The course provides an introduction to linear (futures, forwards, swaps) and nonlinear (options) financial derivatives, written on a range of underlying factors like stock prices, indexes, interest rates, etc. 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.

Knowledge: • main asset classes traded on financial markets and their use in different strategies; • classes of players on financial markets and their role; • 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; • 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. o Probability: random variables, multivariate probability distributions, covariance and correlation, stochastic processes. o Statistics: parameter estimation, hypothesis testing, correlation analysis, and linear regression. 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, content is associated with chapter numbers of the course textbook: • The structure and purpose of derivative markets [Chaps 1, 2] 6 hours • Linear derivatives: Forward contracts, futures, vanilla swaps; pricing and use in hedging strategies [Chaps 3, 4, 5, 6, 7] 14 hours • Securitization and credit issues [Chaps 8, 9] 4 hours • Introduction to options: Basic properties, trading strategies and pricing by binomial lattices [Chaps 10, 11, 12, 13] 10 hours • Stochastic calculus: Wiener process; Markov processes and martingales; stochastic integrals and stochastic differential equations; Itô’s lemma; linking parabolic PDEs and diffusion processes by Feynman-Kaĉ stochastic representation theorem; change of measure and Girsanov theorem [Chaps 14, 28, additional slides] 12 hours • The Black-Scholes-Merton model: BSM pricing formula; extension to indexes and futures options; Greeks [Chaps 15, 17, 18, 19] 16 hours • Numerical methods: Finite differences, Monte Carlo simulation, ad hoc methods, MATLAB implementation [Chaps 21, 27] 9 hours • Interest rate derivatives [Chaps 29, 31, 32] 9 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 shall also consider a couple of HBS (Harvard Business School) business cases.

The course textbook is: • John C. Hull. Options, futures, and other derivatives (10th edition). Pearson, 2018. When the book needs to be integrated by additional material, slides will be provided. The book contains several problems, and a student solution manual is also available. We will also consider HBS cases like G. Chacko, E.P. Strick. Pine Street Capital. HBS Case 201071-PDF-ENG. The the full list will be provided as a coursepack link on Harvard web site.

Modalità di esame: Prova scritta su carta con videosorveglianza dei docenti; Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;

The exam consists of a test (90 minutes), including numerical problems, theoretical questions, simple proofs. 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. A series of 8 open-ended exercises and questions (3 points each) and one problem are proposed. The latter contributes 8/30 to the final grade. The problem and the questions 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, and statistics) 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.

Modalità di esame: Prova scritta (in aula); Prova scritta su carta con videosorveglianza dei docenti;

The exam consists of a written test (90 minutes), including numerical problems, theoretical questions, simple proofs. 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. 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, and statistics) 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.