Mathematical Finance and Probability

The objective of this book is to give a self-contained presentation to the theory underlying the valuation of derivative financial instruments, which

is becoming a standard part of the toolbox of professionals in the financial industry. Although a complete derivation of the Black-Scholes

option pricing formula is given, the focus is on finite-time models. Not going for the greatest possible level of generality is greatly rewarded by

a greater insight into the underlying economic ideas, putting the reader in an excellent position to proceed to the more general continuous-time

theory.

The material will be accessible to students and practitioners having a working knowledge of linear algebra and calculus. All additional material

is developed from the very beginning as needed. In particular, the book also offers an introduction to modern probability theory, albeit mostly

within the context of finite sample spaces.

The styleof presentation will appeal to financial economics students seeking an elementary but rigorous introduction to the subject; mathematics

and physics students looking for an opportunity to become acquainted with this modern applied topic; and mathematicians, physicists or quantitatively inclined economists working in the financial industry.

Texte du rabat

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.

Contenu

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.