一本是有限差分方法，另一本是有限元方法。欢迎业界的朋友发表对两种方法的看法以及它们的优缺点等的比较，哪种方法在业界更常用。谢谢！
1.
Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach
Daniel J. Duffy
ISBN: 978-0-470-85882-0
Hardcover
440 pages
March 2006
The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method. In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature:
Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options
Early exercise features and approximation using front-fixing, penalty and variational methods
Modelling stochastic volatility models using Splitting methods
Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work
Modelling jumps using Partial Integro Differential Equations (PIDE)
Free and moving boundary value problems in QF
Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs. 0 Goals of this Book and Global Overview. PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. 1 An Introduction to Ordinary Differential Equations. 2 An Introduction to Partial Differential Equations. 3 Second-Order Parabolic Differential Equations. 4 An Introduction to the Heat Equation in One Dimension. 5 An Introduction to the Method of Characteristics. PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS. 6 AnIntroduction to the Finite Difference Method. 7 An Introduction to the Method of Lines. 8 General Theory of the Finite Difference Method. 9 Finite Difference Schemes for First-Order Partial Differential Equations. 10 FDM for the One-Dimensional Convection–Diffusion Equation. 11 Exponentially Fitted Finite Difference Schemes. PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING. 12 Exact Solutions and Explicit Finite Difference Method for One-Factor Models. 13 An Introduction to the Trinomial Method. 14 Exponentially Fitted Difference Schemes for Barrier Options. 15 Advanced Issues in Barrier and Lookback Option Modelling. 16 The Meshless (Meshfree) Method in Financial Engineering. 17 Extending the Black–Scholes Model: Jump Processes. PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS. 18 Finite Difference Schemes for Multidimensional Problems. 19 An Introduction to Alternating Direction Implicit and Splitting Methods. 20 Advanced Operator Splitting Methods: Fractional Steps. 21 Modern Splitting Methods. PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING. 22 Options with Stochastic Volatility: The Heston Model. 23 Finite Difference Methods for Asian Options and Other ‘Mixed’ Problems. 24 Multi-Asset Options. 25 Finite Difference Methods for Fixed-Income Problems. PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS. 26 Background to Free and Moving Boundary Value Problems. 27 Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods. 28 Viscosity Solutions and Penalty Methods for American Option Problems. 29 Variational Formulation of American Option Problems. PART VII DESIGN AND IMPLEMENTATION IN C++. 30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem. 31 Design and Implementation of First-Order Problems. 32 Moving to Black–Scholes. 33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs. Appendices. A1 An introduction to integral and partial integro-differential equations. A2 An introduction to the finite element method. Bibliography. Index.

2.
Financial Engineering with Finite Elements
Juergen Topper
ISBN: 978-0-471-48690-9
Hardcover
378 pages
February 2005
Preface. List of Symbols. PART I: PRELIMINARIES. 1. Introduction. 2. Some Prototype Models. 2.1 Optimal Price Policy of a Monopolist. 2.2 The Black-Scholes Option Pricing Model. 2.3 Pricing American Options. 2.4 Multi-Asset Options with Stochastic Correlation. 2.5 The Steady-State Distribution of the Vasicek Interest Rate Process. 2.6 Notes. 3. The Conventional Approach: Finite Differences. 3.1 General Considerations for Numerical Computations. 3.2 Ordinary Initial-Value-Problems. 3.3 Ordinary Two-Point Boundary-Value-Problems. 3.4 Initial-Boundary-Value-Problems. 3.5 Notes. PART II: FINITE ELEMENTS. 4. Static 1D Problems. 4.1 Basic Features of Finite Element Methods. 4.2 The Method of Weighted Residuals - One Element Solutions. 4.3 The Ritz Variational Method. 4.4 The Method of Weighted Residuals - a More General View. 4.5 Multi-Element Solutions. 4.6 Case Studies. 4.7 Convergence. 4.8 Notes. 5. Dynamic 1D Problems. 5.1 Derivation of Element Equations. 5.2 Case Studies. 6. Static 2D Problems. 6.1 Introduction and Overview. 6.2 Construction of a Mesh . 6.3 The Galerkin Method. 6.4 Case Studies. 6.5 Notes. 7. Dynamic 2D Problems. 7.1 Derivation of Element Equations. 7.2 Case Studies. 8. Static 3D Problems. 8.1 Derivation of Element Equations: The Collocation Method. 8.2 Case Studies. 8.3 Notes. 9. Dynamic 3D Problems. 9.1 Derivation of Element Equations: The Collocation Method. 9.2 Case Studies. 10. Nonlinear Problems. 10.1 Introduction. 10.2 Case Studies. 10.3 Notes. PART III: OUTLOOK. 11. Future Directions of Research . PART IV: APPENDICES. A: Some Useful Results from Analysis. A.1 Important Theorems from Calculus. A.2 Basic Numerical Tools. A.3 Differential Equations. A.4 Calculus of Variations. B: Some Useful Results from Stochastics. B.1 Some Important Distributions. B.2 Some Important Processes. B.3 Results. B.4 Notes. C: Some Useful Results from Linear Algebra. C.1 Some Basic Facts. C.2 Errors and Norms. C.3 Ill-Conditioning. C.4 Solving Linear Algebraic Systems. C.5 Notes. D: A Quick Introduction to PDE2D. References. Index .

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