1．Introduction 1
2．Arbitrage,Martingales and Numerical Methods 5
2.1 Arbitrage and Martingales 6
2.1.1 Basic Setup 6
2.1.2 Equivalent Martingale Measure 8
2.1.3 Change of Numeraire Theorem 10
2.1.4 Girsanov's Theorem and It？'s Lemma 11
2.1.5 Application：Black-Scholes Model 12
2.1.6 Application：Foreign-Exchange Options 14
2.2 Numerical Methods 16
2.2.1 Derivation of Black-Scholes Partial Differential Equation 16
2.2.2 Feynman-Kac Formula 17
2.2.3 Numerical Solution of PDE's 18
2.2.4 Monte Carlo Simulation 18
2.2.5 Numerical Integration 20
Part Ⅰ．Spot and Forward Rate Models 23
3．Spot and Forward Rate Models 23
3.1 Vasicek Methodology 23
3.1.1 Spot Interest Rate 23
3.1.2 Partial Differential Equation 24
3.1.3 Calculating Prices 25
3.1.4 Example：Ho-Lee Model 26
3.2 Heath-Jarrow-Morton Methodology 27
3.2.1 Forward Rates 27
3.2.2 Equivalent Martingale Measure 28
3.2.3 Calculating Prices 29
3.2.4 Example：Ho-Lee Model 30
3.3 Equivalence of the Methodologies 30
4．Fundamental Solutions and the Forward-Risk-Adjusted Measure 31
4.1 Forward-Risk-Adjusted Measure 32
4.2 Fundamental Solutions 34
4.3 Obtaining Fundamental Solutions 36
4.4 Example：Ho-Lee Model 37
4.4.1 Radon-Nikodym Derivative 37
4.4.2 Fundamental Solutions 38
4.5 Fundamental Solutions for Normal Models 40
5．The Hull-White Model 45
5.1 Spot Rate Process 46
5.1.1 Partial Differential Equation 47
5.1.2 Transformation of Variables 47
5.2 Analytical Formul？ 48
5.2.1 Fundamental Solutions 49
5.2.2 Option Prices 50
5.2.3 Prices for Other Instruments 51
5.3 Implementation of the Model 52
5.3.1 Fitting the Model to the Initial Tern-Structure 52
5.3.2 Transformation of Variables 53
5.3.3 Trinomial Tree 53
5.4 Performance of the Algorithm 55
5.5 Appendix 57
6．The Squared Gaussian Model 59
6.1 Spot Rate Process 60
6.1.1 Partial Differential Equation 60
6.2 Analytical Formul？ 61
6.2.1 Fundamental Solutions 62
6.2.2 Option Prices 63
6.3 Implementation of the Model 64
6.3.1 Fitting the Model to the Initial Term-Structure 64
6.3.2Trinomial Tree 66
6.4 Appendix A 66
6.5 Appendix B 69
7．An Empirical Comparison of One-Factor Models 71
7.1 Yield-Curve Models 72
7.2 Econometric Approach 74
7.3 Data 77
7.4 Empirical Results 77
7.5 Conclusions 84
Part Ⅱ．Market Rate Models 87
8．LIBOR and Swap Market Models 87
8.1 LIBOR Market Models 88
8.1.1 LIBOR Process 88
8.1.2 Caplet Price 89
8.1.3 Terminal Measure 90
8.2 Swap Market Models 91
8.2.1 Interest Rate Swaps 92
8.2.2 Swaption Price 93
8.2.3 Terminal Measure 95
8.2.4 T1-Forward Measure 96
8.3 Monte Carlo Simulation for LIBOR Market Models 97
8.3.1 Calculating the Numeraire Rebased Payoff 98
8.3.2 Example：Vanilla Cap 99
8.3.3 Discrete Barrier Caps/Floors 100
8.3.4 Discrete Barrier Digital Caps/Floors 102
8.3.5 Payment Stream 103
8.3.6 Ratchets 103
8.4 Monte Carlo Simulation for Swap Market Models 104
8.4.1 Terminal Measure 104
8.4.2 T1-Forward Measure 105
8.4.3 Example：Spread Option 106
9． Markov-Functional Models 109
9.1 Basic Assumptions 110
9.2 LIBOR Markov-Functional Model 111
9.3 Swap Markov-Functional Model 114
9.4 Numerical Implementation 115
9.4.1 Numerical Integration 115
9.4.2 Non-Parametric Implementation 117
9.4.3 Semi-Parametric Implementation 118
9.5 Forward Volatilities and Auto-Correlation 120
9.5.1 Mean-Reversion and Auto-Correlation 120
9.5.2 Auto-Correlation and the Volatility Function 121
9.6 LIBOR Example：Barrier Caps 121
9.6.1 Numerical Calculation 121
9.6.2 Comparison with LIBOR Market Model 123
9.6.3 Impact of Mean-Reversion 124
9.7 LIBOR Example：Chooser-and Auto-Caps 125
9.7.1 Auto-Caps/Floors 125
9.7.2 Chooser-Caps/Floors 125
9.7.3 Auto-and Chooser-Digitals 125
9.7.4 Numerical Implementation 125
9.8 Swap Example：Bermudan Swaptions 127
9.8.1 Early Notification 127
9.8.2 Comparison Between Models 128
10．An Empirical Comparison of Market Models 131
10.1 Data Description 132
10.2 LIBOR Market Model 132
10.2.1 Calibration Methodology 132
10.2.2 Estimation and Pricing Results 134
10.3 Swap Market Model 135
10.3.1 Calibration Methodology 135
10.3.2 Estimation and Pricing Results 135
10.4 Conclusion 136
11．Convexity Correction 139
11.1 Convexity Correction and Change of Numeraire 140
11.1.1 Multi-Currency Change of Numeraire Theorem 140
11.1.2 Convexity Correction 142
11.2 Options on Convexity Corrected Rates 145
11.2.1 Option Price Formula 146
11.2.2 Digital Price Formula 147
11.3 Single Index Products 147
11.3.1 LIBOR in Arrears 147
11.3.2 Constant Maturity Swap 149
11.3.3 Diffed LIBOR 150
11.3.4 Diffed CMS 150
11.4 Multi-Index Products 151
11.4.1 Rate Based Spread Options 151
11.4.2 Spread Digital 153
11.4.3 Other Multi-Index Products 153
11.4.4 Comparison with Market Models 154
11.5 A Warning on Convexity Correction 155
11.6 Appendix：Linear Swap Rate Model 156
12．Extensions and Further Developments 159
12.1 General Philosophy 159
12.2 Multi-Factor Models 160
12.3 Volatility Skews 161
References 163
Index 167