Financial Engineering with Finite Elements


Topper brings together a powerful set of tools for quantitative modeling in finance.  When confronted with challenging numerical PDE problems, I have more than once been told that the recommended recipes are well known by PDE specialists, although not found in books.  Topper's book is now an exception!  Darrel Duffie, Graduate School of Business, Stanford University, USA

"Financial Engineering with Finite Elements is packed with state of the art valuation methods.  Written in a clear and intuitive way makes it a must have for anyone wanting to stay ahead of the game.  The book stands out from the crowd with a lot of information I have never seen published in any other finance book.Espen Gaarder Haug, Trader, J.P. Morgan, New York

"Finite elements have for a long time been a preferred numerical scheme for the solution of differential equations arising in the hard sciences.  Now Jürgen Topper shows us how to apply the technique effectively to solve equations from the financial arena including the pricing of derivatives.  He includes sections on multi-asset options, non-linear equations, exit times and exotics.  A very welcome addition to the literature on numerical analysis in finance."  Paul Wilmott, fund mananger and mathematician, London

"Jürgen Topper fills an important gap in the burgeoning literature on mathematical finance by providing a systematic and accessible description of the Method of Finite Elements and its applications.  Since this important computational technique has a lot of potential in financem the book is very timely.Alexander Lipton, Citadel Investment Group, L.L.C., Chicago

JÜRGEN TOPPER is a Manager of d-fine GmbH, Frankfurt, a company that specialises in financial and commodity risk consulting. Prior to this, he worked for the financial risk management consulting division of Arthur Andersen since 1997.

List of Symbols.


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.


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.


11. Future Directions of Research .


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.


Index .