The Oxford Guide to FINANCIAL MODELING

Applications for Capital Markets, Corporate Finance, Risk Management, and Financial Institutions

Thomas S. Y. Ho Sang Bin Lee

OXFORD UNIVERSITY PRESS 2004 Contents

Model List xxiii

PART I. DERIVATIVES VALUATION 1. Introduction: Discounted Cash Flow Method 3 1.1 Examples of Financial Issues 3 1.2 Financial Models 6 1.3 Basics of Modeling: Present Value and Measures of Risk 9 1.4 Summary 14 2. Equity Market: The Capital Asset Pricing Model 16 2.1 Real and Financial Sectors 17 2.2 Stocks and Stock Markets 18 J$.3 Perfect Capital Market 22 2.4 Efficient Capital Market Hypothesis 24 2.5 Diversification 25 2.6 Capital Asset Pricing Model (CAPM) 29 2.7 Beta—The Systematic Risk 31 2.8 The Stock Model-Dividend Discount Model 33 2.9 An Application of the Capital Asset Pricing Model in Investment Services 35 2.10 Empirical Tests of the Capital Asset Pricing Model 35 2.11 Summary 37 Appendix A. Expectations and Standard Deviations 37 Appendix B. A Summary of the CAPM 38 3. Bond Markets: The Bond Model 43 3.1 Bond Mathematics 44 3.2 Bonds and Bond Markets 46 3.3 Swap Markets 48 3.4 Economics of the Yield Curve 49 3.5 The Bond Model 53 3.6 Forward Prices and Forward Rates 56 3.7 Bond Analysis 62 3.8 Applications of the Bond Analytics 70 3.9 Law of One Price: An Arbitrage Trade and Fair Value Anlaysis 73 3.10 Summary 74

XV xvi CONTENTS

Appendix A. Taylor Expansion 75 Appendix B. The Derivation ofMacaulay Duration and Convexity 76 Appendix C. Duration and Convexity in Measuring Price Sensitivity 77 4. Equity Options: the Black-Scholes Model 80 4.1 Description of an Option 80 4.2 Institutional Framework 82 4.3 Put-Call Parity 83 4.4 The Main Insight of the Black-Scholes Model 85 4.5 Valuation Methods 92 4.6 Relationships of Risk-Neutral and Market Binomial Lattices 96 4.7 Option Behavior and the Sensitivity Analysis 98 4.8 Extensions of the Black-Scholes Model 100 4.9 Option Pricing Procedure and Analytic Framework 102 4.10 Applications of Option Models 103 4.11 Accounting for Employee Stock Options 106 4.12 Intuitive Explanations of the Behavior of an Equity Option 107 4.13 Summary 108 Appendix A. Derivation of the Black—Scholes Continuous Time Model Using Different Numeraires 109 Appendix B. The Relationship Between the Time Decay and the Gamma of a Delta-Neutral Portfolio 113 Appendix C. Pathwise Valuation 114 Appendix D. Derivation of Discrete Time Parameters 115 Appendix E. Monte Carlo Simulation and Finite Difference Method 116 5. Interest Rate Derivatives: Interest Rate Models 123 5.1 Interest Rate Movements: Historical Experiences 124 5.2 The Three-Factor Yield Curve Movement Model 129 5.3 Equilibrium Models 132 5.4 Arbitrage-Free Models 136 5.5 A Comparison of Models 156 5.6 Generalizations of Interest Rate Models 157 5.7 Summary 163 Appendix A. Yield Curve Movements Represented by the Principal Components 164 Appendix B. Derivation ofHo-Lee, Extended Ho-Lee, and Ho-Lee Two-Factor Models 164 6. Implied Volatility Surface: Calibrating the Models 172 6.1 Implied Volatility Surface and Benchmark Securities 173 6.2 Price Quotes of Benchmark Securities 175 6.3 Valuation of Interest Rate Derivatives Using Market Benchmark Prices 182 6.4 Calibration of the Black-Derman-Toy Model 185 CONTENTS xvii

6.5 Calibration of the Ho-Lee Models 186 6.6 Calibration of the Longstaff-Santa-Clara-Schwartz "String" Model 188 6.7 Calibration of the Brace-Gatarek-Musiela/Jamshidian Model (the LIBOR Market Model) 193 6.8 Comparing the Black Models and the Interest Rate Models 195 6.9 Applications of Interest Rate Models 200 6.10 Key Rate Duration and Dynamic Hedging 203 6.11 Pathwise Valuation and Decomposition of an Option 207 6.12 Intuitive Explanation of the Bond Option Valuation 209 6.13 Lecture on Convexity 210 6.14 Summary 211 7. Exotic Options: Bellman's Optimization, the Filtration Model, and the w-Factor Model 214 7.1 Options with Alternative Payoffs at Expiration 215 7.2 Options with Boundary Conditions 216 7.3 Options with the Early Exercise Feature (American) and the Bellman Optimization 218 7.4 Compound Options 223 7.5 Options with Look-back Features (Asian) and the Filtration Model 225 J. 6 Chooser Option 232 7.7 Multiple Risk Sources 233 7.8 Constant Maturity Swap 238 7.9 Interest Rate Spread Option 240 7.10 Options on Forward/Futures Contracts 242 7.11 Commodity Options 243 7.12 An Overview of the Valuation Framework 247 7.13 Summary 248 Appendix A. Optimal Early Exercise Using Simulations 249 Appendix B. N-Factor Lattice Model 249 Appendix C. A Numerical Example of Dynamic Programming 252 PART II. CORPORATE LIABILITIES 8. Investment Grade Corporate Bonds: Option Adjusted Spreads 259 8.1 Describing a Corporate Bond 260 8.2 Valuation of a Bond 265 8.3 Numerical Examples 27'4 8.4 Liquidity (Marketability) Spread 280 8.5 Credit Scoring Approaches 283 8.6 Bond Analysis 284 8.7 Numerical Example: Valuing a Eurobond Issue 286 8.8 Applications of Bond Analytics 288 8.9 Explaining the Concept of the Arbitrage-free Condition on a Solemn Occasion 290 xviii CONTENTS

8.10 Summary 290 Appendix. Callable Bond and Sinking Fund Bond Pricing 292 9. High-Yield Corporate Bonds: The Structural Models 298 9.1 An Example of a High-Yield Bond 299 9.2 Institutional Framework of Bankruptcy and Bankruptcy Proceedings 303 9.3 The Fisher Model 306 9.4 An Actuarial Model 306 9.5 Historical Experience and Estimation of the Parameters of Default Models 308 9.6 The Reduced Form Models 312 9.7 The Structural Model 313 9.8 Valuation of a Debt Package Using a Compound Option Model 317 9.9 Empirical Evidence 321 9.10 A Review of the High Yield Bond Models 322 9.11 Analysis of the McLeodUSA Bond 323 9.12 Analysis of Credit Risk 324 9.13 Summary 327 10. Convertibles, MBS/CMO, and Other Bonds: The Behavioral Models 332 10.1 Convertible Bonds 333 10.2 Mortgage-Backed Securities (Pass-through Certificates) 341 10.3 Collateralized Mortgage Obligations (CMO) 349 10.4 Other Bonds 352 10.5 Credit Derivatives 356 10.6 Managing a CMO Portfolio 358 10.7 Summary 363 11. Financial Institutions' Liabilities: Required Option Adjusted Spread 366 11.1 Balance Sheet Analysis—Book Value 367 11.2 Fair Values 368 11.3 Liability Modeling 374 11.4 Bank Liabilities 376 11.5 Property and Casualty Insurance 380 11.6 Life Insurance Products 385 11.7 Pension Liabilities 404 11.8 Applications of the Financial Models to Liability Management 406 11.9 "Not If But When" 407 11.10 Summary 408

PART III. CORPORATE FINANCE 12. Valuation of a Firm: The Business Model 413 12.1 Descriptions of a Firm 414 12.2 Traditional Firm Valuation Methodologies 417 12.3 Corporate Financial Decisions and Firm Value Maximization 419 CONTENTS xix

12.4 Miller-Modigliani Theories 421 12.5 Free Cash Flow Discount Model 428 12.6 Business Model 430 12.7 Implications of the Valuation Model 440 12.8 Analyses of the Business Model 441 12.9 From the Senior Management Perspective 442 12.10 Summary 443 Appendix A. The MM Propositions 444 Appendix B. The Miller Model 446 13. Strategic Value of a Firm: Real Options 452 13.1 Characteristics of a Growth Company—An Example of Real Options 453 13.2 Salient Features of Real Options 454 13.3 Examples of Real Options in Capital Budgeting 457 13.4 Examples of Businesses with Embedded Options 461 13.5 Strategic Value of a Firm 463 13.6 Analysis of the Real Option Value 465 13.7 A Business Model with Embedded Options: Starbucks Coffee Japan 467 13.8 A Business Model Approach and the Free Cash Flow Discounting Approach—A Comparison 470 *%7 3.9 Empirical Implications of the Business Model 471 13.10 Implications of the Business Model 475 13.11 Empirical Research on Real Options 479 13.12 What is Financial Modeling to Senior Management? 479 13.13 Summary 480 Appendix. The Business Model 481 14. Optimal Corporate Financial Decisions: Corporate Model 493 14.1 Corporate Financial Planning—the DFA Approach 494 14.2 Extensions of the MM Theory 497 14.3 Empirical Evidence on the MM Theory and Its Extensions 500 14.4 The Corporate Model 504 14.5 Specifications of the Corporate Model 506 14.6 A Comparison with Previous Research 512 14.7 New Perspectives in Viewing Firms as Contingent Claims 513 14.8 Principles of Risk Management 516 14.9 Summary 517 Appendix. The Firm Model 518 15. Risk Management 525 15.1 Risk Measurement—Value at Risk 526 15.2 Market Risk 528 15.3 Delta Normal Methodology 529 15.4 Historical Simulation Methodology 535 15.5 Monte Carlo Simulation Methodology 537 xx CONTENTS

15.6 Extreme Value Theory 538 15.7 Credit Risk 540 15.8 Risk Reporting 546 15.9 Risk Monitoring 548 15.10 Risk Management 549 15.11 Boardroom with a View—The Coffin Story 553 15.12 Summary 554 Appendix A. Selected Historical Financial Losses 554 Appendix B. Lessons Learned from the Historical Financial Losses 559 16. Financial Institutions: Applications of Financial Models 565 16.1 An Overview of the Financial Sector 566 16.2 Organization and the Business Model of a Financial Institution 569 16.3 Financial Disclosures on Valuation 572 16.4 Risk Management 575 16.5 Capital Allocation and Risk-Adjusted Performance Measures 576 16.6 Financial Modeling of a Financial Institution 578 16.7 Applications: Asset and Liability Management 592 16.8 Regulatory Issues 594 16.9 Summary 599 Appendix A. What Actions Have Commercial Banks Taken? 600 Appendix B. Capital Requirements and Risk-based Capital 602 17. Structured Finance: Foreign Exchange Models 606 17.1 Background 606 17.2 Economics of the Structure 607 17.3 Deal Structure 608 17.4 Pricing 611 17.5 Simulations 614 17.6 VaR Calculation 615 17.7 Remarks 616 17.8 Concluding Remarks on Financial Modeling 617 Appendix A. Total Return Swap Terms and Conditions as of January 29, 1997 618 Appendix B. Indonesian Rupiah-Linked Notes Final Terms and Conditions618 Appendix C. The Put-Call Parity 619 18. Concluding Thoughts 620 18.1 Conceptual Development of Financial Models 621 18.2 Overview of Valuation Models 622 18.3 Applications of the Financial Models 624 18.4 Financial Modeling as a Process 627 18.5 Looking into the Future 628 18.6 The World of Contingent Claims 631 CONTENTS

19. Technical Matters: Market Model and Binomial Lattices 633 19.1 Building a Market Model in a Binomial Lattice 634 19.2 A General Approach to Valuation 647 19.3 Comparison of the Option Derivation Between Discrete Time and Continuous Time 660 Appendix A. Continuous Time Versions of the Ho—Lee Models 669 Appendix B. Summary of Continuous Time Interest Rate Models 674 Appendix C. Recombining Lattice 677

Glossary of Notations 681 Glossary of Terms 687 Index 705