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Stochastic Calculus and Financial Applications
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Stochastic Calculus and Financial Applications
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Steven Shreve: Stochastic Calculus and Finance
PRASAD CHALASANI
Carnegie Mellon University
chal@cs.cmu.edu
SOMESH JHA
Carnegie Mellon University
sjha@cs.cmu.edu
c
Copyright; Steven E. Shreve, 1996
July 25, 1997
Contents
1 Introduction to Probability Theory 11
1.1 TheBinomialAssetPricingModel.......................... 11
1.2 Finite Probability Spaces ............................... 16
1.3 LebesgueMeasureandtheLebesgueIntegral .................... 22
1.4 General Probability Spaces .............................. 30
1.5 Independence..................................... 40
1.5.1 Independenceofsets............................. 40
1.5.2 Independence of
-algebras ......................... 41
1.5.3 Independence of random variables ...................... 42
1.5.4 Correlationandindependence ........................ 44
1.5.5 Independenceandconditionalexpectation. ................. 45
1.5.6 LawofLargeNumbers............................ 46
1.5.7 CentralLimitTheorem............................ 47
2 Conditional Expectation 49
2.1 ABinomialModelforStockPriceDynamics .................... 49
2.2 Information...................................... 50
2.3 ConditionalExpectation ............................... 52
2.3.1 Anexample.................................. 52
2.3.2 Definition of Conditional Expectation .................... 53
2.3.3 FurtherdiscussionofPartialAveraging ................... 54
2.3.4 PropertiesofConditionalExpectation.................... 55
2.3.5 ExamplesfromtheBinomialModel..................... 57
2.4 Martingales...................................... 58
1
2
3 Arbitrage Pricing 59
3.1 BinomialPricing ................................... 59
3.2 Generalone-stepAPT................................. 60
3.3 Risk-Neutral Probability Measure .......................... 61
3.3.1 PortfolioProcess............................... 62
3.3.2 Self-financing Value of a Portfolio Process
................ 62
3.4 Simple European Derivative Securities ........................ 63
3.5 TheBinomialModelisComplete........................... 64
4 The Markov Property 67
4.1 BinomialModelPricingandHedging ........................ 67
4.2 ComputationalIssues................................. 69
4.3 MarkovProcesses................................... 70
4.3.1 DifferentwaystowritetheMarkovproperty ................ 70
4.4 ShowingthataprocessisMarkov .......................... 73
4.5 ApplicationtoExoticOptions ............................ 74
5 Stopping Times and American Options 77
5.1 AmericanPricing................................... 77
5.2 ValueofPortfolioHedginganAmericanOption................... 79
5.3 Information up to a Stopping Time .......................... 81
6 Properties of American Derivative Securities 85
6.1 Theproperties..................................... 85
6.2 ProofsoftheProperties................................ 86
6.3 Compound European Derivative Securities ...................... 88
6.4 OptimalExerciseofAmericanDerivativeSecurity.................. 89
7 Jensen’s Inequality 91
7.1 Jensen’s Inequality for Conditional Expectations ................... 91
7.2 OptimalExerciseofanAmericanCall........................ 92
7.3 Stopped Martingales ................................. 94
8 Random Walks 97
8.1 FirstPassageTime .................................. 97
3
8.2
isalmostsurelyfinite................................ 97
8.3 The moment generating function for
........................ 99
8.4 Expectation of
.................................... 100
8.5 TheStrongMarkovProperty............................. 101
8.6 GeneralFirstPassageTimes ............................. 101
8.7 Example:PerpetualAmericanPut .......................... 102
8.8 DifferenceEquation.................................. 106
8.9 DistributionofFirstPassageTimes.......................... 107
8.10TheReflectionPrinciple ............................... 109
9 Pricing in terms of Market Probabilities: The Radon-Nikodym Theorem. 111
9.1 Radon-Nikodym Theorem .............................. 111
9.2 Radon-Nikodym Martingales . . . .......................... 112
9.3 TheStatePriceDensityProcess ........................... 113
9.4 Stochastic Volatility Binomial Model ......................... 116
9.5 Another Applicaton of the Radon-Nikodym Theorem . ............... 118
10 Capital Asset Pricing 119
10.1AnOptimizationProblem............................... 119
11 General Random Variables 123
11.1 Law of a Random Variable .............................. 123
11.2 Density of a Random Variable . . .......................... 123
11.3Expectation...................................... 124
11.4 Two random variables ................................. 125
11.5MarginalDensity ................................... 126
11.6ConditionalExpectation ............................... 126
11.7ConditionalDensity.................................. 127
11.8MultivariateNormalDistribution........................... 129
11.9Bivariatenormaldistribution............................. 130
11.10MGF of jointly normal random variables ....................... 130
12 Semi-Continuous Models 131
12.1Discrete-timeBrownianMotion ........................... 131
4
12.2TheStockPriceProcess................................ 132
12.3RemainderoftheMarket............................... 133
12.4Risk-NeutralMeasure................................. 133
12.5Risk-NeutralPricing ................................. 134
12.6Arbitrage ....................................... 134
12.7StalkingtheRisk-NeutralMeasure.......................... 135
12.8PricingaEuropeanCall................................ 138
13 Brownian Motion 139
13.1 Symmetric Random Walk ............................... 139
13.2TheLawofLargeNumbers.............................. 139
13.3CentralLimitTheorem ................................ 140
13.4 Brownian Motion as a Limit of Random Walks ................... 141
13.5BrownianMotion................................... 142
13.6CovarianceofBrownianMotion ........................... 143
13.7Finite-DimensionalDistributionsofBrownianMotion................ 144
13.8 Filtration generated by a Brownian Motion ...................... 144
13.9MartingaleProperty.................................. 145
13.10TheLimitofaBinomialModel............................ 145
13.11StartingatPointsOtherThan0............................ 147
13.12MarkovPropertyforBrownianMotion........................ 147
13.13Transition Density ................................... 149
13.14FirstPassageTime .................................. 149
14 The It
ˆ
o Integral 153
14.1BrownianMotion................................... 153
14.2FirstVariation..................................... 153
14.3QuadraticVariation.................................. 155
14.4 Quadratic Variation as Absolute Volatility ...................... 157
14.5 Construction of the ItˆoIntegral............................ 158
14.6 Itˆointegralofanelementaryintegrand........................ 158
14.7 Properties of the Itˆointegralofanelementaryprocess................ 159
14.8 Itˆointegralofageneralintegrand........................... 162
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