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Financial Modeling : A Backward Stochastic Differential Equations Perspective / by Stéphane Crépey

By: Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Finance | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2013Description: Online-Ressource (XIX, 459 p. 13 illus. in color, digital)ISBN:
  • 9783642371134
Subject(s): Additional physical formats: 9783642371127 | Erscheint auch als: Financial modeling. Druck-Ausgabe Berlin, Heidelberg : Springer, 2013. XIX, 459 S.DDC classification:
  • 332.0151
  • 004
  • 003.3 23
MSC: MSC: *91-02 | 91G80 | 91G60 | 91G20 | 60H35 | 60H10RVK: RVK: QP 700 | SK 980 | SK 820LOC classification:
  • QA71-90
DOI: DOI: 10.1007/978-3-642-37113-4Online resources:
Contents:
Financial Modeling; Preface; Structure of the Book; Outline; Roadmap; The Role of BSDEs; Bibliographic Guidelines; Acknowledgements; Contents; Part I: An Introductory Course in Stochastic Processes; Chapter 1: Some Classes of Discrete-Time Stochastic Processes; 1.1 Discrete-Time Stochastic Processes; 1.1.1 Conditional Expectations and Filtrations; 1.1.1.1 Main Properties; 1.2 Discrete-Time Markov Chains; 1.2.1 An Introductory Example; 1.2.2 Definitions and Examples; 1.2.3 Chapman-Kolmogorov Equations; 1.2.4 Long-Range Behavior; 1.3 Discrete-Time Martingales; 1.3.1 Definitions and Examples
1.3.2 Stopping Times and Optional Stopping Theorem1.3.2.1 Uniform Integrability and Martingales; 1.3.3 Doob's Decomposition; Chapter 2: Some Classes of Continuous-Time Stochastic Processes; 2.1 Continuous-Time Stochastic Processes; 2.1.1 Generalities; 2.1.2 Continuous-Time Martingales; 2.2 The Poisson Process and Continuous-Time Markov Chains; 2.2.1 The Poisson Process; 2.2.2 Two-State Continuous Time Markov Chains; 2.2.3 Birth-and-Death Processes; 2.3 Brownian Motion; 2.3.1 Definition and Basic Properties; 2.3.2 Random Walk Approximation; 2.3.3 Second Order Properties; 2.3.4 Markov Properties
Reflection PrincipleArctan Law and Recurrence; Strong Law of Large Numbers; 2.3.5 First Passage Times of a Standard Brownian Motion; 2.3.6 Martingales Associated with Brownian Motion; 2.3.6.1 Exit Time from a Corridor; 2.3.7 First Passage Times of a Drifted Brownian Motion; 2.3.8 Geometric Brownian Motion; Chapter 3: Elements of Stochastic Analysis; 3.1 Stochastic Integration; 3.1.1 Integration with Respect to a Symmetric Random Walk; Properties Enjoyed by the Stochastic Integral Yn=k=1n zetakDeltaSk; 3.1.2 The Itô Stochastic Integral for Simple Processes
Properties Enjoyed by the Itô Stochastic Integral for Simple Processes Z3.1.3 The General Itô Stochastic Integral; Properties Enjoyed by the (General) Itô Stochastic Integral; 3.1.4 Stochastic Integral with Respect to a Poisson Process; 3.1.5 Semimartingale Integration Theory (*); 3.2 Itô Formula; 3.2.1 Introduction; 3.2.1.1 What about 0t Ws dWs?; 3.2.1.2 What About 0t Ns-dNs ?; 3.2.2 Itô Formulas for Continuous Processes; The First Extension of the Simple Itô Formula; The Second Extension of the Simple Itô Formula; 3.2.2.1 Examples; 3.2.3 Itô Formulas for Processes with Jumps (*)
3.2.4 Brackets (*)3.3 Stochastic Differential Equations (SDEs); 3.3.1 Introduction; 3.3.2 Diffusions; 3.3.2.1 SDEs for Diffusions; 3.3.2.2 Examples; 3.3.2.3 Solving Diffusion SDEs; 3.3.3 Jump-Diffusions (*); 3.4 Girsanov Transformations; 3.4.1 Girsanov Transformation for Gaussian Distributions; 3.4.1.1 Gaussian Random Variables; Hint; 3.4.1.2 Brownian Motion; 3.4.2 Girsanov Transformation for Poisson Distributions; 3.4.2.1 Poisson Random Variables; 3.4.2.2 Poisson Process; 3.4.3 Abstract Bayes Formula; 3.5 Feynman-Kac Formulas (*); 3.5.1 Linear Case
3.5.2 Backward Stochastic Differential Equations (BSDEs)
Summary: Part I: An Introductory Course in Stochastic Processes -- 1.Some classes of Discrete-Time Stochastic Processes.-2.Some Classes of Continuous-Time Stochastic Processes -- 3.Elements of Stochastic Analysis -- Part II: Pricing Equations -- 4.Martingale Modeling -- 5.Benchmark Models -- Part III: Numerical Solutions -- 6.Monte Carlo Methods -- 7.Tree Methods -- 8.Finite Differences -- 9.Callibration Methods -- Part IV: Applications -- 10.Simulation/ Regression Pricing Schemes in Diffusive Setups -- 11.Simulation/ Regression Pricing Schemes in Pure Jump Setups -- Part V: Jump-Diffusion Setup with Regime Switching (**) -- 12.Backward Stochastic Differential Equations -- 13.Analytic Approach -- 14.Extensions -- Part VI: Appendix -- A.Technical Proofs (**) -- B.Exercises -- C.Corrected Problem Sets.Summary: Backward stochastic differential equations (BSDEs) provide a general mathematical framework for solving pricing and risk management questions of financial derivatives. They are of growing importance for nonlinear pricing problems such as CVA computations that have been developed since the crisis. Although BSDEs are well known to academics, they are less familiar to practitioners in the financial industry. In order to fill this gap, this book revisits financial modeling and computational finance from a BSDE perspective, presenting a unified view of the pricing and hedging theory across all asset classes. It also contains a review of quantitative finance tools, including Fourier techniques, Monte Carlo methods, finite differences and model calibration schemes. With a view to use in graduate courses in computational finance and financial modeling, corrected problem sets and Matlab sheets have been provided. Stéphane Crépey’s book starts with a few chapters on classical stochastic processes material, and then... fasten your seatbelt... the author starts traveling backwards in time through backward stochastic differential equations (BSDEs). This does not mean that one has to read the book backwards, like a manga! Rather, the possibility to move backwards in time, even if from a variety of final scenarios following a probability law, opens a multitude of possibilities for all those pricing problems whose solution is not a straightforward expectation. For example, this allows for framing problems like pricing with credit and funding costs in a rigorous mathematical setup. This is, as far as I know, the first book written for several levels of audiences, with applications to financial modeling and using BSDEs as one of the main tools, and as the song says: "it's never as good as the first time". Damiano Brigo, Chair of Mathematical Finance, Imperial College London While the classical theory of arbitrage free pricing has matured, and is now well understood and used by the finance industry, the theory of BSDEs continues to enjoy a rapid growth and remains a domain restricted to academic researchers and a handful of practitioners. Crépey’s book presents this novel approach to a wider community of researchers involved in mathematical modeling in finance. It is clearly an essential reference for anyone interested in the latest developments in financial mathematics. Marek Musiela, Deputy Director of the Oxford-Man Institute of Quantitative Finance.PPN: PPN: 1652500146Package identifier: Produktsigel: ZDB-2-SXMS | ZDB-2-SMA | ZDB-2-SEB
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