Comparison of option pricing between ARMA-GARCH and GARCH-M models

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Yi Xi, The University of Western Ontario
Supervisor: Dr. Reginald J. Kulperger, The University of Western Ontario
A thesis submitted in partial fulfillment of the requirements for the Master of Science degree in
Statistics and Actuarial Sciences
© Yi Xi 2013
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COMPARISON OF OPTION PRICING BETWEEN ARMA-GARCH AND
GARCH-M MODELS
(Thesis format: Monograph)
by
Yi Xi
Graduate Program in Statistics and Actuarial Science
A thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science
The School of Graduate and Postdoctoral Studies
The University of Western Ontario
London, Ontario, Canada
⃝c Yi Xi 2013
Abstract
Option pricing is a major area in financial modeling. Option pricing is sometimes based
on normal GARCH models. Normal GARCH models fail to capture the skewness and the
leptokurtosis in financial data. The variant GARCH-in-mean (GARCH-M) model is widely
used in the option pricing literature. It adds a heteroskedasticity term to the mean equation,
whichisinterpretedasariskpremium,andalsoincorporatesatypeofasymmetry.
OurgoalistocompareoptionvaluationbetweenGARCH-MandARMA-GARCHmodels
with normal and non-normal, z-distributed innovations. The models are fitted to the historical
return data, and risk neutral measures are based on the conditional Esscher transform and the
extended Girsanov principle. We compare European Calls on the S&P 500 with the model
predictions. The TGARCH is best for ARMA-GARCH/GARCH-M models. Neither normal
norzdominatestheother,butoverallz-TGARCH-M(z-innovations)seemstobebest,ARMA-
TGARCHissurprisinglygood.
Keywords: option pricing, ARMA-GARCH, GARCH-in-Mean, z-distribution, Esscher
transform,ExtendedGirsanovprinciple
ii
Acknowlegements
First and foremost, I would like to express my deepest gratitude to my program supervisor
Dr. Reginald J. Kulperger for his generous support, encouragement, patience and valuable
guidance during my time at University of Western Ontario. During these two years spent at
Westernhewasnotonlymyacademicsupervisorbutalsomymentor. Ideeplyindebtedtoyou
forallthatyouhavedoneformeduringalltheseyears.
I was one of recipients of Richard Konrad Ontario Graduate Scholarship. I really appreci-
atedthegeneroussupportfromMr. KonradandtheProvincialGovernment.
I also wish to express my heart-felt thank you to all the faculty, staff, and fellow graduate
students in the Department of Statistical and Actuarial Science who have made my stay here
bothenjoyableandmemorable.
My sincere thanks to Dr. Mark Resser, Dr. Matt Davison and Dr. Rogemar Mamon for
agreeingtoactasexaminers.
I am appreciated the selfless love, devotion, and encouragement from my parents. In addi-
tion, I am grateful to my husband Lijun. Her love, support and words of encouragement have
beenaconstantthatIhavebeenabletocountonthroughoutthisexperience.
iii
Contents
Abstract ii
Acknowlegements iii
ListofFigures vii
ListofTables viii
ListofAppendices x
1 Introduction 1
2 FinancialSeriesandModels 4
2.1 FinancialTimeSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 ARMAModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 GARCHModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 PropertiesoftheGARCH(1,1)Models . . . . . . . . . . . . . . . . . . 7
2.3.2 ExtensionsofGARCHModel . . . . . . . . . . . . . . . . . . . . . . 8
3 EstimationoftheGARCHModels 11
3.1 EstimationMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 MaximumLikelihoodEstimation . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Quasi-MaximumLikelihood . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 EstimatingARMA-GARCHandGARCH-MModels . . . . . . . . . . . . . . 14
3.3 ModelFittingAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 RiskNeutralMeasuresUnderGARCHModels 22
4.1 DefinitionsandNotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 RiskNeutralMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1 ConditionalEsscherTransform . . . . . . . . . . . . . . . . . . . . . . 24
4.2.2 ExtendedGirsanovPrinciple . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 ApplicationsforGARCHModels . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.1 GARCHwithNormalInnovation . . . . . . . . . . . . . . . . . . . . . 27
4.3.2 GARCHwithZ-distributedInnovation . . . . . . . . . . . . . . . . . . 28
5 NumericalExperiments 32
5.1 SimulationSteps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
iv
5.2 OptionDataDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 EmpiricalAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.1 OptionDataSet1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.2 OptionDataSet2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 ConclusionandFutureresearch 52
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 FutureResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliography 55
A TablesofPricingErrorsforExample2 59
CurriculumVitae 68
v
List of Figures
3.1 Daily closing prices and returns of S&P 500 Index from January 02, 1988 to
January06,2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Log-density of observed residuals versus their theoretical density for Normal
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Log-densityofobservedresidualsversustheirtheoreticaldensityforz-distribution
21
5.1 European Call option price evaluated on April 18, 2002 for different models
withthematuritiesT = 22,46,109,173,and234days;theclosingstockprice
onthatdaywasS = $1124.47. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
0
5.1 Continued Table: European Call option price evaluated on April 18, 2002 for
different models with the maturities T = 22, 46, 109, 173, and 234 days; the
closingstockpriceonthatdaywasS = $1124.47. . . . . . . . . . . . . . . . 41
0
5.2 ModelpricingerrorsforEuropeanCalloptionsonApril18,2002. . . . . . . . 42
5.2 Continued Table: Model pricing errors for European Call options on April 18,
2002. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 Implied volatility smiles based on MLE estimates using returns from January
07,2004toDecember29,2004 . . . . . . . . . . . . . . . . . . . . . . . . . 49
vi
List of Tables
3.1 ϵ˜(ϑ)andσ˜ 2(ψ)inGARCHmodels . . . . . . . . . . . . . . . . . . . . . . . 15
t t
3.2 GARCHandTGARCHparametersestimatedbyMLEwithnormalinnovations
usingdailyclosingpricesofS&P500fromJanuary02,1988toJanuary06,2004 18
3.3 GARCH and TGARCH parameters estimated by MLE, which under the as-
sumption on z-distributed noise, using daily closing prices of S&P 500 from
January02,1988toJanuary06,2004 . . . . . . . . . . . . . . . . . . . . . . . 19
5.1 S&P500calloptionpricesfromSchoutens(2003) . . . . . . . . . . . . . . . 35
5.2 NumberofCalloptioncontracts(S&P500Index,January07,2004toDecem-
ber29,2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Average price of Call option contracts (S&P 500 Index, January 07, 2004 to
December29,2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 OverallpricingerrorsforEuropeanCalloptionsonApril18,2002 . . . . . . . 38
5.5 GARCHandTGARCHparametersestimatedbyMLEwithnormaldistribution
usingdailyclosingpricesofS&P500fromJanuary02,1988toApril17,2002 44
5.6 GARCH and TGARCH parameters estimated by MLE, which under the as-
sumption on z-distributed noise, using daily closing prices of S&P 500 from
January02,1988toApril17,2002 . . . . . . . . . . . . . . . . . . . . . . . . 45
5.7 OverallpricingerrorsforEuropeanCalloptionsindataset2 . . . . . . . . . . 46
5.8 Overall pricing errors regarding to moneyness(Call option contracts of S&P
500Index,January07,2004toDecember29,2004) . . . . . . . . . . . . . . . 50
5.9 Overall pricing errors regarding to maturity (Call option contracts of S&P 500
Index,January07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . 51
A.1 PricingerrorsforGARCH-M(CalloptioncontractsofS&P500Index,January
07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.2 Pricing errors for TGARCH-M (Call option contracts of S&P 500 Index, Jan-
uary07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . . . 61
A.3 Pricing errors for ARMA-GARCH (Call option contracts of S&P 500 Index,
January07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . 62
A.4 Pricing errors for ARMA-TGARCH (Call option contracts of S&P 500 Index,
January07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . 63
A.5 Pricing errors for z-GARCH-M (Call option contracts of S&P 500 Index, Jan-
uary07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . . . 64
A.6 Pricing errors for z-TGARCH-M (Call option contracts of S&P 500 Index,
January07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . 65
vii
A.7 Pricingerrorsforz-ARMA-GARCH(CalloptioncontractsofS&P500Index,
January07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . . . 66
A.8 Pricing errors for z-ARMA-TGARCH (Call option contracts of S&P 500 In-
dex,January07,2004toDecember29,2004) . . . . . . . . . . . . . . . . . . 67
viii
List of Appendices
AppendixATablesofPricingErrorsforExample2 . . . . . . . . . . . . . . . . . . . . 59
ix