1 Probability

1.1 Probabilities and Events

1.2 Conditional Probability

1.3 Random Variables and Expected Values

1.4 Covariance and Correlation

1.5 Conditional Expectation

1.6 Exercises

2 Normal Random Variables

2.1 Continuous Random Variables

2.2 Normal Random Variables

2.3 Properties of Normal Random Variables

2.4 The Central Limit Theorem

2.5 Exercises

3 Brownian Motion and Geometric Brownian Motion

3.1 Brownian Motion

3.2 Brownian Motion as a Limit of Simpler Models

3.3 Geometric Brownian Motion

3.3.1 Geometric Brownian Motion as a Limit of Simpler Models

3.4 The Maximum Variable

3.5 The Cameron-Martin Theorem

3.6 Exercises

4 Interest Rates and Present Value Analysis

4.1 Interest Rates

4.2 Present Value Analysis

4.3 Rate of Return

4.4 Continuously Varying Interest Rates

4.5 Exercises

5 Pricing Contracts via Arbitrage

5.1 An Example in Options Pricing

5.2 Other Examples of Pricing via Arbitrage

5.3 Exercises

6 The Arbitrage Theorem

6.1 The Arbitrage Theorem

6.2 The Mulfiperiod Binomial Model

6.3 Proof of the Arbitrage Theorem

6.4 Exercises

7 The Black-Scholes Formula

7.1 Introduction

7.2 The Black-Scholes Formula

7.3 Properties of the Black-Scholes Option Cost

7.4 The Delta Hedging Arbitrage Strategy

7.5 Some Derivations

7.5.1 The Black-Scholes Formula

7.5.2 The Partial Derivatives

7.6 European Put Options

7.7 Exercises

8 Additional Results on Options

8.1 Introduction

8.2 Call Options on Dividend-Paying Securities

8.2.1 The Dividend for Each Share of the Security Is Paid Continuously in Time at a Rate Equal to a Fixed Fraction f of the Price of the Security

8.2.2 For Each Share Owned, a Single Payment of fS(td) Is Made at Time td

8.2.3 For Each Share Owned, a Fixed Amount D Is to Be Paid at Time td

8.3 Pricing American Put Options

8.4 Adding Jumps to Geometric Brownian Motion

8.4.1 When the Jump Distribution Is Lognormal

8.4.2 When the Jump Distribution Is General

8.5 Estimating the Volatility Parameter

8.5.1 Estimating a Population Mean and Variance

8.5.2 The Standard Estimator of Volatility

8.5.3 Using Opening and Closing Data

8.5.4 Using Opening, Closing, and High-Low Data

8.6 Some Comments

8.6.1 When the Option Cost Differs from the Black-Scholes Formula

8.6.2 When the Interest Rate Changes

8.6.3 Final Comments

8.7 Appendix

8.8 Exercises

9 Valuing by Expected Utility

9.1 Limitations of Arbitrage Pricing

9.2 Valuing Investments by Expected Utility

9.3 The Portfolio Selection Problem

9.3.1 Estimating Covariances

9.4 Value at Risk and Conditional Value at Risk

9.5 The Capital Assets Pricing Model

9.6 Rates of Return: Single-Period and Geometric Brownian Motion

9.7 Exercises

10 Stochastic Order Relations

10.1 First-Order Stochastic Dominance

10.2 Using Coupling to Show Stochastic Dominance

10.3 Likelihood Ratio Ordering

10.4 A Single-Period Investment Problem

10.5 Second-Order Dominance

10.5.1 Normal Random Variables

10.5.2 More on Second-Order Dominance

10.6 Exercises

11 Optimization Models

11.1 Introduction

11.2 A Deterministic Optimization Model

11.2.1 A General Solution Technique Based on Dynamic Programming

11.2.2 A Solution Technique for Concave Return Functions

11.2.3 The Knapsack Problem

11.3 Probabilistic Optimization Problems

11.3.1 A Gambling Model with Unknown Win Probabilities

11.3.2 An Investment Allocation Model

11.4 Exercises

12 Stochastic Dynamic Programming

12.1 The Stochastic Dynamic Programming Problem

12.2 Infinite Time Models

12.3 Optimal Stopping Problems

12.4 Exercises

13 Exotic Options

13.1 Introduction

13.2 Barrier Options

13.3 Asian and Lookback Options

13.4 Monte Carlo Simulation

13.5 Pricing Exotic Options by Simulation

13.6 More Efficient Simulation Estimators

13.6.1 Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations

13.6.2 Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations

13.7 Options with Nonlinear Payoffs

13.8 Pricing Approximations via Multiperiod Binomial Models

13.9 Continuous Time Approximations of Barrier and Lookback Options

13.10 Exercises

14 Beyond Geometric Brownian Motion Models

14.1 Introduction

14.2 Crude Oil Data

14.3 Models for the Crude Oil Data

14.4 Final Comments

15 Autoregressive Models and Mean Reversion

15.1 The Autoregressive Model

15.2 Valuing Options by Their Expected Return

15.3 Mean Reversion

15.4 Exercises

Index