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Mathematical Finance and Probability

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Table of Contents

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.

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"This is probably the best written book on discrete-time models of mathematical finance. It is self consistent, all notions used in it are carefully defined. That is a mathematical book - by mathematicians and for mathematicians, which also means that its practical applications are restricted. The bibliography is complete. I strongly recommend that title as an introduction to mathematical finance." -- Darius Gatarek (Control and Cybernetics) "The style of presentation will appeal to anyone who is seeking an elementary but rigorous introduction to the pricing of derivative securities. The book is written carefully and is very readable." --Mathematical Reviews "The book offers a self-contained elementary but rigorous and very clear introduction to the pricing of derivative instruments in discrete time... For the interested reader who has not been exposed to modern probability theory before, the book provides an excellent starting point for studying the theory of derivative pricing. In particular, for a rigorous course on derivative pricing in an economics department or at a business school this introduction seems to be well-suited." --Zentralblatt Math "The book presents the part of mathematical finance devoted to the pricing of derivative instruments; its basic theme is the study of prices in securities markets in an uncertain environment... As the objective of the book is to provide a sound understanding of important issues of modern approaches to mathematical finance, several mathematical models are developed and examined in detail. The focus is on finite-time models and the highest level of generality is frequently sacrificed for the sake of a greater insight into the underlying economic ideas. Even when the problems are approached from the mathematical point of view and almost all results are strictly proved, the financial interpretation is always stressed... The style of presentation is aimed at students of financial economics, mathematics and physics and at mathematicians, physicists and economists working in financial industry." --APPLICATIONS OF MATHEMATICS

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