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Shreve S.E. Stochastic calculus for finance II.pdf

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    Series Title
    Series Contents
    Title
    Date-line
    Preface
    Contents
    Introduction
    Summary of Volume I
    Summary of Volume II
    1 General Probability Theory
    1.1 Infinite 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 ?-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 the Symmetric Random Walk
    3.2.3 Martingale Property for the Symmetric Random Walk
    3.2.4 Quadratic Variation of the Symmetric Random Walk
    3.2.5 Scaled Symmetric Random Walk
    3.2.6 Limiting Distribution of the Scaled Random Walk
    3.2.7 Log-Normal Distribution as the Limit of the 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 Reflection 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 The Greeks
    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 the Brownian Bridge
    4.7.5 Brownian Bridge as a 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 the 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 Coefficients
    5.5.3 Lump Payments of Dividends
    5.5.4 Lump Payments of Dividends with Constant Coefficients
    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 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 a 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 Ito-Doeblin Formula for One Jump Process
    11.5.2 Ito-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 $\sigma$-algebras
    A.3 Random Variable with Neither Density nor Probability Mass Function
    B Existence of Conditional Expectations
    C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing
    References
    Index
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