Quantum finance: path integrals and Hamiltonians for options

334 Pages·2004·1.584 MB·English

by  Belal E. Baaquie

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QUANTUM FINANCE
PathIntegralsandHamiltoniansforOptionsandInterestRates
Thisbookappliesthemathematicsandconceptsofquantummechanicsandquan-
tum field theory to the modelling of interest rates and the theory of options.
ParticularemphasisisplacedonpathintegralsandHamiltonians.
Financialmathematicsatpresentisalmostcompletelydominatedbystochastic
calculus. This book is unique in that it offers a formulation that is completely
independent of that approach. As such many new results emerge from the ideas
developedbytheauthor.
Thispioneeringworkwillbeofinteresttophysicistsandmathematicianswork-
inginthefieldoffinance,toquantitativeanalystsinbanksandfinancefirms,andto
practitionersinthefieldoffixedincomesecuritiesandforeignexchange.Thebook
can also be used as a graduate text for courses in financial physics and financial
mathematics.
BELAL E. BAAQUIE earned his B.Sc. from Caltech and Ph.D. in theoretical
physicsfromCornellUniversity.Hehaspublishedover50papersinleadinginter-
national journals on quantum field theory and related topics, and since 1997 has
regularlypublishedpapersonapplyingquantumfieldtheorytoboththetheoretical
andempiricalaspectsoffinance.HehelpedtolaunchtheInternationalJournalof
TheoreticalandAppliedFinancein1998andcontinuestobeoneoftheeditors.

QUANTUM FINANCE
Path Integrals and Hamiltonians for Options and Interest Rates
BELAL E. BAAQUIE
NationalUniversityofSingapore
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Informationo nthi stitle :www.cambri dge.org/9780521840453
©B.E.Baaquie2004
This publication isin copyright. Subject tostatutory exceptionandto theprovision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2004
ISBN-13 978-0-511-26469-6 eBook (EBL)
ISBN-10 0-511-26469-0 eBook (EBL)
ISBN-13 978-0-521-84045-3 hardback
ISBN-10 0-521-84045-7 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
IdedicatethisbooktomyfatherMohammadAbdulBaaquieandtothememory
ofmymotherBegumAjmeriRoanaqAraBaaquie,fortheirpreciouslifelong
supportandencouragement.

Contents
Foreword page xi
Preface xiii
Acknowledgments xv
1 Synopsis 1
PartI Fundamentalconceptsoffinance
2 Introductiontofinance 7
2.1 Efficientmarket:randomevolutionofsecurities 9
2.2 Financialmarkets 11
2.3 Riskandreturn 13
2.4 Timevalueofmoney 15
2.5 Noarbitrage,martingalesandrisk-neutralmeasure 16
2.6 Hedging 18
2.7 Forwardinterestrates:fixed-incomesecurities 20
2.8 Summary 23
3 Derivativesecurities 25
3.1 Forwardandfuturescontracts 25
3.2 Options 27
3.3 Stochasticdifferentialequation 30
3.4 Itocalculus 31
3.5 Black–Scholesequation:hedgedportfolio 34
3.6 Stockpricewithstochasticvolatility 38
3.7 Merton–Garmanequation 39
3.8 Summary 41
3.9 Appendix:Solutionforstochasticvolatilitywithρ = 041
PartII Systemswithfinitenumberofdegreesoffreedom
4 Hamiltoniansandstockoptions 45
4.1 Essentialsofquantummechanics 45
4.2 Statespace:completenessequation 47
vii
viii Contents
4.3 Operators:Hamiltonian 49
4.4 Black–ScholesandMerton–GarmanHamiltonians 52
4.5 Pricingkernelforoptions 54
4.6 Eigenfunctionsolutionofthepricingkernel 55
4.7 Hamiltonianformulationofthemartingalecondition 59
4.8 Potentialsinoptionpricing 60
4.9 Hamiltonianandbarrieroptions 62
4.10 Summary 66
4.11 Appendix:Two-statequantumsystem(qubit) 66
4.12 Appendix:Hamiltonianinquantummechanics 68
4.13 Appendix:Down-and-outbarrieroption’spricingkernel 69
4.14 Appendix:Double-knock-outbarrieroption’spricingkernel 73
4.15 Appendix:SchrodingerandBlack–Scholesequations 76
5 Pathintegralsandstockoptions 78
5.1 Lagrangianandactionforthepricingkernel 78
5.2 Black–ScholesLagrangian 80
5.3 Pathintegralsforpath-dependentoptions 85
5.4 Actionforoption-pricingHamiltonian 86
5.5 Pathintegralforthesimpleharmonicoscillator 86
5.6 Lagrangianforstockpricewithstochasticvolatility 90
5.7 Pricingkernelforstockpricewithstochasticvolatility 93
5.8 Summary 96
5.9 Appendix:Path-integralquantummechanics 96
5.10 Appendix:Heisenberg’suncertaintyprincipleinfinance 99
5.11 Appendix:Pathintegrationoverstockprice 101
5.12 Appendix:Generatingfunctionforstochasticvolatility 103
5.13 Appendix:Momentsofstockpriceandstochasticvolatility 105
5.14 Appendix:Lagrangianforarbitraryα 107
5.15 Appendix:Pathintegrationoverstockpriceforarbitraryα 108
5.16 Appendix:MonteCarloalgorithmforstochasticvolatility 111
5.17 Appendix:Merton’stheoremforstochasticvolatility 115
6 Stochasticinterestrates’Hamiltoniansandpathintegrals 117
6.1 SpotinterestrateHamiltonianandLagrangian 117
6.2 Vasicekmodel’spathintegral 120
6.3 Heath–Jarrow–Morton(HJM)model’spathintegral 123
6.4 MartingaleconditionintheHJMmodel 126
6.5 PricingofTreasuryBondfuturesintheHJMmodel 130
6.6 PricingofTreasuryBondoptionintheHJMmodel 131
6.7 Summary 133
6.8 Appendix:SpotinterestrateFokker–PlanckHamiltonian 134