Introduction xiii
Part I THEORY 1
1 Mathematics in a Pill 3
1.1 Probability Space and Random Variables 3
1.2 Normal Distributions 4
1.3 Stochastic Processes 4
1.4 Wiener Processes 5
1.5 Geometric Wiener Processes 5
1.6 Markov Processes 6
1.7 Stochastic Integrals and Stochastic Differential Equations 6
1.8 Ito’s Formula 7
1.9 Martingales 7
1.10 Girsanov’s Theorem 7
1.11 Black’s Formula (1976) 8
1.12 Pricing Derivatives and Changing of Numeraire 8
1.13 Pricing of Interest Rate Derivatives and the Forward Measure 9
2 Heath-Jarrow-Morton and Brace-Gatarek-Musiela Models 13
2.1 HJM and BGM Models Under the Spot Measure 13
2.2 Vasiˇcek Model 16
2.3 Cox-Ingersoll-Ross Model 17
2.4 Black-Karasi´nski Model 17
2.5 HJM and BGM Models under the Forward Measures 18
vi Contents
3 Simulation 21
3.1 Simulation of HJM and BGM Models under the Forward Measure 21
3.2 Monte Carlo Simulation of Multidimensional Gaussian
Variables 22
Random Numbers Generation 22
Principal Components Analysis (PCA) 23
Cholesky Decomposition 24
3.3 Trinomial Tree Simulation of Multidimensional Gaussian
Variables 25
4 Swaption Pricing and Calibration 27
4.1 Linear Pricing in the BGM Model 29
4.2 Linear Pricing of Swaptions in the HJM Model 30
4.3 Universal Volatility Function 31
4.4 Time Homogeneous Volatility 33
4.5 Separated Volatility 34
Example of Separated Calibration 36
4.6 Parametrized Volatility 37
4.7 Parametric Calibration to Caps and Swaptions Based on Rebonato
Approach 38
4.8 Semilinear Pricing of Swaptions in the BGM Model 40
4.9 Semilinear Pricing of Swaptions in the HJM Model 41
4.10 Nonlinear Pricing of Swaptions 43
4.11 Examples 43
5 Smile Modelling in the BGM Model 45
5.1 The Shifted BGM Model 46
5.2 Stochastic Volatility for Long Term Options 48
5.3 The Uncertain Volatility Displaced LIBOR Market Model 50
5.4 Mixing the BGM and HJM Models 52
6 Simplified BGM and HJM Models 55
6.1 CMS Rate Dynamics in Single-Factor HJM Model 55
6.2 CMS Rate Dynamics in a Single Factor BGM Model 57
6.3 Calibration 58
6.4 Smile 59
Part II CALIBRATION 63
7 Calibration Algorithms to Caps and Floors 67
7.1 Introduction 67
7.2 Market Data 67
Interpretation of ATM Swaption Quotes 69
7.3 Calibration to Caps 70
7.3.1 Caplet Values 70
7.3.2 ATM Strikes for Caps 71
7.3.3 Stripping Caplet Volatilities from Cap Quotes 74
Contents vii
7.4 Non-Parametric Calibration Algorithms 78
7.4.1 Piecewise Constant Instantaneous Volatilities Depending on
the Time to Maturity 79
7.4.2 Piecewise Constant Instantaneous Volatilities Depending on
the Maturity of the Underlying Forward Rate 84
7.5 Conclusions 86
8 Non-Parametric Calibration Algorithms to Caps and Swaptions 89
8.1 Introduction 89
8.2 The Separated Approach 90
8.3 The Separated Approach with Optimization 109
8.4 The Locally Single Factor Approach 117
8.5 Calibration with Historical Correlations of Forward Rates 120
8.6 Calibration to Co-Terminal Swaptions 125
8.7 Conclusions 129
9 Calibration Algorithms to Caps and Swaptions Based on
Optimization Techniques 131
9.1 Introduction 131
9.2 Non Parametric Calibration to Caps and Swaptions 132
9.3 Parametric Method of Calibration 157
9.3.1 Parametric Calibration to Cap Prices 157
9.3.2 Parametric Calibration to Swaptions 161
9.4 Conclusions 166
Part III SIMULATION 167
10 Approximations of the BGM Model 171
10.1 Euler Approximation 171
10.2 Predictor-Corrector Approximation 171
10.3 Brownian Bridge Approximation 172
10.4 Combined Predictor-Corrector-Brownian Bridge 173
10.5 Single-Dimensional Case 174
10.6 Single-Dimensional Complete Case 175
10.7 Binomial Tree Construction for LAn
t 177
10.8 Binomial Tree Construction for LDN
t 180
10.9 Numerical Example of Binomial Tree Construction 181
10.10 Trinomial Tree Construction for LAN
t 188
10.11 Trinomial Tree Construction for LDN
t 191
10.12 Numerical Results 192
10.13 Approximation of Annuities 192
10.14 Swaption Pricing 195
10.15 Lognormal Approximation 198
10.16 Comparison 200
10.17 Practical Example – Calibration to Co-terminal Swaptions and
Simulation 200
viii Contents
11 The One Factor LIBOR Markov Functional Model 205
11.1 LIBOR Markov Functional Model Construction 205
11.2 Binomial Tree Construction – Approach 1 207
11.3 Binomial Tree Construction – Approach 2 215
12 Optimal Stopping and Pricing of Bermudan Options 219
12.1 Tree/Lattice Pricing 220
12.2 Stochastic Meshes 221
12.3 The Direct Method 221
12.4 The Longstaff-Schwartz Method 222
12.5 Additive Noise 224
12.6 Example of BGM Dynamics 228
12.7 Comparison of Methods 228
13 Using the LSM Approach for Derivatives Valuation 229
13.1 Pricing Algorithms 229
13.2 Numerical Examples of Algorithms 13.1–13.4 234
13.3 Calculation Results 252
13.4 Some Theoretical Remarks on Optimal Stopping Under LSM 253
13.5 Summary 257