Analytical and Numerical Methods for Pricing Financial Derivatives.

327 Pages·2012·4.324 MB·English

by  Mikula, Karol, Sevcovic, Daniel, Stehlikova, Beata

MATHEMATICS RESEARCH DEVELOPMENTS

A N
NALYTICAL AND UMERICAL
M P
ETHODS FOR RICING
FINANCIAL DERIVATIVES

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M R D
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MATHEMATICS RESEARCH DEVELOPMENTS

A N
NALYTICAL AND UMERICAL
M P
ETHODS FOR RICING
FINANCIAL DERIVATIVES

DANIEL SEVCOVIC
BEÁTA STEHLÍKOVÁ
AND
KAROL MIKULA

Nova Science Publishers, Inc.
New York
Copyright © 2011 by Nova Science Publishers, Inc.

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Library of Congress Cataloging-in-Publication Data

Sevcovic, Daniel.
Analytical and numerical methods for pricing financial derivatives /
Daniel Sevcovic.
p. cm.
Includes index.
ISBN 978-1-61761-350-0 (e-book)
1. Derivative securities--Prices--Mathematical models. 2. Options
(Finance)--Prices--Mathematical models. I. Title.
HG6024.A3S46 2010
332.64'57--dc22
2010026267

Published by Nova Science Publishers, Inc. † New York
Contents
Preface xi
Introduction xiii
1 TheRoleofProtectingFinancialPortfolios 1
1.1. StochasticCharacterofFinancialAssets . . . . . . . . . . . . . . . . . . . 2
1.2. UsingDerivativeSecuritiesasaToolforProtectingVolatileStockPortfolios 4
1.2.1. ForwardsandFutures . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2. PlainVanillaCallandPutOptions . . . . . . . . . . . . . . . . . . 5
2 Black–ScholesandMertonModel 9
2.1. StochasticProcessesandStochasticDifferentialCalculus . . . . . . . . . . 9
2.1.1. WienerProcessandGeometricBrownianMotion . . . . . . . . . . 10
2.1.2. Ito¯’sIntegralandIsometry . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3. Ito¯’sLemmaforScalarRandomProcesses . . . . . . . . . . . . . 16
2.1.4. Ito¯’sLemmaforVectorRandomProcesses . . . . . . . . . . . . . 18
2.2. TheBlack–ScholesEquation . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1. AStochasticDifferentialEquationfortheOptionPrice . . . . . . . 20
2.2.2. Self-financingPortfolioManagementwithZeroGrowthofInvestment 21
2.3. TerminalConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1. Pay–offDiagramsforCallandPutOptions . . . . . . . . . . . . . 24
2.3.2. Pay–offDiagramsforCombinedOptionStrategies . . . . . . . . . 24
2.4. BoundaryConditionsforDerivativePrices . . . . . . . . . . . . . . . . . . 28
2.4.1. BoundaryConditionsforCallandPutOptions . . . . . . . . . . . 28
2.4.2. BoundaryConditionsforCombinedOptionStrategies . . . . . . . 29
3 EuropeanStyleofOptions 33
3.1. PricingPlainVanillaCallandPutOptions . . . . . . . . . . . . . . . . . . 33
3.2. PricingPutOptionsUsingCallOptionPricesandForwards,Put-CallParity 38
3.3. PricingCombinedOptionsStrategies: Spreads,Straddles,Condors,Butter-
fliesandDigitalOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4. ComparisonofTheoreticalPricingResultstoRealMarketData . . . . . . 41
3.5. Black–ScholesEquationforPricingIndexOptions . . . . . . . . . . . . . 42
v
vi Contents
4 AnalysisofDependenceofOptionPricesonModelParameters 49
4.1. HistoricalVolatilityofStocks . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1. AUsefulIdentityforBlack–ScholesOptionPrices . . . . . . . . . 51
4.2. ImpliedVolatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3. VolatilitySmile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4. DeltaofanOption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5. GammaofanOption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6. OtherSensitivityFactors: Theta,Vega,Rho . . . . . . . . . . . . . . . . . 60
4.6.1. SensitivitywithRespecttoaChangeintheInterestRate–FactorRho 60
4.6.2. SensitivitytotheTimetoExpiration–FactorTheta . . . . . . . . . 61
4.6.3. SensitivitytoaChangeinVolatility–FactorVega . . . . . . . . . 62
5 OptionPricingunderTransactionCosts 65
5.1. LelandModel,Hoggard,WilmottandWhalleyModel . . . . . . . . . . . . 65
5.2. ModelingOptionBid–AskSpreadsbyUsing
Leland’sModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6 ModelingandPricingExoticFinancialDerivatives 75
6.1. AsianOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.1. APartialDifferentialEquationforPricingAsianOptions . . . . . . 76
6.1.2. Dimension Reduction Method and Numerical Approximation of a
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2. BarrierOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.1. Numerical and Analytical Solutions to the Partial Differential
EquationforPricingBarrierOptions . . . . . . . . . . . . . . . . . 83
6.3. BinaryOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4. CompoundOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5. LookbackOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 ShortInterestRateModeling 93
7.1. One-FactorInterestRateModels . . . . . . . . . . . . . . . . . . . . . . . 93
7.2. TheDensityofIto¯’sStochasticProcessandtheFokker–PlankEquation . . 97
7.2.1. MultidimensionalVersionoftheFokker-PlanckEquation . . . . . . 98
7.3. Two-andMulti-factorInterestRateModels . . . . . . . . . . . . . . . . . 103
7.4. CalibrationofShortRateModels . . . . . . . . . . . . . . . . . . . . . . . 105
7.4.1. Maximum Likelihood Method for Estimation of the Parameters in
theVasicekandCIRModels . . . . . . . . . . . . . . . . . . . . . 105
8 PricingofInterestRateDerivatives 109
8.1. BondsandTermStructuresofInterestRates . . . . . . . . . . . . . . . . . 109
8.1.1. One–FactorEquilibriumModels,VasicekandCIRModels . . . . . 110
8.1.2. Two-factorEquilibriumModels . . . . . . . . . . . . . . . . . . . 117
8.1.3. Non-arbitrageModels,theHo-LeeandHull–WhiteModels . . . . 122
8.2. OtherInterestRateDerivatives . . . . . . . . . . . . . . . . . . . . . . . . 124
Contents vii
9 AmericanStyleofDerivativeSecurities 129
9.1. PricingofAmericanOptionsbySolutionstoFreeBoundaryProblems . . . 131
9.2. Pricing American Style of Options by Solutions to a Linear Complemen-
tarityProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.3. PricingPerpetualOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10 NumericalMethodsforPricingofSimpleDerivatives 141
10.1. Explicit Numerical Finite Difference Method for Solving the Black–
ScholesEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.1.1. DiscreteMethodsBasedonBinomialandTrinomialTrees . . . . . 145
10.2. ImplicitNumericalMethodforSolvingtheBlack–ScholesEquation . . . . 149
10.3. CompendiumofNumericalMethodsforSolvingSystemsofLinearEquations152
10.3.1. LUDecompositionMethod . . . . . . . . . . . . . . . . . . . . . 152
10.3.2. Gauss–SeidelSuccessiveOver-relaxationMethod . . . . . . . . . . 153
10.4. MethodsforSolvingLinearComplementarityProblems . . . . . . . . . . . 156
10.4.1. ProjectedSuccessiveOver-relaxationMethod . . . . . . . . . . . . 156
10.4.2. NumericalSolutionsoftheObstacleProblem . . . . . . . . . . . . 158
10.5. NumericalMethodsforPricingofAmericanStyleOptions . . . . . . . . . 159
10.5.1. IdentificationoftheEarlyExerciseBoundaryforAmericanOptions 161
10.5.2. ImpliedVolatilityforAmericanOptions . . . . . . . . . . . . . . . 164
10.5.3. SourcecodesforNumericalAlgorithms . . . . . . . . . . . . . . . 164
11 NonlinearExtensionsoftheBlack–ScholesPricingModel 167
11.1. Overview of Nonlinear Extensions to the Black–Scholes Option Pricing
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
11.2. RiskAdjustedPricingmethodologyModel . . . . . . . . . . . . . . . . . 170
11.2.1. NonlinearBlack–ScholesEquationfortheRiskAdjustedPricing
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.2.2. DerivationoftheGammaEquation . . . . . . . . . . . . . . . . . 174
11.2.3. PricingofEuropeanStyleofOptionsbytheRAPMModel . . . . . 175
11.2.4. ExplanationoftheVolatilitySmilebytheRAPMModel . . . . . . 177
11.3. ModelingFeedbackEffects . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.3.1. TheCaseoftheStandardBlack–ScholesDeltaHedgingStrategy. . 180
11.3.2. ZeroTrackingErrorStrategyandtheNonlinearBlack–Scholes . . 181
11.3.3. TheGammaEquationforFrey’sNonlinearModel . . . . . . . . . 181
11.4. ModelingInvestor’sPreferences . . . . . . . . . . . . . . . . . . . . . . . 182
11.4.1. TheGammaEquationandaNumericalApproximationScheme . . 188
11.5. JumpingVolatilityModelandLeland’sModel . . . . . . . . . . . . . . . . 189
11.6. FiniteDifferenceSchemeforSolvingtheG Equation . . . . . . . . . . . . 191
12 TransformationMethodsforPricingAmericanOptions 195
12.1. TransformationMethodsforPlainVanillaOptions . . . . . . . . . . . . . . 195
12.1.1. FixedDomainTransformationfortheAmericanCallOption . . . . 196
12.1.2. ReductiontoaNonlinearIntegralEquation . . . . . . . . . . . . . 198
12.1.3. NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . 201
viii Contents
12.1.4. EarlyExerciseBoundaryfortheAmericanPutOption . . . . . . . 204
12.1.5. AnalyticalApproximationValuationFormulabyZhu . . . . . . . . 207
12.2. Transformation Method for a Class of Nonlinear Equations of the Black–
ScholesType . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12.2.1. AlternativeRepresentationoftheEarlyExerciseBoundary . . . . . 211
12.2.2. NumericalIterativeAlgorithmforApproximationoftheEarlyEx-
erciseBoundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
12.2.3. NumericalApproximationsoftheEarlyExerciseBoundary . . . . 214
12.3. EarlyExerciseBoundaryforAmericanStyleofAsianOptions . . . . . . . 218
12.3.1. American-StyleofAsianCallOptions . . . . . . . . . . . . . . . . 221
12.3.2. FixedDomainTransformation . . . . . . . . . . . . . . . . . . . . 221
12.3.3. ANumericalApproximationOperatorSplittingScheme . . . . . . 224
12.3.4. ComputationalExamplesoftheFreeBoundaryApproximation . . 226
13 CalibrationofInterestRateandTermStructureModels 231
13.1. GeneralizedMethodofMoments . . . . . . . . . . . . . . . . . . . . . . . 231
13.2. Nowman’sParameterEstimates . . . . . . . . . . . . . . . . . . . . . . . 233
13.3. MethodBasedonComparisonwithEntireMarketTermStructures . . . . . 236
13.3.1. ParametersReductionPrinciple . . . . . . . . . . . . . . . . . . . 237
13.3.2. TheLossFunctional . . . . . . . . . . . . . . . . . . . . . . . . . 239
13.3.3. Non-linearRegressionProblemfortheLossFunctional . . . . . . . 240
13.3.4. EvolutionStrategies . . . . . . . . . . . . . . . . . . . . . . . . . 241
13.3.5. Calibration Based on Maximization of the Restricted Likelihood
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
13.3.6. QualitativeMeasureofGoodnessofFitandNon-linearR 2 Ratio . 243
13.3.7. ResultsofCalibration . . . . . . . . . . . . . . . . . . . . . . . . 244
13.4. OverviewofOtherMethods . . . . . . . . . . . . . . . . . . . . . . . . . 246
14 AdvancedTopicsintheTermStructureModeling 247
14.1. ApproximateAnalyticalSolutionforaClassofOne-factorModels . . . . . 247
14.1.1. UniquenessofaSolutiontothePDEforBondPrices . . . . . . . . 248
14.1.2. ErrorEstimatesfortheApproximateAnalyticalSolution . . . . . . 251
14.1.3. ImprovedHigherOrderApproximationFormula . . . . . . . . . . 253
14.1.4. Comparison of Approximations to the Exact Solution for the CIR
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
14.2. MathematicalAnalysisoftheTwo-FactorVasicekModel . . . . . . . . . . 255
14.2.1. StatisticalPropertiesofBondPricesandInterestRates . . . . . . . 257
14.2.2. AveragedBondPrice,TermStructuresandTheirConfidenceIntervals261
14.2.3. RelationofAveragedBondPricestoSolutionsofOne-FactorModels262
14.3. TheTwo-FactorCoxIngersollRossModel. . . . . . . . . . . . . . . . . . 264
14.3.1. DistributionofBondPricesandInterestRates . . . . . . . . . . . . 265
14.3.2. Averaged Bond Prices, Term Structures and Their Confidence In-
tervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
14.3.3. RelationofAveragedBondPricestoSolutionsofOne-FactorInter-
estRateModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270