Arbitrage Theory in Continuous Time

Höfundur: Tomas Björk (Útgáfa: 4)
Arbitrage Theory in Continuous Time

Kaup valmöguleikar

The fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus.

It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Björk has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds.

This edition includes an entirely new section presenting dynamic equilibrium theory, covering unit net supply endowments models and the Cox-Ingersoll-Ross equilibrium factor model. Providing two full treatments of arbitrage theory-the classical delta hedging approach and the modern martingale approach-this book is written so that these approaches can be studied independently of each other, thus providing the less mathematically-oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action.

This textbook is a natural choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in the market.

Nánar um bókina

Útgefandi
Oxford University Press Academic UK
ISBN
9780192592453
Print ISBN
9780198851615
Format
ePub
Útgáfa
4
Höfundar
Tomas Björk
Tungumál
English
Útgefið
2019-12-05
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • Cover page
  • Halftitle page
  • Title page
  • Copyright page
  • Dedication page
  • Preface to the Fourth Edition
  • Preface to the First Edition
  • Acknowledgements
  • Contents
  • 1 Introduction
  • 1.1 Problem Formulation
  • Part I Discrete Time Models
  • 2 The Binomial Model
  • 2.1 The One Period Model
  • 2.2 The Multiperiod Model
  • 2.3 Exercises
  • 2.4 Notes
  • 3 A More General One Period Model
  • 3.1 The Model
  • 3.2 Absence of Arbitrage
  • 3.3 Martingale Measures
  • 3.4 Martingale Pricing
  • 3.5 Completeness
  • 3.6 Stochastic Discount Factors
  • 3.7 Exercises
  • Part II Stochastic Calculus
  • 4 Stochastic Integrals
  • 4.1 Introduction
  • 4.2 Information
  • 4.3 Stochastic Integrals
  • 4.4 Martingales
  • 4.5 Stochastic Calculus and the Itô Formula
  • 4.6 Examples
  • 4.7 The Multidimensional Itô Formula
  • 4.8 Correlated Wiener Processes
  • 4.9 Exercises
  • 4.10 Notes
  • 5 Stochastic Differential Equations
  • 5.1 Stochastic Differential Equations
  • 5.2 Geometric Brownian Motion
  • 5.3 The Linear SDE
  • 5.4 The Infinitesimal Operator
  • 5.5 Partial Differential Equations
  • 5.6 The Kolmogorov Equations
  • 5.7 Exercises
  • 5.8 Notes
  • Part III Arbitrage Theory
  • 6 Portfolio Dynamics
  • 6.1 Introduction
  • 6.2 Self-financing Portfolios in Discrete Time
  • 6.3 Self-financing Portfolios in Continuous Time
  • 6.4 Portfolio Weights
  • 7 Arbitrage Pricing
  • 7.1 Introduction
  • 7.2 More on the Bank Account
  • 7.3 Contingent Claims and Arbitrage
  • 7.4 The Black–Scholes Equation
  • 7.5 Risk Neutral Valuation
  • 7.6 The Black–Scholes Formula
  • 7.7 Forward and Futures Contracts
  • 7.8 Volatility
  • 7.9 American Options
  • 7.10 Exercises
  • 7.11 Notes
  • 8 Completeness and Hedging
  • 8.1 Introduction
  • 8.2 Completeness in the Black–Scholes Model
  • 8.3 Completeness—Absence of Arbitrage
  • 8.4 Exercises
  • 8.5 Notes
  • 9 A Primer on Incomplete Markets
  • 9.1 Introduction
  • 9.2 A Scalar Non-priced Underlying Asset
  • 9.3 Summing Up
  • 9.4 Exercises
  • 9.5 Notes
  • 10 Parity Relations and Delta Hedging
  • 10.1 Parity Relations
  • 10.2 The Greeks
  • 10.3 Delta and Gamma Hedging
  • 10.4 Exercises
  • 11 The Martingale Approach to Arbitrage Theory
  • 11.1 The Case of Zero Interest Rate
  • 11.2 Absence of Arbitrage and Martingale Measures
  • 11.3 A Rough Sketch of the Proof
  • 11.4 The General Case
  • 11.5 Completeness
  • 11.6 Pricing Contingent Claims
  • 11.7 Pricing by Replication
  • 11.8 Stochastic Discount Factors
  • 11.9 Summary for the Working Economist
  • 11.10 Notes
  • 12 The Mathematics of the Martingale Approach
  • 12.1 Stochastic Integral Representations
  • 12.2 The Girsanov Theorem_ Heuristics
  • 12.3 The Girsanov Theorem
  • 12.4 The Converse of the Girsanov Theorem
  • 12.5 Girsanov Transformations and Stochastic Differentials
  • 12.6 Maximum Likelihood Estimation
  • 12.7 Exercises
  • 12.8 Notes
  • 13 Black–Scholes from a Martingale Point of View
  • 13.1 Absence of Arbitrage
  • 13.2 Pricing
  • 13.3 Completeness
  • 14 Multidimensional Models: Martingale Approach
  • 14.1 Absence of Arbitrage
  • 14.2 Completeness
  • 14.3 Hedging
  • 14.4 Pricing
  • 14.5 Markovian Models and PDEs
  • 14.6 Market Prices of Risk
  • 14.7 The Stochastic Discount Factor
  • 14.8 The Hansen–Jagannathan Bounds
  • 14.9 Exercises
  • 14.10 Notes
  • 15 Change of Numeraire
  • 15.1 Introduction
  • 15.2 Generalities
  • 15.3 Changing the Numeraire
  • 15.4 Some Examples
  • 15.5 Forward Measures
  • 15.6 A General Option Pricing Formula
  • 15.7 The Numeraire Portfolio
  • 15.8 Exercises
  • 15.9 Notes
  • 16 Dividends
  • 16.1 Discrete Dividends
  • 16.2 Continuous Dividends I: Classical Methods
  • 16.3 Continuous Dividends II: Martingale Methods
  • 16.4 Exercises
  • 17 Forward and Futures Contracts
  • 17.1 Forward Contracts
  • 17.2 Futures Contracts
  • 17.3 Futures Options and Black-76
  • 17.4 Exercises
  • 17.5 Notes
  • 18 Currency Derivatives
  • 18.1 Pure Currency Contracts
  • 18.2 The Martingale Approach
  • 18.3 Domestic and Foreign Equity Markets
  • 18.4 An Extended Black–Scholes Model
  • 18.5 The Siegel Paradox
  • 18.6 Exercises
  • 18.7 Notes
  • 19 Bonds and Interest Rates
  • 19.1 Zero Coupon Bonds
  • 19.2 Interest Rates
  • 19.3 Coupon Bonds, Swaps, and Yields
  • 19.4 Exercises
  • 19.5 Notes
  • 20 Short Rate Models
  • 20.1 Generalities
  • 20.2 The Term Structure Equation
  • 20.3 Martingale Analysis
  • 20.4 Exercises
  • 20.5 Notes
  • 21 Martingale Models for the Short Rate
  • 21.1 Q-Dynamics
  • 21.2 Properties of the Short Rate Models
  • 21.3 Inversion of the Yield Curve
  • 21.4 Affine Term Structures
  • 21.5 Analytical Results for Some Standard Models
  • 21.6 Bond Options in the Hull–White Model
  • 21.7 Exercises
  • 21.8 Notes
  • 22 Forward Rate Models
  • 22.1 The Heath–Jarrow–Morton Framework
  • 22.2 Martingale Modeling
  • 22.3 The General Gaussian Model
  • 22.4 The Musiela Parameterization
  • 22.5 Exercises
  • 22.6 Notes
  • 23 Libor Market Models
  • 23.1 Caps: Definition and Market Practice
  • 23.2 The LIBOR Market Model
  • 23.3 Pricing Caps in the LIBOR Model
  • 23.4 Terminal Measure Dynamics and Existence
  • 23.5 Calibration and Simulation
  • 23.6 The Discrete Savings Account
  • 23.7 Notes
  • 24 Potentials and Positive Interest
  • 24.1 Generalities
  • 24.2 The Flesaker–Hughston Framework
  • 24.3 Changing Base Measure
  • 24.4 Decomposition of a Potential
  • 24.5 The Markov Potential Approach of Rogers
  • 24.6 Exercises
  • 24.7 Notes
  • Part IV Optimal Control and Investment Theory
  • 25 Stochastic Optimal Control
  • 25.1 An Example
  • 25.2 The Formal Problem
  • 25.3 Embedding the Problem
  • 25.4 Time Consistency and the Bellman Principle
  • 25.5 Deriving the Hamilton–Jacobi–Bellman Equation
  • 25.6 Handling the HJB Equation
  • 25.7 The Linear Regulator
  • 25.8 Exercises
  • 25.9 Notes
  • 26 Optimal Consumption and Investment
  • 26.1 A Generalization
  • 26.2 Optimal Consumption and Investment
  • 26.3 The Mutual Fund Theorems
  • 26.4 Exercises
  • 26.5 Notes
  • 27 The Martingale Approach to Optimal Investment
  • 27.1 Generalities
  • 27.2 The Basic Idea
  • 27.3 The Optimal Terminal Wealth
  • 27.4 The Optimal Wealth Process
  • 27.5 The Optimal Portfolio
  • 27.6 Log Utility
  • 27.7 Other Utility Functions
  • 27.8 Optimal Consumption Problems
  • 27.9 Exercises
  • 27.10 Notes
  • 28 Optimal Stopping Theory and American Options
  • 28.1 Introduction
  • 28.2 Generalities
  • 28.3 Some Simple Results
  • 28.4 Discrete Time
  • 28.5 Continuous Time
  • 28.6 American Options
  • 28.7 Exercises
  • 28.8 Notes
  • Part V Incomplete Markets
  • 29 Incomplete Markets
  • 29.1 Introduction
  • 29.2 A Markov Factor Model
  • 29.3 The Independent Factor Markov Model
  • 29.4 Methods to Handle Market Incompleteness
  • 29.5 Notes
  • 30 The Esscher Transform and the Minimal Martingale Measure
  • 30.1 The Esscher Transform
  • 30.2 The Minimal Martingale Measure
  • 30.3 Notes
  • 31 Minimizing f-Divergence
  • 31.1 Definition and Basic Properties
  • 31.2 Minimal Reverse Entropy
  • 31.3 Minimal Entropy in a Factor Model
  • 31.4 Duality
  • 31.5 Notes
  • 32 Portfolio Optimization in Incomplete Markets
  • 32.1 Setup
  • 32.2 The Complete Market Case
  • 32.3 Incomplete Market, Finite Ω
  • 32.4 Incomplete Market, General Ω
  • 32.5 Notes
  • 33 Utility Indifference Pricing and Other Topics
  • 33.1 Global Indifference Pricing
  • 33.2 Marginal Indifference Pricing
  • 33.3 Hedging
  • 33.4 Notes
  • 34 Good Deal Bounds
  • 34.1 General Ideas
  • 34.2 The Model
  • 34.3 The Good Deal Bounds
  • 34.4 The Embedded Optimization Problem
  • 34.5 Relations to the Minimal Martingale Measure
  • 34.6 An Option with Basis Risk
  • 34.7 Notes
  • Part VI Dynamic Equilibrium Theory
  • 35 Equilibrium Theory: A Simple Production Model
  • 35.1 The Model
  • 35.2 Equilibrium
  • 35.3 Introducing a Central Planner
  • 35.4 Exercises
  • 35.5 Notes
  • 36 The Cox–Ingersoll–Ross Factor Model
  • 36.1 The Model
  • 36.2 The Portfolio Problem
  • 36.3 Equilibrium
  • 36.4 The Short Rate and the Risk Premium for F
  • 36.5 The Equilibrium Stochastic Discount Factor
  • 36.6 Risk Neutral Valuation
  • 36.7 Introducing a Central Planner
  • 36.8 Exercises
  • 36.9 Notes
  • 37 The Cox–Ingersoll–Ross Interest Rate Model
  • 37.1 Exercises
  • 37.2 Notes
  • 38 Endowment Equilibrium: Unit Net Supply
  • 38.1 The Model
  • 38.2 The Martingale Approach
  • 38.3 Extending the Model
  • 38.4 Several Endowment Processes
  • 38.5 Exercises
  • 38.6 Notes
  • Appendix A: Measure and Integration
  • A.1 Sets and Mappings
  • A.2 Measures and Sigma-Algebras
  • A.3 Integration
  • A.4 Sigma-Algebras and Partitions
  • A.5 Sets of Measure Zero
  • A.6 The Lp Spaces
  • A.7 Hilbert Spaces
  • A.8 Sigma-Algebras and Generators
  • A.9 Product Measures
  • A.10 The Lebesgue Integral
  • A.11 The Radon–Nikodym Theorem
  • A.12 Exercises
  • A.13 Notes
  • Appendix B: Probability Theory
  • B.1 Random Variables and Processes
  • B.2 Partitions and Information
  • B.3 Sigma-Algebras and Information
  • B.4 Independence
  • B.5 Conditional Expectations
  • B.6 Equivalent Probability Measures
  • B.7 Exercises
  • B.8 Notes
  • Appendix C: Martingales and Stopping Times
  • C.1 Martingales
  • C.2 Discrete Stochastic Integrals
  • C.3 Likelihood Processes
  • C.4 Stopping Times
  • C.5 Exercises
  • Appendix D: Convex Duality
  • D.1 Conjugate Functions
  • D.2 Lagrange Functions and Saddle Points
  • D.3 An Envelope Theorem
  • References
  • Index