Series Representations and Approximation of some Quantile Functions appearing in Finance

164 Pages·2013·1.49 MB·English

Series Representations and
Approximation of some Quantile
Functions appearing in Finance
Asad Ullah Khan Munir
Adissertationsubmittedinpartialfulfillment
oftherequirementsforthedegreeof
DoctorofPhilosophy
ofthe
UniversityofLondon.
DepartmentofMathematics
UniversityCollegeLondon
2013
1
I, Asad Ullah Khan Munir, confirm that the work presented in this thesis is my
own. Where information has been derived from other sources, I confirm that this has
beenindicatedinthethesis.
Abstract
Ithaslongbeenagreedbyacademicsthattheinversionmethodisthemethodofchoice
for generating random variates, given the availability of a cheap but accurate approx-
imation of the quantile function. However for several probability distributions arising
in practice a satisfactory method of approximating these functions is not available.
The main focus of this thesis will be to develop Taylor and asymptotic series rep-
resentations for quantile functions of the following probability distributions; Variance
Gamma, Generalized Inverse Gaussian, Hyperbolic, α-Stable and Snedecor’s F distri-
butions. Asasecondarymatterwebrieflyinvestigatetheproblemofapproximatingthe
entirequantilefunction. Indeedwiththeavailabilityofthesenewanalyticexpressionsa
wholehostofpossibilitiesbecomeavailable. Weoutlineseveralalgorithmsandinpar-
ticular provide a C++ implementation for the variance gamma case. To our knowledge
thisisthefastestavailablealgorithmofitssort.
Acknowledgements
I am especially grateful to my supervisor Prof. William T. Shaw for giving me the
opportunity to do a PhD in mathematics. He has been extremely inspirational to me
in developing the results presented in this thesis. Indeed without my discussions with
Prof. Shaw I fail to see where I would have found the motivation to persevere. I very
much appreciate the financial support and resources provided to me by both the com-
puter science and mathematics departments at UCL. I thank the Knowledge Transfer
Network (KTN) for Industrial Mathematics and the reinsurance group Willis Re for
offering me a research internship. I have only good words to say about both of these
organizations. LastbutnotleastIwouldliketothankmyfamilyinparticularmyloving
parentsfortheirendlesspatience,financialandmoralsupport.
Contents
1 Introduction 7
2 QuantileFunctions 11
3 Applications 15
3.1 ValueatRisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 MonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 ExistingMethods 24
4.1 NumericalTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1 RootFinding . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2 RationalApproximation . . . . . . . . . . . . . . . . . . . . . 25
4.1.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.4 RelationshipbetweenDistributions . . . . . . . . . . . . . . . 26
4.2 Cornish-FisherExpansions . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 OrthogonalExpansions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 SolutionTechniques 32
5.1 Taylor’sMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 RecursiveIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 QuantileMechanics 39
6.1 Beta,FandStudent-tDistributions . . . . . . . . . . . . . . . . . . . . 43
6.2 HyperbolicDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2.1 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2.2 AsymptoticExpansion . . . . . . . . . . . . . . . . . . . . . . 49
6.2.3 ChangeofVariable . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3 VarianceGammaDistribution . . . . . . . . . . . . . . . . . . . . . . 53
6.3.1 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3.2 AsymptoticExpansion . . . . . . . . . . . . . . . . . . . . . . 57
Contents 5
6.3.3 ChangeofVariable . . . . . . . . . . . . . . . . . . . . . . . . 60
6.3.4 Amoreefficientmethod . . . . . . . . . . . . . . . . . . . . . 64
6.4 GeneralizedInverseGaussianDistribution . . . . . . . . . . . . . . . . 66
6.4.1 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4.2 AsymptoticExpansion . . . . . . . . . . . . . . . . . . . . . . 69
6.4.3 ChangeofVariable . . . . . . . . . . . . . . . . . . . . . . . . 72
7 AlternativeRoutes 75
7.1 MomentumSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.1.1 CharacteristicQuantileEquation(C.Q.E.) . . . . . . . . . . . . 76
7.1.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.1.3 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 TheLagrangeApproach . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.2.1 α-StableDistribution. . . . . . . . . . . . . . . . . . . . . . . 86
8 NumericalTechniquesandExamples 91
8.1 RootFinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.2 NumericalIntegrators . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8.3 ContinuedFractionsandPadéApproximants . . . . . . . . . . . . . . 95
8.3.1 ContinuedFractions . . . . . . . . . . . . . . . . . . . . . . . 96
8.3.2 PadéApproximants . . . . . . . . . . . . . . . . . . . . . . . . 98
8.3.3 ErrorandPartitionManagement . . . . . . . . . . . . . . . . . 101
8.4 SequenceTransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.5 ChebyshevSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.5.1 DiscreteFourierTransform . . . . . . . . . . . . . . . . . . . . 107
8.5.2 FromTaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . 109
8.5.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.6 Chebyshev-PadéApproximants . . . . . . . . . . . . . . . . . . . . . . 113
8.7 OsculatoryRationalInterpolation . . . . . . . . . . . . . . . . . . . . . 115
8.8 MinimaxApproximations . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.8.1 TheSecondAlgorithmofRemez . . . . . . . . . . . . . . . . 118
8.8.2 Maehly’sIndirectMethod . . . . . . . . . . . . . . . . . . . . 122
8.9 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
9 Conclusion&FurtherWork 133
A ThePearsonFamilyofDistributions 135
Contents 6
B Variance-MeanMixtureDistributions 137
B.1 GeneralizedInverseGaussian . . . . . . . . . . . . . . . . . . . . . . . 138
B.2 GeneralizedHyperbolicDistribution . . . . . . . . . . . . . . . . . . . 140
C CodeListings 144
Bibliography 155
Chapter 1
Introduction
Analytic expressions for quantile functions have long been sought after. The import-
ance of these functions comes from their widespread use in applications of statistics,
probability theory, finance and econometrics. Therefore much effort has been devoted
intotheirstudy,inparticular sinceclosedformexpressionsforthequantilefunctionof
mostdistributionsarenotknown,severalapproximationsappearintheliterature. These
approximations generally fall into one of four categories, series expansions, functional
approximations,numericalalgorithmsorclosedformexpressionswrittenintermsofa
quantile function of another distribution. The focus of this report is on the former two
categories.
We follow the philosophy of Steinbrecher and Shaw (2008), Shaw et al. (2011)
andShawandMcCabe(2009)closely,thatisto“elevatequantilefunctionstothesame
levelofmanagementasmanyoftheclassicalspecialfunctionsofmathematicalphysics
and applied mathematics”. In particular this requires efficient computation. According
toLozierandOlver(1994),therearethreestagesinthedevelopmentofcomputational
proceduresofspecialfunctions:
1. Derivation of relevant mathematical properties: This stage is primarily an ex-
ercise in applied mathematics, and is concerned with finding representations of
various forms such as, asymptotic expansions, difference and differential equa-
tions,functionalidentities,integralrepresentations,andTaylorseriesexpansions.
2. Development of numerical approximations and algorithms: This stage is an ex-
ercise in numerical analysis and is concerned with finding Chebyshev series ex-
pansions, minimax polynomial and rational approximations, Padé approxima-
tions,numericalquadratureandnumericalsolutionsofdifferenceanddifferential
equations.
3. Construction and testing of robust software: This final stage is an exercise in
computer science, and though highly dependent on stage 2, great benefits can be
8
gainedatthisstagesuchasthepossibilityofparallelizationandortheutilization
ofvariousplatformspecificfeatures.
Thebulkoftheworkwecarryoutinthisreportbelongstostages1and2,inparticular
we will develop Taylor and asymptotic series expansions for the quantile functions of
the following probability distributions; Variance Gamma, Generalized Inverse Gaus-
sian, Hyperbolic, α-Stable and Snedecor’s F. With these analytic expressions in place
weproceedtostage2wherewewillbeconcernedwiththeconstructionofvarioustypes
of approximants such as continued fractions, Chebyshev series and minimax approx-
imations. Finallywebrieflyvisitstage3,whereaC++implementationofanalgorithm
usedtoapproximatethevariancegammaquantileisprovided.
There is no shortage of research articles discussing the approximation of quantile
functions,seeforexampleDagpunar(1989);Derflingeretal.(2009,2010);Lai(2009);
Farnum(1991);LeydoldandHörmann(2011)tonameafew. Thesepapersareprimar-
ilyconcernedwithapplyingnumericaltechniquessuchasrootfindingandinterpolation
to approximate the quantile function. Our approach however is centred around formu-
lating a first order ordinary differential equation problem. We are not the first however
toconsiderthisdifferentialequationapproach,sucharouteisalsotakenbyUlrichand
Watson (1987) and Leobacher and Pillichshammer (2002). Here numerical schemes
such as Runge-Kutta methods are used to seek solutions. On the other hand we will
seek analytic solutions by applying certain well known techniques from applied math-
ematics.
Aside from Shaw’s series of Quantile Mechanics papers which we will discuss in
subsequent chapters, few authors have written about series representations of quantile
functions. Possiblytheearliestknown(atleasttothisauthor)seriesrepresentationisthe
famous Cornish Fisher expansion introduced in (Cornish and Fisher, 1938; Fisher and
Cornish, 1960). The Cornish Fisher expansion expresses the u-quantile of a random
variableX intermsofitscumulantsandtheu-quantileofthestandardGaussiandistri-
bution. TheideawasgeneralizedinHillandDavis(1968)byallowingfornonGaussian
basedistributions. TheCornishFisherexpansionisoftenusedforVARapplicationsin
finance,seeforexampleJaschke(2002).
Another interesting approach was introduced by Takemura (1983). Here a base
quantile function Q is chosen, based on which an orthonormal basis for the set of
B
square integrable functions on the unit interval is formed. Consequently the Fourier
series expansion of the target quantile Q with respect to this basis may be developed.
T
Unlike the Cornish Fisher expansion which is asymptotic in nature Takemura’s ap-
proachyieldsaconvergentseriesintheL2 norm. Notehoweverthecomputationofthe
Fourier coefficients usually requires numerical quadrature and that for approximation
9
purposes the L∞ norm is preferred, see Shaw et al. (2011) for details. We will discuss
theCornishFisherexpansionandTakemura’sapproachinmoredetailinchapter4.
After a brief introduction to the theory of quantile functions in chapter 2 we mo-
tivatethediscussionwithachapteronapplicationsandanotheronexistingmethods. In
chapter 5 we discuss some general techniques used to solve non-linear ordinary differ-
entialequations. Thesewillbeemployedinchapter6,whereweprovideseriesrepres-
entations for some quantile functions not currently available. Key to this approach is
theavailabilityofthedensityfunction. Inchapter7somealternativeapproachesreliant
insteadonthecharacteristicanddistributionfunctionsrespectivelyareinvestigated. As
aresultseriesrepresentationsfortheα-stablequantilefunctionwillbepresented.
The fact that a convergent series for a specific quantile function is available may
seemtoindicatethatthecomputationofsuchafunctionisofnoconcern. Howeverfor
moderncomputingsystemsthisissimplynotthecase,since:
• theyarelimitedinspeed;abigconcernforslowlyconvergingpowerserieswhen
adesiredaccuracyisrequired,
• canonlyrepresentafiniterangeofnumbers;thetermsoftheseriesmaybecome
extremelysmallorlargeleadingtounderflowandoverflowerrorsrespectively,
• and can only perform finite precision arithmetic; which of course is a problem
formostnumericalcomputations.
Itisforthesereasonsseriesaccelerationtechniques(BrezinskiandZaglia,1991)prove
so useful, and we will use them to good affect in our numerical experiments discussed
in chapter 8. Here the reader will find many useful recipes to construct algorithms
for approximating Q. The key here is to apply a change of variable. To this end we
will employ a technique devised by Shaw et al. (2011), albeit from a slightly different
perspective,andconsequentlytackletheproblemofapproximatingQ. Anoverviewof
aC++programimplementingtheseideasispresentedinsection8.9.
Tosummarizethekeycontributionsinthisthesisareprovidedinchapters6and8.
Chapter6willbeconcernedwiththedevelopingthe“ingredients”requiredtoconstruct
numerical recipes for approximating Q. In particular we write down some functional
identities and develop some series representations in this chapter. In chapter 8 we put
togethersomenumericalrecipesbasedontheseingredients,forexample
• wedeviseastrategybasedoncontinuedfractions,
• using the methods of Thacher (1964) and Sidi (1975) we construct Chebyshev-
Padéapproximants,