Financial Engineering with Finite Elements,9780471486909

Financial Engineering with Finite Elements

by
Edition: 1st
ISBN13: 9780471486909
ISBN10: 0471486906
Format: Hardcover
Pub. Date: 2005-04-01
Publisher(s): WILEY
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Summary

The pricing of derivative instruments has always been a highly complex and time-consuming activity. Advances in technology, however, have enabled much quicker and more accurate pricing through mathematical rather than analytical models. In this book, the author bridges the divide between finance and mathematics by applying this proven mathematical technique to the financial markets. Utilising practical examples, the author systematically describes the processes involved in a manner accessible to those without a deep understanding of mathematics. * Explains little understood techniques that will assist in the accurate more speedy pricing of options * Centres on the practical application of these useful techniques * Offers a detailed and comprehensive account of the methods involved and is the first to explore the application of these particular techniques to the financial markets

Author Biography

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.

Table of Contents

Preface xv
List of symbols
xvii
PART I PRELIMINARIES
1(54)
Introduction
3(4)
Some Prototype Models
7(10)
Optimal price policy of a monopolist
7(1)
The Black--Scholes option pricing model
8(2)
Pricing American options
10(2)
Multi-asset options with stochastic correlation
12(2)
The steady-state distribution of the Vasicek interest rate process
14(2)
Notes
16(1)
The Conventional Approach: Finite Differences
17(38)
General considerations for numerical computations
17(5)
Evaluation criteria
17(1)
Turning unbounded domains into bounded domains
18(4)
Ordinary initial value problems
22(24)
Basic concepts
22(1)
Euler's method
23(5)
Taylor methods
28(2)
Runge--Kutta methods
30(3)
The backward Euler method
33(2)
The Crank--Nicolson method
35(1)
Predictor--corrector methods
36(2)
Adaptive techniques
38(1)
Methods for systems of equations
39(7)
Ordinary two-point boundary value problems
46(3)
Introductory remarks
46(1)
Finite difference methods
46(3)
Shooting methods
49(1)
Initial boundary value problems
49(4)
The explicit scheme
49(2)
The implicit scheme
51(1)
The Crank--Nicolson method
52(1)
Integrating early exercise
52(1)
Notes
53(2)
PART II FINITE ELEMENTS
55(198)
Static ID Problems
57(52)
Basic features of finite element methods
57(1)
The method of weighted residuals -- one-element solutions
57(15)
The Ritz variational method
72(2)
The method of weighted residuals -- a more general view
74(1)
Multi-element solutions
75(24)
The Galerkin method with linear elements
76(13)
The Galerkin method with quadratic trial functions
89(4)
The collocation method with cubic Hermite trial functions
93(6)
Case studies
99(7)
The Evans model of a monopolist
99(1)
First exit time of a geometric Brownian motion
99(2)
The steady-state distribution of the Ornstein--Uhlenbeck process
101(1)
Convection-dominated problems
102(4)
Convergence
106(1)
Notes
107(2)
Dynamic 1D Problems
109(52)
Derivation of element equations
109(6)
The Galerkin method
109(5)
The collocation method
114(1)
Case studies
115(46)
Plain vanilla options
115(8)
Hedging parameters
123(9)
Various exotic options
132(10)
The CEV model
142(8)
Some practicalities: Dividends and settlement
150(11)
Static 2D Problems
161(34)
Introduction and overview
161(1)
Construction of a mesh
162(3)
The Galerkin method
165(22)
The Galerkin method with linear elements (triangles)
165(22)
The Galerkin method with linear elements (rectangular elements)
187(1)
Case studies
187(7)
Brownian motion leaving a disk
187(1)
Ritz revisited
188(3)
First exit time in a two-asset pricing problem
191(3)
Notes
194(1)
Dynamic 2D Problems
195(12)
Derivation of element equations
195(2)
Case studies
197(10)
Various rainbow options
197(6)
Modeling volatility as a risk factor
203(4)
Static 3D Problems
207(8)
Derivation of element equations: The collocation method
207(2)
Case studies
209(4)
First exit time of purely Brownian motion
209(2)
First exit time of geometric Brownian motion
211(2)
Notes
213(2)
Dynamic 3D Problems
215(6)
Derivation of element equations: The collocation method
215(1)
Case studies
216(5)
Pricing and hedging a basket option
216(2)
Basket options with barriers
218(3)
Nonlinear Problems
221(32)
Introduction
221(2)
Case studies
223(29)
Penalty methods
223(1)
American options
223(4)
Passport options
227(13)
Uncertain volatility: Best and worst cases
240(8)
Worst-case pricing of rainbow options
248(4)
Notes
252(1)
PART III OUTLOOK
253(4)
Future Directions of Research
255(2)
PART IV APPENDICES
257(86)
A Some Useful Results from Analysis
259(46)
A.1 Important theorems from calculus
259(1)
A.1.1 Various concepts of continuity
259(1)
A.1.2 Taylor's theorem
260(2)
A.1.3 Mean value theorems
262(1)
A.1.4 Various theorems
263(1)
A.2 Basic numerical tools
264(1)
A.2.1 Quadrature
264(4)
A.2.2 Solving nonlinear equations
268(2)
A.3 Differential equations
270(1)
A.3.1 Definition and classification
270(2)
A.3.2 Ordinary initial value problems
272(7)
A.3.3 Ordinary boundary value problems
279(6)
A.3.4 Partial differential equations of second order
285(2)
A.3.5 Parabolic problems
287(8)
A.3.6 Elliptic PDEs
295(1)
A.3.7 Hyperbolic PDEs
296(1)
A.3.8 Hyperbolic conservation laws
297(2)
A.4 Calculus of variations
299(6)
B Some Useful Results from Stochastics
305(24)
B.1 Some important distributions
305(1)
B.1.1 The univariate normal distribution
305(1)
B.1.2 The bivariate normal distribution
305(2)
B.1.3 The multivariate normal distribution
307(1)
B.1.4 The lognormal distribution
307(1)
B.1.5 The Γ distribution
307(2)
B.1.6 The central Χ2 distribution
309(1)
B.1.7 The noncentral Χ2 distribution
310(1)
B.2 Some important processes
310(1)
B.2.1 Basic concepts
310(2)
B.2.2 Wiener process
312(1)
B.2.3 Brownian motion with drift
313(1)
B.2.4 Geometric Brownian motion
313(1)
B.2.5 Ito process
314(1)
B.2.6 Ornstein--Uhlenbeck process
314(1)
B.2.7 A process for commodities
314(1)
B.3 Results
314(1)
B.3.1 The transition probability density Function
314(1)
B.3.2 The backward Kolmogorov equation
315(1)
B.3.3 The forward Kolmogorov equation
316(5)
B.3.4 Steady-state distributions
321(1)
B.3.5 First exit times
322(3)
B.3.6 Ito's lemma
325(1)
B.4 Notes
326(3)
C Some Useful Results from Linear Algebra
329(12)
C.1 Some basic facts
329(2)
C.2 Errors and norms
331(2)
C.3 Ill-conditioning
333(1)
C.4 Solving linear algebraic systems
333(6)
C.5 Notes
339(2)
D A Quick Introduction to PDE2D
341(2)
References 343(8)
Index 351

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