Introduction to Stochastic Finance

406 Pages·2018·3.885 MB·English

by  Jia-An Yan

Universitext
Jia-An Yan
Introduction
to Stochastic
Finance
Universitext
Universitext
Serieseditors
SheldonAxler
SanFranciscoStateUniversity
CarlesCasacuberta
UniversitatdeBarcelona
AngusMacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski
CaseWesternReserveUniversity
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Jia-An Yan
Introduction to Stochastic
Finance
123
Jia-AnYan
AcademyofMathematicsandSystemScience
ChinesesAcademyofSciences
Beijing,China
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-981-13-1656-2 ISBN978-981-13-1657-9 (eBook)
https://doi.org/10.1007/978-981-13-1657-9
LibraryofCongressControlNumber:2018952344
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Preface
The history of financial mathematics can be traced back to the French mathemati-
cian Louis Bachelier’s doctoral dissertation “Théorie de la speculation” in 1900
Bachelier (1900). Bachelier’s work, however, was not known to most economists
until Paul A. Samuelson mentioned it in an article of 1965. In 1969 and 1971,
Robert C. Merton studied the optimal portfolio problem in continuous-time using
the stochastic dynamic programming method. In 1973, Fischer Black and Myron
S. Scholes used stochastic analysis, in particular, Kiyosi Itô’s formula, to derive
the famous Black-Scholes formula for option pricing. Almost at the same time,
Merton(1973a)improvedtheBlack-Scholesmodelanddevelopedanideaofusing
options to evaluate a company’s debt in the so-called contingent claims analysis.
Harrison and Kreps (1979) proposed a martingale method to characterize a no-
arbitrage market and the use of equivalent martingale measure on options pricing
and hedging, which had profound influence on the subsequent development of
financial mathematics. Since 1970s, in order to study the pricing of interest rate
derivatives,manyscholarshaveproposedseveralinterestratetermstructuremodels,
including Vasicek, CIR, HJM, and BGM models, which could reflect the market
trendofthefuturespotrate.
For more than half a century, many scholars worked research on theory and
applications of financial mathematics (also known as mathematical finance) and
published many books in the area. Financial mathematics not only has a direct
impact on the innovation of financial instruments and on the efficient functioning
of financial markets but also is widely used in investment decisions, valuation of
researchanddevelopmentprojects,andinriskmanagement.
This book is intended to give a systematic introduction to the basic theory of
financial mathematics,withanemphasisonapplications ofmartingalemethodsin
pricing and hedging of contingent claims, interest rate term structure models, and
expected utility maximization problems. The book consists of 14 chapters. Chap-
ter1introducesthebasictheoryofprobabilityanddiscretetimemartingales,which
is specially designed for readers who do not have much knowledge of probability
theory.InChap.2,weintroducethetheoryofdiscretetimeportfolioselection(i.e.,
Harry M. Markowitz’s mean-variance analysis), the capital asset pricing model
v
vi Preface
(CAPM), the arbitrage pricing theory (APT), and the multistage mean-variance
analysis. The basic idea of expected utility theory and the consumption-based
asset pricing model are also sketched in this chapter. In Chap.3, we introduce
discrete time financial markets and martingale characterization of arbitrage-free
markets and present the martingale method for the expected utility maximization
and the risk-neutral pricing principle for European contingent claims. Chapter 4
systematically presents Itô’s theory of stochastic analysis (including Itô’s integral
andItô’sformula,Girsanov’stheorem,andthemartingalerepresentationtheorem)
which is the theoretical basis of the martingale method in financial mathematics.
Thischaptercanbeusedseparatelyasaconcisetextforpostgraduatesinprobability
studying Itô’s theory of stochastic analysis. In Chap.5, for the Black-Scholes
model, we introduce the martingale method for pricing and hedging European
contingent claims and derive the Black-Scholes formula for pricing European
options. The pricing of American options is also briefly discussed. In addition,
some examples are given to illustrate applications of the martingale method, and
several modifications of the Black-Scholes model are presented. In Chap.6, we
introduce several commonly used exotic options: barrier options, Asian options,
lookbackoptions,andresetoptions.Thepricingandhedgingoftheseexoticoptions
are investigated with the martingale method and partial differential equations.
Chapter7presentsItôprocessanddiffusionprocessmodels.Themartingalemethod
of contingent claim pricing is presented in detail, including an introduction of
time and scale transformation to give some explicit formulas for option pricing.
Chapter 8 introduces the bond market and interest rate term structure models,
including a variety of single-factor short-term interest rate model, HJM model,
and BGM Model, and studies the pricing of interest rate derivatives. Chapter 9
introduces optimal investment portfolios and investment-consumption strategies
for diffusion process models. Within L2-allowable trading strategies, the risk-
mean portfolio selection problem, expected utility maximization problem, and
the selection of portfolio strategy with consumption are investigated. Chapter 10
introduces the general theory of static risk measures, which includes consistent
risk measures, convex risk measures, comonotonically sub-additive risk measures,
comonotonically convex risk measure, and a variety of distribution invariant risk
measures, as well as their characterizations and representations. In Chap.11, after
a brief overview of semimartingales and stochastic calculus, we introduce some
basicconceptsandresultsonmarketsofsemimartingalemodelandgivenumeraire-
free characterizations of attainable contingent claims. In Chap.12, we give a
survey on convex duality theory for optimal investment and present a numeraire-
free and original probability-based framework for financial markets. The expected
utilitymaximizationandvaluationproblemsinageneralsemimartingalesettingare
studiedinChap.13.ForamarketdrivenbyaLévyprocess,theoptimalportfolioand
therelatedmartingalemeasureareworkedoutexplicitlyforsomeparticulartypesof
utilityfunction.Finally,inChap.14,weintroducethe“optimalgrowthportfolios”
inmarketsofsemimartingalemodelandworkouttheirexpressionsinageometric
Lévyprocessmodelandajump-diffusion-likeprocessmodel.
Preface vii
I have taught from the manuscript of this book in an introductory course of
mathematicalfinanceformyformergraduatestudents.Thankstothemforreporting
misprints and errors. I am grateful to Prof. Jianming Xia for contributing to the
writingofSect.9.2ofChap.9;toDr.YongshengSongforhisPhDthesis,whichis
the basic material for Chap.10 of the book; and to Dr. Jun Yan for typographical
andgrammaticalsuggestions.
Beijing,China Jia-AnYan
May,2018
Contents
1 Foundation of Probability Theory and Discrete-Time
Martingales ................................................................. 1
1.1 BasicConceptsofProbabilityTheory............................... 1
1.1.1 EventsandProbability ..................................... 1
1.1.2 Independence,0-1Law,andBorel-CantelliLemma...... 3
1.1.3 Integrals,(Mathematical)ExpectationsofRandom
Variables .................................................... 5
1.1.4 ConvergenceTheorems .................................... 7
1.2 ConditionalMathematicalExpectation ............................. 9
1.2.1 DefinitionandBasicProperties............................ 9
1.2.2 ConvergenceTheorems .................................... 14
1.2.3 TwoTheoremsAboutConditionalExpectation........... 15
1.3 DualsofSpacesL∞((cid:2),F)andL∞((cid:2),F,m) .................... 17
1.4 FamilyofUniformlyIntegrableRandomVariables ................ 18
1.5 DiscreteTimeMartingales........................................... 22
1.5.1 BasicDefinitions ........................................... 22
1.5.2 BasicTheorems............................................. 24
1.5.3 MartingaleTransforms..................................... 27
1.5.4 SnellEnvelop ............................................... 30
1.6 MarkovSequences ................................................... 31
2 PortfolioSelectionTheoryinDiscrete-Time............................. 33
2.1 Mean-VarianceAnalysis............................................. 34
2.1.1 Mean-Variance Frontier Portfolios Without
Risk-FreeAsset............................................. 35
2.1.2 RevisedFormulationsofMean-VarianceAnalysis
WithoutRisk-FreeAsset ................................... 38
2.1.3 Mean-VarianceFrontierPortfolioswithRisk-Free
Asset......................................................... 43
2.1.4 Mean-VarianceUtilityFunctions .......................... 45
ix
x Contents
2.2 CapitalAssetPricingModel(CAPM) .............................. 47
2.2.1 MarketCompetitiveEquilibriumandMarketPortfolio .. 47
2.2.2 CAPMwithRisk-FreeAsset............................... 49
2.2.3 CAPMWithoutRisk-FreeAsset........................... 52
2.2.4 EquilibriumPricingUsingCAPM ........................ 53
2.3 ArbitragePricingTheory(APT)..................................... 54
2.4 Mean-SemivarianceModel .......................................... 57
2.5 MultistageMean-VarianceModel................................... 58
2.6 ExpectedUtilityTheory ............................................. 62
2.6.1 UtilityFunctions............................................ 63
2.6.2 Arrow-Pratt’sRiskAversionFunctions.................... 64
2.6.3 ComparisonofRiskAversionFunctions.................. 66
2.6.4 PreferenceDefinedbyStochasticOrders.................. 66
2.6.5 MaximizationofExpectedUtilityandInitialPrice
ofRiskyAsset .............................................. 70
2.7 Consumption-BasedAssetPricingModels ......................... 72
3 FinancialMarketsinDiscreteTime...................................... 75
3.1 BasicConceptsofFinancialMarkets ............................... 75
3.1.1 Numeraire................................................... 76
3.1.2 PricingandHedging........................................ 76
3.1.3 Put-CallParity.............................................. 76
3.1.4 IntrinsicValueandTimeValue ............................ 77
3.1.5 Bid-AskSpread............................................. 77
3.1.6 EfficientMarketHypothesis ............................... 78
3.2 BinomialTreeModel ................................................ 78
3.2.1 TheOne-PeriodCase....................................... 78
3.2.2 TheMultistageCase........................................ 79
3.2.3 TheApproximatelyContinuousTradingCase............ 82
3.3 TheGeneralDiscrete-TimeModel.................................. 83
3.3.1 TheBasicFramework...................................... 83
3.3.2 Arbitrage,Admissible,andAllowableStrategies......... 85
3.4 MartingaleCharacterizationofNo-ArbitrageMarkets............. 86
3.4.1 TheFiniteMarketCase .................................... 86
3.4.2 TheGeneralCase:Dalang-Morton-WillingerTheorem.. 87
3.5 PricingofEuropeanContingentClaims............................. 90
3.6 MaximizationofExpectedUtilityandOptionPricing............. 92
3.6.1 GeneralUtilityFunctionCase ............................. 92
3.6.2 HARAUtilityFunctionsandTheirDualityCase......... 94
3.6.3 UtilityFunction-BasedPricing ............................ 96
3.6.4 MarketEquilibriumPricing................................ 99
3.7 AmericanContingentClaimsPricing ............................... 103
3.7.1 Super-HedgingStrategiesinCompleteMarkets.......... 103
3.7.2 Arbitrage-FreePricinginCompleteMarkets ............. 104
3.7.3 Arbitrage-FreePricinginNon-completeMarkets ........ 105