Fixed Income Analysis:Securities, Pricing, and Risk Management

固定收益分析的一本好书

 

 

1  Basic  interest  rate  markets,  concepts,  and  relations  1

1.1 Introduction .  .  1

1.2 Markets for bonds and interest rates   .  . 1

1.3 Discount factors and zero-coupon bonds   .  .  5

1.4 Zero-coupon rates and forward rates   .  .  6

1.4.1  Annual compounding  7

1.4.2  Compounding over other discrete periods { LIBOR rates 8

1.4.3  Continuous compounding   .  .  8

1.4.4  Different ways to represent the term structure of interest rates  .  10

1.5 Determining the zero-coupon yield curve:  Bootstrapping  .  .  . 10

1.6 Determining the zero-coupon yield curve:  Parameterized forms   . 14

1.6.1  Cubic splines   . 14

1.6.2  The Nelson-Siegel parameterization . 18

1.6.3  Additional remarks on yield curve estimation   .  19

1.7 Exercises .  20

2  Fixed  income  securities 22

2.1 Introduction .  .  22

2.2 Floating rate bonds .  . 22

2.3 Forwards on bonds  .  . 23

2.4 Interest rate forwards { forward rate agreements  .  25

2.5 Futures on bonds  . 26

2.6 Interest rate futures { Eurodollar futures .  . 26

2.7 Options on bonds .  .  .  28

2.7.1  Options on zero-coupon bonds  . 29

2.7.2  Options on coupon bonds   .  .  30

2.8 Caps, floors, and collars   .  31

2.8.1  Caps  .  . 31

2.8.2  Floors  .  33

2.8.3  Collars . 33

2.8.4  Exotic caps and floors   .  . 34

2.9 Swaps and swaptions  . 35

i

Contents  ii

2.9.1  Swaps   .  35

2.9.2  Swaptions  . 39

2.9.3  Exotic swap instruments . 40

2.10  Exercises  .  41

3  Stochastic  processes  and  stochastic  calculus  43

3.1 Probability spaces  43

3.2 Stochastic processes  .  44

3.2.1  Different types of stochastic processes   . 44

3.2.2  Basic concepts   .  . 45

3.2.3  Markov processes and martingales   .  46

3.2.4  Continuous or discontinuous paths   .  46

3.3 Brownian motions 47

3.4 Diffusion processes   .  .  51

3.5 Itˆo processes . 52

3.6 Stochastic integrals  .  . 53

3.6.1  Definition and properties of stochastic integrals  .  . 53

3.6.2  Leibnitz’ rule for stochastic integrals  .  .  54

3.7 Itˆo’s Lemma .  .  56

3.8 Important diffusion processes   .  . 57

3.8.1  Geometric Brownian motions   .  . 57

3.8.2  Ornstein-Uhlenbeck processes  .  .  59

3.8.3  Square root processes 62

3.9 Multi-dimensional processes  . 64

3.10  Change of probability measure .  67

3.11  Exercises  .  70

4  Asset  pricing  and  term  structures:  discrete-time  models 71

4.1 Introduction .  .  .  .  71

4.2 A one-period model  . 72

4.2.1  Assets, portfolios, and arbitrage 72

4.2.2  Investors  . 73

4.2.3  State-price vectors and deflators  .  .  .  .  75

4.2.4  Risk-neutral probabilities   .  . 77

4.2.5  Redundant assets .  .  . 79

4.2.6  Complete markets   .  .  80

4.2.7  Equilibrium and representative agents in complete markets   .  82

4.3 A multi-period, discrete-time model 84

4.3.1  Assets, trading strategies, and arbitrage   .  .  85

4.3.2  Investors  . 86

4.3.3  State-price vectors and deflators  .  .  87

4.3.4  Risk-neutral probability measures . 89

4.3.5  Redundant assets . 91

4.3.6  Complete markets   .  .  .  .  92

Contents  iii

4.3.7  Equilibrium and representative agents in complete markets   .  93

4.4 Discrete-time, finite-state models of the term structure  .  .  .  . 94

4.5 Concluding remarks  . 95

4.6 Exercises   .  95

5  Asset  pricing  and  term  structures:  an  introduction  to  continuous-time  models   97

5.1 Introduction .  .  .  .  97

5.2 Asset pricing in continuous-time models   .  .  98

5.2.1  Assets, trading strategies, and arbitrage   .  .  99

5.2.2  Investors  . 101

5.2.3  State-price deflators   .  .  .  102

5.2.4  Risk-neutral probability measures . 103

5.2.5  From no arbitrage to state-price deflators and risk-neutral measures   .  . 105

5.2.6  Market prices of risk  .  105

5.2.7  Complete vs. incomplete markets  .  .  108

5.2.8  Extension to intermediate dividends   .  .  109

5.2.9  Equilibrium and representative agents in complete markets   .  110

5.3 Other probability measures convenient for pricing . 111

5.3.1  A zero-coupon bond as the numeraire { forward martingale measures .  . 113

5.3.2  An annuity as the numeraire { swap martingale measures   .  . 114

5.3.3  A general pricing formula for European options  .  . 114

5.4 Forward prices and futures prices  .  .  116

5.4.1  Forward prices   .  .  116

5.4.2  Futures prices . 117

5.4.3  A comparison of forward prices and futures prices  118

5.5 American-style derivatives  .  .  119

5.6 Diffusion models and the fundamental partial differential equation   .  119

5.6.1  One-factor diffusion models   .  120

5.6.2  Multi-factor diffusion models   .  . 125

5.7 The Black-Scholes-Merton model and Black’s variant  . 127

5.7.1  The Black-Scholes-Merton model  .  .  127

5.7.2  Black’s model  .  129

5.7.3  Problems in applying Black’s model to fixed income securities .  .  133

5.8 An overview of continuous-time term structure models  .  . 134

5.8.1  Overall categorization   .  .  135

5.8.2  Some frequently applied models .  .  .  136

5.9 Exercises .  138

6  The  Economics  of  the  Term  Structure  of  Interest  Rates  139

6.1 Introduction .  .  139

6.2 Real interest rates and aggregate consumption 140

6.3 Real interest rates and aggregate production  . 142

6.4 Equilibrium interest rate models  .  .  145

6.4.1  Production-based models  .  . 145

Contents iv

6.4.2  Consumption-based models   .  146

6.5 Real and nominal interest rates and term structures   .  147

6.5.1  Real and nominal asset pricing   . 148

6.5.2  No real effects of inflation  .  . 151

6.5.3  A model with real effects of money  .  152

6.6 The expectation hypothesis   . 157

6.6.1  Versions of the pure expectation hypothesis  .  . 157

6.6.2  The pure expectation hypothesis and equilibrium  .  158

6.6.3  The weak expectation hypothesis  .  . 159

6.7 Liquidity preference, market segmentation, and preferred habitats  . 160

6.8 Concluding remarks  .  .  . 161

6.9 Exercises .  161

7  One-factor  diffusion  models  163

7.1 Introduction .  .  163

7.2 Affine models  .  164

7.2.1  Bond prices, zero-coupon rates, and forward rates 165

7.2.2  Forwards and futures . 167

7.2.3  European options on coupon bonds:  Jamshidian’s trick . 169

7.3 Merton’s model  .  . 172

7.3.1  The short rate process  .  .  172

7.3.2  Bond pricing   . 173

7.3.3  The yield curve  .  . 173

7.3.4  Forwards and futures . 174

7.3.5  Option pricing   .  . 174

7.4 Vasicek’s model  .  .  .  . 176

7.4.1  The short rate process  .  .  176

7.4.2  Bond pricing   . 178

7.4.3  The yield curve  .  . 181

7.4.4  Forwards and futures . 184

7.4.5  Option pricing   .  . 185

7.5 The Cox-Ingersoll-Ross model  .  .  188

7.5.1  The short rate process  .  .  188

7.5.2  Bond pricing   . 189

7.5.3  The yield curve  .  . 190

7.5.4  Forwards and futures . 191

7.5.5  Option pricing   .  . 192

7.6 Non-affine models .  193

7.7 Parameter estimation and empirical tests . 196

7.8 Concluding remarks  . 199

7.9 Exercises .  199

Contents  v

8  Multi-factor  diffusion  models  201

8.1 What is wrong with one-factor models? 201

8.2 Multi-factor diffusion models of the term structure   .  .  203

8.3 Multi-factor affine diffusion models  .  .  . 205

8.3.1  Two-factor affine diffusion models . 205

8.3.2  n-factor affine diffusion models   .  208

8.3.3  European options on coupon bonds .  209

8.4 Multi-factor Gaussian diffusion models  . 209

8.4.1  General analysis   .  209

8.4.2  A specific example:  the two-factor Vasicek model  .  211

8.5 Multi-factor CIR models  . 212

8.5.1  General analysis   .  212

8.5.2  A specific example:  the Longstaff-Schwartz model 214

8.6 Other multi-factor diffusion models  . 219

8.6.1  Models with stochastic consumer prices   .  . 219

8.6.2  Models with stochastic long-term level and volatility  .  220

8.6.3  A model with a short and a long rate  . 222

8.6.4  Key rate models   .  222

8.6.5  Quadratic models .  223

8.7 Final remarks  .  223

9  Calibration  of  diffusion  models  225

9.1 Introduction .  .  .  .  225

9.2 Time inhomogeneous affine models   . 226

9.3 The Ho-Lee model (extended Merton)   .  228

9.4 The Hull-White model (extended Vasicek)  .  230

9.5 The extended CIR model  232

9.6 Calibration to other market data   .  .  233

9.7 Initial and future term structures in calibrated models  .  . 234

9.8 Calibrated non-affine models  236

9.9 Is a calibrated one-factor model just as good as a multi-factor model?   . 237

9.10  Final remarks  . 238

9.11  Exercises  .  239

10 Heath-Jarrow-Morton  models 240

10.1  Introduction .  . 240

10.2  Basic assumptions  .  .  240

10.3  Bond price dynamics and the drift restriction   . 242

10.4  Three well-known special cases   . 244

10.4.1   The Ho-Lee (extended Merton) model   .  244

10.4.2   The Hull-White (extended Vasicek) model  . 245

10.4.3   The extended CIR model   .  . 246

10.5  Gaussian HJM models  .  .  .  . 247

10.6  Diffusion representations of HJM models .  . 248

Contents vi

10.6.1   On the use of numerical techniques for diffusion and non-diffusion models   . 249

10.6.2   In which HJM models does the short rate follow a diffusion process?  .  .  249

10.6.3   A two-factor diffusion representation of a one-factor HJM model   .  . 252

10.7  HJM-models with forward-rate dependent volatilities  .  253

10.8  Concluding remarks  . 254

11 Market  models 256

11.1  Introduction .  . 256

11.2  General LIBOR market models   . 257

11.2.1   Model description 257

11.2.2   The dynamics of all forward rates under the same probability measure  .  258

11.2.3   Consistent pricing   .  .  263

11.3  The lognormal LIBOR market model .  . 263

11.3.1   Model description 263

11.3.2   The pricing of other securities .  . 265

11.4  Alternative LIBOR market models   . 267

11.5  Swap market models  . 268

11.6  Further remarks  . 270

11.7  Exercises  .  271

12 The  measurement  and  management  of  interest  rate  risk  272

12.1  Introduction .  . 272

12.2  Traditional measures of interest rate risk .  . 272

12.2.1   Macaulay duration and convexity  .  .  272

12.2.2   The Fisher-Weil duration and convexity   .  .  .  . 274

12.2.3   The no-arbitrage principle and parallel shifts of the yield curve   . 275

12.3  Risk measures in one-factor diffusion models .  . 276

12.3.1   Definitions and relations  . 276

12.3.2   Computation of the risk measures in affine models   .  . 279

12.3.3   A comparison with traditional durations  .  . 281

12.4  Immunization  .  282

12.4.1   Construction of immunization strategies  .  . 282

12.4.2   An experimental comparison of immunization strategies   . 284

12.5  Risk measures in multi-factor diffusion models .  288

12.5.1   Factor durations, convexities, and time value   .  288

12.5.2   One-dimensional risk measures in multi-factor models   .  . 290

12.6  Duration-based pricing of options on bonds   .  . 292

12.6.1   The general idea   .  292

12.6.2   A mathematical analysis of the approximation  293

12.6.3   The accuracy of the approximation in the Longstaff-Schwartz model   .  .  294

12.7  Alternative measures of interest rate risk .  .  296

13 Mortgage-backed  securities 299

Contents vii

14 Credit  risky  securities  300

15 Stochastic  interest  rates  and  the  pricing  of  stock  and  currency  derivatives 301

15.1  Introduction .  . 301

15.2  Stock options   . 301

15.2.1   General analysis   . 301

15.2.2   Deterministic volatilities  .  304

15.3  Options on forwards and futures  .  .  306

15.3.1   Forward and futures prices  . 306

15.3.2   Options on forwards   .  307

15.3.3   Options on futures  .  .  308

15.4  Currency derivatives   .  309

15.4.1   Currency forwards   .  .  309

15.4.2   A model for the exchange rate . 310

15.4.3   Currency futures  .  .  . 311

15.4.4   Currency options  .  311

15.4.5   Alternative exchange rate models  .  .  313

15.5  Final remarks  . 313

16 Numerical  techniques  315

A   Results  on  the  lognormal  distribution  316

References 319

 

 

 

 

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