The Price of Fixed Income Market Volatility

The Price of Fixed Income Market Volatility

Preface

Contents

Information About the Authors

Chapter 1: Introduction

1.1 Background

1.2 From Realized to Expected Fixed Income Volatility

1.3 The Right Numéraire and Volatility Pricing

1.3.1 Market Risk and Model-Free Pricing

1.3.2 Getting the Right Volatility with the Right Model

1.3.2.1 Rescaling

1.3.2.2 Naïve Model-Free Methodology

1.3.2.3 Approximations of the Market Probability

1.3.2.4 Maturity Mismatch

1.3.2.5 American Corrections

1.4 Scope and Plan of the Book

Chapter 2: Variance Contracts: Fixed Income Security Design

2.1 Introduction

2.2 Market Numéraires and Volatilities

2.3 Interest Rate Variance Swaps

2.3.1 Contracts and Model-Free Pricing

2.3.2 Log Versus Quadratic Contracts

2.3.3 Hedging

2.3.4 Constant Gamma Exposure

2.4 Implied Volatility Indexes

2.4.1 Model-Free Indexes

2.4.2 Comparisons to Model-Based Log-Normal and Normal Implied Volatility

2.4.2.1 Skews

2.4.2.2 Estimating Expected Volatility from ATM Implied Volatilities

2.4.3 Index Decompositions

2.5 Implementing Basis Point Variance Swaps

2.5.1 Incremental Versus Point-to-Point Realized Variance

2.5.1.1 Basis Point

2.5.1.2 Percentage

2.5.2 Volatility Risk Premiums

2.6 Skew Shifts and the Dynamics of Volatility Indexes

2.6.1 Truncations

2.6.1.1 The Effects on the BP Volatility Index

2.6.1.2 The Percentage Index

2.6.2 Numerical Experiments and Interpretation of Actual Index Behavior

2.7 Jumps

Appendix A: Appendix on Security Design and Volatility Indexing

A.1 Proof of Proposition 2.2

A.2 A Stochastic Multiplier Beyond the Market NumÉraire

A.3 Vega and Gamma in Gaussian Markets

A.4 Proof of Proposition 2.3

A.5 Approximating Indexes

A.6 Jumps

Chapter 3: Interest Rate Swaps

3.1 Introduction

3.2 Risks Regarding Interest Rate Swaps

3.2.1 The Annuity Factor

3.2.1.1 The Forward Swap Rate: De?nition

3.2.1.2 The Risks in the Interest Rate Swap Space

3.2.2 Option-Based Volatility Trading

3.3 Interest Rate Swap Variance Contracts

3.3.1 Risks and Spanning Derivatives

3.3.2 Contract Designs

3.3.3 Pricing

3.3.4 Marking to Market

3.3.5 Hedging

3.3.5.1 Percentage

3.3.5.2 Basis Point

3.3.6 Links to Constant Maturity Swaps

3.3.7 Physical Swap Settlement and Variance Contracts

3.3.8 Multiple Curves

3.4 Trading Strategies

3.4.1 Spot Trading Through IRV Swaps

3.4.2 Spot Trading Through Standardized IRV Swaps

3.4.3 Forward Trading

3.5 Interest Rate Swap Volatility Indexes

3.5.1 Basis Point Volatility Index

3.5.2 Percentage Volatility Index

3.5.3 Experiments

3.5.4 Jumps

3.6 Swap Versus Equity Variance Contracts and Indexes

3.7 Index Implementation

3.7.1 A Numerical Example

3.7.2 Historical Performance

Appendix B: Appendix on Interest Rate Swap Markets

B.1 P&L of Option-Based Volatility Trading

B.2 Spanning IRS Variance Contracts

B.3 Hedging

B.4 Constant Maturity Swaps

B.5 The Contract and Index in the Vasicek Market

Chapter 4: Government Bonds and Time-Deposits

4.1 Introduction

4.2 Government Bonds

4.2.1 Pricing Spot Volatility

4.2.1.1 Government Bond Prices and Volatility

4.2.1.2 The Two Steps to Variance Swap Evaluation

First Step: Pricing Volatility Through Risk-Neutral Evaluation

Second Step: Matching Volatility Through Spanning Contracts

4.2.1.3 Naïve Model-Free Methodology and Pricing Biases

4.2.2 Basis Assets

4.2.3 Percentage Price Volatility

4.2.3.1 Pricing

4.2.3.2 Discussion

4.2.4 Basis Point Price Volatility

4.2.5 Marking to Market

4.2.6 Replication

4.2.6.1 Percentage Variance Swaps

4.2.6.2 Basis Point Variance Swaps

4.2.7 Forward Price Adjustments

4.2.8 Model-Free Measures of Basis Point Yield Volatility

4.2.8.1 Certainty Equivalent Bond Prices

4.2.8.2 Yield Volatility

Duration-Based Yield Volatility

Yield-Based Yield Volatility

4.2.9 Certainty Equivalent Bond Prices as Expectations of Forward Prices

4.2.9.1 Theory

4.2.9.2 One Example

4.2.9.3 Pricing Government Bond Volatility Products: An Introductory Example

4.2.10 Early Exercise and Futures Corrections

4.2.11 Implementation Example

4.2.12 Jumps

4.3 Time Deposits

4.3.1 The Underlying Risks

4.3.2 Variance Contracts and Volatility Indexes

4.3.3 Yield Volatility

4.3.4 American Future Corrections

4.3.5 Implementation Example

4.3.6 LIBOR Variance Contracts and Volatility Indexes

4.4 Maturity Mismatch

4.4.1 Government Bonds

4.4.1.1 Percentage Volatility

4.4.1.2 Basis Point

4.4.1.3 Basis Point Yield Volatility

4.4.1.4 Forward Adjustments

4.4.2 Time Deposits

4.4.2.1 Prices

4.4.2.2 Yields

4.4.2.3 Rates

4.4.2.4 Forward Adjustments

4.4.3 Alternative Characterizations of Variance Contracts and Indexes

4.4.3.1 Percentage

4.4.3.2 Basis Point

4.4.4 Tilting the Variance Payoff

4.5 Index Design with Heterogeneous Market Data

4.5.1 Sandwich Combinations

4.5.2 Rolling Indexes

Appendix C: Appendix on Government Bonds and Time Deposit Markets

C.1 The Equity VIX with Stochastic Interest Rates

C.2 Naïve Model-Free Methodology and Bias in Vasicek's Market

C.3 Marking to Market

C.4 Replication of Variance Swaps

C.5 Estimates Based on Forward Price Approximations

C.6 Certainty Equivalence, and Existence of Basis Point Yield Volatility

C.7 Illustrations with a Stochastic Volatility Model

C.8 The Future Price in Vasicek's Model

C.9 Future and Forward LIBOR Options in Vasicek's Model

C.10 The Impact of Early Exercise Premiums and Maturity Mismatch

Chapter 5: Credit

5.1 Introduction

5.2 Existing Credit Trading Practices

5.2.1 Assumptions

5.2.2 CDS Indexes

5.2.3 CDS Index Options

5.2.3.1 Forward Positions

5.2.3.2 Front-End Protection and Credit Default Options

5.3 Credit Variance Contracts

5.3.1 Percentage

5.3.2 Basis Point

5.3.3 Marking to Market

5.3.3.1 Hedging

5.4 Credit Volatility Indexes

5.4.1 De?nitions

5.4.2 Forward Premium Adjustments

5.4.3 Differences with Respect to Other Fixed Income Volatility Gauges

5.4.4 Implementation Example

5.4.5 Index Design Through Option Cycles

5.5 Post "Big-Bang" Conventions and Index Adjustments

5.5.1 Index Values Under Constant Hazard Rates

5.5.2 Forward Positions

5.5.3 Option Payoffs and Evaluation

5.5.3.1 Modi?ed Market Formula

5.5.3.2 Pedersen's Model

5.5.4 Index Corrections

5.5.4.1 Correction Based on the Modi?ed Black Formula

5.5.4.2 Correction Based on Pedersen's Model

Appendix D: Appendix on Credit Markets

D.1 Preliminary Facts Concerning CDS Indexes

D.2 Spanning Credit Variance Contracts

D.3 Hedging

References