FIXED INCOME SECURITIES

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1 FIXED INCOME SECURITIES Valuation, Risk, and Risk Management Pietro Veronesi University of Chicago WILEY JOHN WILEY & SONS, INC.

2 CONTENTS Preface Acknowledgments PART I BASICS xix xxxiii AN INTRODUCTION TO FIXED INCOME MARKETS Introduction The Complexity of Fixed Income Markets No Arbitrage and the Law of One Price The Government Debt Markets Zero Coupon Bonds Floating Rate Coupon Bonds The Municipal Debt Market The Money Market Federal Funds Rate Eurodollar Rate LIBOR The Repo Market General Collateral Rate and Special Repos What if the T-bond Is Not Delivered? The Mortgage Backed Securities Market and Asset-Backed Securities Market 21

3 VI CONTENTS 1.6 The Derivatives Market Swaps Futures and Forwards Options Roadmap of Future Chapters Summary 28 BASICS OF FIXED INCOME SECURITIES Discount Factors Discount Factors across Maturities Discount Factors over Time Interest Rates Discount Factors, Interest Rates, and Compounding Frequencies The Relation between Discounts Factors and Interest Rates The Term Structure of Interest Rates The Term Structure of Interest Rates over Time Coupon Bonds From Zero Coupon Bonds to Coupon Bonds From Coupon Bonds to Zero Coupon Bonds Expected Return and the Yield to Maturity Quoting Conventions Floating Rate Bonds The Pricing of Floating Rate Bonds Complications Summary Exercises Case Study: Orange County Inverse Floaters Decomposing Inverse Floaters into a Portfolio of Basic Securities Calculating the Term Structure of Interest Rates from Coupon Bonds Calculating the Price of the Inverse Floater Leveraged Inverse Floaters Appendix: Extracting the Discount Factors Z(0, T) from Coupon Bonds Bootstrap Again Regressions Curve Fitting Curve Fitting with Splines 70

4 CONTENTS V» BASICS OF INTEREST RATE RISK MANAGEMENT The Variation in Interest Rates The Savings and Loan Debacle The Bankruptcy of Orange County Duration Duration of a Zero Coupon Bond Duration of a Portfolio Duration of a Coupon Bond Duration and Average Time of Cash Flow Payments Properties of Duration Traditional Definitions of Duration The Duration of Zero Investment Portfolios: Dollar Duration Duration and Value-at-Risk Duration and Expected Shortfall Interest Rate Risk Management Cash Flow Matching and Immunization Immunization versus Simpler Investment Strategies Why Does the Immunization Strategy Work? Asset-Liability Management Summary Exercises Case Study: The 1994 Bankruptcy of Orange County Benchmark: What if Orange County was Invested in Zero Coupon Bonds Only? The Risk in Leverage The Risk in Inverse Floaters The Risk in Leveraged Inverse Floaters What Can We Infer about the Orange County Portfolio? Conclusion Case Analysis: The Ex-Ante Risk in Orange County's Portfolio The Importance of the Sampling Period Conclusion Appendix: Expected Shortfall under the Normal Distribution 111 BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT Convexity The Convexity of Zero Coupon Bonds The Convexity of a Portfolio of Securities The Convexity of a Coupon Bond Positive Convexity: Good News for Average Returns A Common Pitfall 121

5 CONTENTS Convexity and Risk Management Convexity Trading and the Passage of Time Slope and Curvature Implications for Risk Management Factor Models and Factor Neutrality Factor Duration Factor Neutrality Estimation of the Factor Model Summary Exercises Case Study: Factor Structure in Orange County's Portfolio Factor Estimation Factor Duration of the Orange County Portfolio The Value-at-Risk of the Orange County Portfolio with Multiple Factors Appendix: Principal Component Analysis Benefits from PCA ~ The Implementation of PCA 150 INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS Forward Rates and Forward Discount Factors Forward Rates by No Arbitrage The Forward Curve Extracting the Spot Rate Curve from Forward Rates Forward Rate Agreements The Value of a Forward Rate Agreement Forward Contracts A No Arbitrage Argument Forward Contracts on Treasury Bonds The Value of a Forward Contract r Interest Rate Swaps The Value of a Swap The Swap Rate The Swap Curve The LIBOR Yield Curve and the Swap Spread The Forward Swap Contract and the Forward Swap Rate Payment Frequency and Day Count Conventions Interest Rate Risk Management using Derivative Securities Summary Exercises Case Study: PiVe Capital Swap Spread Trades Setting Up the Trade. 191

6 CONTENTS IX The Quarterly Cash Flow Unwinding the Position? Conclusion 196 INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS Interest Rate Futures Standardization Margins and Mark-to-Market The Convergence Property of Futures Prices Futures versus Forwards Hedging with Futures or Forwards? Options Options as Insurance Contracts Option Strategies Put-Call Parity "Hedging with Futures or with Options? Summary Exercises Appendix: Liquidity and the LIBOR Curve 233 INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE The Federal Reserve Monetary Policy, Economic Growth, and Inflation The Tools of Monetary Policy The Federal Funds Rate Predicting the Future Fed Funds Rate Fed Funds Rate, Inflation and Employment Growth Long-Term Fed Funds Rate Forecasts Fed Funds Rate Predictions Using Fed Funds Futures Understanding the Term Structure of Interest Rates Why Does the Term Structure Slope up in Average? The Expectation Hypothesis Predicting Excess Returns Conclusion Coping with Inflation Risk: Treasury Inflation-Protected Securities TIPS Mechanics Real Bonds and the Real Term Structure of Interest Rates Real Bonds and TIPS Fitting the Real Yield Curve The Relation between Nominal and Real Rates 268

7 X CONTENTS 7.5 Summary Exercises Case Study: Monetary Policy during the Subprime Crisis of Problems on the Horizon August 17, 2007: Fed Lowers the Discount Rate September - December 2007: The Fed Decreases Rates and Starts TAF January 2008: The Fed Cuts the Fed Funds Target and Discount Rates March 2008: Beam Stearns Collapses and the Fed Bolsters Liquidity Support to Primary Dealers September - October 2008: Fannie Mae, Freddie Mac, Lehman Brothers, and AIG Collapse Appendix: Derivation of Expected Return Relation BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES Securitization The Main Players in the RMBS Market Private Labels and the Credit Crisis Default Risk and Prepayment in Agency RMBSs Mortgages and the Prepayment Option The Risk in the Prepayment Option Mortgage Prepayment Mortgage Backed Securities Measures of Prepayment Speed Pass-Through Securities The Effective Duration of Pass-Through Securities The Negative Effective Convexity of Pass-Through Securities The TBA Market Collateralized Mortgage Obligations CMO Sequential Structure CMO Planned Amortization Class (PAC) Interest Only and Principal Only Strips Summary^ Exercises Case Study: PiVe Investment Group and the Hedging of Pass-Through Securities Three Measures of Duration and Convexity PSA-Adjusted Effective Duration and Convexity 325

8 CONTENTS Xi Empirical Estimate of Duration and Convexity The Hedge Ratio Appendix: Effective Convexity 330 PART II TERM STRUCTURE MODELS: TREES 9 ONE STEP BINOMIAL TREES A one-step interest rate binomial tree Continuous Compounding The Binomial Tree for a Two-Period Zero Coupon Bond No Arbitrage on a Binomial Tree The Replicating Portfolio Via No Arbitrage Where Is the Probability pi Derivative Pricing as Present Discounted Values of Future Cash Flows Risk Premia in Interest Rate Securities The Market Price of Interest Rate Risk An Interest Rate Security Pricing Formula What If We Do Not Know p? Risk Neutral Pricing Risk Neutral Probability The Price of Interest Rate Securities Risk Neutral Pricing and Dynamic Replication Risk Neutral Expectation of Future Interest Rates Summary Exercises MULTI-STEP BINOMIAL TREES A Two-Step Binomial Tree Risk Neutral Pricing Risk Neutral Pricing by Backward Induction Dynamic Replication Matching the Term Structure Multi-step Trees Building a Binomial Tree from Expected Future Rates Risk Neutral Pricing Pricing and Risk Assessment: The Spot Rate Duration Summary Exercises RISK NEUTRAL TREES AND DERIVATIVE PRICING Risk Neutral Trees The Ho-Lee Model The Simple Black, Derman, and Toy (BDT) Model 383

9 XM CONTENTS Comparison of the Two Models Risk Neutral Trees and Future Interest Rates Using Risk Neutral Trees Intermediate Cash Flows Caps and Floors Swaps Swaptions Implied Volatilities and the Black, Derman, and Toy Model Flat and Forward Implied Volatility Forward Volatility and the Black, Derman, and Toy Model Risk Neutral Trees for Futures Prices Eurodollar Futures T-Note and T-Bond Futures Implied Trees: Final Remarks Summary Exercises AMERICAN OPTIONS Callable Bonds An Application to U.S. Treasury Bonds The Negative Convexity in Callable Bonds The Option Adjusted Spread Dynamic Replication of Callable Bonds American Swaptions ' Mortgages and Residential Mortgage Backed Securities Mortgages and the Prepayment Option The Pricing of Residential Mortgage Backed Securities The Spot Rate Duration of MBS Summary Exercises MONTE CARLO SIMULATIONS ON TREES Monte Carlo Simulations on a One-step Binomial Tree Monte Carlo Simulations on a Two-step Binomial Tree Example: Non-Recombining Trees in Asian Interest Rate Options Monte Carlo Simulations for Asian Interest Rate Options Monte Carlo Simulations on Multi-step Binomial Trees Does This Procedure Work? Illustrative Example: Long-Term Interest Rate Options How Many Simulations are Enough? 472

10 CONTENTS XIII 13.4 Pricing Path Dependent Options Illustrative Example: Long-Term Asian Options Illustrative Example: Index Amortizing Swaps Spot Rate Duration by Monte Carlo Simulations Pricing Residential Mortgage Backed Securities Simulating the Prepayment Decision Additional Factors Affecting the Prepayment Decision Residential Mortgage Backed Securities Prepayment Models Summary Exercises 492 PART III TERM STRUCTURE MODELS: CONTINUOUS TIME 14 INTEREST RATE MODELS IN CONTINUOUS TIME Brownian Motions Properties of the Brownian Motion Notation Differential Equations Continuous Time Stochastic Processes Ito's Lemma Illustrative Examples Summary Exercises Appendix: Rules of Stochastic Calculus NO ARBITRAGE AND THE PRICING OF INTEREST RATE SECURITIES Bond Pricing with Deterministic Interest Rate Interest Rate Security Pricing in the Vasicek Model The Long / Short Portfolio The Fundamental Pricing Equation The Vasicek Bond Pricing Formula Parameter Estimation Derivative Security Pricing Zero Coupon Bond Options Options on Coupon Bonds The Three Steps to Derivative Pricing No Arbitrage Pricing in a General Interest Rate Model The Cox, Ingersoll, and Ross Model Bond Prices under the Cox, Ingersoll, and Ross Model 551

11 XiV CONTENTS 15.5 Summary Exercises Appendix: Derivations Derivation of the Pricing Formula in Equation The Derivation of the Vasicek Pricing Formula The CIR Model DYNAMIC HEDGING AND RELATIVE VALUE TRADES The Replicating Portfolio Rebalancing Application 1: Relative Value Trades on the Yield Curve Relative Pricing Errors Discovery Setting Up the Arbitrage Trade Application 2: Hedging Derivative Exposure Hedging and Dynamic Replication Trading on Mispricing and Relative Value Trades The Theta - Gamma Relation Summary Exercises Case Study: Relative Value Trades on the Yield Curve Finding the Relative Value Trade Setting Up the Trade Does It Work? Simulations Does It Work? Data Conclusion Appendix: Derivation of Delta for Call Options RISK NEUTRAL PRICING AND MONTE CARLO SIMULATIONS Risk Neutral Pricing Feynman-Kac Theorem Application of Risk Neutral Pricing: Monte Carlo Simulations Simulating a Diffusion Process Simulating the Payoff Standard Errors Example: Pricing a Range Floater Hedging with Monte Carlo Simulations Convexity by Monte Carlo Simulations Summary Exercises Case Study: Procter & Gamble / Bankers Trust Leveraged Swap 619

12 CONTENTS XV Parameter Estimates Pricing by Monte Carlo Simulations THE RISK AND RETURN OF INTEREST RATE SECURITIES Expected Return and the Market Price Risk The Market Price of Risk in a General Interest Rate Model Risk Analysis: Risk Natural Monte Carlo Simulations Delta Approximation Errors A Macroeconomic Model of the Term Structure Market Participants Equilibrium Nominal Bond Prices Conclusion Case Analysis: The Risk in the P&G Leveraged Swap Summary, Exercises Appendix: Proof of Pricing Formula in Macroeconomic Model NO ARBITRAGE MODELS AND STANDARD DERIVATIVES No Arbitrage Models The Ho-Lee Model Revisited Consistent Derivative Pricing The Term Structure of Volatility in the Ho-Lee Model The Hull-White Model The Option Price Standard Derivatives under the "Normal" Model Options on Coupon Bonds Caps and Floors Caps and Floors Implied Volatility European Swaptions Swaptions'Implied Volatility The "Lognormal" Model The Black, Derman, and Toy Model The Black and Karasinski Model Generalized Affine Term Structure Models Summary Exercises Appendix: Proofs Proof of the Ho-Lee Pricing Formula Proof of the Expression in Equation Proof of the Hull-White Pricing Formula 682

13 XVi CONTENTS Proof of the Expression in Equation Proof of the Expressions in Equations and THE MARKET MODEL FOR STANDARD DERIVATIVES The Black Formula for Caps and Floors Pricing Flat and Forward Volatilities Extracting Forward Volatilities from Flat Volatilities The Behavior of the Implied Forward Volatility Forward Volatilities and the Black, Derman, and Toy Model The Black Formula for Swaption Pricing Summary Exercises FORWARD RISK NEUTRAL PRICING AND THE LIBOR MARKET MODEL One Difficulty with Risk Neutral Pricing Change of Numeraire and the Forward Risk Neutral Dynamics Two Important Results Generalizations The Option Pricing Formula in "Normal" Models The LIBOR Market Model The Black Formula for Caps and Floors Valuing Fixed Income Securities that Depend on a Single LIBOR Rate The LIBOR Market Model for More Complex Securities Extracting the Volatility of Forward Rates from Caplets' Forward Volatilities Pricing Fixed Income Securities by Monte Carlo Simulations Forward Risk Neutral Pricing and the Black Formula for Swaptions Remarks: Forward Risk Neutral Pricing and No Arbitrage The Heath, Jarrow, and Morton Framework Futures and Forwards Unnatural Lag and Convexity Adjustment Unnatural Lag and Convexity A Convexity Adjustment Summary Exercises Appendix: Derivations Derivation of the Partial Differential Equation in the Forward Risk Neutral Dynamics 740

14 CONTENTS XVM Derivation of the Call Option Pricing Formula (Equations 21.11) Derivation of the Formula in Equations and Derivation of the Formula in Equation Derivation of the Formula in Equation MULTIFACTOR MODELS Multifactor Ito's Lemma with Independent Factors No Arbitrage with Independent Factors A Two-Factor Vasicek Model A Dynamic Model for the Short and Long Yield Long-Term Spot Rate Volatility Options on Zero Coupon Bonds Correlated Factors The Two-Factor Vasicek Model with Correlated Factors Zero Coupon Bond Options The Two-Factor Hull-White Model The Feynman-Kac Theorem Application: Yield Curve Steepener Simulating Correlated Brownian Motions Forward Risk Neutral Pricing Application: Options on Coupon Bonds The Multifactor LIBOR Market Model Level, Slope, and Curvature Factors for Forward Rates Affine and Quadratic Term Structure Models Affine Models Quadratic Models Summary Exercises Appendix The Coefficients of the Joint Process for Short- and Long-Term Rates The Two-Factor Hull-White Model 787 References 789 Index 797

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