Introduction to Probability Theory and Stochastic Processes for Finance
Lecture Notes
Fabio Trojani
Department of Economics, University of St. Gallen, Switzerland

Contents
1 Introduction to Probability Theory 4
1.1 The BinomialModel . . . .  . . . . 4
1.1.1 The Risky Asset . . . .  . . . 4
1.1.2 The Riskless Asset . . . . . . . . . . . . . . . 4
1.1.3 A Basic No Arbitrage Condition . . . .  . . . 5
1.1.4 Some Basic Remarks . . . . . .  . . 5
1.1.5 Pricing Derivatives: a first Example . .  . . . 5
1.2 Finite Probability Spaces . . .. . . . . . . . 7
1.2.1 Measurable Spaces . . . . . . . . 7
1.2.2 Probabilitymeasures . . . . . . . . 11
1.2.3 RandomVariables . . . . . . . . . . . 14
1.2.4 Expected Value of Random Variables Defined on Finite Measurable Spaces 15
1.2.5 Examples of Probability Spaces and Random Variables with Finite Sample Space . . . . . . . . . . . . 16
1.3 General Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Some First Examples of Probability Spaces with non finite Sample Spaces . 18
1.3.2 Continuity Properties of ProbabilityMeasures . . . . . . . . . . . . . . . . 20
1.3.3 RandomVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Expected Value and Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . 25
1.3.5 Some Further Examples of Probability Spaces with uncountable Sample Spaces 28
1.4 Stochastic Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Conditional Expectations and Martingales 33
2.1 The BinomialModel OnceMore . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Sub Sigma Algebras and (Partial) Information . . . . . . . . . . . . . . . . . . . . 34
2.3 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Martingale Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Pricing Principles in the Absence of Arbitrage 44
3.1 Stock Prices, Risk Neutral Probability Measures and Martingales . . . . . . . . . . 45
3.2 Self Financing Strategies, Risk Neutral Probability Measures and Martingales . . . 46
3.3 Existence of Risk Neutral Probability Measures and Derivatives Pricing . . . . . . 48
3.4 Uniqueness of Risk Neutral ProbabilityMeasures and Derivatives Hedging . . . . . 50
3.5 Existence of Risk Neutral Probability Measures and Absence of Arbitrage . . . . . 52
4 Introduction to Stochastic Processes 52
4.1 Basic Definitions . . . . . . . . . . . 52
4.2 Discrete Time BrownianMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Girsanov Theorem: Application to a Semicontinuous PricingModel . . . . . . . . . 57
4.3.1 A Semicontinuous PricingModel . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Risk Neutral Valuation in the SemicontinuousModel . . . . . . . . . . . . . 58
4.3.3 A Discrete Time Formulation of Girsanov Theorem. . . . . . . . . . . . . . 60
4.3.4 A Discrete Time Derivation of Black and Scholes Formula . . . . . . . . . . 64
4.4 Continuous Time BrownianMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Introduction to Stochastic Calculus 71
5.1 Starting Point,Motivation . . . . . . . . . . . . 71
5.2 The Stochastic Integral . . . . . .. . . . . . 73
5.2.1 Some Basic Preliminaries . . . .. . . 74
5.2.2 Simple Integrands . . . . . . .. . . 75
5.2.3 Squared Integrable Integrands . . . . . . . . . . 81
5.2.4 Properties of Stochastic Integrals . . .  . . . . . . 84
5.3 Itô’s Lemma . . . . . . .  . . . 85
5.3.1 Starting Point,Motivation and Some First Examples . . . . . . . . . . . . . 85
5.3.2 A Simplified Derivation of Itô’s Formula . . . . . . . . . . . . . . . . . . . . 88
5.4 An Application of Stochastic Calculus: the Black-ScholesModel . . . . . . . . . . 93
5.4.1 The Black-ScholesMarket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.2 Self Financing Portfolios and Hedging in the Black-ScholesModel . . . . . 93
5.4.3 Probabilistic Interpretation of Black-Scholes Prices: Girsanov Theorem once
more . . . .. . . 95