The concepts and practice of mathematical finance / M.S. Joshi.

Includes bibliographical references (pages 526-532) and index.

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 -- 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 -- 3. Trees and option pricing -- 3.1 A two-world universe -- 3.2 A three-state model -- 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 -- 4. Practicalities -- 4.1 Introduction -- 4.2 Trading volatility -- 4.3 Smiles -- 4.4 The Greeks -- 4.5 Alternative models -- 4.6 Transaction costs -- 5. The Ito calculus -- 5.1 Introduction -- 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 -- 6. Risk neutrality and martingale measures -- 6.1 Plan -- 6.2 Introduction -- 6.3 The existence of risk-neutral measures -- 6.4 The concept of information -- 6.5 Discrete martingale pricing -- 6.6 Continuous martingales and filtrations -- 6.7 Identifying continuous martingales -- 6.8 Continuous martingale pricing -- 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 -- 7. The practical pricing of a European option -- 7.1 Introduction -- 7.2 Analytic formulae -- 7.3 Trees -- 7.4 Numerical integration -- 7.5 Monte Carlo -- 7.6 PDE methods -- 7.7 Replication -- 8. Continuous barrier options -- 8.1 Introduction -- 8.2 The PDE pricing of continuous barrier options -- 8.3 Expectation pricing of continuous barrier options -- 8.4 The reflection principle -- 8.5 Girsanov's theorem revisited -- 8.6 Joint distribution -- 8.7 Pricing continuous barriers by expectation -- 8.8 American digital options -- 9. Multi-look exotic options -- 9.1 Introduction -- 9.2 Risk-neutral pricing for path-dependent options -- 9.3 Weak path dependence -- 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 -- 10. Static replication -- 10.1 Introduction -- 10.2 Continuous barrier options -- 10.3 Discrete barriers -- 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 -- 11. Multiple sources of 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 -- 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 -- 13. Interest rate derivatives -- 13.1 Introduction -- 13.2 The simplest instruments -- 13.3 Caplets and swaptions -- 13.4 Curves and more curves -- 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 -- 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 -- 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 -- 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 -- 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 model to the product -- Appendix A Financial and mathematical jargon -- Appendix B Computer projects -- B.1 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 -- B.9 Project 7: Replication models for continuous barrier options -- B.10 Project 8: Multi-asset options -- B.11 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.1 Definitions -- C.2 Expectations and moments -- C.3 Joint density and distribution functions -- C.4 Covariances and correlations -- Appendix D Order notation -- D.1 Big O -- D.2 Small o.

"An ideal introduction for those starting out as practitioners of mathematical finance, this book provides a clear understanding of the intuition behind derivatives pricing, how models are implemented, and how they are used and adapted in practice. Strengths and weaknesses of different models, e.g. Black-Scholes, stochastic volatility, jump-diffusion and variance gamma, are examined. Both the theory and the implementation of the industry-standard LIBOR market model are considered in detail. Each pricing problem is approached using multiple techniques including the well-known PDE and martingale approaches. This second edition contains many more worked examples and over 200 exercises with detailed solutions. Extensive appendices provide a guide to jargon, a recap of the elements of probability theory, and a collection of computer projects. The author brings to this book a blend of practical experience and rigorous mathematical background and supplies here the working knowledge needed to become a good quantitative analyst."--Publisher.

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