The concepts and practice of mathematical finance /
M.S. Joshi.
- Second edition.
- xviii, 539 pages : illustrations ; 26 cm.
- Mathematics, finance and risk .
- Mathematics, finance, and risk. .
Includes bibliographical references (pages 526-532) and index.
Risk -- What is risk? -- Market efficiency -- The most important assets -- Risk diversification and hedging -- The use of options -- Classifying market participants -- Pricing methodologies and arbitrage -- Some possible methodologies -- Delta hedging -- What is arbitrage? -- The assumptions of mathematical finance -- An example of arbitrage-free pricing -- The time value of money -- Mathematically defining arbitrage -- Using arbitrage to bound option prices -- Conclusion -- Trees and option pricing -- A two-world universe -- A three-state model -- Multiple time steps -- Many time steps -- A normal model -- Putting interest rates in -- A log-normal model -- Consequences -- Summary -- Practicalities -- Introduction -- Trading volatility -- Smiles -- The Greeks -- Alternative models -- Transaction costs -- The Ito calculus -- Introduction -- Brownian motion -- Quadratic variation -- Stochastic processes -- Ito's lemma -- Applying Ito's lemma -- An informal derivation of the Black-Scholes equation -- Justifying the derivation -- Solving the Black-Scholes equation -- Dividend-paying assets -- Risk neutrality and martingale measures -- Plan -- Introduction -- The existence of risk-neutral measures -- The concept of information -- Discrete martingale pricing -- Continuous martingales and filtrations -- Identifying continuous martingales -- Continuous martingale pricing -- Equivalence to the PDE method -- Hedging -- Time-dependent parameters -- Completeness and uniqueness -- Changing numeraire -- Dividend-paying assets -- Working with the forward -- The practical pricing of a European option -- Introduction -- Analytic formulae -- Trees -- Numerical integration -- Monte Carlo -- PDE methods -- Replication -- Continuous barrier options -- Introduction -- The PDE pricing of continuous barrier options -- Expectation pricing of continuous barrier options -- The reflection principle -- Girsanov's theorem revisited -- Joint distribution -- Pricing continuous barriers by expectation -- American digital options -- Multi-look exotic options -- Introduction -- Risk-neutral pricing for path-dependent options -- Weak path dependence -- Path generation and dimensionality reduction -- Moment matching -- Trees, PDEs and Asian options -- Practical issues in pricing multi-look options -- Greeks of multi-look options -- Static replication -- Introduction -- Continuous barrier options -- Discrete barriers -- Path-dependent exotic options -- The up-and-in put with barrier at strike -- Put-call symmetry -- Conclusion and further reading -- Multiple sources of risk -- Introduction -- Higher-dimensional Brownian motions -- The higher-dimensional Ito calculus -- The higher-dimensional Girsanov theorem -- Practical pricing -- The Margrabe option -- Quanto options -- Higher-dimensional trees -- Options with early exercise features -- Introduction -- The tree approach -- The PDE approach to American options -- American options by replication -- American options by Monte Carlo -- Upper bounds by Monte Carlo -- Interest rate derivatives -- Introduction -- The simplest instruments -- Caplets and swaptions -- Curves and more curves -- The pricing of exotic interest rate derivatives -- Introduction -- Decomposing an instrument into forward rates -- Computing the drift of a forward rate -- The instantaneous volatility curves -- The instantaneous correlations between forward rates -- Doing the simulation -- Rapid pricing of swaptions in a BGM model -- Automatic calibration to co-terminal swaptions -- Lower bounds for Bermudan swaptions -- Upper bounds for Bermudan swaptions -- Factor reduction and Bermudan swaptions -- Interest-rate smiles -- Incomplete markets and jump-diffusion processes -- Introduction -- Modelling jumps with a tree -- Modelling jumps in a continuous framework -- Market incompleteness -- Super- and sub-replication -- Choosing the measure and hedging exotic options -- Matching the market -- Pricing exotic options using jump-diffusion models -- Does the model matter? -- Log-type models -- Stochastic volatility -- Introduction -- Risk-neutral pricing with stochastic-volatility models -- Monte Carlo and stochastic volatility -- Hedging issues -- PDE pricing and transform methods -- Stochastic volatility smiles -- Pricing exotic options -- Variance Gamma models -- The Variance Gamma process -- Pricing options with Variance Gamma models -- Pricing exotic options with Variance Gamma models -- Deriving the properties -- Smile dynamics and the pricing of exotic options -- Introduction -- Smile dynamics in the market -- Dynamics implied by models -- Matching the smile to the model -- Hedging -- Matching the model to the product -- Financial and mathematical jargon -- Computer projects -- Introduction -- Two important functions -- Project 1: Vanilla options in a Black-Scholes world -- Project 2: Vanilla Greeks -- Project 3: Hedging -- Project 4: Recombining trees -- Project 5: Exotic options by Monte Carlo -- Project 6: Using low-discrepancy numbers -- Project 7: Replication models for continuous barrier options -- Project 8: Multi-asset options -- Project 9: Simple interest-rate derivative pricing -- Project 10: LIBOR-in-arrears -- Project 11: BGM -- Project 12: Jump-diffusion models -- Project 13: Stochastic volatility -- Project 14: Variance Gamma -- Elements of probability theory -- Definitions -- Expectations and moments -- Joint density and distribution functions -- Covariances and correlations -- Order notation -- Big O -- Small o. 1. 1.1 1.2 1.3 1.4 1.5 1.6 2. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4. 4.1 4.2 4.3 4.4 4.5 4.6 5. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 6. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 7. 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8. 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 9. 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 10. 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11. 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 12. 12.1 12.2 12.3 12.4 12.5 12.6 13. 13.1 13.2 13.3 13.4 14. 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 15. 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 16. 16.1 16.2 16.3 16.4 16.5 16.6 16.7 17. 17.1 17.2 17.3 17.4 18. 18.1 18.2 18.3 18.4 18.5 18.6 Appendix A Appendix B B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 B.12 B.13 B.14 B.15 B.16 Appendix C C.1 C.2 C.3 C.4 Appendix D D.1 D.2
"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.