Lecture Notes on discrete time finance

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Lecture Notes on discrete time finance
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Lecture Notes on Discrete-time Finance

 Chuan shu Ji

Contents

1ModelSpeci?cations1
1.1Assetpricedynamics...............................1
1.2Tradingstrategies................................2
1.3Valueprocesses,gainprocessesandself-?nancingstrategies.........2
1.4Discountedprices.................................3
2BinomialTrees:anExample5
2.1IllustrationofconceptsintroducedinLecture1................5
2.2Whatisafairprice?...............................8
2.3Riskneutralprobabilities............................12
3ArbitrageandRiskNeutralProbabilityMeasures13
3.1Someeconomicconsiderations..........................13
3.2ProofofTheorem3.1:su?ciencyΓ=Γ?....................1
3.3ProofofTheorem3.1:necessityΓflΓ?.....................1
4RiskNeutralValuationofContingentClaims17
4.1Lawofonepriceandriskneutralvaluationprinciple.............17
4.2Completemarkets................................18
5BinomialTrees:aGeneralSetting20
5.1Thebasicbinomialtreemodel.........................20
5.2Optionpricingusingbinomialtrees.......................21
ii

6TheBlack-ScholesOptionPricingFormula23
7AmericanOptionsasOptimalStoppingProblems26
7.1Aspecialcase:ΓflAmerican=EuropeanΓ?....................
7.2Optimalstopping.................................27
8MoreonValuationofAmericanOptions32
8.1ΓflAmericancalls=EuropeancallsΓ?.......................
8.2Optionsonadividend-payingstock.......................33
9ReturnandRisk35
9.1Returnprocesses.................................35
9.2Riskpremium(singleperiod)..........................36
10OptimalPortfolios39
10.1Optimalportfolios................................39
10.2Computationviadynamicprogramming....................40
11OptimizationviaEMMs42
11.1Basicapproach..................................42
11.2Examples.....................................43
12TheBinomialCapitalAssetPricingModel45
13CashFlowsandForwardPrices48
13.1Dividendsandreturns..............................48
13.2Forwardcontractsandprices..........................49
14FuturesContracts53
14.1Futuresvsforwardcontracts..........................53
14.2Futuresprices...................................54
14.3Optionsonfutures................................57
iii

15Zero-couponBonds,YieldsandForwardRates59
16ExamplesofTermStructureModels62
17SpotRateModellingviaMarkovChainsandStochasticDi?erenceEqua-
tions 68
17.1SpotrateMarkovchains.............................68
17.2Stochasticdi?erenceequations.........................70
18Vasicek,Cox-Ingersoll-Ross,andHull-WhiteModels72
18.1VasicekandCIRmodels.............................72
18.2Hull-Whitemodel.................................73
18.3Re?nedlatticeandSDE.............................73
19Heath-Jarrow-MortonApproachandHo-LeeModel75
19.1HJMsetting....................................75
19.2Ho-Leemodel...................................77
19.3SpotratemodelsfromHo-Lee..........................79
20ForwardRiskAdjustedProbabilityMeasures80
21BondOptionsandCouponBonds83
22Swaps,CapsandFloors86
22.1Swapsandswaptions...............................86
22.2Capsand?oors..................................88
23ImpliedVolatility90
23.1InvertingtheBlack-Scholesformula.......................90
23.2Volatilitysmile..................................92
24ImpliedVolatilityTrees94
24.1Constructionofimpliedvolatilitytreesviaforwardinduction........94
iv

24.2Speci?cationof V viaArrow-Debreusecurities...............96
put
24.3Howtodealwithpossiblebadprobabilities?..................97

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