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Interest Rate Models: Theory and Practice By: Damiano Brigo, Fabio Mercurio Normal price: 50 Our price: 45 + postage Bargain: save £5 ! Product code: 19984 ISBN: 3540221492 981 pages Format: Hb Published by: Springer Verlag, 2006, 2nd edition There are no reviews available for this book Description of Interest Rate Models: Theory and Practice The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. The fast-growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives - mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments. Interest Rate Models: Theory and Practice - Chapter headings Preface Abbreviations and Notation PART I: MODELS: THEORY AND IMPLEMENTATION 1. Definitions and notations 1.1 The bank account and the short rate 1.2 Zero coupon bonds and spot interest rates 1.3 Fundamental interest rate curves 1.4 Forward rates 1.5 Interest rate swaps and forward swap rates 1.6 Interest rate caps/floors and swaptions 2. No-arbitrage pricing and numeraire change 2.1 No-arbitrage in continuous time 2.2 The change-of-numeraire technique 2.3 A change-of-numeraire toolkit 2.4 The choice of a convenient numeraire 2.5 The forward measure 2.6 The fundamental pricing formulas 2.6.1 The pricing of caps and floors 2.7 Pricing claims with deferred payoffs 2.8 Pricing claims with multiple payoffs 2.9 Foreign markets and numeraire change 3. One-factor short-rate models 3.1 Introduction and guided tour 3.2 Classical time-homogeneous short-rate models 3.2.1 The Vasicek (1977) model 3.2.2 The Dothan (1978) model 3.2.3 The Cox, Ingersoll and Ross (1985) (CIR) model 3.2.4 Affine term-structure models 3.2.5 The Exponential-Vasicek (EV) model 3.3 The Hull-White extended Vasicek model 3.3.1 The short-rate dynamics 3.3.2 Bond and option pricing 3.3.3 The construction of a trinomial tree 3.4 Possible extensions of the CIR model 3.5 The Black-Karasinski (1991) model 3.5.1 The short-rate dynamics 3.5.2 The construction of a trinomial tree 3.6 Volatility structures in one-factor short rate models 3.7 Humped-volatility short-rate models 3.8A general deterministic-shift extension 3.8.1 The basic assumptions 3.8.2 Fitting the initial term structure of interest rates 3.8.3 Explicit formulas for European options 3.8.4 The Vasicek case 3.9 The CIR++ model 3.9.1 The construction of a trinomial tree 3.9.2 The positivity of rates and fitting quality 3.10 Deterministic-shift extension of lognormal models 3.11 Some further remarks on derivatives pricing 3.11.1 Pricing European options on a coupon bearing bond 3.11.2 The Monte Carlo simulation 3.11.3 Pricing early-exercise derivatives with a tree 3.11.4 A fundamental case of early exercise: Bermudan-style swaptions 3.12 Implied cap volatility curves 3.12.1 The Black and Karasinski (1991) model 3.12.2 The CIR++ model 3.12.3 The Extended Exponential Vasicek model 3.13 Implied swaption volatility surfaces 3.13.1 The Black and Karasinski (1991) model 3.13.2 The Extended Exponential Vasicek model 3.14 An example of calibration to real-market data 4. Two-factor short-rate models 4.1 Introduction and motivation 4.2 The two-additive-factor Gaussian model G2++ 4.2.1 The short rate dynamics 4.2.2 The pricing of a zero coupon bond 4.2.3 Volatility and correlation structures in two-factor models 4.2.4 The pricing of a European option on a zero coupon bond The pricing of caplets and floorlets The pricing of caps and floors The pricing of European swaptions 4.2.5 The analogy with the Hull-White (1994c) two-factor model 4.2.6 The construction of an approximating tree The binomial trees for x and y The approximating tree for r 4.3 The two-additive-factor extended CIR/LS model CIR2++ 4.3.1 The basic two-factor CIR2 model 4.3.2 Relationship with the Longstaff and Schwartz (1992b) model (LS) 4.3.3 Forward measure dynamics and option pricing for CIR2 4.3.4 The CIR2++ model and option pricing 5.The Heath-Jarrow-Morton (HJM) framework 5.1 The HJM forward-rate dynamics 5.2 Markovianity of the short-rate process 5.3 The Ritchken and Sankararamanian (1995) framework 5.4 The Mercurio and Moraleda (2000a) model 6.The Libor and swap market models (LFM and LSM) 6.1 Introduction 6.2 Market models: A guided tour 6.3 The lognormal forward Libor model (LFM) 6.3.1 Some Specifications of the Instantaneous Volatility of Forward Rates 6.3.2 Forward-Rate Dynamics under Different Numeraires 6.4 Calibration of the LFM to caps and floors prices 6.4.1 Piecewise-Constant Instantaneous-Volatility Structures 6.4.2 Parametric Volatility Structures 6.4.3 Cap Quotes in the Market 6.5 The term structure of volatility 6.5.1 Piecewise-Constant Instantaneous Volatility Structures 6.5.2 Parametric Volatility Structures 6.6 Instantaneous correlation and terminal correlation 6.7 Swaptions and the lognormal forward--swap model (LSM) 6.7.1 Cash-Settled Swaptions}{226} 6.8 Incompatibility between the LFM and LSM market models 6.9 The structure of instantaneous correlations 6.10 Montecarlo pricing of swaptions with the LFM model 6.11 Rank-1 analytical swaption prices 6.12 Rank-r analytical swaption prices 6.13 A simpler LFM formula for swaptions volatilities 6.14 A formula for terminal correlations of forward rates 6.15 Calibration to swaptions prices 6.16 Connecting caplets' and S x 1 swaptions' volatilities 6.17 Forward and spot rates over non-standard periods 6.17.1 Drift interpolation 6.17.2 The bridging technique 6.18 Including the caplets smiles in the LFM model 6.18.1 A mini-tour on the smile problem 6.18.2 Modelling the smile 6.18.3 The shifted-lognormal case 6.18.4 The Constant Elasticity of Variance (CEV) model 6.18.5 A mixture of lognormals model 6.18.6 Shifting the lognormal-mixture dynamics 7. Cases of calibration of the Libor market model 7.1 The inputs 7.2 Joint calibration with piecewise constant volatilities as in Table 5 7.2.1 Instantaneous correlations: Narrowing the angles 7.2.2 Instantaneous correlations: Fixing the angles to typical values 7.2.3 Instantaneous correlations: fixing the angles to atypical values 7.2.4 Instantaneous correlations: collapsing to one factor 7.3 Joint calibration with parameterized volatilities as in Formulation 7 7.3.1 Formulation 7: Narrowing the angles 7.3.2 Formulation 7: Calibrating only to swaptions volatilities 7.4 Exact Swaptions Calibration with Volatilities as in Table 1 7.4.1 Some numerical results 7.5 Conclusions: Where now? 8. Monte Carlo tests for LFM analytical approximations 8.1 The specification of rates 8.2 The ``testing plan" for volatilities 8.3 Test Results for volatilities: MC method vs analytical approximations 8.3.1 Case (1): Constant instantaneous volatilities 8.3.2 Case (2): Volatilities as functions of time-to maturity 8.3.3 Case (3): Humped and maturity-adjusted instantaneous volatilities depending only on time to maturity, typical rank-two correlations 8.4 The ``testing plan" for terminal correlations 8.5 Test Results for terminal correlations: MC vs analytical approximations 8.5.1 Case (i): Humped and maturity-adjusted instantaneous volatilities depending only on time to maturity, typical rank-two correlations 8.5.2 Case (ii): Constant instantaneous volatilities, typical rank-two correlations. 8.5.3 Case (iii): Humped and maturity-adjusted instantaneous volatilities depending only on time to maturity, some negative rank-two correlations. 8.5.4 Case (iv): Constant instantaneous volatilities, some negative rank-two correlations. 8.5.5 Case (v): Constant instantaneous volatilities, perfect correlations, upwardly shifted Phi's 8.6 Test results: Stylized conclusions 9. Other interest-rate models 9.1 Brennan and Schwartz's model 9.2 Balduzzi, Das, Foresi and Sundaram's model 9.3 Flesaker and Hughston's model 9.4 Roger's potential approach 9.5 Markov Functional models PART II: PRICING DERIVATIVES IN PRACTICE 10. Pricing derivatives on a single interest rate curve 10.1 In advance swaps 10.2 In advance caps 10.2.1 A First Analytical Formula (LFM) 10.2.2 A Second Analytical Formula (G2++) 10.3 Autocaps 10.4 Caps with deferred caplets 10.4.1 A First Analytical Formula (LFM) 10.4.2 A Second Analytical Formula (G2++) 10.5 Ratchets (one way floaters) 10.6 Constant maturity swaps (CMS) 10.6.1 CMS with the LFM model 10.6.2 CMS with the G2++ model 10.7 The convexity adjustment and applications to CMS 10.7.1 Natural and unnatural time lags 10.7.2 The convexity adjustment technique 10.7.3 Deducing a simple lognormal dynamics from the adjustment 10.7.4 Application to CMS 10.7.5 Forward Rate Resetting Unnaturally and Average-Rate Swaps 10.8 Captions and floortions 10.9 Eurodollar futures 10.9.1 The Shifted Two-Factor Vasicek G2++ Model 10.9.2 LFM market model 10.10 LFM pricing with ``in-between" spot rates 10.10.1 Accrual swaps 10.10.2 Trigger swaps 10.11 LFM pricing with early exercise and possible path dependence 10.12 LFM: Pricing Bermudan swaptions 10.12.1 Bermudan swaptions: Longstaff and Schwartz 10.12.2 Bermudan swaptions: Carr and Yang 10.12.3 Bermudan swaptions: Andersen 11. Pricing derivatives on two interest-rate curves 11.1 The attractive features of G2++ for multi-curve payoffs 11.1.1 The model 11.1.2 Interaction between models of the two curves "1" and "2" 11.1.3 The two-models dynamics under a unique convenient forward measure 11.2 Quanto Constant--maturity swaps 11.2.1 Quanto--CMS: The contract 11.2.2 Quanto--CMS: The G2++ model Quanto-CMS: Monte Carlo pricing and examples 11.2.3 Quanto--CMS: Quanto adjustment 11.3 Differential swaps 11.3.1 The contract 11.3.2 Differential swaps with the G2++ model 11.3.3 A market-like formula 11.4 Market formulas for basic quanto derivatives 11.4.1 The pricing of quanto caplets/floorlets 11.4.2 The pricing of quanto caps/floors 11.4.3 The pricing of differential swaps 11.4.4 The pricing of quanto swaptions The pricing of a spread option The pricing of a quanto swaption 12. Pricing equity derivatives under stochastic rates 12.1 The short rate and asset price dynamics 12.1.1 The dynamics under the forward measure 12.2 The pricing of a European option on the given asset 12.3 A more general model 12.3.1 The construction of an approximating tree for r 12.3.2 The approximating tree for S 12.3.3 The two-dimensional tree PART III: APPENDICES A. A Crash Introduction to Stochastic Differential Equations A.1 From Deterministic to Stochastic Differential Equations A.2 Ito's Formula A.3 Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes A.4 Examples A.5 Two important Theorems B. A useful calculation C. Approximating diffusions with trees D. Talking to the traders