Mathematical finance and probability : a discrete introduction
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書誌事項
Mathematical finance and probability : a discrete introduction
Birkhäuser, c2003
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注記
Bibliography: p. [321]-324
Includes index
内容説明・目次
内容説明
This self-contained book presents the theory underlying the valuation of derivative financial instruments, which is becoming a standard part of the professional toolbox in the financial industry. It provides great insight into the underlying economic ideas in a very readable form, putting the reader in an excellent position to proceed to the more general continuous-time theory.
目次
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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