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.
This book evolved from the first ten years of the Carnegie Mellon professional Master's program in Computational Finance. 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 refined through classroom experience with this material, are provided throughout the book. Volume I introduces the fundamental concepts in a discrete-time setting and Volume II builds on this foundation to develop stochastic calculus, martingales, risk-neutral pricing, exotic options, and term structure models, all in continuous time.
"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 should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." --SIAM
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.
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 Itˆo's Integral for Simple Integrands
4.2.1 Construction of the Integral
4.2.2 Properties of the Integral
4.3 Itˆo's Integral for General Integrands
4.4 Itˆo-Doeblin Formula
4.4.1 Formula for Brownian Motion
4.4.2 Formula for Itˆo 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 Stochastic Calculus
4.6.1 Multiple Brownian Motions
4.6.2 Itˆo-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
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 Num´eraire
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 Num´eraire
9.1 Introduction
9.2 Num´eraire
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 Num´eraire
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 Stochastic Calculus for Jump Processes
11.5.1 Itˆo-Doeblin Formula for One Jump Process
11.5.2 Itˆo-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
References