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2009 | Buch

Continuous-Path Random Processes: Mathematical Prerequisites Kapitel lesen Erstes Kapitel lesen

Mathematical Methods for Financial Markets

verfasst von: Monique Jeanblanc, Marc Yor, Marc Chesney

Verlag: Springer London

Buchreihe : Springer Finance

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Mathematical finance has grown into a huge area of research which requires a large number of sophisticated mathematical tools. This book simultaneously introduces the financial methodology and the relevant mathematical tools in a style that is mathematically rigorous and yet accessible to practitioners and mathematicians alike. It interlaces financial concepts such as arbitrage opportunities, admissible strategies, contingent claims, option pricing and default risk with the mathematical theory of Brownian motion, diffusion processes, and Lévy processes. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes.

The extensive bibliography comprises a wealth of important references and the author index enables readers quickly to locate where the reference is cited within the book, making this volume an invaluable tool both for students and for those at the forefront of research and practice.

Inhaltsverzeichnis

Frontmatter

Continuous Path Processes

Frontmatter
1. Continuous-Path Random Processes: Mathematical Prerequisites
Abstract
Historically, in mathematical finance, continuous-time processes have been considered from the very beginning, e.g., Bachelier [39, 41] deals with Brownian motion, which has continuous paths. This may justify making our starting point in this book to deal with continuous-path random processes, for which, in this first chapter, we recall some well-known facts. We try to give all the definitions and to quote all the important facts for further use. In particular, we state, without proofs, results on stochastic calculus, change of probability and stochastic differential equations.
Monique Jeanblanc, Marc Yor, Marc Chesney
2. Basic Concepts and Examples in Finance
Abstract
In this chapter, we present briefly the main concepts in mathematical finance as well as some straightforward applications of stochastic calculus for continuous-path processes. We study in particular the general principle for valuation of contingent claims, the Feynman-Kac approach, the Ornstein-Uhlenbeck and Vasicek processes, and, finally, the pricing of European options.
Monique Jeanblanc, Marc Yor, Marc Chesney
3. Hitting Times: A Mix of Mathematics and Finance
Abstract
In this chapter, we establish well known results on first hitting times of levels for Brownian motion, Brownian motion with drift and geometric Brownian motion, and we study barrier and lookback options. In the last part of the chapter, we present applications to the structural approach of default risk and real options theory and we give a short presentation of American options.
Monique Jeanblanc, Marc Yor, Marc Chesney
4. Complements on Brownian Motion
Abstract
In the first part of this chapter, we present the definition of local time and the associated Tanaka formulae, first for Brownian motion, then for more general continuous semi-martingales. In the second part, we give definitions and basic properties of Brownian bridges and Brownian meander. This is motivated by the fact that, in order to study complex derivative instruments, such as passport options or Parisian options, some knowledge of local times, bridges and excursions with respect to BM in particular and more generally for diffusions, is useful. We give some applications to exotic options, in particular to Parisian options.
Monique Jeanblanc, Marc Yor, Marc Chesney
5. Complements on Continuous Path Processes
Abstract
In this chapter, we present the important notion of time change, which will be crucial when studying applications to finance in a Lévy process setting. We then introduce the operation of dual predictable projection, which will be an important tool when working with the reduced form approach in the default risk framework (of course, it has many other applications as will appear clearly in subsequent chapters). We present important facts about general homogeneous diffusions, in particular concerning their Green functions, scale functions and speed measures. These three quantities are of great interest when valuing options in a general setting. We study applications related to last passage times. A section is devoted to enlargements of filtrations, an important subject when dealing with insider trading.
Monique Jeanblanc, Marc Yor, Marc Chesney
6. A Special Family of Diffusions: Bessel Processes
Abstract
Bessel processes are intensively used in finance, to model the dynamics of asset prices, of the spot rate and of the stochastic volatility, or as a computational tool. In particular, we show that computations for the celebrated Cox-Ingersoll-Ross and Constant Elasticity Variance models can be carried out using Bessel processes.
Monique Jeanblanc, Marc Yor, Marc Chesney

Jump Processes

Frontmatter
7. Default Risk: An Enlargement of Filtration Approach
Abstract
In this chapter, our goal is to present results that cover the reduced form methodology of credit risk modelling. In the first part, we provide a detailed analysis of the relatively simple case where the flow of information available to an agent reduces to observations of the random time which models the default event. The focus is on the evaluation of conditional expectations with respect to the filtration generated by a default time by means of the hazard function. In the second part, we study the case where an additional information flow is present; we then use the conditional survival probability, also called the hazard process. We present the intensity approach and discuss the link between both approaches. After a short introduction to Credit Default Swap’s, we end the chapter with a study of hedging defaultable claims.
Monique Jeanblanc, Marc Yor, Marc Chesney
8. Poisson Processes and Ruin Theory
Abstract
We give in this chapter the main results on Poisson processes, which are basic examples of jump processes. Despite their elementary properties they are building blocks of jump process theory. We present various generalizations such as inhomogeneous Poisson processes and compound Poisson processes. We end this chapter with two sections about point processes and marked point processes.
Monique Jeanblanc, Marc Yor, Marc Chesney
9. General Processes: Mathematical Facts
Abstract
In this chapter, we consider studies involving càdlàg processes. We pay particular attention to semi-martingales with respect to a given filtration; these processes will always be taken with càdlàg paths. We present the definition of stochastic integrals with respect to a square integrable martingale, and we extend the definition to stochastic integrals with respect to a local martingale. Then, we introduce semi-martingales, quadratic covariation processes for semi-martingales and some general versions of Itô’s formula and Girsanov’s theorem. We give necessary and sufficient conditions for the existence of an equivalent martingale measure. We end the chapter with a brief survey of valuation in an incomplete market.
Monique Jeanblanc, Marc Yor, Marc Chesney
10. Mixed Processes
Abstract
In this chapter, we present stochastic calculus for mixed processes (also often called jump-diffusions), i.e., loosely speaking they are processes whose dynamics are driven by a pair of processes consisting of a Brownian motion and a compound Poisson process. We give some applications to finance.
Monique Jeanblanc, Marc Yor, Marc Chesney
11. Lévy Processes
Abstract
In this chapter, we present briefly Lévy processes and some of their applications to finance.
Monique Jeanblanc, Marc Yor, Marc Chesney
Backmatter
Metadaten
Titel
Mathematical Methods for Financial Markets
verfasst von
Monique Jeanblanc
Marc Yor
Marc Chesney
Copyright-Jahr
2009
Verlag
Springer London
Electronic ISBN
978-1-84628-737-4
Print ISBN
978-1-85233-376-8
DOI
https://doi.org/10.1007/978-1-84628-737-4