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Stochastic Calculus for Finance II : Continuous-Time Models :  Continuous-Time Models - Steven E. Shreve

Stochastic Calculus for Finance II : Continuous-Time Models

Continuous-Time Models

Hardcover

Published: June 2004
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Stochastic Calculus for Finance evolved from the first ten years of the Carnegie Mellon Professional Master's program in Computational Finance. The content of this book has been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs, but more importantly intuitive explanations developed and refine through classroom experience with this material are provided. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties. Advanced topics include foreign exchange models, forward measures, and jump-diffusion processes.

This book is being published in two volumes. This second volume develops stochastic calculus, martingales, risk-neutral pricing, exotic options and term structure models, all in continuous time.

Master's level students and researchers in mathematical finance and financial engineering will find this book useful.

From the reviews of the first edition:

"Steven Shreve's comprehensive two-volume Stochastic Calculus for Finance may well be the last word, at least for a while, in the flood of Master's level books.... a detailed and authoritative reference for "quants" (formerly known as "rocket scientists"). The books are derived from lecture notes that have been available on the Web for years and that have developed a huge cult following among students, instructors, and practitioners. The key ideas presented in these works involve the mathematical theory of securities pricing based upon the ideas of classical finance.
...the beauty of mathematics is partly in the fact that it is self-contained and allows us to explore the logical implications of our hypotheses. The material of this volume of Shreve's text is a wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions.
In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach. It is accessible to a broad audience and has been developed after years of teaching the subject. It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." (SIAM, 2005)

"The contents of the book have been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise Statements of results, plausibility arguments, and even some proofs. But more importantly, intuitive explanations, developed and refine through classroom experience with this material are provided throughout the book." (Finanz Betrieb, 7:5, 2005)

"The origin of this two volume textbook are the well-known lecture notes on Stochastic Calculus ... . The first volume contains the binomial asset pricing model. ... The second volume covers continuous-time models ... . This book continues the series of publications by Steven Shreve of highest quality on the one hand and accessibility on the other end. It is a must for anybody who wants to get into mathematical finance and a pleasure for experts ... ." (www.mathfinance.de, 2004)

"This is the latter of the two-volume series evolving from the author's mathematics courses in M.Sc. Computational Finance program at Carnegie Mellon University (USA). The content of this book is organized such as to give the reader precise statements of results, plausibility arguments, mathematical proofs and, more importantly, the intuitive explanations of the financial and economic phenomena. Each chapter concludes with summary of the discussed matter, bibliographic notes, and a set of really useful exercises." (Neculai Curteanu, Zentralblatt MATH, Vol. 1068, 2005)

General Probability Theory
In.nite Probability Spaces
Random Variables and Distributions
Expectations
Convergence of Integrals
Computation of Expectations
Change of Measure
Summary
Notes
Exercises
Information and Conditioning
Information and s-algebras
Independence
General Conditional Expectations
Summary
Notes
Exercises
Brownian Motion
Introduction
Scaled Random Walks
Symmetric Random Walk
Increments of Symmetric Random Walk
Martingale Property for Symmetric Random Walk
Quadratic Variation of Symmetric Random Walk
Scaled Symmetric Random Walk
Limiting Distribution of Scaled Random Walk
Log-Normal Distribution as Limit of Binomial Model
Brownian Motion
Definition of Brownian Motion
Distribution of Brownian Motion
Filtration for Brownian Motion
Martingale Property for Brownian Motion
Quadratic Variation
First-Order Variation
Quadratic Variation
Volatility of Geometric Brownian Motion
Markov Property
First Passage Time Distribution
Re.ection Principle
Reflection Equality
First Passage Time Distribution
Distribution of Brownian Motion and Its Maximum
Summary
Notes
Exercises
Stochastic Calculus
Introduction
Ito's Integral for Simple Integrands
Construction of the Integral
Properties of the Integral
Ito's Integral for General Integrands
Ito-Doeblin Formula
Formula for Brownian Motion
Formula for Ito Processes
Examples
Black-Scholes-Merton Equation
Evolution of Portfolio Value
Evolution of Option Value
Equating the Evolutions
Solution to the Black-Scholes-Merton Equation
The Greeks
Put-Call Parity
Multivariable Stochastic Calculus
Multiple Brownian Motions
Ito-Doeblin Formula for Multiple Processes
Recognizing a Brownian Motion
Brownian Bridge
Gaussian Processes
Brownian Bridge as a Gaussian Process
Brownian Bridge as a Scaled Stochastic Integral
Multidimensional Distribution of Brownian Bridge
Brownian Bridge as Conditioned Brownian Motion
Summary
Notes
Exercises
Risk-Neutral Pricing
Introduction
Risk-Neutral Measure
Girsanov's Theorem for a Single Brownian Motion
Stock Under the Risk-Neutral Measure
Value of Portfolio Process Under the Risk-Neutral Measure
Pricing Under the Risk-Neutral Measure
Deriving the Black-Scholes-Merton Formula
Martingale Representation Theorem
Martingale Representation with One Brownian Motion
Hedging with One Stock
Fundamental Theorems of Asset Pricing
Girsanov and Martingale Representation Theorems
Multidimensional Market Model
Existence of Risk-Neutral Measure
Uniqueness of the Risk-Neutral Measure
Dividend-Paying Stocks
Continuously Paying Dividend
Continuously Paying Dividend with Constant Coeffcients
Lump Payments of Dividends
Lump Payments of Dividends with Constant Coeffcients
Forwards and Futures
Forward Contracts
Futures Contracts
Forward-Futures Spread
Summary
Notes
Exercises
Connections with Partial Differential Equations
Introduction
Stochastic Differential Equations
The Markov Property
Partial Differential Equations
Interest Rate Models
Multidimensional Feynman-Kac Theorems
Summary
Notes
Exercises
Exotic Options
Introduction
Maximum of Brownian Motion with Drift
Knock-Out Barrier Options
Up-and-Out Call
Black-Scholes-Merton Equation
Computation of the Price of the Up-and-Out Call
Lookback Options
Floating Strike Lookback Option
Black-Scholes-Merton Equation
Reduction of Dimension
Computation of the Price of the Lookback Option
Asian Options
Fixed-Strike Asian Call
Augmentation of the State
Change of Num'eraire
Summary
Notes
Exercises
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780387401010
ISBN-10: 0387401016
Series: Springer Finance
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 550
Published: June 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.8
Weight (kg): 2.16