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Author: Donny Lesmana
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Languages : en
Pages :
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[Truncated abstract] This thesis develops the numerical methods and their mathematical analysis for solving nonlinear partial and integral-partial differential equations and inequalities arising from the valuation of European and American option with transaction costs. The models can hardly be solvable analytically. Therefore, in practice, approximate solutions to such a model are always sought. In this thesis, we discuss two models for the asset price movements: the geometric Brownian motion and jump diffusion process. For the valuation of European options with transaction costs when the underlying asset price follows a geometric Brownian motion, the classical Black-Scholes model becomes a nonlinear partial differential equation. To approximately solve this, we use an upwind finite difference scheme for the spatial discretization and a fully implicit time-stepping scheme. We prove that the system matrix from this scheme is an M-matrix and that the approximate solution converges unconditionally to the exact one by proving that the scheme is consistent, monotone and unconditionally stable. The discretized nonlinear system is then solved using a Newton iterative algorithm.
Author: Donny Lesmana
Publisher:
ISBN:
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Languages : en
Pages :
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[Truncated abstract] This thesis develops the numerical methods and their mathematical analysis for solving nonlinear partial and integral-partial differential equations and inequalities arising from the valuation of European and American option with transaction costs. The models can hardly be solvable analytically. Therefore, in practice, approximate solutions to such a model are always sought. In this thesis, we discuss two models for the asset price movements: the geometric Brownian motion and jump diffusion process. For the valuation of European options with transaction costs when the underlying asset price follows a geometric Brownian motion, the classical Black-Scholes model becomes a nonlinear partial differential equation. To approximately solve this, we use an upwind finite difference scheme for the spatial discretization and a fully implicit time-stepping scheme. We prove that the system matrix from this scheme is an M-matrix and that the approximate solution converges unconditionally to the exact one by proving that the scheme is consistent, monotone and unconditionally stable. The discretized nonlinear system is then solved using a Newton iterative algorithm.
Author: Matthias Ehrhardt
Publisher: Springer
ISBN: 3319612824
Category : Mathematics
Languages : en
Pages : 599
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Book Description
This book discusses the state-of-the-art and open problems in computational finance. It presents a collection of research outcomes and reviews of the work from the STRIKE project, an FP7 Marie Curie Initial Training Network (ITN) project in which academic partners trained early-stage researchers in close cooperation with a broader range of associated partners, including from the private sector. The aim of the project was to arrive at a deeper understanding of complex (mostly nonlinear) financial models and to develop effective and robust numerical schemes for solving linear and nonlinear problems arising from the mathematical theory of pricing financial derivatives and related financial products. This was accomplished by means of financial modelling, mathematical analysis and numerical simulations, optimal control techniques and validation of models. In recent years the computational complexity of mathematical models employed in financial mathematics has witnessed tremendous growth. Advanced numerical techniques are now essential to the majority of present-day applications in the financial industry. Special attention is devoted to a uniform methodology for both testing the latest achievements and simultaneously educating young PhD students. Most of the mathematical codes are linked into a novel computational finance toolbox, which is provided in MATLAB and PYTHON with an open access license. The book offers a valuable guide for researchers in computational finance and related areas, e.g. energy markets, with an interest in industrial mathematics.
Author: Yves Achdou
Publisher: SIAM
ISBN: 0898715733
Category : Technology & Engineering
Languages : en
Pages : 308
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This book allows you to understand fully the modern tools of numerical analysis in finance.
Author: Swathi Amarala
Publisher:
ISBN:
Category :
Languages : en
Pages : 154
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Book Description
Multigrid methods are numerical solvers for partial differential equations (PDEs) that systematically exploit the relationship between approximate solutions on multiple grids to arrive at a solution whose accuracy is consistent with the finest grid but for considerably less work. These methods converge in a small number of constant iterations independent of the grid size and hence, are often dramatically more efficient than others. In this thesis, we develop multigrid methods for three different classes of PDEs. In addition, we also develop discretization schemes for two model problems. First, we propose multigrid methods based on upwind interpolation and restriction techniques for computing the steady state solutions for systems of one and two-dimensional nonlinear hyperbolic conservation laws. We prove that the two-grid method is total variation diminishing and the multigrid methods are consistent and convergent for one-dimensional linear systems. Second, we propose a fully implicit, positive coefficient discretization that converges to the viscosity solution for a two-dimensional system of Hamilton-Jacobi-Bellman (HJB) PDEs resulting from dynamic Bertrand duopoly. Furthermore, we develop fast multigrid methods for solving the systems of discrete nonlinear HJB PDEs. The new multigrid methods are general and can be applied to other systems of HJB and HJB-Isaacs (HJBI) PDEs resulting from American options under regime switching and American options with unequal lending/borrowing rates and stock borrowing fees under regime switching, respectively. We provide a theoretical analysis for the smoother, restriction and interpolation operators of the multigrid methods. Finally, we develop a fully implicit, unconditionally monotone finite difference numerical scheme, that converges to the viscosity solution of the three-dimensional PDE to price European options under a two-factor stochastic volatility model. The presence of cross derivative terms in high dimensional PDEs makes the construction of monotone discretization schemes challenging. We develop a wide stencil discretization based on a local coordinate transformation to eliminate the cross derivative terms. But, wide stencil discretization is first order accurate and computationally expensive compared to the second order fixed stencil discretization. Therefore, we use a hybrid stencil in which fixed stencil is used as much as possible and a wide stencil when the fixed stencil discretization does not satisfy the positive coefficient condition. We also develop fast multigrid methods to solve the discrete linear system.
Author: S.D. Howison
Publisher: CRC Press
ISBN: 9780412630705
Category : Mathematics
Languages : en
Pages : 164
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Book Description
Mathematical Models in Finance compiles papers presented at the Royal Society of London discussion meeting. Topics range from the foundations of classical theory to sophisticated, up-to-date mathematical modeling and analysis. In the wake of the increased level of mathematical awareness in the financial research community, attention has focused on fundamental issues of market modelling that are not adequately allowed for in the standard analyses. Examples include market anomalies and nonlinear coupling effects, and demand new synthesis of mathematical and numerical techniques. This line of inquiry is further stimulated by ever tightening profits due to increased competition. Several papers in this volume offer pointers to future developments in this area.
Author: Fernando Manuel Rodrigues Ferreira Gonçalves
Publisher:
ISBN:
Category :
Languages : en
Pages : 114
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Author: Mitsuhiro T. Nakao
Publisher: Springer Nature
ISBN: 9811376697
Category : Mathematics
Languages : en
Pages : 469
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Book Description
In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information. The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense. In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.
Author: Gunter H Meyer
Publisher: World Scientific
ISBN: 9814619698
Category : Business & Economics
Languages : en
Pages : 286
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Book Description
Few financial mathematical books have discussed mathematically acceptable boundary conditions for the degenerate diffusion equations in finance. In The Time-Discrete Method of Lines for Options and Bonds, Gunter H Meyer examines PDE models for financial derivatives and shows where the Fichera theory requires the pricing equation at degenerate boundary points, and what modifications of it lead to acceptable tangential boundary conditions at non-degenerate points on computational boundaries when no financial data are available.Extensive numerical simulations are carried out with the method of lines to examine the influence of the finite computational domain and of the chosen boundary conditions on option and bond prices in one and two dimensions, reflecting multiple assets, stochastic volatility, jump diffusion and uncertain parameters. Special emphasis is given to early exercise boundaries, prices and their derivatives near expiration. Detailed graphs and tables are included which may serve as benchmark data for solutions found with competing numerical methods.
Author: Song Wang
Publisher: Springer
ISBN: 9789811595578
Category : Mathematics
Languages : en
Pages : 94
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Book Description
This book contains mostly the author’s up-to-date research results in the area. Option pricing has attracted much attention in the past decade from applied mathematicians, statisticians, practitioners and educators. Many partial differential equation-based theoretical models have been developed for valuing various options. These models do not have any practical use unless their solutions can be found. However, most of these models are far too complex to solve analytically and numerical approximations have to be sought in practice. The contents of the book consist of three parts: (i) basic theory of stochastic control and formulation of various option pricing models, (ii) design of finite volume, finite difference and penalty-based algorithms for solving the models and (iii) stability and convergence analysis of the algorithms. It also contains extensive numerical experiments demonstrating how these algorithms perform for practical problems. The theoretical and numerical results demonstrate these algorithms provide efficient, accurate and easy-to-implement numerical tools for financial engineers to price options. This book is appealing to researchers in financial engineering, optimal control and operations research. Financial engineers and practitioners will also find the book helpful in practice.
Author: Fernando Manuel Rodrigues Ferreira Gonçalves
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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