Stochastic Calculus Models for Finance II Continuous-Time Models

ISBN-10: 0387401016
ISBN-13: 9780387401010
Edition: 2004
Authors: Steven E. Shreve
List price: $74.95 Buy it from $47.09
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Description: 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  More...

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Book details

List price: $74.95
Copyright year: 2004
Publisher: Springer
Publication date: 12/13/2010
Binding: Hardcover
Pages: 550
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 1.584

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. Masters level students and researchers in mathematical finance and financial engineering will find this book useful. Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.

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

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