Calibration and Parameterization Methods for the Libor Market Model

69 Pages·2014·3.363 MB·English

by  Christoph Hackl (auth.)

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Christoph Hackl
Calibration and
Parameterization
Methods for the
Libor Market Model
Christoph Hackl
Vienna, Austria
Masterthesis, University of Applied Sciences (bfi ) Vienna, Austria
ISBN 978-3-658-04687-3 ISBN 978-3-658-04688-0 (eBook)
DOI 10.1007/978-3-658-04688-0
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Foreword
QuantitativeFinanceisatopicwhichhasbecomequitepopularinthelastdecade,com-
biningtechniquesoutofthedisciplinesofmathematics,finance,statisticsandcomputer
science. Modellingofinterestratesisahotpartofmodernquantitativefinanceandhas
verywideapplicationforfinancialinstrumentspricingandriskmeasurement.
The Libor Market Model is a mathematical no-arbitrage interest rate model which re-
quiresastrongbackgroundinmathematical/statisticalfieldsandfinance. Inaddition,its
applicationrequiresasoundknowledgeinthefieldofcomputersciencestoimplementthe
wholecomputationallydemandingmodel. Itisalreadyquitealotofacademicresearch
availablewhichdealwithspecialtopicsaboutMarketModelsbutishardtofindabook
or paper which covers the full picture starting with the mathematical background up
to building a model and calibrate it to market data with a special focus on speed and
efficiencyofthemodel.
Thisthesisstartswithanextensiveintroductiontothestatisticaltheoryunderlyingmar-
ketmodels. Next,efficientrandomnumbergeneratorsaredescribed. Inthemainpart,a
fullframeworktobuildandcalibrateaLiborMarketModelisexplained. Thisincludes
the theoretical presentation of the model, computational formulas to calibrate model
volatilities and correlations, factor reduction methods. In the analysis two calibration
schemas are presented. One is based on cap volatilities and the second one uses swap-
tionsascalibrationinstruments. Bothschemasarevalidatedintermsofcorrespondence
betweenmodelled(simulated)andrealmarketpricesofcalibrationinstruments.
Thedatausedintheanalysisbelongstostandardmarketinstrumentsandcanbeeasily
obtainedfromanyfinancialinformationprovider.
The calibration of interest rate models and especially the Libor Model is an open topic
V
Foreword
nowadays in banking practice. Most crucial is the speed of this calibration and re-
calibration for any comprehensive simulation framework. This paper addresses exactly
thisissueaboutefficientsimulatorsandmostprecisecalibrationinstruments. Themod-
ernparameterizationandcalibrationmethodoftheLiborMarketModelareimplemented
inR.The quantitative results are comparedwithrespect to computationalandapplied
marketdataefficiency.
I hope that the findings of this analysis will bring their input into banking practice to
improve the understanding and efficiency of both calibration and simulations from the
LiborMarketModel.
Dr. TatjanaMiazhynskaia
VI
Contents
Foreword V
1. Introduction 1
1.1. Researchtopicanditsrelevance . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Researchquestion(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Researchmethods/approach . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4. ThesisStructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Foundations of Mathematical Finance and Stochastic Calculus 5
2.1. InterestRatesandDerivatives . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. StochasticCalculusandNo-ArbitragePricing . . . . . . . . . . . . . . . 11
2.3. MonteCarloSimulationandComputationalAspects . . . . . . . . . . . 16
2.3.1. MonteCarloMethods . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2. RandomNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3. Quasi-randomnumbersandAntithetics . . . . . . . . . . . . . . . 19
3. The Libor Market Model 21
3.1. LiborMarketModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2. Sensitivities(Greeks)intheLiborMarketModel. . . . . . . . . . . . . . 23
3.3. TermStructureInterpolationMethod . . . . . . . . . . . . . . . . . . . . 25
4. Volatility and Correlation in the Libor Market Model 31
4.1. ModelVolatilityCalibration . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2. ModelCorrelationCalibration . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3. FactorReduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5. Applications and Results 39
5.1. ParameterizationandCalibrationResults. . . . . . . . . . . . . . . . . . 39
5.1.1. Calibrationtocapvolatilities . . . . . . . . . . . . . . . . . . . . 39
5.1.2. Calibrationtoswaptionvolatilities . . . . . . . . . . . . . . . . . 42
VII
Contents
5.2. PricingApplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.1. Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.2. Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3. Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4. Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4.1. Validationofcappricing . . . . . . . . . . . . . . . . . . . . . . . 49
5.4.2. Validationofswaptionpricing . . . . . . . . . . . . . . . . . . . . 50
5.4.3. ConvergenceSpeed . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6. Conclusion 55
7. Bibliography 57
A. Appendix: Proofs 59
B. Appendix: Graphs 63
VIII
List of Abbreviations
LiborMarketModel-LMM
StochasticDifferentialEquation-SDE
EquivalentMartingaleMeasure-EMM
Nelson-SiegelSvensson-NSS
Atthemoney-ATM
IX