Derivatives and Internal Models

686 Pages·2004·5.025 MB·English

by  Dr Hans-Peter Deutsch (auth.)

Derivatives and
Internal Models
Third Edition
Hans-Peter Deutsch
Derivatives and Internal Models
Third Edition
This page intentionally left blank
Derivatives and Internal Models
Third Edition
Dr Hans-Peter Deutsch
©Hans-PeterDeutsch2004
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Preface
Thephilosophyofthisbookistoprovideanintroductiontothevaluationandrisk
management of modern (cid:31)nancial instruments formulated in precise (and mathemat-
ically correct) expressions, covering all pertinent topics with a consistent and exact
notation and with a depth of detail su(cid:33)cient to give the reader a truly sound under-
standingofthematerial. Anunderstandingwhichevenplacesthereaderinaposition
toindependentlydevelop pricingandriskmanagementalgorithms(includingactually
writing computer programs), should this be necessary. Such tasks will greatly be fa-
cilitatedbytheCD-ROMaccompanyingthebook. ThisCD-ROMcontainsMicrosoft
ExcelTM workbooks presenting concrete realizations of the concepts discussed in the
bookintheformofexecutablealgorithms. Ofcourse,thereaderhasfullaccesstoall
source codes of the Visual BasicTM modules as well as to all calculations done in the
spread sheet cells. The CD-ROM thus contains a collection of literally thousands of
examplesprovidingthereaderwithvaluableassistanceinunderstandingthecomplex
materialandservingasthepotentialbasisforthefurtherdevelopmentofthereader’s
ownparticularpricingandriskmanagementprocedures.
Thebookshouldequipthereaderwithawidearrayoftoolsneededforallessential
topics in the (cid:31)eld of modern market risk management. The reader is not expected
to have previous knowledge of (cid:31)nance, but rather a sound mathematical and analyt-
ical background typical of scientists, mathematicians, computer scientists, engineers,
etc.Thenoviceisnotevenrequiredtobefamiliarwithideasasfundamentalascom-
pounding interest. The book, however, is certainly also of interestto the experienced
risk manager or (cid:31)nancial engineer, since the concepts introduced are widely elabo-
rated upon and analyzed down to the very foundations, making a comprehension of
the material possible which goes signi(cid:31)cantly beyond the level held to be “common
knowledge”inthis(cid:31)eld.
Since the beauty of a room behind a closed door is of little use if the door itself
cannotbefound,emphasishasbeenplacedonprovidinganeasyentryintotheanalysis
of each of the various topics. As the author does not wish to lose the reader at the
outset, or expect the reader to (cid:31)rst engage in the study of quoted literature before
proceeding, the book is practically self-contained. An explanation of almost every
expressionornotionneededcanbefoundinthebookitself,rangingfromcompounding
interesttotermstructuremodels,fromexpectationtoValueatRisk,fromtimeseries
analysistoGARCHmodels,fromarbitragetodi(cid:30)erentialequationsandexoticoptions,
fromthenormaldistributiontomartingales,andsoon.
v
vi
The selection of the topics and the nature of their presentation result to a great
extent frommy personal experience as a consultant in the world of (cid:31)nancial services;
(cid:31)rst with the Financial Risk Consulting division of Arthur Andersen in Germany,
which it has been my pleasure to establish and direct for many years, and later with
the company d-(cid:31)ne, which is in fact this former Financial Risk Consulting division,
nowoperatingasacompanyofitsown. Inthesefunctions,Ihavebeeninapositionto
observeandidentifyexactlywhatknowledgeandmethodsarerequiredinthe(cid:31)nancial
worldaswellastoseewhattoolsareindispensableforanewcomertothisworld.
Iwouldliketotakethisopportunitytothankmanyofthe(inpartformer)members
oftheFinancialRiskConsulting teamandofd-(cid:31)ne fortheirvaluableinputandmany
fruitfuldiscussionsnotonlyconcerningthisbook,butalsoinourdaytodayconsulting
work.
Wiesbaden,October2003
Hans-PeterDeutsch
Contents
I Fundamentals 1
1 Introduction 3
2 Legal Environment 7
3 Fundamental Risk Factors of Financial Markets 11
3.1 InterestRates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 DayCountConventions . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2 BusinessDayConventions . . . . . . . . . . . . . . . . . . . . . . 14
3.1.3 DiscountFactors . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.4 CompoundingMethods . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.5 SpotRates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.6 ForwardRates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 MarketPrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 AnIntuitiveModelforFinancialRiskFactors . . . . . . . . . . . . . . . 24
3.3.1 RandomWalksastheBasisforPricingandRiskModels. . . . . 24
3.3.2 RiskFactorsasRandomWalks . . . . . . . . . . . . . . . . . . . 27
3.4 ItoProcessesandStochasticAnalysis . . . . . . . . . . . . . . . . . . . 34
3.4.1 GeneralDi(cid:30)usionProcesses . . . . . . . . . . . . . . . . . . . . . 34
3.4.2 Ito’sLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.3 TransitionProbabilities,ForwardandBackwardEquation . . . . 39
3.4.4 ForwardandBackwardEquationintheBlack-ScholesWorld . . 46
4 Financial Instruments: A System of Derivatives and Underlyings 47
4.1 SpotTransactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 MoneyMarketSecurities . . . . . . . . . . . . . . . . . . . . . . 48
4.1.2 CapitalMarketSecurities . . . . . . . . . . . . . . . . . . . . . . 52
4.1.3 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 ForwardTransactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
vii
viii CONTENTS
II Methods 61
5 Overview of the Assumptions 63
6 Present Value Methods, Yields and Traditional Risk Measures 67
6.1 PresentValueandYieldtoMaturity . . . . . . . . . . . . . . . . . . . . 67
6.2 InternalRateofReturnandNetPresentValue . . . . . . . . . . . . . . 69
6.3 AccruedInterest,ResidualDebtandParRates . . . . . . . . . . . . . . 72
6.4 TraditionalSensitivitiesofInterestRateInstruments . . . . . . . . . . . 75
6.4.1 AverageLifetimeandMacaulayDuration . . . . . . . . . . . . . 75
6.4.2 Modi(cid:31)edDurationandConvexity. . . . . . . . . . . . . . . . . . 75
6.4.3 SummationofTraditionalSensitivities . . . . . . . . . . . . . . . 79
7 Arbitrage 81
7.1 ForwardContracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.1.1 ForwardPriceandCash&CarryArbitrage . . . . . . . . . . . . 81
7.1.2 TheStochasticProcessfortheForwardPrice . . . . . . . . . . . 83
7.1.3 ForwardPositions . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1.4 FuturePositionsandBasisRisk. . . . . . . . . . . . . . . . . . . 84
7.2 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2.1 UpperandLowerBoundsforOptionPrices . . . . . . . . . . . 85
7.2.2 EarlyExerciseofAmericanOptions . . . . . . . . . . . . . . . . 86
7.2.3 RelationshipsbetweenPutsandCalls . . . . . . . . . . . . . . . 87
8 The Black-Scholes Di(cid:30)erential Equation 89
8.1 TheBlack-ScholesEquationfromArbitrageArguments . . . . . . . . . 90
8.1.1 TheBlack-ScholesEquationforEuropeanOptions . . . . . . . . 90
8.1.2 TheBlack-ScholesInequalityforAmericanOptions . . . . . . . 92
8.1.3 AFirstContactwiththeRisk-NeutralWorld . . . . . . . . . . . 95
8.2 TheBlack-ScholesEquationandtheBackwardEquation. . . . . . . . . 95
8.2.1 ASecondContactwiththeRisk-NeutralWorld . . . . . . . . . . 98
8.3 TheRelationshiptotheHeatEquation. . . . . . . . . . . . . . . . . . . 98
9 Integral Forms and Analytic Solutions in the Black-Scholes World 103
9.1 OptionPricesasSolutionsoftheHeatEquation . . . . . . . . . . . . . 103
9.2 OptionPricesandTransitionProbabilities . . . . . . . . . . . . . . . . . 105
9.3 CompilationofBlack-ScholesOptionPricesforDi(cid:30)erentUnderlyings. . 108
9.3.1 OptionsontheSpotPrice . . . . . . . . . . . . . . . . . . . . . . 108
9.3.2 OptionsontheForwardPrice . . . . . . . . . . . . . . . . . . . . 108
9.3.3 OptionsonInterestRates . . . . . . . . . . . . . . . . . . . . . . 110
10 Numerical Solutions Using Finite Di(cid:30)erences 113
10.1 DiscretizingtheBlack-ScholesEquation . . . . . . . . . . . . . . . . . . 114
10.1.1 TheExplicitMethod . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.1.2 TheImplicitMethod . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.1.3 CombinationsofExplicitandImplicitMethods(Crank-Nicolson) 115
10.1.4 SymmetricFiniteDi(cid:30)erencesoftheUnderlyingPrice. . . . . . . 117
CONTENTS ix
10.2 Di(cid:30)erenceSchemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.2.1 InitialConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.2.2 DirichletBoundaryConditions . . . . . . . . . . . . . . . . . . . 123
10.2.3 NeumannBoundaryCondition . . . . . . . . . . . . . . . . . . . 127
10.2.4 Unspeci(cid:31)edBoundaryConditions. . . . . . . . . . . . . . . . . . 132
10.2.5 FreeBoundaryConditionsforAmericanOptions . . . . . . . . . 135
10.3 ConvergenceCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.3.1 ImprovingtheConvergenceProperties . . . . . . . . . . . . . . . 141
10.4 DiscreteDividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11Binomial and Trinomial Trees 149
11.1 GeneralTrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
11.1.1 EvolutionoftheUnderlyingandtheReplicatingPortfolio . . . . 149
11.1.2 EvolutionoftheDerivative . . . . . . . . . . . . . . . . . . . . . 150
11.1.3 ForwardContracts . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.2 RecombinantTrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.2.1 TheUnderlying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.2.2 TheBinomialDistributionforEuropeanDerivatives . . . . . . . 153
11.2.3 AThirdContactwiththeRisk-NeutralWorld . . . . . . . . . . 157
11.3 TheRelationshipbetweenRandomWalkandBinomialParameters . . . 159
11.4 TheBinomialModelwithIn(cid:31)nitesimalSteps . . . . . . . . . . . . . . . 162
11.4.1 ComponentsoftheBlack-ScholesOptionPricingFormula . . . . 163
11.5 TrinomialTrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
11.5.1 TheTrinomialTreeasanImprovedBinomialTree . . . . . . . . 167
11.5.2 RelationshiptotheExplicitFiniteDi(cid:30)erenceMethod . . . . . . 167
12Monte Carlo Simulations 169
12.1 ASimpleExample: TheAreaofaDisk . . . . . . . . . . . . . . . . . . 171
12.2 TheGeneralApproachtoMonteCarloSimulations . . . . . . . . . . . . 174
12.3 MonteCarloSimulationofRiskFactors . . . . . . . . . . . . . . . . . . 175
12.3.1 SimulationoftheEvolutionofaSingle RiskFactor. . . . . . . . 175
12.3.2 SimulationofSeveral CorrelatedRiskFactors . . . . . . . . . . . 178
12.4 Pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
13Hedging 183
13.1 ReplicatingPortfoliosasSyntheticDerivatives . . . . . . . . . . . . . . 183
13.2 HedgingDerivativeswithSpotTransactions . . . . . . . . . . . . . . . . 183
13.2.1 ForwardsandFuturesasDerivatives . . . . . . . . . . . . . . . . 185
13.3 HedgingDerivativeswithForwardContracts . . . . . . . . . . . . . . . 186
13.3.1 HedgingwithForwards . . . . . . . . . . . . . . . . . . . . . . . 187
13.3.2 HedgingwithFutures . . . . . . . . . . . . . . . . . . . . . . . . 189
13.3.3 TheDi(cid:30)erentialEquationforDerivativesonFutures . . . . . . . 190
13.4 Hedge-RatiosforArbitraryCombinationsofFinancialInstruments . . . 191
13.5 “Greek”RiskManagementwithSensitivities . . . . . . . . . . . . . . . 193
13.5.1 SensitivitiesandaPortfolio’sChangeinValue . . . . . . . . . . 193