Finance is one of the fastest growing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for novel and relevant 'real-world' mathematics. In this book the authors describe the modelling of financial derivative products from an applied mathematician's viewpoint, from modelling through analysis to elementary computation. A unified approach to modelling derivative products as partial differential equations is presented, using numerical solutions where appropriate. Some mathematics is assumed, but clear explanations are provided for material beyond elementary calculus, probability, and algebra. Over 140 exercises are included. This volume will become the standard introduction to this exciting new field for advanced undergraduate students.
Part I. Basic Option Theory: 1. An introduction to options and markets; 2. Asset price random walks; 3. The Black-Scholes model; 4. Partial differential equations; 5. The Black-Scholes formulae; 6. Variations on the Black-Scholes model; 7. American options; Part II. Numerical Methods: 8. Finite-difference methods; 9. Methods for American options; 10. Binomial methods; Part III. Further Option Theory: 11. Exotic and path-dependent options; 12. Barrier options; 13. A unifying framework for path-dependent options; 14. Asian options; 15. Lookback options; 16. Options with transaction costs; Part IV. Interest Rate Derivative Products: 17. Interest rate derivatives; 18. Convertible bonds; Hints to selected exercises; Bibliography; Index.