Tools for Computational Finance (Universitext) (Paperback)byRüdiger U. Seydel(Author)


This book is very easy to read and one can gain a quick snapshot of computational issues arising in financial mathematics. Researchers or students of the mathematical sciences with an interest in finance will find this book a very helpful and gentle guide to the world of financial engineering. SIAM review (46, 2004).

The third edition is thoroughly revised and significantly extended. The largest addition is a new section on analytic methods with main focus on interpolation approach and quadratic approximation. New sections and subsections are among others devoted to risk-neutrality, early-exercise curves, multidimensional Black-Scholes models, the integral representation of options and the derivation of the Black-Scholes equation.

New figures, more exercises, more background material make this guide to the world of financial engineering a real must-to-have for everyone working in FE.

contents

Prefaces ....................................................V
Contents ....................................................XIII
Notation ....................................................XVII
Chapter1ModelingToolsforFinancialOptions ..........1
1.1Options.............................................1
1.2ModeloftheFinancialMarket.........................8
1.3NumericalMethods...................................10
1.4TheBinomialMethod.................................12
1.5Risk-NeutralValuation................................21
1.6StochasticProcesses..................................25
1.6.1WienerProcess.................................26
1.6.2StochasticIntegral..............................28
1.7StochasticDifferentialEquations.......................31
1.7.1Itˆ oProcess....................................31
1.7.2ApplicationtotheStockMarket..................33
1.7.3Risk-NeutralValuation..........................36
1.7.4MeanReversion................................37
1.7.5Vector-ValuedSDEs............................39
1.8ItˆoLemmaandImplications...........................40
1.9JumpProcesses......................................45
NotesandComments......................................48
Exercises.................................................52
Chapter2GeneratingRandomNumberswithSpecified
Distributions ...............................................61
2.1UniformDeviates.....................................62
2.1.1LinearCongruentialGenerators..................62
2.1.2QualityofGenerators...........................63
2.1.3RandomVectorsandLatticeStructure............64
2.1.4FibonacciGenerators...........................67
2.2TransformedRandomVariables.........................69
2.2.1Inversion......................................69
2.2.2TransformationsinIR1..........................70
2.2.3TransformationinIRn ...........................72
2.3NormallyDistributedRandomVariables.................72
2.3.1MethodofBoxandMuller.......................72
2.3.2VariantofMarsaglia............................73
2.3.3CorrelatedRandomVariables....................75
2.4MonteCarloIntegration...............................77
2.5SequencesofNumberswithLowDiscrepancy.............79
2.5.1Discrepancy....................................79
2.5.2ExamplesofLow-DiscrepancySequences..........82
NotesandComments......................................85
Exercises.................................................87
Chapter3SimulationwithStochasticDifferential
Equations ...................................................91
3.1ApproximationError..................................92
3.2StochasticTaylorExpansion...........................95
3.3ExamplesofNumericalMethods........................98
3.4IntermediateValues...................................102
3.5MonteCarloSimulation...............................102
3.5.1IntegralRepresentation..........................103
3.5.2TheBasicVersionforEuropeanOptions...........104
3.5.3Bias..........................................107
3.5.4VarianceReduction.............................108
3.5.5AmericanOptions..............................111
3.5.6FurtherHints..................................116
NotesandComments......................................117
Exercises.................................................119
Chapter4StandardMethodsforStandardOptions .......123
4.1Preparations.........................................124
4.2FoundationsofFinite-DifferenceMethods................126
4.2.1DifferenceApproximation........................126
4.2.2TheGrid......................................127
4.2.3ExplicitMethod................................128
4.2.4Stability.......................................130
4.2.5AnImplicitMethod.............................133
4.3Crank-NicolsonMethod...............................135
4.4BoundaryConditions..................................138
4.5AmericanOptionsasFreeBoundaryProblems...........140
4.5.1Early-ExerciseCurve............................141
4.5.2FreeBoundaryProblems........................143
4.5.3Black-ScholesInequality.........................146
4.5.4ObstacleProblems..............................148
4.5.5LinearComplementarityforAmericanPutOptions.151ContentsXV
4.6ComputationofAmericanOptions......................152
4.6.1DiscretizationwithFiniteDifferences.............152
4.6.2IterativeSolution...............................154
4.6.3AnAlgorithmforCalculatingAmericanOptions....157
4.7OntheAccuracy.....................................161
4.7.1ElementaryErrorControl.......................162
4.7.2Extrapolation..................................165
4.8AnalyticMethods.....................................165
4.8.1ApproximationBasedonInterpolation............167
4.8.2QuadraticApproximation........................169
4.8.3AnalyticMethodofLines........................172
4.8.4MethodsEvaluatingProbabilities.................173
NotesandComments......................................174
Exercises.................................................178
Chapter5Finite-ElementMethods .......................183
5.1WeightedResiduals...................................184
5.1.1ThePrincipleofWeightedResiduals..............184
5.1.2ExamplesofWeightingFunctions.................186
5.1.3ExamplesofBasisFunctions.....................187
5.2GalerkinApproachwithHatFunctions..................188
5.2.1HatFunctions..................................189
5.2.2Assembling....................................191
5.2.3ASimpleApplication...........................192
5.3ApplicationtoStandardOptions.......................194
5.4ErrorEstimates......................................198
5.4.1ClassicalandWeakSolutions....................199
5.4.2ApproximationonFinite-DimensionalSubspaces...201
5.4.3C´ ea’sLemma..................................202
NotesandComments......................................205
Exercises.................................................206
Chapter6PricingofExoticOptions ......................209
6.1ExoticOptions.......................................210
6.2OptionsDependingonSeveralAssets...................211
6.3AsianOptions........................................214
6.3.1ThePayoff.....................................214
6.3.2ModelingintheBlack-ScholesFramework.........215
6.3.3ReductiontoaOne-DimensionalEquation.........216
6.3.4DiscreteMonitoring.............................220
6.4NumericalAspects....................................222
6.4.1Convection-DiffusionProblems...................222
6.4.2VonNeumannStabilityAnalysis.................225
6.5UpwindSchemesandOtherMethods....................226
6.5.1UpwindScheme................................226
6.5.2Dispersion.....................................230
6.6High-ResolutionMethods..............................231
6.6.1TheLax-WendroffMethod.......................231
6.6.2TotalVariationDiminishing......................232
6.6.3NumericalDissipation...........................233
NotesandComments......................................235
Exercises.................................................237
Appendices .................................................239
AFinancialDerivatives..................................239
A1InvestmentandRisk............................239
A2FinancialDerivatives............................240
A3ForwardsandtheNo-ArbitragePrinciple..........243
A4TheBlack-ScholesEquation......................244
A5Early-ExerciseCurve............................249
BStochasticTools......................................253
B1EssentialsofStochastics.........................253
B2AdvancedTopics...............................257
B3State-PriceProcess.............................260
CNumericalMethods...................................265
C1BasicNumericalTools...........................265
C2IterativeMethodsfor Ax = b .....................270
C3FunctionSpaces................................272
DComplementaryMaterial..............................277
D1BoundsforOptions.............................277
D2ApproximationFormula.........................279
D3Software.......................................281
References ..................................................283
Index .......................................................293