Introduction and Preface
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 Multiperiod Binomial Model
6.3 Proof of the Arbitrage Theorem
6.4 Exercises
7 The Black-Scboles 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
fStd 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
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