A Simulation-Based lntroduction Using Excel

Transcription

1 Quantitative Finance A Simulation-Based lntroduction Using Excel Matt Davison University of Western Ontario London, Canada CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK

2 Contents Preface, xvii Author, xix CHARTER 1 Introduction 1 CHARTER 2 Intuition about Uncertainty and Risk CHARTER SUMMARY INTRODUCTION INDIVIDUAL ATTITUDES TOWARD RISK THE ST. PETERSBURG PARADOX Resolving the Paradox with Utilities Resolving the Petersburg Paradox with Risk Exposure Other Ways to Resolve the Paradox LOOKING FORWARD TO CHAPTER 3 13 EXERCISES 14 FURTHER READING 15 CHARTER 3 The Classical Approach to Decision Making under Uncertainty CHAPTER SUMMARY MAP TO THE FUTURE 21 EXERCISE 21 FURTHER READING 21

3 vi Contents CHARTER 4 Valuing Investment Opportunities: The Discounted Cash Flow Method CHARTER SUMMARY DISCOUNTED CASH FLOW METHOD FOR EVALUATING INVESTMENT OPPORTUNITIES Example of a Discounted Cash Flow Technique Choosing the Discount Rate P Philosophical Problems with DCF Why Ulis Is a Good Approach Despite Its Uncertain Philosophical Status CONCLUSIONS 26 EXERCISES 26 FURTHER READING 27 CHARTER 5 Repaying Loans over Time CHAPTER SUMMARY INTRODUCTION REPAYING A LOAN OVER TIME: EXCEL REPAYING A LOAN OVER TIME: MATH EM ATICS FIRST-ORDER DIFFERENCE EQUATIONS SOLVING THE LOAN REPAYMENT DIFFERENCE EQUATION Loan Repaid "Quickly" Loan Repaid Continuously MORE EXAMPLES OF USING DIFFERENCE EQUATIONS TO FIND LOAN PAYMENTS WRITING THE DIFFERENCE EQUATION IN FORWARD VERSUS BACKWARD FORMS BRIDGESTOTHE FUTURE 46 EXERCISES 46 FURTHER READING 46

4 Contents vii CHARTER 6 Bond Pricing with Default: Using Simulations CHAPTER SUMMARY MODELING A DEFAULTABLE BOND OR LOAN FINANCIAL INSIGHTS SIMULATING LOAN PORTFOLIOS WHAT HAPPENS IF THERE ARE A LARGE NUMBER OF INDEPENDENT LOANS? BRIDGE TO THE FUTURE 66 EXERCISES 66 FURTHER READING 67 CHARTER 7 Bond Pricing with Default: Using Difference Equations CHAPTER SUMMARY RISKY BONDS USING DIFFERENCE EQUATIONS TO FIND C EXPLORING THE INSIGHTS ARISING FROM EQUATION DETERMINING RECOVERY RATES DETERMINING THE PROBABILITY OF DEFAULT A BRIDGE TO THE FUTURE 76 EXERCISES 76 FURTHER READING 76 CHARTER 8 Difference Equations for Life Annuities CHAPTER SUMMARY INTRODUCTION 79 EXERCISES 87 FURTHER READING 88 CHARTER 9 Tranching and Col lateral ized Debt Obligations CHAPTER SUMMARY COLLATERALIZED DEBT OBLIGATIONS 90

5 viii Contents 9.3 TRANCHED PORTFOLIOS THE DETAILED CALCULATION Pricing a Bond with Two Default Events Finding the "Fair" Coupon for Tranche B CORRELATION OF TWO IDENTICAL BONDS CONCLUSION 102 EXERCISES 102 FURTHER READING 103 CHARTER 10 Bond CDOs: More than Two Bonds, Correlation, and Simulation CHAPTER SUMMARY INTRODUCTION USING AN EXCEL SIMULATION TO ANALYZE CDOs WITH MORE THAN TWO BONDS COLLATERALIZED DEBT OBLIGATIONS: AN EXAMPLE OF FINANCIAL ENGINEERING THE BINOMIAL SIMPLIFICATION CORRELATED DEFAULTS 114 EXERCISES 118 FURTHER READING 119 CHARTER 11 Fundamentals of Fixed Income Markets CHAPTER SUMMARY WHAT ARE BONDS? Debt Markets versus Borrowing from Small Number of Large Entities Different Types of Bonds GETTING DOWN TO QUANTITATIVE DETAILS Interest Rate Conventions Discount Factor Conventions SIMPLEST BOND PRICING EQUATION 126

6 Contents ix 11.5 HOW BONDS ARE TRADED IN CANADA BondAuctions After Auction Trading of Bonds CLEAN AND DIRTY BOND PRICES Day Count Convention (or, the Dirty Secret of Clean Prices) Dirty Price, Clean Price, and Invoice Price CONCLUSION AND BRIDGE TO THE NEXT CHAPTER 133 EXERCISES 134 FURTHER READING 134 CHARTER 12 Yield Curves and Bond Risk Measures CHAPTER SUMMARY INTRODUCTION Computing Yield to Maturity from Bond Prices Other Yield Measures CONSTRUCTING YIELD CURVES FROM BOND PRICES Linear Interpolation BOND PRICE SENSITIVITIES TO THE YIELD Example Duration Calculation for a Zero Coupon Bond Curvatures or "Convexities" 145 EXERCISES 146 FURTHER READING 148 CHARTER 13 Forward Rates CHAPTER SUMMARY INTRODUCTION RELATIONSHIPS BETWEEN FORWARD RATES AND THE YIELD CURVE 150

7 x Contents 13.4 YIELD CURVES, DISCOUNT FACTORS, AND FORWARD RATES INTERPRETING FORWARD CURVES 154 EXERCISES 155 FURTHER READING 155 CHARTER 14. Modeling Stock Prices CHAPTER SUMMARY WH AT ARE STOCKS? SIMPLE STATISTICAL ANALYSIS OF REAL STOCK DATA 157 EXERCISES 161 FURTHER READING 162 CHARTER 15 Mean Variance Portfolio Optimization CHAPTER SUMMARY SELECTING PORTFOLIOS Basic Model Assumptions Turning Our Model Setting into an Optimization Problem Studying the Formula in a Spreadsheet Data Requirements CAPM AND MARKOWITZ 174 EXERCISES 177 FURTHER READING 180 CHARTER 16 A Qualitative Introduction to Options CHAPTER SUMMARY STOCK OPTION DEFINITIONS USES FOR PUT AND CALL OPTIONS QUALITATIVE BEHAVIOR OF PUTS AND CALLS 184 EXERCISES 185 FURTHER READING 185

8 Contents xi CHARTER 17. Value at Risk CHAPTER SUMMARY INTRODUCTION TO VALUE AT RISK PITFALLS OF VaR SUMMARY 191 EXERCISES 191 FURTHER READING 192 CHARTER 18 Pricing Qptions Using Binomial Trees CHAPTER SUMMARY INTRODUCTION BINOMIAL MODEL SINGLE-PERIOD BINOMIAL TREE MODEL FOR OPTION PRICING EXTENDING THE BINOMIAL MODEL TO MULTIPLE TIME STEPS Numerical Example: Pricing a Two-Period Binomial Put Option MULTIPLE-STEP BINOMIAL TREES SUMMARY 215 EXERCISES 216 FURTHER READING 217 CHARTER 19 Random Walks CHAPTER SUMMARY INTRODUCTION DERIVING THE DIFFUSION PARTIAL DIFFERENTIAL EQUATION 222 EXERCISES 224 FURTHER READING 225 CHARTER 20 - Basic Stochastic Calculus CHAPTER SUMMARY BASICS OF STOCHASTIC CALCULUS 227

9 xii Contents 20.3 STOCHAST1C INTEGRATION BY EXAMPLES Review of the Left Endpoint Rule of Introductory Calculus Ito Integration Ito Isometry Introduction to Ordinary Differential Equations Solution of SDEs Arithmetic Brownian Motion Geometrie Brownian Motion Ornstein-Uhlenbeck Process CONCLUSIONS AND BRIDGE TO NEXT CHARTERS 249 EXERCISES 249 FURTHER READING 252 CHARTER 21. Simulating Geometrie Brownian Motion SIMULATING GBM STOCK PRICES AT A SINGLE FUTURE TIME SIMULATING A TIME SEQUENCE OF GBM STOCK PRICES SUMMARY 258 EXERCISES 258 FURTHER READING 259 CHARTER 22 Black Scholes PDE for Pricing Options in Continuous Time 261^ 22.1 CHAPTER SUMMARY INTRODUCTION HEDGING ARGUMENT CALL PRICE SOLUTION OF THE BLACK SCHOLES EQUATION WHY SHORT SELLING IS SO DANGEROUS SUMMARY AND BRIDGE TO THE FUTURE 265 EXERCISES 265 FURTHER READING 266

10 Contents xiii CHARTER 23 Solving the Black Scholes PDE CHAPTER SUMMARY SOLVING THE BLACK SCHOLES PARTIAL PDE FOR A EUROPEAN CALL GENERAL EUROPEAN OPTION PAYOFFS: RISK- NEUTRAL PRICING SUMMARY 284 EXERCISES 285 CHARTER 24 Pricing Put Options Using Put Call Parity CHAPTER SUMMARY SUMMARY 290 EXERCISES 290 FURTHER READING 290 CHARTER 25 Some Approximate Values of the Black Scholes Call Formula INTRODUCTION APPROXIMATE CALL FORMULAS AT-THE-MONEY APPROXIMATE CALL VALUES NEAR-THE-MONEY APPROXIMATE CALL VALUES FAR-FROM- THE-MONEY 296 EXERCISES 304 FURTHER READING 304 CHARTER 26 Simulating Delta Hedging CHAPTER SUMMARY INTRODUCTION HOW DOES DELTA HEDGING REALLY WORK? UNDERSTANDING THE RESULTS OF THE DELTA HEDGING PROCESS THE IMPACT OF TRANSACTION COSTS A HEDGERS PERSPECTIVE ON OPTION GAMMA OR, "BIG GAMMA" = "BIG MONEY" 317

11 xiv Contents 26.7 BRIDGE TO THE FUTURE 318 EXERCISES 319 FURTHER READING 319 CHARTER 27 Black Scholes with Dividends 321_ 27.1 CHAPTER SUMMARY MODELING DIVIDENDS "Tailed Stock Positions" THE BLACK SCHOLES PDF FOR THE CONTINUOUSLY PAID DIVIDEND GASE PRICING THE PREPAID FORWARD ON A J CONTINUOUS DIVIDEND PAYING STOCK MORE COMPLICATED DERIVATIVES ON UNDERLYINGS PAYING CONTINUOUS DIVIDENDS 327 EXERCISES 329 FURTHER READING 329 CHARTER 28 - American Options 331 _ 28.1 CHAPTER SUMMARY 331 \ 28.2 INTRODUCTION AND BINOMIAL PRICING 331 \ 28.3 AMERICAN PUTS 337 \ 28.4 AMERICAN CALLS 340 \ EXERCISES 342 FURTHER READING 343 CHARTER 29. Pricing the Perpetua! American Put and Call CHAPTER SUMMARY PERPETUAL OPTIONS: UNDERLYING PAYS NO DIVIDENDS Basic Perpetual American Put BASIC PERPETUAL AMERICAN CALL PERPETUAL AMERICAN CALL/PUT MODEL WITH DIVIDENDS 352

12 Contents xv 29.5 THE PERPETUAL AMERICAN CALL, CONTINUOUS DIVIDENDS 357 EXERCISE 362 FURTHER READING 362 CHARTER 30. Options on Multiple Underlying Assets INTRODUCTION EXCHANGE OPTIONS 365 EXERCISE 373 FURTHER READING 373 CHARTER 31 Interest Rate Models CHAPTER SUMMARY SETTING THE STAGE FOR STOCHASTIC INTEREST RATE MODELS PRICING WHEN YOU CANNOT TRADE THE UNDERLYING ASSET HEDGING BONDS IN CONTINUOUS TIME SOLVING THE BOND PRICING PDE VASICEK MODEL SUMMARY 395 EXERCISES 395 FURTHER READING 396 CHARTER 32. Incomplete Markets CHAPTER SUMMARY INTRODUCTION TO INCOMPLETE MARKETS TRYING TO HEDGE OPTIONS ON A TRINOMIAL TREE Review of the Standard Binomial Tree Model Extension to a Trinomial Tree Model MINIMUM VARIANCE HEDGING OFA EUROPEAN OPTION WITH DEFAULT Binomial Tree Model for Option Pricing 408

13 xvi Contents 32.5 BINOMIAL TREE MODEL WITH DEFAULT RISK EXERCISE FURTHER READING APPENDIX 1: PROBABILITY THEORY BASICS, 415 APPENDIX 2: PROOF OF DEMOIVRE-LAPLACE THEOREM, 475 APPENDIX 3: NAMING VARIABLES IN EXCEL, 481 APPENDIX 4: BUILDING VBA MACROS FROM EXCEL, 485 INDEX, 501