Introduction

• The course builds on a foundational model with a single risk-free bond and a risky asset. It aims to introduce several theoretical concepts, such as no-arbitrage pricing, risk-neutral valuation, and quantitative finance (Q-finance). Most of the course is conducted in a discrete-time framework, establishing the connection between portfolio replication and risk-neutral valuation. The course concludes by extending the discrete-time paradigm into continuous time, culminating in the derivation of the Black-Scholes pricing equation.
• Additionally, the course explores the mean-variance paradigm, delving into essential concepts such as the risk-reward trade-off, diversification, and optimization. In terms of derivatives, the course covers the fundamentals of forwards and futures before focusing on plain vanilla options, such as European options, while distinguishing between European and American options
• Practicality is emphasized throughout the course by leveraging open-source R code for numerical examples, making abstract concepts more applied. No prior coding experience is expected, as assignments can also be completed using Excel. The mathematical rigor is maintained at an introductory level, tailored for senior undergraduate students.

Pre-requisites
Prerequisites include elementary calculus, probability, and some linear algebra. For calculus, students are expected to have experience with derivatives and partial derivatives, finding maxima or minima of differentiable functions of one or more variables, the Taylor formula, and integrals. Topics in probability include random variables and probability distributions, in particular, the binomial and normal distributions, expectation, variance and covariance, conditional probability, and independence. Familiarity with the Central Limit Theorem would be a bonus. In linear algebra, the student should be able to solve (numerically at least) systems of linear equations, add, multiply, transpose, and invert matrices, and compute determinants. For a reference in probability theory, see Introduction to Probability by Charles M. Grinstead and Laurie Snell (publicly available)

Course Outcomes

After successful completion of this course, students will be able to

• Understand basic financial concepts in FE, e.g., time value of money and no-arb pricing
• Build discrete-time models, e.g., binomial trees
• Develop continuous-time models, e.g., Brownian motion
• Value different asset classes and derivatives
• Perform statistical and numerical analysis

Textbook

Mathematics for Finance: An Introduction to Financial Engineering 2nd ed. 2011 Edition by Marek Capiński and Tomasz Zastawniak (CZ)

Additional References

1. . Practical Methods of Financial Engineering and Risk Management: Tools for Modern Financial Professionals by Rupak Chatterjee
2. Options, Futures, and Other Derivatives (Global Edition) 11th Edition by John Hull
3. Introduction to Probability by Charles M. Grinstead and Laurie Snell (publicly available)
4. Mathematics and Statistics for Financial Risk Management by Michael B. Miller

Grading Policies