Mathematical Finance and Probability: A Discrete Introduction

1 Introduction. - 2 A Short Primer on Finance. - 2.1 A One-Period Model with Two States and Two Securities. - 2.2 Law of One Price, Completeness and Fair Value. - 2.3 Arbitrage and Positivity of the Pricing Functional. - 2.4 Risk-Adjusted Probability Measures. - 2.5 Equivalent Martingale Measures. - 2.6 Options and Forwards. - 3 Positive Linear Functionals. - 3.1 Linear Functionals. - 3.2 Positive Linear Functionals Introduced. - 3.3 Separation Theorems. - 3.4 Extension of Positive Linear Functionals. - 3.5 Optimal Positive Extensions*. - 4 Finite Probability Spaces. - 4.1 Finite Probability Spaces. - 4.2 Laplace Experiments. - 4.3 Elementary Combinatorial Problems. - 4.4 Conditioning. - 4.5 More on Urn Models. - 5 Random Variables. - 5.1 Random Variables and their Distributions. - 5.2 The Vector Space of Random Variables. - 5.3 Positivity on L(S2). - 5.4 Expected Value and Variance. - 5.5 Two Examples. - 5.6 The L2-Structure on L(S2). - 6 General One-Period Models. - 6.1 The Elements of the Model. - 6.2 Attainability and Replication. - 6.3 The Law of One Price and Linear Pricing Functionals. - 6.4 Arbitrage and Strongly Positive Pricing Functionals. - 6.5 Completeness. - 6.6 The Fundamental Theorems of Asset Pricing. - 6.7 Fair Value in Incomplete Markets*. - 7 Information and Randomness. - 7.1 Information, Partitions and Algebras. - 7.2 Random Variables and Measurability. - 7.3 Linear Subspaces of L(S2) and Measurability. - 7.4 Random Variables and Information. - 7.5 Information Structures and Flow of Information. - 7.6 Stochastic Processes and Information Structures. - 8 Independence. - 8.1 Independence of Events. - 8.2 Independence of Random Variables. - 8.3 Expectations, Variance and Independence. - 8.4 Sequences of Independent Experiments. - 9 Multi-Period Models: The Main Issues. - 9.1 The Elements of the Model. - 9.2 Portfolios and Trading Strategies. - 9.3 Attainability and Replication. - 9.4 The Law of One Price and Linear Pricing Functionals. - 9.5 No-Arbitrage and Strongly Positive Pricing Functionals. - 9.6 Completeness. - 9.7 Strongly Positive Extensions of the Pricing Functional. - 9.8 Fair Value in Incomplete Markets*. - 10 Conditioning and Martingales. - 10.1 Conditional Expectation. - 10.2 Conditional Expectations and L2-Orthogonality. - 10.3 Martingales. - 11 The Fundamental Theorems of Asset Pricing. - 11.1 Change of Numeraire and Discounting. - 11.2 Martingales and Asset Prices. - 11.3 The Fundamental Theorems of Asset Pricing. - 11.4 Risk-Adjusted and Forward-Neutral Measures. - 12 The Cox-Ross-Rubinstein Model. - 12.1 The Cox-Ross-Rubinstein Economy. - 12.2 Parametrizing the Model. - 12.3 Equivalent Martingale Measures: Uniqueness. - 12.4 Equivalent Martingale Measures: Existence. - 12.5 Pricing in the Cox-Ross-Rubinstein Economy. - 12.6 Hedging in the Cox-Ross-Rubinstein Economy. - 12.7 European Call and Put Options. - 13 The Central Limit Theorem. - 13.1 Motivating Example. - 13.2 General Probability Spaces. - 13.3 Random Variables. - 13.4 Weak Convergence of a Sequence of Random Variables. - 13.5 The Theorem of de Moivre-Laplace. - 14 The Black-Scholes Formula. - 14.1 Limiting Behavior of a Cox-Ross-Rubinstein Economy. - 14.2 The Black-Scholes Formula. - 15 Optimal Stopping. - 15.1 Stopping Times Introduced. - 15.2 Sampling a Process by a Stopping Time. - 15.3 Optimal Stopping. - 15.4 Markov Chains and the Snell Envelope. - 16 American Claims. - 16.1 The Underlying Economy. - 16.2 American Claims Introduced. - 16.3 The Buyer's Perspective: Optimal Exercise. - 16.4 The Seller's Perspective: Hedging. - 16.5 The Fair Value of an American Claim. - 16.6 Comparing American to European Options. - 16.7 Homogeneous Markov Processes. - A Euclidean Space and Linear Algebra. - A.1 Vector Spaces. - A.2 Inner Product and Euclidean Spaces. - A.3 Topology in Euclidean Space. - A.4 Linear Operators. - A.5 Linear Equations. - B Proof of the Theorem of de Moivre-Laplace. - B.1 Preliminary results. - B.2 Proof of the Theorem of de Moivre-Laplace.