Fixed Income Securities: Valuation, Risk, and Risk Management

By Pietro Veronesi · Wiley · 2010

A textbook on the forces that affect the valuation, risk and return of fixed income securities and their derivatives, published by Wiley in January 2010. Its publisher describes it as a thorough discussion of these complex securities, the forces affecting their prices, their risks and the appropriate risk management practices, and as providing a methodology rather than a shopping list: examples and methodologies that can be applied quite universally once the basic concepts are understood.

On the syllabus

Exam CP-351 — Chapter 5, Interest Rate Derivatives: Forwards and Swaps, and Chapter 6, Interest Rate Derivatives: Futures and Options; outcome 3f.

Preface

Acknowledgments

Part I Basics

1 An Introduction to Fixed Income Markets

1.1 Introduction
- 1.1.1 The Complexity of Fixed Income Markets
- 1.1.2 No Arbitrage and the Law of One Price
1.2 The Government Debt Markets
- 1.2.1 Zero Coupon Bonds
- 1.2.2 Floating Rate Coupon Bonds
- 1.2.3 The Municipal Debt Market
1.3 The Money Market
- 1.3.1 Federal Funds Rate
- 1.3.2 Eurodollar Rate
- 1.3.3 LIBOR
1.4 The Repo Market
- 1.4.1 General Collateral Rate and Special Repos
- 1.4.2 What if the T-bond Is Not Delivered?
1.5 The Mortgage Backed Securities Market and Asset-Backed Securities Market
1.6 The Derivatives Market
- 1.6.1 Swaps
- 1.6.2 Futures and Forwards
- 1.6.3 Options
1.7 Roadmap of Future Chapters
1.8 Summary

2 Basics of Fixed Income Securities

2.1 Discount Factors
- 2.1.1 Discount Factors across Maturities
- 2.1.2 Discount Factors over Time
2.2 Interest Rates
- 2.2.1 Discount Factors, Interest Rates, and Compounding Frequencies
- 2.2.2 The Relation between Discounts Factors and Interest Rates
2.3 The Term Structure of Interest Rates
- 2.3.1 The Term Structure of Interest Rates over Time
2.4 Coupon Bonds
- 2.4.1 From Zero Coupon Bonds to Coupon Bonds
- 2.4.2 From Coupon Bonds to Zero Coupon Bonds
- 2.4.3 Expected Return and the Yield to Maturity
- 2.4.4 Quoting Conventions
2.5 Floating Rate Bonds
- 2.5.1 The Pricing of Floating Rate Bonds
- 2.5.2 Complications
2.6 Summary
2.7 Exercises
2.8 Case Study: Orange County Inverse Floaters
- 2.8.1 Decomposing Inverse Floaters into a Portfolio of Basic Securities
- 2.8.2 Calculating the Term Structure of Interest Rates from Coupon Bonds
- 2.8.3 Calculating the Price of the Inverse Floater
- 2.8.4 Leveraged Inverse Floaters
2.9 Appendix: Extracting the Discount Factors Z(0, T) from Coupon Bonds
- 2.9.1 Bootstrap Again
- 2.9.2 Regressions
- 2.9.3 Curve Fitting
- 2.9.4 Curve Fitting with Splines

3 Basics of Interest Rate Risk Management

3.1 The Variation in Interest Rates
- 3.1.1 The Savings and Loan Debacle
- 3.1.2 The Bankruptcy of Orange County
3.2 Duration
- 3.2.1 Duration of a Zero Coupon Bond
- 3.2.2 Duration of a Portfolio
- 3.2.3 Duration of a Coupon Bond
- 3.2.4 Duration and Average Time of Cash Flow Payments
- 3.2.5 Properties of Duration
- 3.2.6 Traditional Definitions of Duration
- 3.2.7 The Duration of Zero Investment Portfolios: Dollar Duration
- 3.2.8 Duration and Value-at-Risk
- 3.2.9 Duration and Expected Shortfall
3.3 Interest Rate Risk Management
- 3.3.1 Cash Flow Matching and Immunization
- 3.3.2 Immunization versus Simpler Investment Strategies
- 3.3.3 Why Does the Immunization Strategy Work?
3.4 Asset-Liability Management
3.5 Summary
3.6 Exercises
3.7 Case Study: The 1994 Bankruptcy of Orange County
- 3.7.1 Benchmark: What if Orange County was Invested in Zero Coupon Bonds Only?
- 3.7.2 The Risk in Leverage
- 3.7.3 The Risk in Inverse Floaters
- 3.7.4 The Risk in Leveraged Inverse Floaters
- 3.7.5 What Can We Infer about the Orange County Portfolio?
- 3.7.6 Conclusion
3.8 Case Analysis: The Ex-Ante Risk in Orange County’s Portfolio
- 3.8.1 The Importance of the Sampling Period
- 3.8.2 Conclusion
3.9 Appendix: Expected Shortfall under the Normal Distribution

4 Basic Refinements in Interest Rate Risk Management

4.1 Convexity
- 4.1.1 The Convexity of Zero Coupon Bonds
- 4.1.2 The Convexity of a Portfolio of Securities
- 4.1.3 The Convexity of a Coupon Bond
- 4.1.4 Positive Convexity: Good News for Average Returns
- 4.1.5 A Common Pitfall
- 4.1.6 Convexity and Risk Management
- 4.1.7 Convexity Trading and the Passage of Time
4.2 Slope and Curvature
- 4.2.1 Implications for Risk Management
- 4.2.2 Factor Models and Factor Neutrality
- 4.2.3 Factor Duration
- 4.2.4 Factor Neutrality
- 4.2.5 Estimation of the Factor Model
4.3 Summary
4.4 Exercises
4.5 Case Study: Factor Structure in Orange County’s Portfolio
- 4.5.1 Factor Estimation
- 4.5.2 Factor Duration of the Orange County Portfolio
- 4.5.3 The Value-at-Risk of the Orange County Portfolio with Multiple Factors
4.6 Appendix: Principal Component Analysis
- 4.6.1 Benefits from PCA
- 4.6.2 The Implementation of PCA

5 Interest Rate Derivatives: Forwards and Swaps

5.1 Forward Rates and Forward Discount Factors
- 5.1.1 Forward Rates by No Arbitrage
- 5.1.2 The Forward Curve
- 5.1.3 Extracting the Spot Rate Curve from Forward Rates
5.2 Forward Rate Agreements
- 5.2.1 The Value of a Forward Rate Agreement
5.3 Forward Contracts
- 5.3.1 A No Arbitrage Argument
- 5.3.2 Forward Contracts on Treasury Bonds
- 5.3.3 The Value of a Forward Contract
5.4 Interest Rate Swaps
- 5.4.1 The Value of a Swap
- 5.4.2 The Swap Rate
- 5.4.3 The Swap Curve
- 5.4.4 The LIBOR Yield Curve and the Swap Spread
- 5.4.5 The Forward Swap Contract and the Forward Swap Rate
- 5.4.6 Payment Frequency and Day Count Conventions
5.5 Interest Rate Risk Management using Derivative Securities
5.6 Summary
5.7 Exercises
5.8 Case Study: PiVe Capital Swap Spread Trades
- 5.8.1 Setting Up the Trade
- 5.8.2 The Quarterly Cash Flow
- 5.8.3 Unwinding the Position?
- 5.8.4 Conclusion

6 Interest Rate Derivatives: Futures and Options

6.1 Interest Rate Futures
- 6.1.1 Standardization
- 6.1.2 Margins and Mark-to-Market
- 6.1.3 The Convergence Property of Futures Prices
- 6.1.4 Futures versus Forwards
- 6.1.5 Hedging with Futures or Forwards?
6.2 Options
- 6.2.1 Options as Insurance Contracts
- 6.2.2 Option Strategies
- 6.2.3 Put-Call Parity
- 6.2.4 Hedging with Futures or with Options?
6.3 Summary
6.4 Exercises
6.5 Appendix: Liquidity and the LIBOR Curve

7 Inflation, Monetary Policy, and the Federal Funds Rate

7.1 The Federal Reserve
- 7.1.1 Monetary Policy, Economic Growth, and Inflation
- 7.1.2 The Tools of Monetary Policy
- 7.1.3 The Federal Funds Rate
7.2 Predicting the Future Fed Funds Rate
- 7.2.1 Fed Funds Rate, Inflation and Employment Growth
- 7.2.2 Long-Term Fed Funds Rate Forecasts
- 7.2.3 Fed Funds Rate Predictions Using Fed Funds Futures
7.3 Understanding the Term Structure of Interest Rates
- 7.3.1 Why Does the Term Structure Slope up in Average?
- 7.3.2 The Expectation Hypothesis
- 7.3.3 Predicting Excess Returns
- 7.3.4 Conclusion
7.4 Coping with Inflation Risk: Treasury Inflation-Protected Securities
- 7.4.1 TIPS Mechanics
- 7.4.2 Real Bonds and the Real Term Structure of Interest Rates
- 7.4.3 Real Bonds and TIPS
- 7.4.4 Fitting the Real Yield Curve
- 7.4.5 The Relation between Nominal and Real Rates
7.5 Summary
7.6 Exercises
7.7 Case Study: Monetary Policy during the Subprime Crisis of 2007 - 2008
- 7.7.1 Problems on the Horizon
- 7.7.2 August 17, 2007: Fed Lowers the Discount Rate
- 7.7.3 September - December 2007: The Fed Decreases Rates and Starts TAF
- 7.7.4 January 2008: The Fed Cuts the Fed Funds Target and Discount Rates
- 7.7.5 March 2008: Bearn Stearns Collapses and the Fed Bolsters Liquidity Support to Primary Dealers
- 7.7.6 September – October 2008: Fannie Mae, Freddie Mac, Lehman Brothers, and AIG Collapse
7.8 Appendix: Derivation of Expected Return Relation

8 Basics of Residential Mortgage Backed Securities

8.1 Securitization
- 8.1.1 The Main Players in the RMBS Market
- 8.1.2 Private Labels and the 2007 - 2009 Credit Crisis
- 8.1.3 Default Risk and Prepayment in Agency RMBSs
8.2 Mortgages and the Prepayment Option
- 8.2.1 The Risk in the Prepayment Option
- 8.2.2 Mortgage Prepayment
8.3 Mortgage Backed Securities
- 8.3.1 Measures of Prepayment Speed
- 8.3.2 Pass-Through Securities
- 8.3.3 The Effective Duration of Pass-Through Securities
- 8.3.4 The Negative Effective Convexity of Pass-Through Securities 302 8.3.5 The TBA Market
8.4 Collateralized Mortgage Obligations
- 8.4.1 CMO Sequential Structure
- 8.4.2 CMO Planned Amortization Class (PAC)
- 8.4.3 Interest Only and Principal Only Strips.
8.5 Summary
8.6 Exercises
8.7 Case Study. PiVe Investment Group and the Hedging of Pass-Through Securities
- 8.7.1 Three Measures of Duration and Convexity
- 8.7.2 PSA-Adjusted Effective Duration and Convexity
- 8.7.3 Empirical Estimate of Duration and Convexity
- 8.7.4 The Hedge Ratio
8.8 Appendix: Effective Convexity

Part II Term Structure Models: Trees

9 One Step Binomial Trees

9.1 A one-step interest rate binomial tree
- 9.1.1 Continuous Compounding
- 9.1.2 The Binomial Tree for a Two-Period Zero Coupon Bond
9.2 No Arbitrage on a Binomial Tree
- 9.2.1 The Replicating Portfolio Via No Arbitrage
- 9.2.2 Where Is the Probability p?
9.3 Derivative Pricing as Present Discounted Values of Future Cash Flows
- 9.3.1 Risk Premia in Interest Rate Securities
- 9.3.2 The Market Price of Interest Rate Risk
- 9.3.3 An Interest Rate Security Pricing Formula
- 9.3.4 What If We Do Not Know p?
9.4 Risk Neutral Pricing
- 9.4.1 Risk Neutral Probability
- 9.4.2 The Price of Interest Rate Securities
- 9.4.3 Risk Neutral Pricing and Dynamic Replication
- 9.4.4 Risk Neutral Expectation of Future Interest Rates
9.5 Summary
9.6 Exercises

10 Multi-Step Binomial Trees

10.1 A Two-Step Binomial Tree
10.2 Risk Neutral Pricing
- 10.2.1 Risk Neutral Pricing by Backward Induction
- 10.2.2 Dynamic Replication
10.3 Matching the Term Structure
10.4 Multi-step Trees
- 10.4.1 Building a Binomial Tree from Expected Future Rates
- 10.4.2 Risk Neutral Pricing
10.5 Pricing and Risk Assessment: The Spot Rate Duration
10.6 Summary
10.7 Exercises

11 Risk Neutral Trees and Derivative Pricing

11.1 Risk Neutral Trees
- 11.1.1 The Ho-Lee Model
- 11.1.2 The Simple Black, Derman, and Toy (BDT) Model
- 11.1.3 Comparison of the Two Models
- 11.1.4 Risk Neutral Trees and Future Interest Rates
11.2 Using Risk Neutral Trees
- 11.2.1 Intermediate Cash Flows
- 11.2.2 Caps and Floors
- 11.2.3 Swaps
- 11.2.4 Swaptions
11.3 Implied Volatilities and the Black, Derman, and Toy Model
- 11.3.1 Flat and Forward Implied Volatility
- 11.3.2 Forward Volatility and the Black, Derman, and Toy Model
11.4 Risk Neutral Trees for Futures Prices
- 11.4.1 Eurodollar Futures
- 11.4.2 T-Note and T-Bond Futures
11.5 Implied Trees: Final Remarks
11.6 Summary
11.7 Exercises

12 American Options

12.1 Callable Bonds
- 12.1.1 An Application to U.S. Treasury Bonds
- 12.1.2 The Negative Convexity in Callable Bonds
- 12.1.3 The Option Adjusted Spread
- 12.1.4 Dynamic Replication of Callable Bonds
12.2 American Swaptions
12.3 Mortgages and Residential Mortgage Backed Securities
- 12.3.1 Mortgages and the Prepayment Option
- 12.3.2 The Pricing of Residential Mortgage Backed Securities
- 12.3.3 The Spot Rate Duration of MBS
12.4 Summary
12.5 Exercises

13 Monte Carlo Simulations on Trees

13.1 Monte Carlo Simulations on a One-step Binomial Tree
13.2 Monte Carlo Simulations on a Two-step Binomial Tree
- 13.2.1 Example: Non-Recombining Trees in Asian Interest Rate Options
- 13.2.2 Monte Carlo Simulations for Asian Interest Rate Options
13.3 Monte Carlo Simulations on Multi-step Binomial Trees
- 13.3.1 Does This Procedure Work?
- 13.3.2 Illustrative Example: Long-Term Interest Rate Options
- 13.3.3 How Many Simulations are Enough?
13.4 Pricing Path Dependent Options
- 13.4.1 Illustrative Example: Long-Term Asian Options
- 13.4.2 Case Study: Banc One AIRS
13.5 Spot Rate Duration by Monte Carlo Simulations
13.6 Pricing Residential Mortgage Backed Securities
- 13.6.1 Simulating the Prepayment Decision
- 13.6.2 Additional Factors Affecting the Prepayment Decision
- 13.6.3 Residential Mortgage Backed Securities
- 13.6.4 Prepayment Models
13.7 Summary
13.8 Exercises

Part III Term Structure Models: Continuous Time

14 Interest Rate Models in Continuous Time

14.1 Brownian Motions
- 14.1.1 Properties of the Brownian Motion
- 14.1.2 Notation
14.2 Differential Equations
14.3 Continuous Time Stochastic Processes
14.4 Ito’s Lemma
14.5 Illustrative Examples
14.6 Summary
14.7 Exercises
14.8 Appendix: Rules of Stochastic Calculus

15 No Arbitrage and the Pricing of Interest Rate Securities

15.1 Bond Pricing with Deterministic Interest Rate
15.2 Interest Rate Security Pricing in the Vasicek Model
- 15.2.1 The Long / Short Portfolio
- 15.2.2 The Fundamental Pricing Equation
- 15.2.3 The Vasicek Bond Pricing Formula
- 15.2.4 Parameter Estimation
15.3 Derivative Security Pricing
- 15.3.1 Zero Coupon Bond Options
- 15.3.2 Options on Coupon Bonds
- 15.3.3 The Three Steps to Derivative Pricing
15.4 No Arbitrage Pricing in a General Interest Rate Model
- 15.4.1 The Cox, Ingersoll, and Ross Model
- 15.4.2 Bond Prices under the Cox, Ingersoll, and Ross Model
15.5 Summary
15.6 Exercises
15.7 Appendix: Derivations
- 15.7.1 Derivation of the Pricing Formula in Equation 15.4
- 15.7.2 The Derivation of the Vasicek Pricing Formula
- 15.7.3 The CIR Model

16 Dynamic Hedging and Relative Value Trades

16.1 The Replicating Portfolio
16.2 Rebalancing
16.3 Application 1: Relative Value Trades on the Yield Curve
- 16.3.1 Relative Pricing Errors Discovery
- 16.3.2 Setting Up the Arbitrage Trade
16.4 Application 2: Hedging Derivative Exposure
- 16.4.1 Hedging and Dynamic Replication
- 16.4.2 Trading on MisPricing and Relative Value Trades
16.5 The Theta - Gamma Relation
16.6 Summary
16.7 Exercises
16.8 Case Study: Relative Value Trades on the Yield Curve
- 16.8.1 Finding the Relative Value Trade
- 16.8.2 Setting Up the Trade
- 16.8.3 Does It Work? Simulations
- 16.8.4 Does It Work? Data
- 16.8.5 Conclusion
16.9 Appendix: Derivation of Delta for Call Options

17 Risk Neutral Pricing and Monte Carlo Simulations

17.1 Risk Neutral Pricing
17.2 Feynman-Kac Theorem
17.3 Application of Risk Neutral Pricing: Monte Carlo Simulations
- 17.3.1 Simulating a Diffusion Process
- 17.3.2 Simulating the Payoff
- 17.3.3 Standard Errors
17.4 Example: Pricing a Range Floater
17.5 Hedging with Monte Carlo Simulations
17.6 Convexity by Monte Carlo Simulations
17.7 Summary
17.8 Exercises
17.9 Case Study: Procter & Gamble / Bankers Trust Leveraged Swap
- 17.9.1 Parameter Estimates
- 17.9.2 Pricing by Monte Carlo Simulations

18 The Risk and Return of Interest Rate Securities

18.1 Expected Return and the Market Price Risk
- 18.1.1 The Market Price of Risk in a General Interest Rate Model
18.2 Risk Analysis: Risk Natural Monte Carlo Simulations
- 18.2.1 Delta Approximation Errors
18.3 A Macroeconomic Model of the Term Structure
- 18.3.1 Market Participants
- 18.3.2 Equilibrium Nominal Bond Prices
- 18.3.3 Conclusion
18.4 Case Analysis: The Risk in the P&G Leveraged Swap
18.5 Summary
18.6 Exercises
18.7 Appendix: Proof of Pricing Formula in Macroeconomic Model

19 No Arbitrage Models and Standard Derivatives

19.1 No Arbitrage Models
19.2 The Ho-Lee Model Revisited
- 19.2.1 Consistent Derivative Pricing
- 19.2.2 The Term Structure of Volatility in the Ho-Lee Model
19.3 The Hull-White Model
- 19.3.1 The Option Price
19.4 Standard Derivatives under the “Normal” Model
- 19.4.1 Options on Coupon Bonds
- 19.4.2 Caps and Floors
- 19.4.3 Caps and Floors Implied Volatility
- 19.4.4 European Swaptions
- 19.4.5 Swaptions’ Implied Volatility
19.5 The “Lognormal” Model
- 19.5.1 The Black, Derman, and Toy Model
- 19.5.2 The Black and Karasinski Model
19.6 Generalized Affine Term Structure Models
19.7 Summary
19.8 Exercises
19.9 Appendix: Proofs
- 19.9.1 Proof of the Ho-Lee Pricing Formula
- 19.9.2 Proof of the Expression in Equation 19.13
- 19.9.3 Proof of the Hull-White Pricing Formula
- 19.9.4 Proof of the Expression in Equation 19.28
- 19.9.5 Proof of the Expressions in Equations 19.41 and 19.42

20 The Market Model for Standard Derivatives

20.1 The Black Formula for Caps and Floors Pricing
- 20.1.1 Flat and Forward Volatilities
- 20.1.2 Extracting Forward Volatilities from Flat Volatilities
- 20.1.3 The Behavior of the Implied Forward Volatility
- 20.1.4 Forward Volatilities and the Black, Derman, and Toy Model
20.2 The Black Formula for Swaption Pricing
20.3 Summary
20.4 Exercises

21 Forward Risk Neutral Pricing and the LIBOR Market Model

21.1 One Difficulty with Risk Neutral Pricing
21.2 Change of Numeraire and the Forward Risk Neutral Dynamics
- 21.2.1 Two Important Results
- 21.2.2 Generalizations
21.3 The Option Pricing Formula in “Normal” Models
21.4 The LIBOR Market Model
- 21.4.1 The Black Formula for Caps and Floors
- 21.4.2 Valuing Fixed Income Securities that Depend on a Single LIBOR rate
- 21.4.3 The LIBOR Market Model for More Complex Securities
- 21.4.4 Extracting the Volatility of Forward Rates from Caplets’ Forward Volatilities
- 21.4.5 Pricing Fixed Income Securities by Monte Carlo Simulations
21.5 Forward Risk Neutral Pricing and the Black Formula for Swaptions
- 21.5.1 Remarks: Forward Risk Neutral Pricing and No Arbitrage
21.6 The Heath, Jarrow, and Morton Framework
- 21.6.1 Futures and Forwards
21.7 Unnatural Lag and Convexity Adjustment
- 21.7.1 Unnatural Lag and Convexity
- 21.7.2 A Convexity Adjustment
21.8 Summary
21.9 Exercises
21.10 Appendix: Derivations
- 21.10.1 Derivation of the Partial Differential Equation in the Forward Risk Neutral Dynamics
- 21.10.2 Derivation of the Call Option Pricing Formula (Equation 21.11)
- 21.10.3 Derivation of the Formula in Equation 21.27 and 21.31
- 21.10.4 Derivation of the Formula in Equation 21.21
- 21.10.5 Derivation of the Formula in Equation 21.37

22 Multifactor Models

22.1 Multifactor Ito’s Lemma with Independent Factors
22.2 No Arbitrage with Independent Factors
- 22.2.1 A Two-Factor Vasicek Model
- 22.2.2 A Dynamic Model for the Short and Long Yield
- 22.2.3 Long-Term Spot Rate Volatility
- 22.2.4 Options on Zero Coupon Bonds
22.3 Correlated Factors
- 22.3.1 The Two-Factor Vasicek Model with Correlated Factors
- 22.3.2 Zero Coupon Bond Options
- 22.3.3 The Two-Factor Hull and White Model
22.4 The Feynman-Kac Theorem
- 22.4.1 Application: Yield Curve Steepener
- 22.4.2 Simulating Correlated Brownian Motions
22.5 Forward Risk Neutral Pricing
- 22.5.1 Application: Options on Coupon Bonds
22.6 The Multifactor LIBOR Market Model
- 22.6.1 Level, Slope and Curvature Factors for Forward Rates
22.7 Affine and Quadratic Term Structure Models
- 22.7.1 Affine Models
- 22.7.2 Quadratic Models
22.8 Summary
22.9 Exercises
22.10 Appendix
- 22.10.1 The Coefficients of the Joint Process for Short- and Long- Term Rates
- 22.10.2 The Two-Factor Hull-White Model

References

Index

Sources

Fixed Income Securities: Valuation, Risk, and Risk Management, 1st Edition (Wiley) — the title, author, January 2010 publication, the hardcover ISBN 978-0-470-10910-6 and the description for the lead
Fixed Income Securities, front matter and Chapters 1-3 (Wiley excerpt) — the table of contents, with every chapter, section and subsection title and number; the excerpt is a pre-publication proof whose title page reads "Valuation, Risk and Risk Management" and carries a 2011 copyright line, so the chapter titles follow the publisher's own list on its product page, and the printed typographical errors in section titles are kept as printed
CP 351 Introductory Study Note, Nov 2026/Mar 2027/Jul 2027 (SOA) — the full title as the SOA gives it and the publisher, John Wiley & Sons
CP 351 Syllabus, Nov 2026/Mar 2027/Jul 2027 (SOA) — the year and the chapters assigned, with their learning outcome