The course provides students in Mathematical/Management Engineering with:
• The ability to leverage their quantitative and computational skills within the fields of financial markets, completing their statistical and probabilistic background with a working knowledge of financial model building under uncertainty.
• The possibility of a career in the following industries:
o banking, insurance, mutual/hedge/pension funds;
o high-profile consulting firms;
o software industry for financial and insurance applications;
o risk management offices within large non-financial corporations (e.g., handling foreign exchange and interest rate risk).
Despite a non-negligible contraction, this sector of the job market remains one of the most rewarding and best remunerated ones, and it paves the way for careers abroad. Emphasis is on financial derivatives, but the course also includes a sizable section on stochastic modeling, whose scope goes beyond the specific application field.

Knowledge:
• risk measurement and management (interest rate risk, market risk, exchange rate risk, model risk);
• derivative pricing and use in hedging strategies;
• model building in quantitative finance.
Skills:
• ability to understand the structure and dynamics of financial markets;
• use of derivative assets (futures/forward, swaps, options) for hedging and risk management;
• ability to use advanced mathematical models to represent reality, taking their limitations into proper account.

The course is based on the application of quantitative modeling and computational mathematics. Hence, it is absolutely necessary to have a deep and solid quantitative background including:
o Calculus and linear algebra: Taylor expansion for multivariable functions, differential equations, convex functions, matrix algebra (including eigenanalysis and quadratic forms), and linear spaces (including inner product spaces).
o Numerical analysis: conditioning of a problem and stability of an algorithm; solving systems of linear and nonlinear equations.; numerical integration
o Probability: random variables, multivariate probability distributions, covariance and correlation, stochastic processes.
o Statistics: parameter estimation, hypothesis testing, correlation analysis, and linear regression.
o Operations research: LP model building, elements of nonlinear programming (constrained optimization and Lagrange multipliers).
More generally, a high-level of mathematical maturity is an essential prerequisite, which is not only related with mathematical dexterity, but also with the practical ability of building and solving mathematical models autonomously.

We shall be (partially) covering chapters 1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15 of the course textbook, plus some hints about other parts.
• Introduction to financial markets and related mathematical models. Financial assets and the relevant risk factors.
• The mathematics of interest rates and security pricing. Arbitrage theory.
• Financial derivatives (forward/futures, options).
• Discrete time stochastic models and pricing by binomial lattices. The link between pricing, hedging, and replication.
• Continuous-time stochastic processes, stochastic calculus (stochastic integrals and differential equations).
• Risk-neutral pricing of European-style derivatives. American-style options.
• Interest rate derivatives. Interest rate modeling and risk management.

The course consists of lectures, integrated by the solution of sample exam problems.
We will also learn about financial databases like Thomson-Reuters Eikon for Office.

The course textbook is: G. Campolieti, R.N. Makarov. Financial Mathematics: A Comprehensive Treatment. CRC Press, 2014.

The required background is covered, e.g., in:
• P. Brandimarte. Quantitative methods: An introduction for business management. Wiley, 2011.

Written exam (90 minutes), including numerical problems, theoretical questions, as well as simple proofs and the construction of optimization models. The exam is closed book. Three problems are proposed, and each one contributes 10/30 to the final grade. The problems are not the simple repetition of what is shown in class: The passive understanding of the theory is not sufficient to pass the exam, as a deep understanding is required, of both the mathematics involved and the financial problems we tackle, as well as the ability to apply known concepts to new problems. The grading is based on concrete problem solving ability.