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 Pricing
3.4 Completeness
3.5 Stochastic Discount Factors
3.6 Exercises
4 Stochastic Integrals
4.1 Introduction
4.2 Information
4.3 Stochastic Integrals
4.4 Martingales
4.5 Stochastic Calculus and the It8 Formula
4.6 Examples
4.7 The Multidimensional It6 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
1 5.7 Exercises
Y 5.8 Notes
6 Portfolio Dynamics
6.1 Introduction
6.2 Self-financing Portfolios
'-p ' 6.3 Dividends 6.4 Exercise
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
1 8.4 Exercises
8.5 Notes
i 9 Parity Relations and Delta Hedging
- - 9.1 Parity Relations
9.2 The Greeks
9.3 Delta and Gamma Hedging
1 E 9.4 Exercises
I
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 4 6
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 RRducing 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
16 Dividends
16.1 Discrete Dividends
16.1.1 Price Dynamics and Dividend Structm
16.1.2 Pricing Contingent Claims
16.2 Continuous Dividends
16.2.1 Continuous Dividend Yield
16.2.2 The General Case
16.3 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
18 Barrier Options
, 18.1 Mathematical Background
, 18.2 Out Contracts
18.2.1 Down-and-out Contracts
18.2.2 Upand-Out Contracts
18.2.3 Examples
18.5 Lookbacks
18.6 Exercises
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 Bonds and Interest Rates
20.1 Zero Coupon Bonds
20.2 Interest h t e s
20.2.1 Definitions
20.2.2 Relations between d f (t, T), dp(t, T), and dr(t)
20.2.3 An Alternative View of the Money Account
20.3 Coupon Bonds, Swaps, and Yields
20.3.1 Fixed Coupon Bonds
20.3.2 Floating Rate Bonds
20.3.3 Interest Rate Swaps
20.3.4 Yield and Duration
20.4 Exercises
20.5 Notes
I 21 Short Rate Models
21.1 Generalities
21.2 The Term Structure Equation
2 1.3 Exercises
21.4 Notes
I 22 Martingale Models for the Short Rate
22.1 Q-dynamics
22.2 Inversion of the Yield Curve
22.3 Affine Term Structures
22.3.1 Definition and Existence
22.3.2 A Probabilistic Discussion
22.4 Some Standard Models
22.4.1 The VasiEek Model
22.4.2 The Ho-Lee Model
22.4.3 The CIR Model
22.4.4 The Hull-White Model
22.5 Exercises
I 22.6 Notes
23 Forward Rate Models
23.1 The Heath-Jarrow-Morton Framework
23.2 Martingale Modeling
23.3 The Musiela Parameterization
23.4 Exercises
23.5 Notes
24 Change of Numeraire*
24.1 Introduction

24.2 Generalities
I1 24.3 Changing the Nurneraire
1
24.4.1 Using the T-bond as Numeraire
24.4.2 An Expectation Hypothesis
24.5 A General Option Pricing Formula
24.6 The Hull-White Model
i , 24.7 The General Gaussian Model , 24.8 Caps and Floors
24.10 Notes
25 LIBOR and Swap Market Models
25.1 Caps: Definition and Market Practice
25.2 The LIBOR Market Model
25.3 Pricing Caps in the LIBOR Model
25.4 Terminal Measure Dynamics and Existence
25.5 Calibration and Simulation
i 25.6 The Discrete Savings Account
25.7 Swaps
25.8 Swaptions: Definition and Market Practice
25.9 The Swap Market Models
25.10 Pricing Swaptions in the Swap Market Model
25.11 Drift Conditions for the Regular Swap Market Model
25.12 Concluding Comment
25.13 Exercises
25.14 Notes
( 26 Forwards and Futures
26.1 Forward Contracts
26.2 Futures Contracts
26.3 Exercises
26.4 Notes
A Measure and Integration*
A.l 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-Nikodyrn Theorem
A. 12 Exercises
A.13 Notes
B Probability Theory*
B.l Random Variables and Processes
B.2 Partitions and Information
B.3 Sigmaalgebras and Information
B.4 Independence
B.5 Conditional Expectations
B.6 Equivalent Probability Measures
B. 7 Exercises
B.8 Notes
I C Martingales and Stopping Times*
I C. 1 Martingales
C.2 Discrete Stochastic Integrals
C.3 Likelihood Processes
C.4 Stopping Times
C. 5 Exercises
References
Index