Robust Libor Modelling and Pricing of Derivative Products (Chapman and Hall/CRC Financial Mathematics Series)

One of Riskbook.com's Best of 2005 - Top Ten Finance Books The Libor market model remains one of the most popular and advanced tools for modelling interest rates and interest rate derivatives, but finding a useful procedure for calibrating the model has been a perennial problem. Also the respective pricing of exotic derivative products such as Bermudan callable structures is considered highly non-trivial. In recent studies, author John Schoenmakers and his colleagues developed a fast and robust implied method for calibrating the Libor model and a new generic procedure for the pricing of callable derivative instruments in this model. Within a compact, self-contained review of the requisite mathematical theory on interest rate modelling, Robust Libor Modelling and Pricing of Derivative Products introduces the author's new approaches and their impact on Libor modelling and derivative pricing. Discussions include economically sensible parametrisations of the Libor market model, stability issues connected to direct least-squares calibration methods, European and Bermudan style exotics pricing, and lognormal approximations suitable for the Libor market model. A look at the available literature on Libor modelling shows that the issues surrounding instabilty of calibration and its consequences have not been well documented, and an effective general approach for treating Bermudan callable Libor products has been missing. This book fills these gaps and with clear illustrations, examples, and explanations, offers new methods that surmount some of the Libor model's thornier obstacles. Robust Libor Modelling and Pricing of Derivative Products 3 Preface 6 Contents 9 List of Tables 13 List of Figures 15 Chapter 1: Arbitrage-Free Modelling of Effective Interest Rates 17 1.1 Elements of Arbitrage Theory and Derivative Pricing 17 1.1.1 Arbitrage-free Systems, Self-Financing Trading Strategies, Complete Markets 17 1.1.2 Derivative Claim Pricing in Di.erent Measures 23 1.2 Modelling of E.ective Forward Rates 25 1.2.1 Libor Rate Processes and Measures 25 1.2.2 Swap Rate Processes and Measures 31 1.3 Pricing of Caps and Swaptions in Libor and Swap Market Models 34 1.3.1 Libor Caps and Caplets 34 1.3.2 Swaptions in a Swap Market Model 35 1.3.3 Approximating Swaptions in a Libor Market Model 36 1.3.4 Smile/Skew Extensions of the Libor Market Model 39 Chapter 2: Parametrisation of the Libor Market Model 41 2.1 General Volatility Structures 41 2.2 (Quasi) Time-Shift Homogeneous Models 43 2.2.1 Correlation Structures from Correlation Functions 43 2.2.2 Finitely Decomposable Correlation Functions 43 2.2.3 Ratio Correlation Structures and Functions 44 2.2.4 LMM with Piece-wise Constant Volatility Structure 46 2.2.5 Modi.ed Hull-White Volatility Structure 48 2.2.6 Parametric Scalar Volatility Function 50 2.3 Parametrisation of Correlation Structures 50 2.3.1 A Disadvantage of Low Factor Models 50 2.3.2 Semi-parametric Framework for Libor Correlations 52 2.3.3 Representation Theorems 54 2.3.4 Generation of Low Parametric Structures 58 2.3.5 Parametric Low Rank Structures 61 2.4 Some Possible Applications of Parametric Structures 63 2.4.1 Smoothing Historical Libor Correlations (sketch) 63 2.4.2 Calibration to Caps and Swaptions by Using Historical Correlations ( sketch) 64 Chapter 3: Implied Calibration of a Libor Market Model to Caps and Swaptions 65 3.1 Orientation and General Aspects 65 3.2 Assessment of the Calibration Problem 67 3.2.1 Straightforward Least Squares, Stability Problems 68 3.2.2 Stability Problems in the Laboratory 69 3.3 LSq Calibration and Stability Issues in Practice 72 3.3.1 Transformation of Market Data and LSq Implementation 73 3.3.2 LSq Calibration Studies, Stability Problems in Practice 75 3.4 Regularisation via a Collateral Market Criterion 80 3.4.1 Market Swaption Formula (MSF) 80 3.4.2 Incorporation of the MSF in the Objective Function 83 3.4.3 MSF Calibration Tests, Regularisation of the Volatility Function 86 3.4.4 Calibration of Low Factor Models 90 3.4.5 Implied Calibration to Swaptions Only 90 3.5 Calibration of a Time-Shift Homogeneous LMM 92 3.5.1 Volatility Structure of Hull-White 92 3.5.2 Quasi Time-Shift Homogeneous Volatility Structure 97 Chapter 4: Pricing of Exotic European Style Products 103 4.1 Exotic European Style Products 103 4.1.1 Libor Trigger Swap 103 4.1.2 Ratchet Cap 104 4.1.3 Sticky Cap 105 4.1.4 Auto-.ex Cap 105 4.1.5 Callable Reverse Floater 106 4.2 Factor Dependence of Exotic Products 107 4.3 Case Studies 110 Chapter 5: Pricing of Bermudan Style Libor Derivatives 119 5.1 Orientation 119 5.2 The Bermudan Pricing Problem 120 5.3 Backward Construction of the Exercise Boundary 122 5.4 Iterative Construction of the Optimal Stopping Time 125 5.4.1 A One Step Improvement upon a Given Family of Stopping Times 125 5.4.2 Iterating to the Optimal Stopping Time and the Snell Envelope 128 5.4.3 On the Implementation of the Iterative Procedure 132 5.5 Duality; From Tight Lower Bounds to Tight Upper Bounds 134 5.5.1 Dual Approach 134 5.5.2 Converging Upper Bounds from Lower Bounds 135 5.6 Monte Carlo Simulation of Upper Bounds 138 5.7 Numerical Evaluation of Bermudan Swaptions by Di . erent Methods 140 5.8 E.cient Monte Carlo Construction of Upper Bounds 142 5.8.1 Alternative Estimators for the Target Upper Bound 142 5.8.2 Two Canonical Approximative Processes 146 5.8.3 Numerical Upper Bounds for Bermudan Swaptions 147 5.9 Multiple Callable Structures 154 5.9.1 The Multiple Stopping Problem 154 5.9.2 Iterative Algorithm for Multiple Bermudan Products 156 Chapter 6: Pricing Long Dated Products via Libor Approximations 161 6.1 Introduction 161 6.2 Di.erent Lognormal Approximations 162 6.2.1 More Lognormal Approximations 165 6.2.2 Simulation Analysis of Di.erent Libor Approximations 168 6.3 Direct Simulation of Lognormal Approximations 170 6.3.1 Random Field Simulation of the (g)-approximation 170 6.3.2 Simulation of the (g1 ), (g1), and (g2)-approximation 178 6.3.3 Cost Analysis of Euler SDE Simulation and Direct Simulation Methods 179 6.4 E.ciency Gain with Respect to SDE Simulation; an Optimal Simulation Program 184 6.4.1 Simulation Alternatives 184 6.4.2 An Optimal Simulation Program 187 6.5 Practical Simulation Examples 188 6.5.1 European Swaption 188 6.5.2 Trigger Swap 189 6.5.3 Callable Reverse Floater 190 6.6 Summarisation and Final Remarks 191 Appendix 193 A.1 Glossary of Stochastic Calculus 193 A.1.1 Stochastic Processes in Continuous Time 193 A.1.2 Martingales and Stopping Times 194 A.1.3 Quadratic Variation and the It o Stochastic Integral 195 A.1.4 Regular Conditional Probability 199 A.2 Minimum Search Procedures 200 A.2.1 Halton Quasi-random Numbers 200 A.3 Additional Proofs 203 A.3.1 Covariance of Two Black-Scholes Models 203 A.3.2 Proof of Equality (5.26) 204 A.3.3 Proof of Proposition 5.2 205 A.3.4 Proof of Proposition 5.3 206 References 209 Blank Page 40 1-58488-441-X

One of Riskbook.com's Best of 2005 - Top Ten Finance Books

The Libor market model remains one of the most popular and advanced tools for modelling interest rates and interest rate derivatives, but finding a useful procedure for calibrating the model has been a perennial problem. Also the respective pricing of exotic derivative products such as Bermudan callable structures is considered highly non-trivial. In recent studies, author John Schoenmakers and his colleagues developed a fast and robust implied method for calibrating the Libor model and a new generic procedure for the pricing of callable derivative instruments in this model.

Within a compact, self-contained review of the requisite mathematical theory on interest rate modelling, Robust Libor Modelling and Pricing of Derivative Products introduces the author's new approaches and their impact on Libor modelling and derivative pricing. Discussions include economically sensible parametrisations of the Libor market model, stability issues connected to direct least-squares calibration methods, European and Bermudan style exotics pricing, and lognormal approximations suitable for the Libor market model.

A look at the available literature on Libor modelling shows that the issues surrounding instabilty of calibration and its consequences have not been well documented, and an effective general approach for treating Bermudan callable Libor products has been missing. This book fills these gaps and with clear illustrations, examples, and explanations, offers new methods that surmount some of the Libor model's thornier obstacles.

ARBITRAGE-FREE MODELLING OF EFFECTIVE INTEREST RATESElements of Arbitrage Theory and Derivative Pricing Modelling of Effective Forward RatesPricing of Caps and Swaptions in Libor and Swap Market ModelsPARAMETRISATION OF THE LIBOR MARKET MODELGeneral Volatility Structures(Quasi) Time-Shift Homogeneous ModelsParametrisation of Correlation StructuresSome Possible Applications of Parametric StructuresIMPLIED CALIBRATION OF A LIBOR MARKET MODEL TO CAPS AND SWAPTIONSOrientation and General AspectsAssessment of the Calibration Problem LSq Calibration and Stability Issues in PracticeRegularisation via