The course provides students in Mathematical/Management Engineering with:
• The ability to leverage their quantitative and computational skills within the fields of financial markets and insurance, completing their statistical and probabilistic background with a working knowledge of optimization model building and solving 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. The course also includes a sizable section on optimization under uncertainty, whose scope goes well 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;
• convex, stochastic, and robust optimization .
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;
• ability to build robust strategies dealing with significant uncertainty in the data characterizing a decision problem.

The course is based on a 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.

Furthermore, we will use MATLAB in the course, so it is assumed that students have a working knowledge of this tool and suitable programming skills.

• Introduction to financial markets and related mathematical models. Financial assets (stock shares, bonds, forward/futures contracts, options) and the relevant risk factors; mathematical models for asset pricing, portfolio optimization, and risk measurement/management. [10 hours]
• Stochastic representation of uncertainty. Static and dynamic stochastic models: time series, binomial models; stochastic differential equations and stochastic calculus.. [10 hours]
• Models to price derivatives: no-arbitrage condition and risk-neutral measures in complete and incomplete markets. [15 hours]
• Optimization model building: linear, integer, stochastic and robust optimization models. Applications to portfolio optimization and hedging. Model building in AMPL and CVX. [15 hours]
• Optimization model solving. Basics of convex and non-convex optimization (linear and nonlinear programming; duality theory; integer programming; interior point methods); conic optimization (second-order conic programming; semidefinite programming; conic duality); stochastic programming with recourse and dynamic programming (decomposition methods; decision rules); robust optimization. [30 hours]

The course consists of lectures, integrated by the solution of sample exam problems. We will also analyze scripts and functions written in MATLAB/CVX and AMPL/CPLEX for the practical solution of the optimization models proposed. There is no group work or laboratory.

We will also use financial databases like Thomson-Reuters Eikon for Office.

There is no single course textbook, but course slides will be posted online.

The following ones are useful reference texts, and most of the course material is based on them:
• P. Brandimarte. Financial Markets: A Quantitative Introduction. Wiley, to be published.
• S. Boyd, L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
• G. Cornuejols, R. Tütüncü. Optimization Methods in Finance. Cambridge University Press, 2007.

The required background is covered, e.g., in:
• P. Brandimarte. Quantitative methods: An introduction for business management. Wiley, 2011.
• C.P. Simon, L.E. Blume. Mathematics for economists. W.W. Norton & Company, 1994.

The CVX software may be downloaded from http://cvxr.com/cvx/
The AMPL software will be distributed, and material can be found on http://ampl.com/

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.