Contents -- 1 Introduction -- 1.1 Problem Formulation -- 2 The Binomial Model -- 2.1 The One Period Model -- 2.1.1 Model Description -- 2.1.2 Portfolios and Arbitrage -- 2.1.3 Contingent Claims -- 2.1.4 Risk Neutral Valuation -- 2.2 The Multiperiod Model -- 2.2.1 Portfolios and Arbitrage -- 2.2.2 Contingent Claims -- 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 -- 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 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 -- 6 Portfolio Dynamics -- 6.1 Introduction -- 6.2 Self-financing Portfolios -- 6.3 Dividends -- 6.4 Exercises -- 7 Arbitrage Pricing -- 7.1 Introduction -- 7.2 Contingent Claims and Arbitrage -- 7.3 The Black-Scholes Equation -- 7.4 Risk Neutral Valuation -- 7.5 The Black-Scholes Formula -- 7.6 Options on Futures -- 7.6.1 Forward Contracts -- 7.6.2 Futures Contracts and the Black Formula -- 7.7 Volatility -- 7.7.1 Historic Volatility -- 7.7.2 Implied Volatility -- 7.8 American Options -- 7.9 Exercises -- 7.10 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 Parity Relations and Delta Hedging -- 9.1 Parity Relations -- 9.2 The Greeks.

9.3 Delta and Gamma Hedging -- 9.4 Exercises -- 10 The Martingale Approach to Arbitrage Theory* -- 10.1 The Case with Zero Interest Rate -- 10.2 Absence of Arbitrage -- 10.2.1 A Rough Sketch of the Proof -- 10.2.2 Precise Results -- 10.3 The General Case -- 10.4 Completeness -- 10.5 Martingale Pricing -- 10.6 Stochastic Discount Factors -- 10.7 Summary for the Working Economist -- 10.8 Notes -- 11 The Mathematics of the Martingale Approach* -- 11.1 Stochastic Integral Representations -- 11.2 The Girsanov Theorem: Heuristics -- 11.3 The Girsanov Theorem -- 11.4 The Converse of the Girsanov Theorem -- 11.5 Girsanov Transformations and Stochastic Differentials -- 11.6 Maximum Likelihood Estimation -- 11.7 Exercises -- 11.8 Notes -- 12 Black-Scholes from a Martingale Point of View* -- 12.1 Absence of Arbitrage -- 12.2 Pricing -- 12.3 Completeness -- 13 Multidimensional Models: Classical Approach -- 13.1 Introduction -- 13.2 Pricing -- 13.3 Risk Neutral Valuation -- 13.4 Reducing the State Space -- 13.5 Hedging -- 13.6 Exercises -- 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 Stochastic Discount Factors -- 14.8 The Hansen-Jagannathan Bounds -- 14.9 Exercises -- 14.10 Notes -- 15 Incomplete Markets -- 15.1 Introduction -- 15.2 A Scalar Nonpriced Underlying Asset -- 15.3 The Multidimensional Case -- 15.4 A Stochastic Short Rate -- 15.5 The Martingale Approach* -- 15.6 Summing Up -- 15.7 Exercises -- 15.8 Notes -- 16 Dividends -- 16.1 Discrete Dividends -- 16.1.1 Price Dynamics and Dividend Structure -- 16.1.2 Pricing Contingent Claims -- 16.2 Continuous Dividends -- 16.2.1 Continuous Dividend Yield -- 16.2.2 The General Case -- 16.3 The Martingale Approach* -- 16.3.1 The Bank Account as Numeraire.

16.3.2 An Arbitrary Numeraire -- 16.4 Exercises -- 17 Currency Derivatives -- 17.1 Pure Currency Contracts -- 17.2 Domestic and Foreign Equity Markets -- 17.3 Domestic and Foreign Market Prices of Risk -- 17.4 The Martingale Approach* -- 17.5 Exercises -- 17.6 Notes -- 18 Barrier Options -- 18.1 Mathematical Background -- 18.2 Out Contracts -- 18.2.1 Down-and-out Contracts -- 18.2.2 Up-and-out Contracts -- 18.2.3 Examples -- 18.3 In Contracts -- 18.4 Ladders -- 18.5 Lookbacks -- 18.6 Exercises -- 18.7 Notes -- 19 Stochastic Optimal Control -- 19.1 An Example -- 19.2 The Formal Problem -- 19.3 The Hamilton-Jacobi-Bellman Equation -- 19.4 Handling the HJB Equation -- 19.5 The Linear Regulator -- 19.6 Optimal Consumption and Investment -- 19.6.1 A Generalization -- 19.6.2 Optimal Consumption -- 19.7 The Mutual Fund Theorems -- 19.7.1 The Case with No Risk Free Asset -- 19.7.2 The Case with a Risk Free Asset -- 19.8 Exercises -- 19.9 Notes -- 20 The Martingale Approach to Optimal Investment* -- 20.1 Generalities -- 20.2 The Basic Idea -- 20.3 The Optimal Terminal Wealth -- 20.4 The Optimal Portfolio -- 20.5 Power Utility -- 20.5.1 The Optimal Terminal Wealth Profile -- 20.5.2 The Optimal Wealth Process -- 20.5.3 The Optimal Portfolio -- 20.6 The Markovian Case -- 20.7 Log Utility -- 20.8 Exponential Utility -- 20.8.1 The Optimal Terminal Wealth -- 20.8.2 The Optimal Wealth Process -- 20.8.3 The Optimal Portfolio -- 20.9 Exercises -- 20.10 Notes -- 21 Optimal Stopping Theory and American Options* -- 21.1 Introduction -- 21.2 Generalities -- 21.3 Some Simple Results -- 21.4 Discrete Time -- 21.4.1 The General Case -- 21.4.2 Markovian Models -- 21.4.3 Infinite Horizon -- 21.5 Continuous Time -- 21.5.1 General Theory -- 21.5.2 Diffusion Models -- 21.5.3 Connections to the General Theory -- 21.6 American Options -- 21.6.1 The American Call Without Dividends.

21.6.2 The American Put Option -- 21.6.3 The Perpetual American Put -- 21.7 Exercises -- 21.8 Notes -- 22 Bonds and Interest Rates -- 22.1 Zero Coupon Bonds -- 22.2 Interest Rates -- 22.2.1 Definitions -- 22.2.2 Relations between df (t, T), dp(t, T) and dr(t) -- 22.2.3 An Alternative View of the Money Account -- 22.3 Coupon Bonds, Swaps and Yields -- 22.3.1 Fixed Coupon Bonds -- 22.3.2 Floating Rate Bonds -- 22.3.3 Interest Rate Swaps -- 22.3.4 Yield and Duration -- 22.4 Exercises -- 22.5 Notes -- 23 Short Rate Models -- 23.1 Generalities -- 23.2 The Term Structure Equation -- 23.3 Exercises -- 23.4 Notes -- 24 Martingale Models for the Short Rate -- 24.1 Q-dynamics -- 24.2 Inversion of the Yield Curve -- 24.3 Affine Term Structures -- 24.3.1 Definition and Existence -- 24.3.2 A Probabilistic Discussion -- 24.4 Some Standard Models -- 24.4.1 The Vasiček Model -- 24.4.2 The Ho-Lee Model -- 24.4.3 The CIR Model -- 24.4.4 The Hull-White Model -- 24.5 Exercises -- 24.6 Notes -- 25 Forward Rate Models -- 25.1 The Heath-Jarrow-Morton Framework -- 25.2 Martingale Modeling -- 25.3 The Musiela Parameterization -- 25.4 Exercises -- 25.5 Notes -- 26 Change of Numeraire* -- 26.1 Introduction -- 26.2 Generalities -- 26.3 Changing the Numeraire -- 26.4 Forward Measures -- 26.4.1 Using the T-bond as Numeraire -- 26.4.2 An Expectation Hypothesis -- 26.5 A General Option Pricing Formula -- 26.6 The Hull-White Model -- 26.7 The General Gaussian Model -- 26.8 Caps and Floors -- 26.9 The Numeraire Portfolio -- 26.10 Exercises -- 26.11 Notes -- 27 LIBOR and Swap Market Models -- 27.1 Caps: Definition and Market Practice -- 27.2 The LIBOR Market Model -- 27.3 Pricing Caps in the LIBOR Model -- 27.4 Terminal Measure Dynamics and Existence -- 27.5 Calibration and Simulation -- 27.6 The Discrete Savings Account -- 27.7 Swaps -- 27.8 Swaptions: Definition and Market Practice.

27.9 The Swap Market Models -- 27.10 Pricing Swaptions in the Swap Market Model -- 27.11 Drift Conditions for the Regular Swap Market Model -- 27.12 Concluding Comment -- 27.13 Exercises -- 27.14 Notes -- 28 Potentials and Positive Interest -- 28.1 Generalities -- 28.2 The Flesaker-Hughston Framework -- 28.3 Changing Base Measure -- 28.4 Decomposition of a Potential -- 28.5 The Markov Potential Approach of Rogers -- 28.6 Exercises -- 28.7 Notes -- 29 Forwards and Futures -- 29.1 Forward Contracts -- 29.2 Futures Contracts -- 29.3 Exercises -- 29.4 Notes -- 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 L[sup(p)] 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 -- 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 -- C Martingales and Stopping Times* -- C.1 Martingales -- C.2 Discrete Stochastic Integrals -- C.3 Likelihood Processes -- C.4 Stopping Times -- C.5 Exercises -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.