基本説明
Contents: A Short Primer on Finance - Positive Linear Functionals - Finite Probability Spaces - The Black Scholes Formula - Optimal Stopping - American Claims; and more.
Full Description
On what grounds can one reasonably expect that a complex financial contract solving a complex real-world issue does not deserve the same thorough scientific treatment as an aeroplane wing or a micro-proces sor? Only ignorance would suggest such an idea. E. Briys and F. De Varenne The objective of this book is to give a self-contained presentation of that part of mathematical finance devoted to the pricing of derivative instruments. During the past two decades the pricing of financial derivatives - or more generally: mathematical finance - has steadily won in importance both within the financial services industry and within the academic world. The complexity of the mathemat ics needed to master derivatives techniques naturally resulted in a high demand for quantitatively oriented professionals (mostly mathematicians and physicists) in the banking and insurance world. This in turn triggered a demand for university courses on the relevant topics and at the same time confronted the mathematical community with an interesting field of application for many techniques that had originally been developed for other purposes. Most probably this development was accelerated by an ever more applied orientation of the mathematics curriculum and the fact that finance institutions were often willing to generously support research in this field.
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.- 13.5The 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.
