable, we modify the dynamic model into a Markovian model by modelling the intensity integral directly, which greatly reduces the computational time and increases model fit in calibration. We also show that, when stochastic recovery rates are involved, there is a third no-arbitrage condition for the expected loss process that needs to be built into the Markovian model. For all binomial models, we adopt a unique optimisation algorithm for model calibration - the Cross Entropy method. It is particularly advantageous in solving large-scale non-linear optimsation problems with multiple local extrema, as encountered in our model.

This paper defines generalized binomial coefficients and shows that they can be used to generate generalized Pascal's Triangles and have properties analogous to binomial coefficients. It uses the generalized binomial coefficients to compute the Dilworth number and the Sperner number of certain rings.

The Fibonomial numbers are defined by \[ \begin{bmatrix}n \\ k \end{bmatrix} = \frac{\prod_{i=n-k+1} ^{n} F_i}{\prod_{j=1}^{k} F_j} \] where $F_i$ is the $i$th Fibonacci number, defined by the recurrence $F_n=F_{n-1}+F_{n-2}$ with initial conditions $F_0=0,F_1=1$. In the past year, Sagan and Savage have derived a combinatorial interpretation for these Fibonomial numbers, an interpretation that relies upon tilings of a partition and its complement in a given grid.In this thesis, I investigate previously proven theorems for the Fibonomial numbers and attempt to reinterpret and reprove them in light of this new combinatorial description. I also present combinatorial proofs for some identities I did not find elsewhere in my research and begin the process of creating a general mapping between the two different Fibonomial interpretations. Finally, I provide a discussion of potential directions for future work in this area.

On introduit une nouvelle structure algébrique qui formalise bien les problèmes de reconstruction, assortie d’une conjecture qui permettrait de traiter directement des symétries. Le cadre fournit par cette étude permet de plus d’engendrer des relations qui ont lieu entre les nombres de sous-structures, et d’une certaine façon, la conjecture formulée affirme qu’on les obtient toutes. De plus, la généralisation des résultats précédemment obtenus pour la reconstruction permet de chercher `a en apprécier les limites en recherchant des cas où ces relations sont optimales. Ainsi, on montre que les théorèmes de V.Müller et de L.Lovasz sont les meilleurs possibles en exhibant des cas limites. Cette généralisation aux algèbres d’invariants, déjà effectuée par P.J.Cameron et V.B.Mnukhin, permet de placer les problèmes de reconstruction en tenaille entre d’une part des relations (fournies) que l’on veut exploiter, et des exemples qui établissent l’optimalité du résultat. Ainsi, sans aucune donnée sur le groupe, le résultat de L.Lovasz est le meilleur possible, et si l’on considère l’ordre du groupe, le résultat de V.Müller est le meilleur possible. / A new algebraic structure is described, that is a useful framework in whichreconstruction problems and results can be expressed. A conjecture is madewhich would, provided it is true, help to address the problem of symmetries.A consequence of the abstract language in which the theory is formulated isthe expression of relations between the numbers of substructures of a structure(for example, the number of subgraphs of a given type in a graph).Moreover, a generalisation similar to the one achieved by P.J.Cameron andV.B.Mnukhin of the results of edge reconstruction to invariant algebras isstated. Examples are then provided to show that the result of L.Lovasz isbest possible if one knows nothing about the underlying group, and that theresult of V.Müller is best possible if one knows only the order of the group.Thus, reconstruction problems are set in a theory that generates relationsto address them, and at the same time, provides examples establishing thesharpness of the theorems.