Fourier and Laplace Transform Inversion with Applications in Finance

294 Pages·2014·1.88 MB·English

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FOURIER AND LAPLACE TRANSFORM INVERSION WITH APPLICATIONS IN FINANCE 322 P.W. DEN ISEGER
It is becoming standard for modeling and analysis to include algorithms for computing
probability distributions of interest. Therefore, advances in computational probability increase
the value of stochastic models in among others quantitative finance, queueing, reliability, and Fourier and Laplace
inventory management. Several tools have been developed for this purpose, of which one of
the most powerful tools in numerical Fourier transform inversion. The reason is that proba-
bility distributions can be characterized in terms of Laplace transforms, or more generaly, Transform Inversion with
Fourier transforms. Many results in among others quantitative finance, queueing and relia-
bility theory, are given in the form of transforms and become amenable for practical Applications in Finance
computations once fast and accurate methods for numerical Laplace and Fourier transform
inversion are available.
This thesis presents advanced numerical inversion techniques of Laplace and Fourier
transforms and their innovative application in quantitative finance. Although our applications
are mainly focused on quantitative finance, the presented inversion techniques also tackles
important problems from logistics, queuing, probability theory.
The Erasmus Research Institute of Management (ERIM) is the Research School (Onder-
zoeks chool) in the field of management of the Erasmus University Rotterdam. The founding
participants of ERIM are the Rotterdam School of Management (RSM), and the Erasmus
School of Econom ics (ESE). ERIM was founded in 1999 and is officially accred ited by the
Royal Netherlands Academy of Arts and Sciences (KNAW). The research undert aken by
ERIM is focused on the management of the firm in its environment, its intra- and interfirm
relations, and its busin ess processes in their interdependent connections.
The objective of ERIM is to carry out first rate research in managem ent, and to offer an
adv anced doctoral prog ramme in Research in Management. Within ERIM, over three
hundred senior researchers and PhD candidates are active in the different research pro-
grammes. From a variety of academ ic backgrounds and expertises, the ERIM commun ity is
united in striving for excellence and working at the foref ront of creating new business
knowledge.
ERIM PhD Series
Research in Management
Erasmus Research Institute of Management - Tel. [phone]
Rotterdam School of Management (RSM) Fax [phone]
Erasmus School of Economics (ESE) E-mail
Fourier and Laplace Transform
Inversion with Application in Finance
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Fourier and Laplace Transform Inversion with
Application in Finance
Het inverteren van Fourier en Laplace getransformeerden met toepassingen in de
financiering
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Erasmus Universiteit Rotterdam
op gezag van de rector magnificus
Prof.dr. H.A.P. Pols
en volgens het besluit van het College voor Promoties.
De openbare verdediging zal plaatsvinden op
donderdag 9 oktober 2014 om 13.30 uur
door
PETER DEN ISEGER
geboren te Delft
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Promotiecommissie
Promotor: Prof.dr.ir. R. Dekker
Overige leden: Prof.dr. D.J.C. van Dijk
Prof.dr. M.R.H. Mandjes
Prof.dr. R. Paap
Erasmus Research Institute of Management - ERIM
Rotterdam School of Management (RSM)
Erasmus School of Economics (ESE)
Erasmus University Rotterdam
Internet: http://www.erim.eur.nl
ERIM Electronic Series Portal: http://hdl.handle.net/1765/1
ERIM PhD Series in Research in Management, 322
Reference number ERIM: EPS-2014-322-LIS
ISBN 978-90-5892-375-2
© 2014, Peter den Iseger
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All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any
means electronic or mechanical, including photocopying, recording, or by any information storage and
retrieval system, without permission in writing from the author.
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Preface
First of all, I would like to express my gratitude to Rommert Dekker, for being my pro-
motor. I also would like to express my gratitude to Henk Tijms for his confidence in
my research. This was always an important motivation to me.
I would like to thank Emoke Oldenkamp for our collaboration during the years. We
start many years ago with our research in stochastic operation research and in inven-
tory management. Together we make the switch to quantitative finance, and had a
fruitful cooperation at Cardano Risk Management. You co-authored many papers that
also results in a number of chapters in this thesis.
I would like to thank Michel Mandjes and Paul Gruntjes for our fruitful cooperation
when we applied the techniques in this thesis to the numerical computation of Wiener-
Hopf factorizations. I further would like to thank Sander de Groot and Hans van der
Weide for our nice collaboration that results in chapter 7 of this thesis. I also would
like to express my gratitude to the other members of the committee.
I would like to thank Szabols Gaa´l for our nice collaboration that results in chapter
ten of this thesis. Beside this work I also enjoyed working together at Cardano Risk
Management.
Further, I would like express my gratitude to my brothers Willem and Taco den Iseger
and to my mother Janny den Iseger for their support during the years.
Peter den Iseger
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Contents
List of Figures xi
List of Tables xii
1 Introduction 1
2 Fourier transforms inversion and the Poisson summation formula 9
2.1 Fourier transforms and their properties 9
2.2 The Poisson Summation Formula 14
2.3 The Euler Algorithm of Abate en Whitt 15
2.4 Inverting z-Transforms 18
3 Numerical Transform Inversion Using Gaussian Quadrature 21
3.1 Introduction 21
3.2 Preliminaries 23
3.3 Outline of the method 24
3.4 A simple Laplace transform inversion algorithm 27
3.5 Modifications for non-smooth functions 31
3.6 The Fourier Transform 42
3.7 Proof of theorem 3.3.3 44
3.8 The computation of the {μυk} and {αυk} 45
3.9 Error analysis 47
3.10 On the inversion of z-transforms 54
3.11 Quadrature nodes and weights 56
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4 Laplace Transform Inversion on the entire line 59
4.1 Introduction 59
4.2 Preliminaries: tools for the algorithm 60
4.3 The backbone of the method 64
4.4 Inner product preserving properties of the method 68
4.5 Implementation issues of the method 71
4.6 Error analysis 78
4.7 Modifications for non-smooth functions 79
4.8 Numerical test results 80
4.9 Computational issues 82
4.10 Proof of theorem 4.4.3 86
4.11 Detailed proof of the error theorem 4.6.1 88
5 Pricing guaranteed return rate products and discretely sampledAsian options 93
5.1 Introduction 93
5.2 Discretely sampled Asian options and guaranteed rate products: Pricing
concept 95
5.3 The recursion algorithm for determining G 96
5.4 Greeks 100
5.5 A detailed description of Algorithm 5.3.1 112
5.6 Computing the put option price 120
6 Computing Greeks 129
6.1 Introduction 129
6.2 European style plain – vanilla options 130
6.3 Path dependent options 134
6.4 Computational results 139
7 The Fast Convolution Transform 147
7.1 Introduction 147
7.2 The ladder framework 149
7.3 The fast convolution algorithm 153
7.4 Implementational issues of the ladder framework 158
7.5 Applications 164
7.6 Conclusion 170
7.7 Technical note: The probability density fX
n+1(x) 171
7.8 Windowed Laplace transform 171
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8 Cliquet options 177
8.1 Introduction 177
8.2 Pricing in the constant and stochastic volatility models 178
8.3 Greeks in lognormal and jump diffusion models 184
8.4 Computational details 194
8.5 Preliminary: Plancharel’s identity and derived approximations 194
9 High dimensional transformation algorithms 199
9.1 Introduction 199
9.2 High dimensional transforms in practice:
A (geometric) basket option example 200
9.3 The basis: a brief review of the one–dimensional Laplace and inverse
Laplace transformation algorithms 202
9.4 Multi–dimensional transformation algorithms 205
9.5 Matrix representations with Kronecker products 209
9.6 Computational issues 212
10 A fast Laplace inversion method 221
10.1 Introduction 221
10.2 Barrier options in the Black-Scholes setting 223
10.3 The Helmholtz equation 226
10.4 Scaling 230
10.5 Implementation issues 236
10.6 The numerical algorithm 239
10.7 Numerical results 241
10.8 Conclusion 242
10.9 Computational issues 243
11 High dimensional Fourier transform inversion with lattice quadrature rules 245
11.1 Introduction 245
11.2 Extending the quadrature rule to higher dimensions 245
11.3 Lattice rules 247
11.4 Reproducing kernels 248
11.5 The inverse Fourier transform 250
11.6 Numerical experiments for the inverse Fourier transform 251
Bibliography 255
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