Table Of ContentLecture Notes in Economics
and Mathematical Systems 607
FoundingEditors:
M.Beckmann
H.P.Künzi
ManagingEditors:
Prof.Dr.G.Fandel
FachbereichWirtschaftswissenschaften
FernuniversitätHagen
Feithstr.140/AVZII,58084Hagen,Germany
Prof.Dr.W.Trockel
InstitutfürMathematischeWirtschaftsforschung(IMW)
UniversitätBielefeld
Universitätsstr.25,33615Bielefeld,Germany
EditorialBoard:
A.Basile,A.Drexl,H.Dawid,K.Inderfurth,W.Kürsten
Markus Bouziane
Pricing Interest-Rate
Derivatives
A Fourier-Transform Based Approach
123
Dr.MarkusBouziane
LandesbankBaden-Württemberg
AmHauptbahnhof2
70173Stuttgart
Germany
[email protected]
ISBN978-3-540-77065-7 e-ISBN978-3-540-77066-4
DOI10.1007/978-3-540-77066-4
LectureNotesinEconomicsandMathematicalSystemsISSN0075-8442
LibraryofCongressControlNumber:2008920679
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To Sabine
Foreword
In a hypothetical conversation between a trader in interest-rate derivatives
and a quantitative analyst, Brigo and Mercurio (2001) let the trader answer
about the pros and cons of short rate models: ”... we should be careful in
thinking marketmodelsarethe finalandcompletesolutiontoallproblemsin
interest rate models ... and who knows, maybe short rate models will come
back one day...”
In his dissertation Dr. Markus Bouziane contributes to this comeback of
short rate models. Using Fourier Transform methods he develops a modu-
lar framework for the pricing of interest-rate derivatives within the class of
exponential-affinejump-diffusions. Basedona technique introducedby Lewis
(2001)forequity options,the payoffsandthe stochasticdynamics ofinterest-
rate derivatives are transformed separately. This not only simplifies the ap-
plicationofthe residuecalculusbut improvesthe efficiency ofnumericaleval-
uation schemes considerably. Dr. Bouziane introduces a refined Fractional
InverseFastFourierTransformationalgorithmwhichisabletocalculatethou-
sands of prices within seconds for a given strike range. The potential of this
method is demonstrated for severalone- and two-dimensional models.
Asaresulttheapplicationofjump-enhancedshortratemodelsforinterest-
ratederivativesisontheagendaagain.Ihope,Dr.Bouziane’smonographwill
stimulate further research in this direction.
Tu¨bingen, November 2007 Rainer Sch¨obel
Acknowledgements
This book is based on my Ph.D. thesis titled ”Pricing Interest-Rate Deriva-
tives with Fourier Transform Techniques” accepted at the Eberhard Karls
University of Tu¨bingen, Germany. Writing the dissertation, I am indebted to
many people which contributed academic and personal development. Since
any list would be insufficient, I mention only those who bear in my opinion
the closest relation to this work.
Firstofall,IwouldliketothankmyacademicteacherandsupervisorProf.
Dr.-Ing.RainerSch¨obel.He gaveme valuableadvice andsupportthroughout
the completion of my thesis. Furthermore, I would also express my grati-
tude to Prof. Dr. Joachim Grammig for being the co-referent of this thesis.
Further thanks go to my colleagues from the faculty of Economics and Busi-
ness Administration, especially Svenja Hager, Robert Frontczak, Wolfgang
Kispert, Stefan Rostek and Martin Weiss for fruitful discussions and a pleas-
ant working atmosphere. I very much enjoyed my time at the faculty. Finan-
cial support from the Stiftung Landesbank Baden-Wu¨rttemberg is gratefully
acknowledged.
My deepest gratitude goes to my wife Sabine, my parents Ursula and
Laredj Bouziane, and Norbert Gutbrod for their enduring support and en-
couragement.
Tu¨bingen, November 2007 Markus Bouziane
Contents
List of Abbreviations and Symbols.............................XV
List of Tables ..................................................XIX
List of Figures.................................................XXI
1 Introduction............................................... 1
1.1 Motivation and Objectives................................ 1
1.2 Structure of the Thesis................................... 4
2 A General Multi-Factor Model of the Term Structure
of Interest Rates and the Principles of Characteristic
Functions.................................................. 7
2.1 An Extended Jump-Diffusion Term-Structure Model ......... 7
2.2 Technical Preliminaries................................... 11
2.3 The Risk-Neutral Pricing Approach........................ 13
2.3.1 Arbitrage and the Equivalent Martingale Measure ..... 15
2.3.2 Derivation of the Risk-Neutral Coefficients............ 16
2.4 The Characteristic Function .............................. 21
3 Theoretical Prices of European Interest-Rate Derivatives .. 31
3.1 Overview............................................... 31
3.2 Derivatives with Unconditional Payoff Functions............. 32
3.3 Derivatives with Conditional PayoffFunctions............... 38
XII Contents
4 Three Fourier Transform-Based Pricing Approaches ....... 45
4.1 Overview............................................... 45
4.2 Heston Approach........................................ 49
4.3 Carr-MadanApproach ................................... 55
4.4 Lewis Approach ......................................... 60
5 Payoff Transformations and the Pricing of European
Interest-Rate Derivatives .................................. 69
5.1 Overview............................................... 69
5.2 Unconditional Payoff Functions ........................... 70
5.2.1 General Results ................................... 70
5.2.2 Pricing Unconditional Interest-Rate Contracts ........ 79
5.3 Conditional PayoffFunctions.............................. 81
5.3.1 General Results ................................... 82
5.3.2 Pricing of Zero-Bond Options and Interest-Rate Caps
and Floors........................................ 87
5.3.3 Pricing of Coupon-Bond Options and Yield-Based
Swaptions ........................................ 90
6 Numerical Computation of Model Prices .................. 95
6.1 Overview............................................... 95
6.2 Contracts with Unconditional Exercise Rights............... 96
6.3 Contracts with Conditional Exercise Rights................. 97
6.3.1 Calculating Option Prices with the IFFT............. 97
6.3.2 Refinement of the IFFT Pricing Algorithm ...........101
6.3.3 Determination of the Optimal Parameters for the
Numerical Scheme.................................103
7 Jump Specifications for Affine Term-Structure Models.....111
7.1 Overview...............................................111
7.2 Exponentially Distributed Jumps..........................115
7.3 Normally Distributed Jumps..............................117
7.4 Gamma Distributed Jumps ...............................120
8 Jump-Enhanced One-Factor Interest-Rate Models .........125
8.1 Overview...............................................125
8.2 The Ornstein-Uhlenbeck Model ...........................126
Contents XIII
8.2.1 Derivation of the Characteristic Function.............126
8.2.2 Numerical Results .................................128
8.3 The Square-Root Model..................................136
8.3.1 Derivation of the Characteristic Function.............136
8.3.2 Numerical Results .................................138
9 Jump-Enhanced Two-Factor Interest-Rate Models.........145
9.1 Overview...............................................145
9.2 The Additive OU-SR Model ..............................146
9.2.1 Derivation of the Characteristic Function.............146
9.2.2 Numerical Results .................................148
9.3 The Fong-Vasicek Model .................................159
9.3.1 Derivation of the Characteristic Function.............159
9.3.2 Numerical Results .................................163
10 Non-Affine Term-Structure Models and Short-Rate
Models with Stochastic Jump Intensity ....................171
10.1 Overview...............................................171
10.2 Quadratic Gaussian Models...............................171
10.3 Stochastic Jump Intensity ................................174
11 Conclusion ................................................175
A Derivation of the Complex-Valued Coefficients for the
Characteristic Function in the Square-Root Model.........179
B Derivation of the Complex-Valued Coefficients for the
Characteristic Function in the Fong-Vasicek Model ........183
References.....................................................187
List of Abbreviations and Symbols
δ(x) Dirac delta function
Γ(x) Gamma function
√
ı imaginary unit, −1
ι diag[I ]
M M
Fx[...],F−1[...] Fourier Transformation w.r.t. x and inverse
Transformationoperator
1A indicator function for the event A
C the set of complex-valued numbers
E[...],VAR[...] expectation and variance operator
P,Q real-worldand equivalent martingale measure
R the set of real-valued numbers
F information set available up to time t
t
diag[...] operator returning the diagonal elements of a
quadratic matrix
FFT[...] Fast Fourier Transformationoperator
FRFT[...,ζ] FractionalFourierTransformationoperatorwith
parameter ζ
IFFT[...] inverse Fast Fourier Transformationoperator
Res[...] residue operator
Re[z],Im[z] realandimaginarypartofthecomplex-valued
variable z
RMSE root mean-squared error
RMSEa approximate root mean-squared error
tr[...] trace operator
