Author | Medina, Pablo Koch. author |
---|---|
Title | Mathematical Finance and Probability [electronic resource] : A Discrete Introduction / by Pablo Koch Medina, Sandro Merino |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2003 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8041-1 |
Descript | X, 328 p. 1 illus. online resource |
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 Buyerโs Perspective: Optimal Exercise -- 16.4 The Sellerโs 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