The concepts and practice of mathematical finance

THE CONCEPTS AND PRACTICE OF MATHEMATICAL FINANCE Second Edition

M. s. JOSHI University of Melbourne

CAMBRIDGE UNIVERSITY PRESS

Contents

Preface Acknowledgements 1 Risk 1.1 What is risk? 1.2 Market efficiency 1.3 The most important assets 1.4 Risk diversification and hedging 1.5 The use of options 1.6 Classifying market participants 1.7 Key points 1.8 Further reading 1.9 Exercises 2 Pricing methodologies and arbitrage , 2.1 Some possible methodologies 2.2 Delta hedging 2.3 What is arbitrage? 2.4 The assumptions of mathematical finance 2.5 An example of arbitrage-free pricing 2.6 The time value of money 2.7 Mathematically defining arbitrage 2.8 Using arbitrage to bound option prices 2.9 Conclusion 2.10 Key points 2.11 Further reading 2.12 Exercises 3 Trees and option pricing 3.1 A two-world universe 3.2 A three-state model vu

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3.3 Multiple time steps 3.4 Many time steps 3.5 A normal model 3.6 Putting interest rates in 3.7 A log-normal model 3.8 Consequences 3.9 Summary 3.10 Key points 3.11 Further reading 3.12 Exercises Practicalities 4.1 Introduction 4.2 Trading volatility 4.3 Smiles 4.4 The Greeks 4.5 Alternative models 4.6 Transaction costs 4.7 Key points 4.8 Further reading 4.9 Exercises The Ito calculus Introduction 5.1 5.2 Brownian motion 5.3 Quadratic variation 5.4 Stochastic processes 5.5 Ito's lemma 5.6 Applying Ito's lemma 5.7 An informal derivation of the Black-Scholes equation 5.8 Justifying the derivation 5.9 Solving the Black-Scholes equation 5.10 Dividend-paying assets 5.11 Key points 5.12 Further reading 5.13 Exercises Risk neutrality and martingale measures Plan 6.1 6.2 Introduction 6.3 The existence of risk-neutral measures 6.4 The concept of information 6.5 Discrete martineale oricine

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Contents

6.6 Continuous martingales and nitrations Identifying continuous martingales 6.7 Continuous martingale pricing 6.8 6.9 Equivalence to the PDE method 6.10 Hedging 6.11 Time-dependent parameters 6.12 Completeness and uniqueness 6.13 Changing numeraire 6.14 Dividend-paying assets 6.15 Working with the forward 6.16 Key points 6.17 Further reading 6.18 Exercises The practical pricing of a European option Introduction 7.1 Analytic formulae 7.2 7.3 Trees 7.4 Numerical integration Monte Carlo 7.5 PDE methods 7.6 7.7 Replication Key points 7.8 7.9 Further reading 7.10 Exercises Continuous barrier options Introduction 8.1 The PDE pricing of continuous barrier options 8.2 8.3 Expectation pricing of continuous barrier options The reflection principle 8.4 Girsanov's theorem revisited 8.5 8.6 Joint distribution Pricing continuous barriers by expectation 8.7 American digital options 8.8 Key points 8.9 8.10 Further reading 8.11 Exercises Multi-look exotic options Introduction 9.1 9.2 Risk-neutral pricing for path-dependent options 9.3 Weak path dependence

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Contents 9.4 Path generation and dimensionality reduction 9.5 Moment matching 9.6 Trees, PDEs and Asian options 9.7 Practical issues in pricing multi-look options 9.8 Greeks of multi-look options 9.9 Key points Further reading 9.10 9.11 Exercises 10 Static replication ireplication 10.1 Introduction Continuous barrier options i0.2 Discrete barriers 10.3 10.4 Path-dependent exotic options 10.5 The up-and-in put with barrier at strike 10.6 Put-call symmetry 10.7 Conclusion and further reading 10.8 Key points 10.9 Exercises 11 Multiple sources of risk risk 11.1 Introduction 11.2 Higher-dimensional Brownian motions 11.3 The higher-dimensional Ito calculus 11.4 The higher-dimensional Girsanov theorem 11.5 Practical pricing 11.6 The Margrabe option, 11.7 Quanto options 11.8 Higher-dimensional trees 11.9 Key points 11.10 Further reading 11.11 Exercises 12 Options with early exercise features 12.1 Introduction 12.2 The tree approach 12.3 The PDE approach to American options 12.4 American options by replication 12.5 American options by Monte Carlo 12.6 Upper bounds by Monte Carlo 12.7 Key points 12.8 Further reading 12.9 Exercises

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Contents

13 Interest rate derivatives 13.1 Introduction 13.2 The simplest instruments 13.3 Caplets and swaptions 13.4 Curves and more curves 13.5 Key points 13.6 Further reading 13.7 Exercises 14 The pricing of exotic interest rate derivatives 14.1 Introduction 14.2 Decomposing an instrument into forward rates 14.3 Computing the drift of a forward rate 14.4 The instantaneous volatility curves 14.5 The instantaneous correlations between forward rates 14.6 Doing the simulation 14.7 Rapid pricing of swaptions in a BGM model 14.8 Automatic calibration to co-terminal swaptions 14.9 Lower bounds for Bermudan swaptions 14.10 Upper bounds for Bermudan swaptions 14.11 Factor reduction and Bermudan swaptions 14.12 Interest-rate smiles 14.13 Key points 14.14 Further reading 14.15 Exercises 15 Incomplete markets and jump-diffusion processes 15.1 Introduction 15.2 Modelling jumps with a tree 15.3 Modelling jumps in a continuous framework 15.4 Market incompleteness 15.5 Super- and sub-replication 15.6 Choosing the measure and hedging exotic options 15.7 Matching the market 15.8 Pricing exotic options using jump-diffusion models 15.9 Does the model matter? 15.10 Log-type models 15.11 Key points 15.12 Further reading 15.13 Exercises

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16 Stochastic volatility 16.1 Introduction 16.2 Risk-neutral pricing with stochastic-volatility models 16.3 Monte Carlo and stochastic volatility 16.4 Hedging issues 16.5 PDE pricing and transform methods 16.6 Stochastic volatility smiles 16.7 Pricing exotic options 16.8 Key points 16.9 Further reading 16.10 Exercises 17 Variance Gamma models 17.1 The Variance Gamma process 17.2 Pricing options with Variance Gamma models 17.3 Pricing exotic options with Variance Gamma models 17.4 Deriving the properties 17.5 Key points 17.6 Further reading 17.7 Exercises 18 Smile dynamics and the pricing of exotic options 18.1 Introduction 18.2 Smile dynamics in the market 18.3 Dynamics implied by models 18.4 Matching the smile to the model 18.5 Hedging 18.6 Matching the modelto the product 18.7 Key points 18.8 Further reading Appendix A Financial and mathematical jargon Appendix B Computer projects B.I Introduction B.2 Two important functions B.3 Project 1: Vanilla options in a Black-Scholes world B.4 Project 2: Vanilla Greeks B.5 Project 3: Hedging B.6 Project 4: Recombining trees B.7 Project 5: Exotic options by Monte Carlo B.8 Project 6: Using low-discrepancy numbers Project 7: Replication models for continuous barrier options B.9 B.10 Proiect 8: Multi-asset options

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Contents B.I 1 Project 9: Simple interest-rate derivative pricing B.12 Project 10: LIBOR-in-arrears B.13 Project 11: BGM B.14 Project 12: Jump-diffusion models B.15 Project 13: Stochastic volatility B.16 Project 14: Variance Gamma Appendix C Elements of probability theory C.I Definitions C.2 Expectations and moments C.3 Joint density and distribution functions C.4 Covariances and correlations Appendix D Order notation D.I BigO D.2 Small o Appendix E Hints and answers to exercises References Index

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