The course is strongly quantitative and merges tools from Applied Mathematics with problems in Financial Economics, in order to provide students 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 models, but the course also includes a sizable section on stochastic and optimization modeling, as well as numerical methods, whose scope goes beyond this specific application field.

Knowledge: • main asset classes traded on financial markets and their use in different strategies; • classes of players on financial markets and their role; • risk measurement and management (interest rate risk, market risk, exchange rate risk, model risk); • fixed-income and equity portfolio management; • derivative pricing and use in hedging strategies; • model building in quantitative finance; • computational finance. Skills: • ability to understand the structure and dynamics of financial markets and interpreting real-world cases; • decision-making under uncertainty, trading off risk and reward for both equity and fixed-income markets, and objective assessment of the implied risk; • use of derivative assets (futures/forward, swaps, options) for hedging and risk management; • devising and implementing numerical methods for financial engineering applications; • ability to use advanced mathematical models to represent reality, taking their limitations into proper account; • autonomous problem solving ability for hard quantitative problems involving multiple risk factors.

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 (including the classical heat PDE), 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. It is important to realize that, in the past, the lack of a strong quantitative background, at the level of any high-profile engineering school, has been the single most important reason for failure at exams, together with the inability to understand the practical side of financial markets and to correctly interpret the exam sheet problem statements. The ability to promptly link practical problems and formal models is assumed, as it is a common engineering background.

In the following, [PB] refers to the first textbook (P. Brandimarte, Financial markets) and [CM] to the second one (Campolieti, Makarov, Financial mathematics): • Introduction to financial markets and related mathematical models. Financial assets and the relevant risk factors. The fundamental problems: Portfolio choice, risk management, asset pricing. [PB, Chapters 1, 2]. 8 hours. • The mathematics of arbitrage: Complete and incomplete markets. Equivalent martingale measures. Fundamental theorems of asset pricing. [PB, Chapter 2], [CM, Chapter 5]. 10 hours. • Fixed income securities and markets: Theory of interest rates. Spot and forward rates. Term structures. Bond pricing. Elementary fixed-income derivatives (FRAs and Swaps). Interest rate risk management. [PB, Chapters 3, 4, 5, 6]. 8 hours. • Equity portfolio choice and decision making under uncertainty: Utility theory. Coherent risk measures and efficiency in mean-risk models. Classical mean-variance portfolio optimization. Advanced portfolio optimization models. [PB, Chapters 7, 8, 15]. 10 hours. • Factor models and equilibrium in equity markets: Capital Asset Pricing Model and Arbitrage Pricing Theory. Behavioral finance. [PB, Chapters 9, 10]. 6 hours. • Dynamic models of uncertainty: Discrete-time and continuous-time stochastic processes. Markov processes and martingales. The role of information and filtrations. Stochastic calculus, stochastic integrals and stochastic differential equations, Itô’s lemma. Linking stochastic diffusion processes and parabolic partial differential equations: Feynman-Kaĉ stochastic representation theorem. [PB Chapter 11], [CM Chapters 6, 10, 11]. 14 hours. • Pricing linear derivatives: Forward and futures contracts. Minimum variance hedging [PB, Chapter 12]. 4 hours. • Option pricing: Trading strategies with options. Risk-neutral pricing. Martingale measures and Girsanov theorem. Optimal stopping problems and American-style options. Sensitivity measures and risk management. Complete vs. incomplete markets. Model calibration. [PB, Chapters 13, 14]. 14 hours. • Numerical methods for option pricing: Monte Carlo and finite difference methods. [CM, Chapters 17, 18]. 6 hours.

The course consists of lectures, integrated by the solution of sample exam problems. To provide students with practical computational skills, MATLAB code will be discussed in class (but MATLAB programming is not required for the exam). We will also consider real-life cases (e.g., Orange County, Lehman Brothers, MetallGesellschaft) in order to illustrate the economic impact of financial mismanagement. We will also learn about financial databases like Thomson-Reuters Eikon.

The course textbooks are: • P. Brandimarte. An introduction to financial markets: A quantitative approach. Wiley, 2018. • G. Campolieti, R.N. Makarov. Financial mathematics: A comprehensive treatment. CRC Press, 2014. Slides will be provided for most topics, but they must be integrated by reading the assigned sections on the textbooks. The required background is covered, e.g., in: • P. Brandimarte. Quantitative methods: An introduction for business management. Wiley, 2011.

Modalità di esame: Prova scritta (in aula);

Exam: Written test;

... The exam consists of a written test (90 minutes), including numerical problems, theoretical questions, simple proofs, as well as the construction of optimization models and the outline of numerical solution algorithms (e.g., high-level pseudocode). The exam is strictly closed book and you are only allowed to use a non-programmable calculator. When necessary, statistical tables (probabilities of standard normal distribution) will be provided. You are only allowed to ask questions about the problem statements, not the solution itself, and the number of questions you ask will be recorded. 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 mainly based on concrete problem solving, as well as the ability to generalize the acquired skills to new (but related) problems. Knowledge of background material (e.g., calculus, linear algebra, probability, statistics, optimization theory, numerical analysis) is also investigated, in a problem-oriented way. No oral exam or additional test will be carried out, so the exam is passed when the result of the written test is at least 18/30. About one week after the exam, online solutions will be provided and you will be able to view your exam sheet. The resulting grade must be recorded immediately: it is not allowed to freeze grades and wait for later exam sessions.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.