Financial Derivatives in Theory and Practice

Preface to revised edition.Preface. Acknowledgements.Part I: Theory.1 Single-Period Option Pricing.1.1 Option pricing in a nutshell.1. 2 The simplest setting.1. 3 General one-period economy.1. 4 A two-period example.2 Brownian Motion.2. 1 Introduction.2. 2 Definition and existence.2. 3 Basic properties of Brownian motion.2. 4 Strong Markov property.3 Martingales.3.1 Definition and basic properties.3. 2 Classes of martingales.3. 3 Stopping times and the optional sampling theorem.3. 4 Variation, quadratic variation and integration.3. 5 Local martingales and semimartingales.3. 6 Supermartingales and the Doob-Meyer decomposition.4 Stochastic Integration.4. 1 Outline.4. 2 Predictable processes.4. 3 Stochastic integrals: the L2 theory.4. 4 Properties of the stochastic integral.4. 5 Extensions via localization.4. 6 Stochastic calculus: Ito's formula.5 Girsanov and Martingale Representation.5.1 Equivalent probability measures and the Radon-Nikodym derivative.5.1.1 Basic results and properties.5. 2 Girsanov's theorem.5. 3 Martingale representation theorem.6 Stochastic Differential Equations.6. 1 Introduction.6. 2 Formal definition of an SDE.6. 3 An aside on the canonical set-up.6. 4 Weak and strong solutions.6. 5 Establishing existence and uniqueness: Ito theory.6. 6 Strong Markov property.6. 7 Martingale representation revisited.7 Option Pricing in Continuous Time.7.1 Asset price processes and trading strategies.7.2 Pricing European options.7.3 Continuous time theory.7. 4 Extensions.8 Dynamic Term Structure Models.8. 1 Introduction.8. 2 An economy of pure discount bonds.8. 3 Modelling the term structure.Part II: Practice.9 Modelling in Practice.9. 1 Introduction.9. 2 The real world is not a martingale measure.9. 3 Product-based modelling.9. 4 Local versus global calibration.10 Basic Instruments and Terminology.10. 1 Introduction.10. 2 Deposits.10. 3 Forward rate agreements.10. 4 Interest rate swaps.10. 5 Zero coupon bonds.10. 6 Discount factors and valuation.11 Pricing Standard Market Derivatives.11.1 Introduction.11. 2 Forward rate agreements and swaps.11. 3 Caps and floors.11. 4 Vanilla swaptions.11. 5 Digital options.12 Futures Contracts.12. 1 Introduction.12. 2 Futures contract definition.12. 3 Characterizing the futures price process.12. 4 Recovering the futures price process.12. 5 Relationship between forwards and futures.Orientation: Pricing Exotic European Derivatives.13 Terminal Swap-Rate Models.13.1 Introduction.13. 2 Terminal time modelling.13. 3 Example terminal swap-rate models.13. 4 Arbitrage-free property of terminal swap-rate models.13. 5 Zero coupon swaptions.14 Convexity Corrections.14. 1 Introduction.14. 2 Valuation of 'convexity-related' products.14. 3 Examples and extensions.15 Implied Interest Rate Pricing Models.15.1 Introduction.15. 2 Implying the functional form DTS.15. 3 Numerical implementation.15. 4 Irregular swaptions.15. 5 Numerical comparison of exponential and implied swap-rate models.16 Multi-Currency Terminal Swap-Rate Models.16. 1 Introduction.16. 2 Model construction.16. 3 Examples.16. 3.1 Spread options.Orientation: Pricing Exotic American and Path-Dependent Derivatives.17 Short-Rate Models.17.1 Introduction.17.2 Well-known short-rate models.17.3 Parameter fitting within the Vasicek-Hull-White model.17. 4 Bermudan swaptions via Vasicek-Hull-White.18 Market Models.18. 1 Introduction.18. 2 LIBOR market models.18. 3 Regular swap-market models.18. 4 Reverse swap-market models.19 Markov-Functional Modelling.19. 1 Introduction.19. 2 Markov-functional models.19. 3 Fitting a one-dimensional Markov-functional model to swaption prices.19. 4 Example models.19. 5 Multidimensional Markov-functional models.19. 5.1 Log-normally driven Markov-functional models.19. 6 Relationship to market models.19. 7 Mean reversion, forward volatilities and correlation.19. 7.1 Mean reversion and correlation.19. 7.2 Mean reversion and forward volatilities.19. 7.3 Mean reversion within the Markov-functional LIBOR model.19. 8 Some numerical results.20 Exercises and Solutions.Appendix 1: The Usual Conditions.Appendix 2: L2 Spaces.Appendix 3: Gaussian Calculations.References. Index.