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This book provides a first, basic introduction into the valuation of financial options via the numerical solution of partial differential equations (PDEs). It provides readers with an easily accessible text explaining main concepts, models, methods and results that arise in this approach. In keeping with the series style, emphasis is placed on intuition as opposed to full rigor, and a relatively basic understanding of mathematics is sufficient. The book provides a wealth of examples, and ample numerical experiments are givento illustrate the theory. The main focus is on one-dimensional financial PDEs, notably the Black-Scholes equation. The book concludes with a detailed discussion of the important step towards two-dimensional PDEs in finance.

Finance. --- Financial engineering. --- Financial Engineering. --- Business mathematics. --- Differential equations, Partial. --- Partial differential equations --- Arithmetic, Commercial --- Business --- Business arithmetic --- Business math --- Commercial arithmetic --- Finance --- Mathematics --- Computational finance --- Engineering, Financial

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This monograph presents a novel numerical approach to solving partial integro-differential equations arising in asset pricing models with jumps, which greatly exceeds the efficiency of existing approaches. The method, based on pseudo-differential operators and several original contributions to the theory of finite-difference schemes, is new as applied to the Lévy processes in finance, and is herein presented for the first time in a single volume. The results within, developed in a series of research papers, are collected and arranged together with the necessary background material from Lévy processes, the modern theory of finite-difference schemes, the theory of M-matrices and EM-matrices, etc., thus forming a self-contained work that gives the reader a smooth introduction to the subject. For readers with no knowledge of finance, a short explanation of the main financial terms and notions used in the book is given in the glossary. The latter part of the book demonstrates the efficacy of the method by solving some typical problems encountered in computational finance, including structural default models with jumps, and local stochastic volatility models with stochastic interest rates and jumps. The author also adds extra complexity to the traditional statements of these problems by taking into account jumps in each stochastic component while all jumps are fully correlated, and shows how this setting can be efficiently addressed within the framework of the new method. Written for non-mathematicians, this book will appeal to financial engineers and analysts, econophysicists, and researchers in applied numerical analysis. It can also be used as an advance course on modern finite-difference methods or computational finance.

Mathematics. --- Partial differential equations. --- Economics, Mathematical. --- Computer mathematics. --- Mathematical models. --- Quantitative Finance. --- Mathematical Modeling and Industrial Mathematics. --- Computational Science and Engineering. --- Partial Differential Equations. --- Pseudodifferential operators. --- Finite differences. --- Differences, Finite --- Finite difference method --- Operators, Pseudodifferential --- Pseudo-differential operators --- Numerical analysis --- Operator theory --- Finance. --- Computer science. --- Differential equations, partial. --- Partial differential equations --- Informatics --- Science --- Funding --- Funds --- Economics --- Currency question --- Economics, Mathematical . --- Computer mathematics --- Electronic data processing --- Mathematics --- Models, Mathematical --- Simulation methods --- Mathematical economics --- Econometrics --- Methodology