Arbitrage Theory in Continuous Time

Transcription

1 Arbitrage Theory in Continuous Time THIRD EDITION TOMAS BJORK Stockholm School of Economics OXTORD UNIVERSITY PRESS

2 1 Introduction Problem Formulation i 1 v. 2 The Binomial Model The One Period Model Model Description Portfolios and Arbitrage Contingent Claims Risk Neutral Valuation The Multiperiod Model Portfolios and Arbitrage Contingent Claims Exercises Notes 25 3 A More General One Period Model The Model Absence of Arbitrage Martingale Measures Martingale Pricing Completeness Stochastic Discount Factors Exercises 39 4 Stochastic Integrals Introduction Information Stochastic Integrals Martingales Stochastic Calculus and the Ito Formula Examples The Multidimensional Ito Formula Correlated Wiener Processes Exercises Notes 65 5 Differential Equations Stochastic Differential Equations Geometric Brownian Motion The Linear SDE The Infinitesimal Operator 71

3 5.5 Partial Differential Equations The Kolmogorov Equations Exercises Notes 83 6 Portfolio Dynamics Introduction Self-financing Portfolios ; Dividends V. _ Exercises 91 7 Arbitrage Pricing Introduction Contingent Claims and Arbitrage The Black-Scholes Equation Risk Neutral Valuation The Black-Scholes Formula Options on Futures Forward Contracts Futures Contracts and the Black Formula Volatility Historic Volatility Implied Volatility American Options Exercises Notes Completeness and Hedging Introduction Completeness in the Black-Scholes Model Completeness Absence of Arbitrage Exercises Notes Parity Relations and Delta Hedging Parity Relations The Greeks Delta and Gamma Hedging Exercises ' The Martingale Approach to Arbitrage Theory* The Case with Zero Interest Rate Absence of Arbitrage A Rough Sketch of the Proof Precise Results The General Case 146

4 10.4 Completeness Martingale Pricing Stochastic Discount Factors Summary for the Working Economist Notes The Mathematics of the Martingale Approach* Stochastic Integral Representations The Girsanov Theorem: Heuristics v The Girsanov Theorem The Converse of the Girsanov Theorem Girsanov Transformations and Stochastic Differentials Maximum Likelihood Estimation Exercises Notes Black-Scholes from a Martingale Point of View* Absence of Arbitrage Pricing Completeness Multidimensional Models: Classical Approach Introduction Pricing Risk Neutral Valuation Reducing the State Space Hedging Exercises Multidimensional Models: Martingale Approach* Absence of Arbitrage Completeness Hedging Pricing Markovian Models and PDEs Market Prices of Risk Stochastic Discount Factors The Hansen-Jagannathan Bounds Exercises Notes Incomplete Markets Introduction A Scalar Nonpriced Underlying Asset The Multidimensional Case A Stochastic Short Rate 222

5 15.5 The Martingale Approach* Summing Up Exercises Notes Dividends Discrete Dividends Price Dynamics and Dividend Structure Pricing Contingent Claims K Continuous Dividends Continuous Dividend Yield The General Case The Martingale Approach* The Bank Account as Numeraire An Arbitrary Numeraire Exercises Currency Derivatives Pure Currency Contracts Domestic and Foreign Equity Markets Domestic and Foreign Market Prices of Risk The Martingale Approach* Exercises Notes Barrier Options Mathematical Background Out Contracts Down-and-out Contracts Up-and-out Contracts Examples In Contracts Ladders Lookbacks Exercises Notes Stochastic Optimal Control An Example The Formal Problem The Hamilton-Jacobi-Bellman Equation Handling the HJB Equation The Linear Regulator Optimal Consumption and Investment A Generalization Optimal Consumption 299

6 19.7 The Mutual Fund Theorems The Case with No Risk Free Asset The Case with a Risk Free Asset Exercises Notes The Martingale Approach to Optimal Investment* Generalities The Basic Idea v.. _ The Optimal Terminal Wealth The Optimal Portfolio Power Utility The Optimal Terminal Wealth Profile The Optimal Wealth Process The Optimal Portfolio The'Markovian Case Log Utility Exponential Utility The Optimal Terminal Wealth The Optimal Wealth Process The Optimal Portfolio Exercises Notes Optimal Stopping Theory and American Options* Introduction Generalities Some Simple Results Discrete Time The General Case Markovian Models Infinite Horizon Continuous Time General Theory Diffusion Models Connections to the General Theory American Options The American Call Without Dividends The American Put Option The Perpetual American Put Exercises Notes Bonds and Interest Rates Zero Coupon Bonds 350

7 22.2 Interest Rates Definitions Relations between df(t,t), dp(t,t) and dr(t) An Alternative View of the Money Account Coupon Bonds, Swaps and Yields Fixed Coupon Bonds Floating Rate Bonds Interest Rate Swaps [ Yield and Duration ' ' Exercises Notes ' Short Rate Models Generalities The Term Structure Equation Exercises Notes Martingale Models for the Short Rate Q-dynamics Inversion of the Yield Curve Affine Term Structures Definition and Existence A Probabilistic Discussion Some Standard Models The Vasicek Model The Ho-Lee Model The CIR Model The Hull-White Model Exercises Notes Forward Rate Models The Heath-Jarrow-Morton Framework Martingale Modeling The Musiela Parameterization Exercises Notes, Change of Numeraire* Introduction Generalities Changing the Numeraire Forward Measures Using the T-bond as Numeraire An Expectation Hypothesis 405

8 26.5 A General Option Pricing Formula The Hull-White Model The General Gaussian Model Caps and Floors The Numeraire Portfolio Exercises Notes LIBOR and Swap Market Models K Caps: Definition and Market Practice The LIBOR Market Model Pricing Caps in the LIBOR Model Terminal Measure Dynamics and Existence Calibration and Simulation The Discrete Savings Account Swaps Swaptions: Definition and Market Practice The Swap Market Models Pricing Swaptions in the Swap Market Model Drift Conditions for the Regular Swap Market Model Concluding Comment Exercises Notes Potentials and Positive Interest Generalities The Flesaker-Hughston Framework Changing Base Measure Decomposition of a Potential The Markov Potential Approach of Rogers Exercises Notes Forwards and Futures Forward Contracts Futures Contracts Exercises Notes, 457 A Measure and Integration* 458 A.I Sets and Mappings 458 A.2 Measures and Sigma Algebras 460 A.3 Integration 462 A.4 Sigma-Algebras and Partitions 467 A.5 Sets of Measure Zero 468 A.6 The U> Spaces 469

9 A. 7 Hilbert Spaces 470 A.8 Sigma-Algebras and Generators 473 A.9 Product Measures 476 A.10 The Lebesgue Integral 477 A. 11 The Radon-Nikodym Theorem 478 A. 12 Exercises 482 A. 13 Notes 483 i B Probability Theory* V _- 484 B.I Random Variables and Processes 484 B.2 Partitions and Information 487 B.3 Sigma-algebras and Information 489 B.4 Independence 492 B.5 Conditional Expectations 493 B.6 Equivalent Probability Measures 500 B.7 Exercises 502 B.8 Notes 503 C Martingales and Stopping Times* 504 C.I Martingales 504 C.2 Discrete Stochastic Integrals 507 C.3 Likelihood Processes 508 C.4 Stopping Times 509 C.5 Exercises 512 References 514 Index 521