Robust Libor Modelling and Pricing of Derivative Products, Vol. 3 Book

	List of Tables
	List of Figures
1	Arbitrage-Free Modelling of Effective Interest Rates	1
1.1	Elements of Arbitrage Theory and Derivative Pricing	1
1.1.1	Arbitrage-free Systems, Self-Financing Trading Strategies, Complete Markets	1
1.1.2	Derivative Claim Pricing in Different Measures	7
1.2	Modelling of Effective Forward Rates	9
1.2.1	Libor Rate Processes and Measures	9
1.2.2	Swap Rate Processes and Measures	15
1.3	Pricing of Caps and Swaptions in Libor and Swap Market Models	18
1.3.1	Libor Caps and Caplets	18
1.3.2	Swaptions in a Swap Market Model	19
1.3.3	Approximating Swaptions in a Libor Market Model	20
1.3.4	Smile/Skew Extensions of the Libor Market Model	23
2	Parametrisation of the Libor Market Model	25
2.1	General Volatility Structures	25
2.2	(Quasi) Time-Shift Homogeneous Models	27
2.2.1	Correlation Structures from Correlation Functions	27
2.2.2	Finitely Decomposable Correlation Functions	27
2.2.3	Ratio Correlation Structures and Functions	28
2.2.4	LMM with Piece-wise Constant Volatility Structure	30
2.2.5	Modified Hull-White Volatility Structure	32
2.2.6	Parametric Scalar Volatility Function	34
2.3	Parametrisation of Correlation Structures	34
2.3.1	A Disadvantage of Low Factor Models	34
2.3.2	Semi-parametric Framework for Libor Correlations	36
2.3.3	Representation Theorems	38
2.3.4	Generation of Low Parametric Structures	42
2.3.5	Parametric Low Rank Structures	45
2.4	Some Possible Applications of Parametric Structures	47
2.4.1	Smoothing Historical Libor Correlations (sketch)	47
2.4.2	Calibration to Caps and Swaptions by Using Historical Correlations (sketch)	48
3	Implied Calibration of a Libor Market Model to Caps and Swaptions	49
3.1	Orientation and General Aspects	49
3.2	Assessment of the Calibration Problem	51
3.2.1	Straightforward Least Squares, Stability Problems	52
3.2.2	Stability Problems in the Laboratory	53
3.3	LSq Calibration and Stability Issues in Practice	56
3.3.1	Transformation of Market Data and LSq Implementation	57
3.3.2	LSq Calibration Studies, Stability Problems in Practice	59
3.4	Regularisation via a Collateral Market Criterion	64
3.4.1	Market Swaption Formula (MSF)	64
3.4.2	Incorporation of the MSF in the Objective Function	67
3.4.3	MSF Calibration Tests, Regularisation of the Volatility Function	70
3.4.4	Calibration of Low Factor Models	74
3.4.5	Implied Calibration to Swaptions Only	74
3.5	Calibration of a Time-Shift Homogeneous LMM	76
3.5.1	Volatility Structure of Hull-White	76
3.5.2	Quasi Time-Shift Homogeneous Volatility Structure	81
4	Pricing of Exotic European Style Products	87
4.1	Exotic European Style Products	87
4.1.1	Libor Trigger Swap	87
4.1.2	Ratchet Cap	88
4.1.3	Sticky Cap	89
4.1.4	Auto-flex Cap	89
4.1.5	Callable Reverse Floater	90
4.2	Factor Dependence of Exotic Products	91
4.3	Case Studies	94
5	Pricing of Bermudan Style Libor Derivatives	103
5.1	Orientation	103
5.2	The Bermudan Pricing Problem	104
5.3	Backward Construction of the Exercise Boundary	106
5.4	Iterative Construction of the Optimal Stopping Time	109
5.4.1	A One Step Improvement upon a Given Family of Stopping Times	109
5.4.2	Iterating to the Optimal Stopping Time and the Snell Envelope	112
5.4.3	On the Implementation of the Iterative Procedure	116
5.5	Duality; From Tight Lower Bounds to Tight Upper Bounds	118
5.5.1	Dual Approach	118
5.5.2	Converging Upper Bounds from Lower Bounds	119
5.6	Monte Carlo Simulation of Upper Bounds	122
5.7	Numerical Evaluation of Bermudan Swaptions by Different Methods	124
5.8	Efficient Monte Carlo Construction of Upper Bounds	126
5.8.1	Alternative Estimators for the Target Upper Bound	126
5.8.2	Two Canonical Approximative Processes	130
5.8.3	Numerical Upper Bounds for Bermudan Swaptions	131
5.9	Multiple Callable Structures	138
5.9.1	The Multiple Stopping Problem	138
5.9.2	Iterative Algorithm for Multiple Bermudan Products	140
6	Pricing Long Dated Products via Libor Approximations	145
6.1	Introduction	145
6.2	Different Lognormal Approximations	146
6.2.1	More Lognormal Approximations	149
6.2.2	Simulation Analysis of Different Libor Approximations	152
6.3	Direct Simulation of Lognormal Approximations	154
6.3.1	Random Field Simulation of the (g)-approximation	154
6.3.2	Simulation of the (g1'), (g1), and (g2)-approximation	162
6.3.3	Cost Analysis of Euler SDE Simulation and Direct Simulation Methods	163
6.4	Efficiency Gain with Respect to SDE Simulation; an Optimal Simulation Program	168
6.4.1	Simulation Alternatives	168
6.4.2	An Optimal Simulation Program	171
6.5	Practical Simulation Examples	172
6.5.1	European Swaption	172
6.5.2	Trigger Swap	173
6.5.3	Callable Reverse Floater	174
6.6	Summarisation and Final Remarks	175
A	Appendix	177
A.1	Glossary of Stochastic Calculus	177
A.1.1	Stochastic Processes in Continuous Time	177
A.1.2	Martingales and Stopping Times	178
A.1.3	Quadratic Variation and the Ito Stochastic Integral	179
A.1.4	Regular Conditional Probability	183
A.2	Minimum Search Procedures	184
A.2.1	Halton Quasi-random Numbers	184
A.3	Additional Proofs	187
A.3.1	Covariance of Two Black-Scholes Models	187
A.3.2	Proof of Equality (5.26)	188
A.3.3	Proof of Proposition 5.2	189
A.3.4	Proof of Proposition 5.3	190
	References	193
	Index	199