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2012 | Buch

Modeling Tools for Financial Options Kapitel lesen Erstes Kapitel lesen

Tools for Computational Finance

verfasst von: Rüdiger U. Seydel

Verlag: Springer London

Buchreihe : Universitext

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

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Über dieses Buch

The disciplines of financial engineering and numerical computation differ greatly, however computational methods are used in a number of ways across the field of finance. It is the aim of this book to explain how such methods work in financial engineering; specifically the use of numerical methods as tools for computational finance. By concentrating on the field of option pricing, a core task of financial engineering and risk analysis, this book explores a wide range of computational tools in a coherent and focused manner and will be of use to the entire field of computational finance. Starting with an introductory chapter that presents the financial and stochastic background, the remainder of the book goes on to detail computational methods using both stochastic and deterministic approaches.

Now in its fifth edition, Tools for Computational Finance has been significantly revised and contains:

A new chapter on incomplete markets which links to new appendices on Viscosity solutions and the Dupire equation;Several new parts throughout the book such as that on the calculation of sensitivities (Sect. 3.7) and the introduction of penalty methods and their application to a two-factor model (Sect. 6.7)Additional material in the field of analytical methods including Kim’s integral representation and its computationGuidelines for comparing algorithms and judging their efficiencyAn extended chapter on finite elements that now includes a discussion of two-asset optionsAdditional exercises, figures and references

Written from the perspective of an applied mathematician, methods are introduced as tools within the book for immediate and straightforward application. A ‘learning by calculating’ approach is adopted throughout this book enabling readers to explore several areas of the financial world.

Interdisciplinary in nature, this book will appeal to advanced undergraduate students in mathematics, engineering and other scientific disciplines as well as professionals in financial engineering.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Modeling Tools for Financial Options
Abstract
Basic types of options are explained. The binomial method is described as a first and widely applicable method for pricing options. Stochastic background for modeling is introduced, with a focus on diffusion models, which include geometric Brownian motion and mean reversion stochastic processes. The Ito-lemma is applied and jump diffusion is discussed. The chapter ends with reflections on calibration.
Rüdiger U. Seydel
Chapter 2. Generating Random Numbers with Specified Distributions
Abstract
Chapter 2 is devoted to generating random numbers. Standard generators provide uniform deviates; their properties are clarified. Random variables from other distributions are obtained via inversion or transformation methods. Specially the chapter explains how to generate normally distributed variables. Finally, fully deterministic low-discrepancy numbers are introduced.
Rüdiger U. Seydel
Chapter 3. Monte Carlo Simulation with Stochastic Differential Equations
Abstract
This chapter presents methods for pricing options using the Monte Carlo approach. To provide tools for simulation, the chapter starts with methods for integrating stochastic differential equations. Illustrated with European-type options, a basic Monte Carlo approach is explained, together with methods of variance reduction (antithetic and control variates). Monte Carlo is applied to a two-asset binary put. Then the chapter describes two Monte Carlo approaches for American-style options, namely, parametric methods and regression methods. The chapter ends with the calculation of sensitivities such as Greeks. Pathwise sensitivities and the adjoint method are introduced.
Rüdiger U. Seydel
Chapter 4. Standard Methods for Standard Options
Abstract
The Chapter 4 elaborates the numerical solution of the Black-Scholes equation for European plain-vanilla options, and of the corresponding inequalities for the American-style case. Following the Black-Scholes model, this chapter is confined to constant coefficients. This allows to solve an equivalent partial-differential equation of the simplest parabolic type. Several finite-difference schemes are explained, as well as numerical stability. Boundary conditions are introduced, which lead to obstacle problems in the American-style case and to a formulation as linear complementarity problem. The solution of the free boundary problem is tackled by the Brennan-Schwartz approach and by a projected SOR-method. Error control and accuracy are discussed. The final part of Chapter 4 is devoted to analytic methods. This includes the interpolation method, the quadratic approximation, the analytic method of lines, and quadrature methods for an integral representation. Finally we discuss criteria for the comparison of different methods and for judging their efficiency.
Rüdiger U. Seydel
Chapter 5. Finite-Element Methods
Abstract
Finite element methods provide more flexibility in approximating functions and domains. These advantages can be exploited for the pricing of options, which is explained in this chapter. It starts with an elementary introduction to finite element methods. Then the ideas are applied to standard options based on a single asset. The two-asset case is treated next, with an example of a basket with double barrier. The chapter ends with a proof of quadratic convergence for a standard scenario.
Rüdiger U. Seydel
Chapter 6. Pricing of Exotic Options
Abstract
This chapter is devoted to exotic options, which include multifactor options and Asian options. Non-constant coefficients require numerical methods for more general PDEs than those discussed in Chapter 4. Upwind schemes, stability issues and total variation diminishing are discussed. The final part of the chapter is devoted to penalty methods, here applied to a two-asset option.
Rüdiger U. Seydel
Chapter 7. Beyond Black and Scholes
Abstract
Chapter 7 goes beyond the Black and Scholes model, now turning to incomplete markets. Nonlinear models are discussed, with a focus on considering transaction costs. Numerical schemes for nonlinear PDEs require monotonicity for convergence. This is applied to an uncertain-volatility model with a barrier call. Next, Levy jump processes are briefly introduced, which lead to partial integro-differential equations (PIDEs). This is exemplified by solving numerically the PIDE for Merton’s jump diffusion. The final section introduces the application of the Fourier transform for option pricing.
Rüdiger U. Seydel
Backmatter
Metadaten
Titel
Tools for Computational Finance
verfasst von
Rüdiger U. Seydel
Copyright-Jahr
2012
Verlag
Springer London
Electronic ISBN
978-1-4471-2993-6
Print ISBN
978-1-4471-2992-9
DOI
https://doi.org/10.1007/978-1-4471-2993-6

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