Stochastic Calculus for Finance II: Continuous-Time Models / Edition 1

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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. Masters level students and researchers in mathematical finance and financial engineering will find this book useful.

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Editorial Reviews

From the Publisher

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 … ." (, 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)

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Product Details

  • ISBN-13: 9780387401010
  • Publisher: Springer New York
  • Publication date: 6/1/2004
  • Series: Springer Finance / Springer Finance Textbooks Series
  • Edition description: 1st ed. 2004. Corr. 2nd printing 2010
  • Edition number: 1
  • Pages: 570
  • Sales rank: 244,437
  • Product dimensions: 1.25 (w) x 9.21 (h) x 6.14 (d)

Meet the Author

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.

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Table of Contents

1 General Probability Theory
1.1 In.nite Probability Spaces
1.2 Random Variables and Distributions
1.3 Expectations
1.4 Convergence of Integrals
1.5 Computation of Expectations
1.6 Change of Measure
1.7 Summary
1.8 Notes
1.9 Exercises

2 Information and Conditioning
2.1 Information and s-algebras
2.2 Independence
2.3 General Conditional Expectations
2.4 Summary
2.5 Notes
2.6 Exercises

3 Brownian Motion
3.1 Introduction
3.2 Scaled Random Walks
3.2.1 Symmetric Random Walk
3.2.2 Increments of Symmetric Random Walk
3.2.3 Martingale Property for Symmetric Random Walk
3.2.4 Quadratic Variation of Symmetric Random Walk
3.2.5 Scaled Symmetric Random Walk
3.2.6 Limiting Distribution of Scaled Random Walk
3.2.7 Log-Normal Distribution as Limit of Binomial Model
3.3 Brownian Motion
3.3.1 Definition of Brownian Motion
3.3.2 Distribution of Brownian Motion
3.3.3 Filtration for Brownian Motion
3.3.4 Martingale Property for Brownian Motion
3.4 Quadratic Variation
3.4.1 First-Order Variation
3.4.2 Quadratic Variation
3.4.3 Volatility of Geometric Brownian Motion
3.5 Markov Property
3.6 First Passage Time Distribution
3.7 Re.ection Principle
3.7.1 Reflection Equality
3.7.2 First Passage Time Distribution
3.7.3 Distribution of Brownian Motion and Its Maximum
3.8 Summary
3.9 Notes
3.10 Exercises

4 Shastic Calculus
4.1 Introduction
4.2 Itôo's Integral for Simple Integrands
4.2.1 Construction of the Integral
4.2.2 Properties of the Integral
4.3 Itôo's Integral for General Integrands
4.4 Itôo-Doeblin Formula
4.4.1 Formula for Brownian Motion
4.4.2 Formula for Itôo Processes
4.4.3 Examples
4.5 Black-Scholes-Merton Equation
4.5.1 Evolution of Portfolio Value
4.5.2 Evolution of Option Value
4.5.3 Equating the Evolutions
4.5.4 Solution to the Black-Scholes-Merton Equation
4.5.5 The Greeks
4.5.6 Put-Call Parity
4.6 Multivariable Shastic Calculus
4.6.1 Multiple Brownian Motions
4.6.2 Itôo-Doeblin Formula for Multiple Processes
4.6.3 Recognizing a Brownian Motion
4.7 Brownian Bridge
4.7.1 Gaussian Processes
4.7.2 Brownian Bridge as a Gaussian Process
4.7.3 Brownian Bridge as a Scaled Shastic Integral
4.7.4 Multidimensional Distribution of Brownian Bridge
4.7.5 Brownian Bridge as Conditioned Brownian Motion
4.8 Summary
4.9 Notes
4.10 Exercises

5 Risk-Neutral Pricing
5.1 Introduction
5.2 Risk-Neutral Measure
5.2.1 Girsanov's Theorem for a Single Brownian Motion
5.2.2 Sk Under the Risk-Neutral Measure
5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure
5.2.4 Pricing Under the Risk-Neutral Measure
5.2.5 Deriving the Black-Scholes-Merton Formula
5.3 Martingale Representation Theorem
5.3.1 Martingale Representation with One Brownian Motion
5.3.2 Hedging with One Sk
5.4 Fundamental Theorems of Asset Pricing
5.4.1 Girsanov and Martingale Representation Theorems
5.4.2 Multidimensional Market Model
5.4.3 Existence of Risk-Neutral Measure
5.4.4 Uniqueness of the Risk-Neutral Measure
5.5 Dividend-Paying Sks
5.5.1 Continuously Paying Dividend
5.5.2 Continuously Paying Dividend with Constant Coeffcients
5.5.3 Lump Payments of Dividends
5.5.4 Lump Payments of Dividends with Constant Coeffcients
5.6 Forwards and Futures
5.6.1 Forward Contracts
5.6.2 Futures Contracts
5.6.3 Forward-Futures Spread
5.7 Summary
5.8 Notes
5.9 Exercises

6 Connections with Partial Differential Equations
6.1 Introduction
6.2 Shastic Differential Equations
6.3 The Markov Property
6.4 Partial Differential Equations
6.5 Interest Rate Models
6.6 Multidimensional Feynman-Kac Theorems
6.7 Summary
6.8 Notes
6.9 Exercises

7 Exotic Options
7.1 Introduction
7.2 Maximum of Brownian Motion with Drift
7.3 Knock-Out Barrier Options
7.3.1 Up-and-Out Call
7.3.2 Black-Scholes-Merton Equation
7.3.3 Computation of the Price of the Up-and-Out Call
7.4 Lookback Options
7.4.1 Floating Strike Lookback Option
7.4.2 Black-Scholes-Merton Equation
7.4.3 Reduction of Dimension
7.4.4 Computation of the Price of the Lookback Option
7.5 Asian Options
7.5.1 Fixed-Strike Asian Call
7.5.2 Augmentation of the State
7.5.3 Change of Numeraire
7.6 Summary
7.7 Notes
7.8 Exercises

8 American Derivative Securities
8.1 Introduction
8.2 Stopping Times
8.3 Perpetual American Put
8.3.1 Price under Arbitrary Exercise
8.3.2 Price under Optimal Exercise
8.3.3 Analytical Characterization of the Put Price
8.3.4 Probabilistic Characterization of the Put Price
8.4 Finite-Expiration American Put
8.4.1 Analytical Characterization of the Put Price
8.4.2 Probabilistic Characterization of the Put Price
8.5 American Call
8.5.1 Underlying Asset Pays No Dividends
8.5.2 Underlying Asset Pays Dividends
8.6 Summary
8.7 Notes
8.8 Exercises

9 Change of Numeraire
9.1 Introduction
9.2 Numeraire
9.3 Foreign and Domestic Risk-Neutral Measures
9.3.1 The Basic Processes
9.3.2 Domestic Risk-Neutral Measure
9.3.3 Foreign Risk-Neutral Measure
9.3.4 Siegel's Exchange Rate Paradox
9.3.5 Forward Exchange Rates
9.3.6 Garman-Kohlhagen Formula
9.3.7 Exchange Rate Put-Call Duality
9.4 Forward Measures
9.4.1 Forward Price
9.4.2 Zero-Coupon Bond as Numeraire
9.4.3 Option Pricing with Random Interest Rate
9.5 Summary
9.6 Notes
9.7 Exercises

10 Term Structure Models
10.1 Introduction
10.2 Affine-Yield Models
10.2.1 Two-Factor Vasicek Model
10.2.2 Two-Factor CIR Model
10.2.3 Mixed Model
10.3 Heath-Jarrow-Morton Model
10.3.1 Forward Rates
10.3.2 Dynamics of Forward Rates and Bond Prices
10.3.3 No-Arbitrage Condition
10.3.4 HJM Under Risk-Neutral Measure
10.3.5 Relation to Affine-Yield Models
10.3.6 Implementation of HJM
10.4 Forward LIBOR Model
10.4.1 The Problem with Forward Rates
10.4.2 LIBOR and Forward LIBOR
10.4.3 Pricing a Backset LIBOR Contract
10.4.4 Black Caplet Formula
10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities
10.4.6 A Forward LIBOR Term Structure Model
10.5 Summary
10.6 Notes
10.7 Exercises

11 Introduction to Jump Processes
11.1 Introduction
11.2 Poisson Process
11.2.1 Exponential Random Variables
11.2.2 Construction of a Poisson Process
11.2.3 Distribution of Poisson Process Increments
11.2.4 Mean and Variance of Poisson Increments
11.2.5 Martingale Property
11.3 Compound Poisson Process
11.3.1 Construction of a Compound Poisson Process
11.3.2 Moment Generating Function
11.4 Jump Processes and Their Integrals
11.4.1 Jump Processes
11.4.2 Quadratic Variation
11.5 Shastic Calculus for Jump Processes
11.5.1 Itôo-Doeblin Formula for One Jump Process
11.5.2 Itôo-Doeblin Formula for Multiple Jump Processes
11.6 Change of Measure
11.6.1 Change of Measure for a Poisson Process
11.6.2 Change of Measure for a Compound Poisson Process
11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion
11.7 Pricing a European Call in a Jump Model
11.7.1 Asset Driven by a Poisson Process
11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process
11.8 Summary
11.9 Notes
11.10 Exercises

A Advanced Topics in Probability Theory A.1 Countable Additivity A.2 Generating s-algebras A.3 A Random Variable with Neither a Density nor a Probability Mass Function B Existence of Conditional Expectations C Completion of Proof of Second Fundamental Theorem of Asset Pricing


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