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An Elementary Introduction to Mathematical Finance, Third Edition

Book Description

This textbook on the basics of option pricing is accessible to readers with limited mathematical training. It is for both professional traders and undergraduates studying the basics of finance. Assuming no prior knowledge of probability, Sheldon M. Ross offers clear, simple explanations of arbitrage, the Black-Scholes option pricing formula, and other topics such as utility functions, optimal portfolio selections, and the capital assets pricing model. Among the many new features of this third edition are new chapters on Brownian motion and geometric Brownian motion, stochastic order relations and stochastic dynamic programming, along with expanded sets of exercises and references for all the chapters.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Introduction and Preface
  8. 1. Probability
    1. 1.1 Probabilities and Events
    2. 1.2 Conditional Probability
    3. 1.3 Random Variables and Expected Values
    4. 1.4 Covariance and Correlation
    5. 1.5 Conditional Expectation
    6. 1.6 Exercises
  9. 2. Normal Random Variables
    1. 2.1 Continuous Random Variables
    2. 2.2 Normal Random Variables
    3. 2.3 Properties of Normal Random Variables
    4. 2.4 The Central Limit Theorem
    5. 2.5 Exercises
  10. 3. Brownian Motion and Geometric Brownian Motion
    1. 3.1 Brownian Motion
    2. 3.2 Brownian Motion as a Limit of Simpler Models
    3. 3.3 Geometric Brownian Motion
      1. 3.3.1 Geometric Brownian Motion as a Limit of Simpler Models
    4. 3.4 *The Maximum Variable
    5. 3.5 The Cameron-Martin Theorem
    6. 3.6 Exercises
  11. 4. Interest Rates and Present Value Analysis
    1. 4.1 Interest Rates
    2. 4.2 Present Value Analysis
    3. 4.3 Rate of Return
    4. 4.4 Continuously Varying Interest Rates
    5. 4.5 Exercises
  12. 5. Pricing Contracts via Arbitrage
    1. 5.1 An Example in Options Pricing
    2. 5.2 Other Examples of Pricing via Arbitrage
    3. 5.3 Exercises
  13. 6. The Arbitrage Theorem
    1. 6.1 The Arbitrage Theorem
    2. 6.2 The Multiperiod Binomial Model
    3. 6.3 Proof of the Arbitrage Theorem
    4. 6.4 Exercises
  14. 7. The Black–Scholes Formula
    1. 7.1 Introduction
    2. 7.2 The Black–Scholes Formula
    3. 7.3 Properties of the Black–Scholes Option Cost
    4. 7.4 The Delta Hedging Arbitrage Strategy
    5. 7.5 Some Derivations
      1. 7.5.1 The Black–Scholes Formula
      2. 7.5.2 The Partial Derivatives
    6. 7.6 European Put Options
    7. 7.7 Exercises
  15. 8. Additional Results on Options
    1. 8.1 Introduction
    2. 8.2 Call Options on Dividend-Paying Securities
      1. 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
      2. 8.2.2 For Each Share Owned, a Single Payment of fS(td) Is Made at Time td
      3. 8.2.3 For Each Share Owned, a Fixed Amount D Is to Be Paid at Time td
    3. 8.3 Pricing American Put Options
    4. 8.4 Adding Jumps to Geometric Brownian Motion
      1. 8.4.1 When the Jump Distribution Is Lognormal
      2. 8.4.2 When the Jump Distribution Is General
    5. 8.5 Estimating the Volatility Parameter
      1. 8.5.1 Estimating a Population Mean and Variance
      2. 8.5.2 The Standard Estimator of Volatility
      3. 8.5.3 Using Opening and Closing Data
      4. 8.5.4 Using Opening, Closing, and High–Low Data
    6. 8.6 Some Comments
      1. 8.6.1 When the Option Cost Differs from the Black–Scholes Formula
      2. 8.6.2 When the Interest Rate Changes
      3. 8.6.3 Final Comments
    7. 8.7 Appendix
    8. 8.8 Exercises
  16. 9. Valuing by Expected Utility
    1. 9.1 Limitations of Arbitrage Pricing
    2. 9.2 Valuing Investments by Expected Utility
    3. 9.3 The Portfolio Selection Problem
      1. 9.3.1 Estimating Covariances
    4. 9.4 Value at Risk and Conditional Value at Risk
    5. 9.5 The Capital Assets Pricing Model
    6. 9.6 Rates of Return: Single-Period and Geometric Brownian Motion
    7. 9.7 Exercises
  17. 10. Stochastic Order Relations
    1. 10.1 First-Order Stochastic Dominance
    2. 10.2 Using Coupling to Show Stochastic Dominance
    3. 10.3 Likelihood Ratio Ordering
    4. 10.4 A Single-Period Investment Problem
    5. 10.5 Second-Order Dominance
      1. 10.5.1 Normal Random Variables
      2. 10.5.2 More on Second-Order Dominance
    6. 10.6 Exercises
  18. 11. Optimization Models
    1. 11.1 Introduction
    2. 11.2 A Deterministic Optimization Model
      1. 11.2.1 A General Solution Technique Based on Dynamic Programming
      2. 11.2.2 A Solution Technique for Concave Return Functions
      3. 11.2.3 The Knapsack Problem
    3. 11.3 Probabilistic Optimization Problems
      1. 11.3.1 A Gambling Model with Unknown Win Probabilities
      2. 11.3.2 An Investment Allocation Model
    4. 11.4 Exercises
  19. 12. Stochastic Dynamic Programming
    1. 12.1 The Stochastic Dynamic Programming Problem
    2. 12.2 Infinite Time Models
    3. 12.3 Optimal Stopping Problems
    4. 12.4 Exercises
  20. 13. Exotic Options
    1. 13.1 Introduction
    2. 13.2 Barrier Options
    3. 13.3 Asian and Lookback Options
    4. 13.4 Monte Carlo Simulation
    5. 13.5 Pricing Exotic Options by Simulation
    6. 13.6 More Efficient Simulation Estimators
      1. 13.6.1 Control and Antithetic Variables in the Simulation of Asian and Lookback Option Valuations
      2. 13.6.2 Combining Conditional Expectation and Importance Sampling in the Simulation of Barrier Option Valuations
    7. 13.7 Options with Nonlinear Payoffs
    8. 13.8 Pricing Approximations via Multiperiod Binomial Models
    9. 13.9 Continuous Time Approximations of Barrier and Lookback Options
    10. 13.10 Exercises
  21. 14. Beyond Geometric Brownian Motion Models
    1. 14.1 Introduction
    2. 14.2 Crude Oil Data
    3. 14.3 Models for the Crude Oil Data
    4. 14.4 Final Comments
  22. 15. Autoregressive Models and Mean Reversion
    1. 15.1 The Autoregressive Model
    2. 15.2 Valuing Options by Their Expected Return
    3. 15.3 Mean Reversion
    4. 15.4 Exercises
  23. Index