Financial Engineering with Finite Elements

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 .