Mathematical Finance and Probability

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.

"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 modernprobability 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

Texte du rabat

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.

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.- 13.5The 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 Formu…