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Stochastic Calculus for Finance


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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 Stochastic Calculus 4.1 Introduction 4.2 Ito's Integral for Simple Integrands 4.2.1 Construction of the Integral 4.2.2 Properties of the Integral 4.3 Ito's Integral for General Integrands 4.4 Ito-Doeblin Formula 4.4.1 Formula for Brownian Motion 4.4.2 Formula for Ito 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 TheGreeks 4.5.6 Put-Call Parity 4.6 Multivariable Stochastic Calculus 4.6.1 Multiple Brownian Motions 4.6.2 Ito-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 Stochastic 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 Stock 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 Stock 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 Stocks 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 Stochastic 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

About 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.


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)

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