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Quantitative Finance

Book Description

Teach Your Students How to Become Successful Working Quants

Quantitative Finance: A Simulation-Based Introduction Using Excel provides an introduction to financial mathematics for students in applied mathematics, financial engineering, actuarial science, and business administration. The text not only enables students to practice with the basic techniques of financial mathematics, but it also helps them gain significant intuition about what the techniques mean, how they work, and what happens when they stop working.

After introducing risk, return, decision making under uncertainty, and traditional discounted cash flow project analysis, the book covers mortgages, bonds, and annuities using a blend of Excel simulation and difference equation or algebraic formalism. It then looks at how interest rate markets work and how to model bond prices before addressing mean variance portfolio optimization, the capital asset pricing model, options, and value at risk (VaR). The author next focuses on binomial model tools for pricing options and the analysis of discrete random walks. He also introduces stochastic calculus in a nonrigorous way and explains how to simulate geometric Brownian motion. The text proceeds to thoroughly discuss options pricing, mostly in continuous time. It concludes with chapters on stochastic models of the yield curve and incomplete markets using simple discrete models.

Accessible to students with a relatively modest level of mathematical background, this book will guide your students in becoming successful quants. It uses both hand calculations and Excel spreadsheets to analyze plenty of examples from simple bond portfolios. The spreadsheets are available on the book’s CRC Press web page.

Table of Contents

  1. Preface
  2. Author
  3. Chapter 1 - Introduction
  4. Chapter 2 - Intuition about Uncertainty and Risk
    1. 2.1 CHAPTER SUMMARY
    2. 2.2 Introduction
    3. 2.3 Individual Attitudes toward Risk
      1. Question 1
      2. Question 2 (Bet)
      3. Question 3A
      4. Question 3B
      5. Question 4
      6. Question 5A
      7. Question 5B
      8. Question 6A
      9. Question 6B
    4. 2.4 The St. Petersburg Paradox
      1. 2.4.1 Resolving the Paradox with Utilities
      2. 2.4.2 Resolving the Petersburg Paradox with Risk Exposure
      3. 2.4.3 Other Ways to Resolve the Paradox
    5. 2.5 Looking Forward to Chapter 3
    6. Exercises
    7. FURTHER READING
  5. Chapter 3 - The Classical Approach to Decision Making under Uncertainty
    1. 3.1 CHAPTER SUMMARY
    2. 3.2 Map to the Future
    3. Exercise
    4. FURTHER READING
  6. Chapter 4 - Valuing Investment Opportunities: The Discounted Cash Flow Method
    1. 4.1 CHAPTER SUMMARY
    2. 4.2 Discounted Cash Flow Method for Evaluating Investment Opportunities
      1. 4.2.1 Example of a Discounted Cash Flow Technique
      2. 4.2.2 Choosing the Discount Rate P
      3. 4.2.3 Philosophical Problems with DCF
      4. 4.2.4 Why This Is a Good Approach Despite Its Uncertain Philosophical Status
    3. 4.3 Conclusions
    4. Exercises
    5. FURTHER READING
  7. Chapter 5 - Repaying Loans over Time
    1. 5.1 CHAPTER SUMMARY
    2. 5.2 Introduction
      1. Question
    3. 5.3 Repaying a Loan over Time: Excel
    4. 5.4 Repaying a Loan over Time: Mathematics
    5. 5.5 First-Order Difference Equations
    6. 5.6 Solving the Loan Repayment Difference Equation
      1. 5.6.1 Loan Repaid “Quickly”
      2. 5.6.2 Loan Repaid Continuously
    7. 5.7 More Examples of Using Difference Equations to Find Loan Payments
      1. Solution
    8. 5.8 Writing the Difference Equation in Forward versus Backward Forms
    9. 5.9 Bridges to the Future
    10. Exercises
    11. FURTHER READING
  8. Chapter 6 - Bond Pricing with Default: Using Simulations
    1. 6.1 CHAPTER SUMMARY
    2. 6.2 Modeling a Defaultable Bond or Loan
    3. 6.3 Financial Insights
    4. 6.4 Simulating Loan Portfolios
    5. 6.5 What Happens if There Are a Large Number of Independent Loans?
    6. 6.6 Bridge to the Future
    7. Exercises
    8. FURTHER READING
  9. Chapter 7 - Bond Pricing with Default: Using Difference Equations
    1. 7.1 CHAPTER SUMMARY
    2. 7.2 Risky Bonds
    3. 7.3 Using Difference Equations to Find C
    4. 7.4 Exploring the Insights Arising from Equation 7.5
    5. 7.5 Determining Recovery Rates
    6. 7.6 Determining the Probability of Default
    7. 7.7 A Bridge to the Future
    8. Exercises
    9. FURTHER READING
  10. Chapter 8 - Difference Equations for Life Annuities
    1. 8.1 CHAPTER SUMMARY
    2. 8.2 Introduction
    3. Exercises
    4. FURTHER READING
  11. Chapter 9 - Tranching and Collateralized Debt Obligations
    1. 9.1 CHAPTER SUMMARY
    2. 9.2 Collateralized Debt Obligations
    3. 9.3 Tranched Portfolios
    4. 9.4 The Detailed Calculation
      1. 9.4.1 Pricing a Bond with Two Default Events
      2. 9.4.2 Finding the “Fair” Coupon for Tranche B
    5. 9.5 Correlation of Two Identical Bonds
    6. 9.6 Conclusion
    7. Exercises
    8. FURTHER READING
  12. Chapter 10 - Bond CDOs: More than Two Bonds, Correlation, and Simulation
    1. 10.1 CHAPTER SUMMARY
    2. 10.2 Introduction
    3. 10.3 Using an Excel Simulation to Analyze CDOs with More Than Two Bonds
    4. 10.4 Collateralized Debt Obligations: An Example of Financial Engineering
    5. 10.5 The Binomial Simplification
    6. 10.6 Correlated Defaults
    7. Exercises
    8. FURTHER READING
  13. Chapter 11 - Fundamentals of Fixed Income Markets
    1. 11.1 CHAPTER SUMMARY
    2. 11.2 What Are Bonds?
      1. 11.2.1 Debt Markets versus Borrowing from Small Number of Large Entities
      2. 11.2.2 Different Types of Bonds
    3. 11.3 Getting Down to Quantitative Details
      1. 11.3.1 Interest Rate Conventions
      2. 11.3.2 Discount Factor Conventions
    4. 11.4 Simplest Bond Pricing Equation
    5. 11.5 How Bonds Are Traded in Canada
      1. 11.5.1 Bond Auctions
      2. 11.5.2 After Auction Trading of Bonds
    6. 11.6 Clean and Dirty Bond Prices
      1. 11.6.1 Day Count Convention (or, the Dirty Secret of Clean Prices)
      2. 11.6.2 Dirty Price, Clean Price, and Invoice Price
    7. 11.7 Conclusion and Bridge to the Next Chapter
    8. Exercises
    9. FURTHER READING
  14. Chapter 12 - Yield Curves and Bond Risk Measures
    1. 12.1 CHAPTER SUMMARY
    2. 12.2 Introduction
      1. 12.2.1 Computing Yield to Maturity from Bond Prices
      2. 12.2.2 Other Yield Measures
    3. 12.3 Constructing Yield Curves from Bond Prices
      1. 12.3.1 Linear Interpolation
    4. 12.4 Bond Price Sensitivities to the Yield
      1. 12.4.1 Example Duration Calculation for a Zero Coupon Bond
      2. 12.4.2 Curvatures or “Convexities”
    5. Exercises
    6. FURTHER READING
  15. Chapter 13 - Forward Rates
    1. 13.1 CHAPTER SUMMARY
    2. 13.2 Introduction
    3. 13.3 Relationships between Forward Rates and the Yield Curve
    4. 13.4 Yield Curves, Discount Factors, and Forward Rates
    5. 13.5 Interpreting Forward Curves
    6. Exercises
    7. FURTHER READING
  16. Chapter 14 - Modeling Stock Prices
    1. 14.1 CHAPTER SUMMARY
    2. 14.2 What Are Stocks?
    3. 14.3 Simple Statistical Analysis of Real Stock Data
    4. Exercises
    5. FURTHER READING
  17. Chapter 15 - Mean Variance Portfolio Optimization
    1. 15.1 CHAPTER SUMMARY
    2. 15.2 Selecting Portfolios
      1. 15.2.1 Basic Model Assumptions
      2. 15.2.2 Turning Our Model Setting into an Optimization Problem
      3. 15.2.3 Studying the Formula in a Spreadsheet
      4. 15.2.4 Data Requirements
    3. 15.3 CAPM and Markowitz
    4. Exercises
    5. FURTHER READING
  18. Chapter 16 - A Qualitative Introduction to Options
    1. 16.1 CHAPTER SUMMARY
    2. 16.2 Stock Option Definitions
    3. 16.3 Uses for Put and Call Options
    4. 16.4 Qualitative Behavior of Puts and Calls
    5. Exercises
    6. FURTHER READING
  19. Chapter 17 - Value at Risk
    1. 17.1 CHAPTER SUMMARY
    2. 17.2 Introduction to Value at Risk
    3. 17.3 Pitfalls of VaR
    4. 17.4 SUMMARY
    5. Exercises
    6. FURTHER READING
  20. Chapter 18 - Pricing Options Using Binomial Trees
    1. 18.1 CHAPTER SUMMARY
    2. 18.2 Introduction
    3. 18.3 Binomial Model
    4. 18.4 Single-Period Binomial Tree Model for Option Pricing
    5. 18.5 Extending the Binomial Model to Multiple Time Steps
      1. 18.5.1 Numerical Example: Pricing a Two-Period Binomial Put Option
    6. 18.6 Multiple-Step Binomial Trees
    7. 18.7 SUMMARY
    8. Exercises
    9. FURTHER READING
  21. Chapter 19 - Random Walks
    1. 19.1 CHAPTER SUMMARY
    2. 19.2 Introduction
    3. 19.3 Deriving the Diffusion Partial Differential Equation
    4. Exercises
    5. FURTHER READING
  22. Chapter 20 - Basic Stochastic Calculus
    1. 20.1 CHAPTER SUMMARY
    2. 20.2 Basics of Stochastic Calculus
    3. 20.3 Stochastic Integration by Examples
      1. 20.3.1 Review of the Left Endpoint Rule of Introductory Calculus
      2. 20.3.2 Itô Integration
      3. 20.3.3 Itô Isometry
      4. 20.3.4 Introduction to Ordinary Differential Equations
      5. 20.3.5 Solution of SDEs
        1. 20.3.5.1 Arithmetic Brownian Motion
        2. 20.3.5.2 Geometric Brownian Motion
        3. 20.3.5.3 Ornstein–Uhlenbeck Process
    4. 20.4 Conclusions and Bridge to Next Chapters
    5. Exercises
    6. FURTHER READING
  23. Chapter 21 - Simulating Geometric Brownian Motion
    1. 21.1 Simulating GBM Stock Prices at a Single Future Time
    2. 21.2 Simulating a Time Sequence of GBM Stock Prices
    3. 21.3 Summary
    4. EXERCISES
    5. FURTHER READING
  24. Chapter 22 - Black Scholes PDE for Pricing Options in Continuous Time
    1. 22.1 CHAPTER SUMMARY
    2. 22.2 Introduction
    3. 22.3 Hedging Argument
    4. 22.4 Call Price Solution of the Black Scholes Equation
    5. 22.5 Why Short Selling Is So Dangerous
    6. 22.6 Summary and Bridge to the Future
    7. Exercises
    8. FURTHER READING
  25. Chapter 23 - Solving the Black Scholes PDE
    1. 23.1 CHAPTER SUMMARY
    2. 23.2 Solving the Black Scholes Partial PDE for a European Call
    3. 23.3 General European Option Payoffs: Risk-Neutral Pricing
    4. 23.4 SUMMARY
    5. Exercises
  26. Chapter 24 - Pricing Put Options Using Put Call Parity
    1. 24.1 CHAPTER SUMMARY
    2. 24.2 SUMMARY
    3. Exercises
    4. FURTHER READING
  27. Chapter 25 - Some Approximate Values of the Black Scholes Call Formula
    1. 25.1 INTRODUCTION
    2. 25.2 Approximate Call Formulas at-the-Money
    3. 25.3 Approximate Call Values Near-the-Money
    4. 25.4 Approximate Call Values Far-from-the-Money
    5. Exercises
    6. FURTHER READING
  28. Chapter 26 - Simulating Delta Hedging
    1. 26.1 CHAPTER SUMMARY
    2. 26.2 Introduction
    3. 26.3 How Does Delta Hedging Really Work?
    4. 26.4 Understanding the Results of the Delta Hedging Process
    5. 26.5 The Impact of Transaction Costs
    6. 26.6 A Hedgers Perspective on Option Gamma or, “Big Gamma” = “Big Money”
    7. 26.7 Bridge to the Future
    8. Exercises
    9. FURTHER READING
  29. Chapter 27 - Black Scholes with Dividends
    1. 27.1 CHAPTER SUMMARY
    2. 27.2 Modeling Dividends
      1. 27.2.1 “Tailed Stock Positions”
    3. 27.3 The Black Scholes PDE for the Continuously Paid Dividend Case
    4. 27.4 Pricing the Prepaid Forward on a Continuous Dividend Paying Stock
    5. 27.5 More Complicated Derivatives on Underlyings Paying Continuous Dividends
    6. Exercises
    7. FURTHER READING
  30. Chapter 28 - American Options
    1. 28.1 CHAPTER SUMMARY
    2. 28.2 Introduction and Binomial Pricing
    3. 28.3 American Puts
    4. 28.4 American Calls
    5. Exercises
    6. FURTHER READING
  31. Chapter 29 - Pricing the Perpetual American Put and Call
    1. 29.1 CHAPTER SUMMARY
    2. 29.2 Perpetual Options: Underlying Pays No Dividends
      1. 29.2.1 Basic Perpetual American Put
    3. 29.3 Basic Perpetual American Call
    4. 29.4 Perpetual American Call/Put Model with Dividends
    5. 29.5 The Perpetual American Call, Continuous Dividends
    6. Exercise
    7. further reading
  32. Chapter 30 - Options on Multiple Underlying Assets
    1. 30.1 INTRODUCTION
    2. 30.2 Exchange Options
    3. Exercise
    4. FURTHER READING
  33. Chapter 31 - Interest Rate Models
    1. 31.1 CHAPTER SUMMARY
    2. 31.2 Setting the Stage for Stochastic Interest Rate Models
    3. 31.3 Pricing When You CANNOT Trade the Underlying Asset
    4. 31.4 Hedging Bonds in Continuous Time
    5. 31.5 Solving the Bond Pricing PDE
    6. 31.6 Vasicek Model
    7. 31.7 Summary
    8. Exercises
    9. FURTHER READING
  34. Chapter 32 - Incomplete Markets
    1. 32.1 CHAPTER SUMMARY
    2. 32.2 Introduction to Incomplete Markets
    3. 32.3 Trying to Hedge Options on a Trinomial Tree
      1. 32.3.1 Review of the Standard Binomial Tree Model
      2. 32.3.2 Extension to a Trinomial Tree Model
    4. 32.4 Minimum Variance Hedging of a European Option with Default
      1. 32.4.1 Binomial Tree Model for Option Pricing
    5. 32.5 Binomial Tree Model with Default Risk
    6. EXERCISE
    7. FURTHER READING
  35. Appendix 1: Probability Theory Basics
    1. A1.1 Introduction
    2. A1.2 Conditional probability
      1. A1.2.1 Practical Example
      2. A1.2.2 Some Notations
      3. A1.2.3 Application to Dice Rolling Example
      4. A1.2.4 Calculating Unconditional Probabilities from Conditional
    3. A1.3 Independence
      1. A1.3.1 Calculations Using Independence
    4. A1.4 Factorials, “choose” notation, and Stirling’s formula
      1. A1.4.1 Factorial Notation
    5. A1.5 Binomial random variables
      1. A1.5.1 Discrete Random Variables
      2. A1.5.2 Random Variables
      3. A1.5.3 Probability Density Function
      4. A1.5.4 Bernoulli Random Variables
      5. A1.5.5 Binomial Random Variables
      6. A1.5.6 Cumulative Distribution Function
      7. A1.5.7 Example to Introduce Expected Value
    6. A1.6 Mean and variance
      1. A1.6.1 Definition of Expected Value
      2. A1.6.2 Definition of Variance
      3. A1.6.3 Continuous Random Variables
    7. A1.7 SOME USEFUL CONTINUOUS PDFs
    8. A1.8 NEW RANDOM VARIABLES FROM OLD: LINEAR TRANSFORMATIONS
      1. A1.8.1 Variance
    9. A1.9 JOINT DENSITIES
      1. A1.9.1 Marginal Densities for Continuous RVs
      2. A1.9.2 Independence of Two Random Variables
    10. A1.10 Combining random variables
      1. A1.10.1 Creating New Random Variables from Old Ones
      2. A1.10.2 More about Expected Value and Variance
      3. A1.10.3 Conditional Densities
      4. A1.10.4 Conditional Probabilities
      5. A1.10.5 Conditional Probabilities for Discrete Random Variables
      6. A1.10.6 Conditional Densities for Continuous Random Variables
    11. A1.11 Moment-generating functions
      1. A1.11.1 Using MGFs to Study the Sums of Random Variables
    12. A1.12 Poisson distribution
      1. A1.12.1 Parameter Estimation for Poisson Random Variables
    13. A1.13 Relationship between the Poisson, binomial, and exponential RVs
    14. A1.14 MGFs and THE NORMAL RANDOM VARIABLE
      1. A1.14.1 A Moment Generating Function Proof That the Sum of Two Normal Variables Is Normal
    15. Further Reading
  36. Appendix 2: Proof of DeMoivre–Laplace Theorem
  37. Appendix 3: Naming Variables in Excel
  38. Appendix 4: Building VBA Macros from Excel
    1. A4.1 Set Up Macro Interface
    2. A4.2 Brief Introduction to VBA Code
      1. A4.2.1 Cells (Number of Rows and Number of Columns)
      2. A4.2.2 Range (Cells 1 and 2)
      3. A4.2.3 Cells.Find and Range.Find
      4. A4.2.4 Loops and Condition
    3. A4.3 Example Code