The Price of Fixed Income Market Volatility (eBook)

Antonio Mele holds a Senior Chair at the Swiss Finance Institute, and is a full Professor of Finance at the University of Lugano, after having been a tenured faculty at the London School of Economics & Political Science for a decade. He is also a Research Fellow for the Financial Economics program at the Centre for Economic Policy Research (CEPR) in London. He holds a PhD in Economics from the University of Paris.His academic expertise spans a variety of fields in financial economics, pertaining to capital market volatility, interest rates and credit markets, macro-finance, capital markets and business cycles, and information in securities markets. His research has been published by top journals in Finance and Economics such as the Journal of Financial Economics, the Review of Economic Studies, the Review of Financial Studies, and the Journal of Monetary Economics.His work outside academia includes developing fixed income volatility indexes for Chicago Board Options Exchange. He is the co-inventor of the CBOE Interest Rate Swap Volatility Index (CBOE-SRVX℠) - the first standardized volatility measure in the interest-rate swap market, designed to standardize and simplify swap-rate volatility trading much in the spirit of the CBOE-VIX® index in the equity market.Yoshiki Obayashi is a managing director at Applied Academics LLC in New York, specialized in developing and commercializing ideas emanating from a growing think-tank of academic researchers selected for their work's relevance to practice in the finance industry. His most recent projects range from running systematic trading strategies for funds to developing fixed income volatility indexes for Chicago Board Options Exchange. Yoshiki Obayashi previously managed US and Asian credit portfolios for a proprietary fixed-income trading group at an investment bank. He holds a PhD in Finance and Economics from Columbia Business School.

The Price of Fixed Income Market Volatility 4
Preface 6
Contents 8
Information About the Authors 12
Chapter 1: Introduction 13
1.1 Background 13
1.2 From Realized to Expected Fixed Income Volatility 16
1.3 The Right Numéraire and Volatility Pricing 20
1.3.1 Market Risk and Model-Free Pricing 21
1.3.2 Getting the Right Volatility with the Right Model 23
1.3.2.1 Rescaling 23
1.3.2.2 Naïve Model-Free Methodology 23
1.3.2.3 Approximations of the Market Probability 24
1.3.2.4 Maturity Mismatch 26
1.3.2.5 American Corrections 27
1.4 Scope and Plan of the Book 28
Chapter 2: Variance Contracts: Fixed Income Security Design 30
2.1 Introduction 30
2.2 Market Numéraires and Volatilities 32
2.3 Interest Rate Variance Swaps 33
2.3.1 Contracts and Model-Free Pricing 33
2.3.2 Log Versus Quadratic Contracts 37
2.3.3 Hedging 39
2.3.4 Constant Gamma Exposure 41
2.4 Implied Volatility Indexes 42
2.4.1 Model-Free Indexes 42
2.4.2 Comparisons to Model-Based Log-Normal and Normal Implied Volatility 43
2.4.2.1 Skews 43
2.4.2.2 Estimating Expected Volatility from ATM Implied Volatilities 44
2.4.3 Index Decompositions 45
2.5 Implementing Basis Point Variance Swaps 46
2.5.1 Incremental Versus Point-to-Point Realized Variance 46
2.5.1.1 Basis Point 46
2.5.1.2 Percentage 47
2.5.2 Volatility Risk Premiums 49
2.6 Skew Shifts and the Dynamics of Volatility Indexes 51
2.6.1 Truncations 52
2.6.1.1 The Effects on the BP Volatility Index 53
2.6.1.2 The Percentage Index 54
2.6.2 Numerical Experiments and Interpretation of Actual Index Behavior 54
2.7 Jumps 58
Appendix A: Appendix on Security Design and Volatility Indexing 60
A.1 Proof of Proposition 2.2 60
A.2 A Stochastic Multiplier Beyond the Market NumÉraire 62
A.3 Vega and Gamma in Gaussian Markets 62
A.4 Proof of Proposition 2.3 65
A.5 Approximating Indexes 66
A.6 Jumps 68
Chapter 3: Interest Rate Swaps 70
3.1 Introduction 70
3.2 Risks Regarding Interest Rate Swaps 72
3.2.1 The Annuity Factor 72
3.2.1.1 The Forward Swap Rate: De?nition 72
3.2.1.2 The Risks in the Interest Rate Swap Space 73
3.2.2 Option-Based Volatility Trading 74
3.3 Interest Rate Swap Variance Contracts 77
3.3.1 Risks and Spanning Derivatives 78
3.3.2 Contract Designs 79
3.3.3 Pricing 81
3.3.4 Marking to Market 82
3.3.5 Hedging 83
3.3.5.1 Percentage 84
3.3.5.2 Basis Point 85
3.3.6 Links to Constant Maturity Swaps 87
3.3.7 Physical Swap Settlement and Variance Contracts 87
3.3.8 Multiple Curves 88
3.4 Trading Strategies 89
3.4.1 Spot Trading Through IRV Swaps 89
3.4.2 Spot Trading Through Standardized IRV Swaps 92
3.4.3 Forward Trading 93
3.5 Interest Rate Swap Volatility Indexes 94
3.5.1 Basis Point Volatility Index 94
3.5.2 Percentage Volatility Index 95
3.5.3 Experiments 95
3.5.4 Jumps 98
3.6 Swap Versus Equity Variance Contracts and Indexes 98
3.7 Index Implementation 100
3.7.1 A Numerical Example 101
3.7.2 Historical Performance 104
Appendix B: Appendix on Interest Rate Swap Markets 108
B.1 P& L of Option-Based Volatility Trading
B.2 Spanning IRS Variance Contracts 114
B.3 Hedging 118
B.4 Constant Maturity Swaps 122
B.5 The Contract and Index in the Vasicek Market 124
Chapter 4: Government Bonds and Time-Deposits 136
4.1 Introduction 136
4.2 Government Bonds 140
4.2.1 Pricing Spot Volatility 140
4.2.1.1 Government Bond Prices and Volatility 141
4.2.1.2 The Two Steps to Variance Swap Evaluation 141
First Step: Pricing Volatility Through Risk-Neutral Evaluation 142
Second Step: Matching Volatility Through Spanning Contracts 143
4.2.1.3 Naïve Model-Free Methodology and Pricing Biases 143
4.2.2 Basis Assets 145
4.2.3 Percentage Price Volatility 147
4.2.3.1 Pricing 147
4.2.3.2 Discussion 148
4.2.4 Basis Point Price Volatility 149
4.2.5 Marking to Market 150
4.2.6 Replication 150
4.2.6.1 Percentage Variance Swaps 150
4.2.6.2 Basis Point Variance Swaps 151
4.2.7 Forward Price Adjustments 152
4.2.8 Model-Free Measures of Basis Point Yield Volatility 153
4.2.8.1 Certainty Equivalent Bond Prices 153
4.2.8.2 Yield Volatility 154
Duration-Based Yield Volatility 154
Yield-Based Yield Volatility 155
4.2.9 Certainty Equivalent Bond Prices as Expectations of Forward Prices 156
4.2.9.1 Theory 156
4.2.9.2 One Example 156
4.2.9.3 Pricing Government Bond Volatility Products: An Introductory Example 158
4.2.10 Early Exercise and Futures Corrections 159
4.2.11 Implementation Example 164
4.2.12 Jumps 168
4.3 Time Deposits 168
4.3.1 The Underlying Risks 168
4.3.2 Variance Contracts and Volatility Indexes 169
4.3.3 Yield Volatility 171
4.3.4 American Future Corrections 172
4.3.5 Implementation Example 176
4.3.6 LIBOR Variance Contracts and Volatility Indexes 179
4.4 Maturity Mismatch 181
4.4.1 Government Bonds 181
4.4.1.1 Percentage Volatility 182
4.4.1.2 Basis Point 184
4.4.1.3 Basis Point Yield Volatility 185
4.4.1.4 Forward Adjustments 186
4.4.2 Time Deposits 187
4.4.2.1 Prices 187
4.4.2.2 Yields 190
4.4.2.3 Rates 190
4.4.2.4 Forward Adjustments 192
4.4.3 Alternative Characterizations of Variance Contracts and Indexes 193
4.4.3.1 Percentage 193
4.4.3.2 Basis Point 194
4.4.4 Tilting the Variance Payoff 195
4.5 Index Design with Heterogeneous Market Data 196
4.5.1 Sandwich Combinations 197
4.5.2 Rolling Indexes 199
Appendix C: Appendix on Government Bonds and Time Deposit Markets 200
C.1 The Equity VIX with Stochastic Interest Rates 200
C.2 Naïve Model-Free Methodology and Bias in Vasicek's Market 202
C.3 Marking to Market 204
C.4 Replication of Variance Swaps 204
C.5 Estimates Based on Forward Price Approximations 206
C.6 Certainty Equivalence, and Existence of Basis Point Yield Volatility 208
C.7 Illustrations with a Stochastic Volatility Model 210
C.8 The Future Price in Vasicek's Model 213
C.9 Future and Forward LIBOR Options in Vasicek's Model 213
C.10 The Impact of Early Exercise Premiums and Maturity Mismatch 217
Chapter 5: Credit 221
5.1 Introduction 221
5.2 Existing Credit Trading Practices 223
5.2.1 Assumptions 224
5.2.2 CDS Indexes 224
5.2.3 CDS Index Options 225
5.2.3.1 Forward Positions 225
5.2.3.2 Front-End Protection and Credit Default Options 226
5.3 Credit Variance Contracts 228
5.3.1 Percentage 228
5.3.2 Basis Point 230
5.3.3 Marking to Market 231
5.3.3.1 Hedging 232
5.4 Credit Volatility Indexes 235
5.4.1 De?nitions 235
5.4.2 Forward Premium Adjustments 236
5.4.3 Differences with Respect to Other Fixed Income Volatility Gauges 236
5.4.4 Implementation Example 237
5.4.5 Index Design Through Option Cycles 240
5.5 Post "Big-Bang" Conventions and Index Adjustments 240
5.5.1 Index Values Under Constant Hazard Rates 241
5.5.2 Forward Positions 242
5.5.3 Option Payoffs and Evaluation 243
5.5.3.1 Modi?ed Market Formula 244
5.5.3.2 Pedersen's Model 244
5.5.4 Index Corrections 246
5.5.4.1 Correction Based on the Modi?ed Black Formula 246
5.5.4.2 Correction Based on Pedersen's Model 246
Appendix D: Appendix on Credit Markets 247
D.1 Preliminary Facts Concerning CDS Indexes 247
D.2 Spanning Credit Variance Contracts 248
D.3 Hedging 252
References 256