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This is a masters-level overview of the mathematical concepts needed to fully grasp the art of derivatives pricing, and a must-have for anyone considering a career in quantitative finance in industry or academia. Starting from the foundations of probability, this textbook allows students with limited technical background to build a solid knowledge of the most important principles. It offers a unique compromise between intuition and mathematics, even when discussing abstract ideas such as change of measure. Mathematical concepts are introduced initially using toy examples, before moving on to examples of finance cases, both in discrete and continuous time. Throughout, numerical applications and simulations illuminate the analytical results. The end-of-chapter exercises test students' understanding, with solved exercises at the end of each part to aid self-study. Additional resources are available online, including slides, code and an interactive app.
List of contents
Foreword; General Introduction; Part I. Probability Theory: 1. Probability space; 2. Random variables and distributions; 3. Moments and measure changes; 4. Dealing with partial information; 5. Sampling and Monte Carlo simulation; 6. Solved exercises; Part II. Pricing by Risk-Neutral Expectation: 7. Stochastic process and related concepts; 8. The random walk; 9. Derivative pricing using CRR; 10. The Brownian motion; 11. Derivative pricing using GBM; 12. Solved exercises; Part III. Pricing by Dynamic Replication: 13. Stochastic integrals; 14. Stochastic differential equations; 15. Itô calculus; 16. The Black-Scholes-Merton equation; 17. Solved exercises; Part IV. Hedging and Beyond: 18. Replication and hedging; 19. Fundamental theorems of asset pricing; 20. Pricing via change of numéraire; 21. Beyond Black-Scholes-Merton; 22. Solved exercises; Part V. Appendices: Appendix A. Short-selling in a nutshell; Appendix B. Important functions of distributions; Appendix C. Covergence of random variables; Appendix D. Quadratic variation of smooth functions; Appendix E. Connections between CRR and GBM; Appendix F. Pricing Asian options via Monte Carlo; Appendix G. Itô vs Stratanovich integrals; Appendix H. Itô's lemma: sketch of proof; Appendix I. Acronyms; Bibliography; Index.
About the author
Frédéric D. Vrins has been a quantitative finance professor at the Louvain School of Management (UCLouvain) since 2014, where he coordinates the Financial Engineering track. Previously, he was Senior Quant in the trading room of a systemic bank. His research includes mathematical finance, credit risk and portfolio optimization.
