Displaying theses 110 of 184 total
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I.C. Bodin 
Master programme: Stochastics and Financial Mathematics  August 31st, 2017  
Institute: UvA / Other  Research group: Kortewegde Vries Institute for Mathematics  Graduation thesis  Supervisor: Peter Spreij 
Pricing and Hedging of Mortgage Option This thesis provides different pricing methods for four mortgage options, namely the pipelineoption, the meeneemoption,the LtVoption and rentemiddeling. A theoretical approach is present for each option and also the results of numerical experiments are presented. Common techniques are Monte Carlo, trinomial trees and Lévy theory. Real data is used to calibrate the models to guarantee the most relevant results. These turn out to be in line with the result of the currently used behaviour models in case of European options. American options are differently priced and the bank should be aware of the possible consequences. 

Scientific abstract (pdf 1K) Full text (pdf 6498K) 
M.J.M. Derksen 
Master programme: Stochastics and Financial Mathematics  August 29th, 2017  
Institute: KdVI  Research group: Stochastics and Financial Mathematics  Graduation thesis  Supervisor: Peter Spreij 
Pricing of Contingent Convertible Bonds In this thesis different pricing models are studied for the pricing of Contingent Convertible bonds (CoCos). These are special type of bonds, which convert into equity or are written down, when the capital of the issuing bank becomes too low. In this way, outstanding debt is reduced and capital is raised, to strengthen the capital position of the bank. In practice, this conversion is typically triggered by the capital ratio of the issuing bank falling below some threshold or by a regulator calling for conversion. However, in all of the existing pricing models the conversion of CoCos is triggered by a market value, like a stock price, falling below some threshold. In this thesis a model is proposed, in which the market cannot observe the true asset value process, but it only has access to noisy accounting reports, which are only published at discrete moments in time. In this way, the price of CoCos can only be based on the information from the accounting reports, not on the true asset process. 

Scientific abstract (pdf 1K) Full text (pdf 811K) 
I.S. Liesker 
Master programme: Stochastics and Financial Mathematics  August 22nd, 2017  
Institute: KdVI  Research group: Stochastics and Financial Mathematics  Graduation thesis  Supervisor: Peter Spreij 

Affine and quadratic interest rate models: A theoretical and empirical comparison In the financial world people try to speculate about the financial market. There are many variables that are unknown and that one wants to describe by, for example, stochastic models. These models help to get insight in the financial variables and are sometimes even used to predict the future development of the variable in order to do proper investments or protect themselves against risk. The latter, in form of interest rate risk modeling, is studied in this thesis. One of the popular interest rates models is the affine model. Affine models are becoming increasingly popular due to their analytical and computational tractability. Affine processes have a nice pricing formula for multiple financial products. Quadratic processes are, to some extent, an extension of affine models and have similar properties as affine models. This thesis compares these affine and quadratic models on a theoretical and an empirical level. For the theoretical level, the mathematics of affine and quadratic interest rate models is explained. For both affine and quadratic models analytical ('nice') formulas for some financial products are provided using admissible parameters and Riccati equations. Also, using the analytical bond prices, a small empirical comparison is performed where some computational examples are discussed. 

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R.Q. Riksen 
Master programme: Stochastics and Financial Mathematics  August 22nd, 2017  
Institute: KdVI  Research group: Stochastics and Financial Mathematics  Graduation thesis  Supervisor: Peter Spreij 

Using Artificial Neural Networks in the Calculation of Mortgage Prepayment Risk A client with a mortgage loan has the possibility to pay back part of his mortgage before the end of the contract. Because this poses a risk to the bank due to the loss of future interest payments, it is very important to predict the probability that a client will prepay on his mortgage. There are many parameters that can influence these mortgage prepayments in a complicated way. Artificial neural networks are used as approximators. A network consists of many connected nodes, that are grouped into layers. Each node takes a weighted sum of all the input it receives, applies a certain function to it and sends it on to all neurons in the next layer. The key to making a neural network approximate the target function, is to make it `learn' the correct weights. It gets to see a lot of input values and makes predictions. If the prediction was incorrect, all weights are changed a little in the direction that will make the network give a better prediction next time. This way, the network learns by making mistakes. In this thesis at ABN AMRO, we explore how we can use artificial neural networks to predict prepayment behaviour. 

Scientific abstract (pdf 1K) Full text (pdf 1396K) 
F.S. Labib 
Master programme: Mathematical Physics  July 11th, 2017  
Institute: KdVI  Research group: Mathematical Physics  Graduation thesis  Supervisor: Sergey Shadrin 
Moduli space of curves and tautological relations We start first with the construction of the moduli space of rational curves with marked points and quickly move on to curves of arbitrary genera with marked points. We then define the tautological ring and also the importantψ andκclasses. A set of additive generators is known for the tautological ring and the recent paper [PPZ16] by Pandharipande, Pixton and Zvonkine shows an algorithm to compute relations between these additive generators. We will look into this and compute several already known relations. Also attached to this thesis is a paper [KLLS17] written by R. Kramer, D. Lewanski, S. Shadrin and me where we exploit some polynomiality in the PPZrelations to give a new proof of Looijenga’s result [Loo95] (RH^{g−1}(M_{g,1})=Q and RH^{>g−1}(M_{g,1}) = 0) and also a more recent result by A. Buryak, S. Shadrin and D.Zvonkine [BSZ16] (RH^{g−1}(M_{g,n})=Q^n and RH^{>g−1}(M_{g,n}) = 0). Also we give abound for the dimension of the tautological ring in lower degree. 

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J.H. Westerdiep 
Master programme: Mathematics  June 27th, 2017  
Institute: KdVI  Research group: Dynamical Systems and Numerical Analysis  Graduation thesis  Supervisor: Rob Stevenson 

Twodimensional hpadaptive finite elements in theory and practice Partial differential equations (PDEs) describe many processes in nature, from the flow of water to the shape of a soap bubble. Often, it is hard (or even impossible) to find the function that solves such a PDE. In such cases, one looks for numerical solutions that approximate the true solution. In this thesis, we look at a finite element method: The domain of the function is partitioned into a large number of elements—in our twodimensional case, we will subdivide a polygon into triangular elements. Endowing each triangle with a fixed polynomial degree, our finite element method aids in finding an approximate solution to the PDE that is continuous globally, and a polynomial on each triangle locally. Given such an approximate solution, we often want to refine some of the triangles into smaller ones, so that we may construct a better solution on this refined grid. In this thesis, we analyse a novel algorithm for an even more complex case—hpadaptive finite elements—where we allow increasing the polynomial degree on each triangle separately. We will prove that, under mild circumstances, the size of the global error will decay exponentially in the total number of degrees of freedom. 

Scientific abstract (pdf 1K) For more info or full text, mail to: r.p.stevenson@uva.nl 
M. Sensi 
Master programme: Mathematics  June 21st, 2017  
Institute: UvA / Other  Research group: Kortewegde Vries Institute for Mathematics  Graduation thesis  Supervisor: Ale Jan Homburg 

Homoclinic vegetation stripes in a KlausmeierGrayScott model In this thesis, the background topic is the process of desertification: in particular, the formation of patterns which may hint that such a process is underway. The focus will be on homoclinic orbits. As C. A. Klausmeier noted, vegetation in semiarid regions tends to be extremely patterned: in particular, regular stripes tend naturally to form on hillsides. He proposed a system of two PDEs to describe the laws underlying the interaction of water precipitation U and vegetation V on a sloped terrain. We studied this model, focusing on existence and cardinality of nloops homoclinic orbits to an equilibrium which models a desert state using tools from geometric singular perturbation theory, namely Fenichel's theorems and Melnikov's method. We then started the construction of a connection from this model to the much more widely studied generalised KlausmeierGrayScott model, without completing it, but suggesting a path to follow. 

Scientific abstract (pdf 0K) For more info or full text, mail to: a.j.homburg@uva.nl 
J.M.H. Box 
Master programme: Mathematics  June 14th, 2017  
Institute: KdVI  Research group: Algebraic Geometry  Graduation thesis  Supervisor: Sander Dahmen 

Height Bounds for Mordell Equations Using Modularity Suppose that you have 8 cubic blocks and you build the 2x2x2 cube. Then you add a 9th block and you rearrange the blocks into a 3x3 square. Can this construction be repeated? In other words, can we build a larger cube, such that, after adding one extra block, the blocks can be reagganged into a square? In terms of symbols: which integers x and y satisfy the equation y^2=x^3+1? In this thesis, we study integral solutions of the more general equation y^2=x^3+a, where a is a fixed nonzero integer, called Mordell's equation. We show that each such solution (x,y) must have a bounded xcoordinate: x leq (1728a)^(1310a). In particular, this means that each Mordell equation admits finitely many solutions and that there is, in theory, an algorithm to determine all of these by verifying all numbers below the upper bound. Proving the above upper bound requires advanced mathematical knowledge, which is studied as well. 

Scientific abstract (pdf 140K) Full text (pdf 818K) 
T.P. Veltman 
Master programme: Mathematical Physics  April 11th, 2017  
Institute: KdVI  Research group: Mathematical Physics  Graduation thesis  Supervisor: Jasper Stokman 
Heisenberg Spin Chains with Boundaries and Quantum Groups A mathematical description of the onedimensional Heisenberg spin chain: a finite set of particles in a row, each with a certain spin value. This study takes into account both interactions between neighbouring particles and interactions with the two boundaries on the sides. 

Scientific abstract (pdf 133K) Full text (pdf 464K) 
C. Pizzigoni 
Master programme: Mathematics  March 9th, 2017  
Institute: KdVI  Research group: Dynamical Systems and Numerical Analysis  Graduation thesis  Supervisor: Rob Stevenson 

Numerical Methods for Elliptic Partial Differential Equations with Random Coefficients This thesis analyses the stochastic collocation method for approximating the solution of elliptic partial differential equations with random coefficients. The method consists of a finite element approximation in the spatial domain and a collocation at the zeros of suitable tensor product orthogonal polynomials in the probability space and naturally leads to the solution of uncoupled deterministic problems. The computational cost of the method depends on the choice of the collocation points and thus we compare few possible constructions. Although the random fields describing the coefficients of the problem are in general infinitedimensional, an approximation with certain optimality properties is obtained by truncating the KarhunenLoéve expansion of these random fields. We estimate the convergence rates of the method, depending on the regularity of the random coefficients. In particular we prove exponential convergence in probability space. Numerical examples illustrate the theoretical results. 

Scientific abstract (pdf 41K) Full text (pdf 1158K) 
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