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; 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; • 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: • Calculus and linear algebra: Taylor expansion for multivariable functions, differential equations (including the classical heat PDE), convex functions. • Probability: random variables, multivariate probability distributions, covariance and correlation, stochastic processes. • 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 main course textbook: • The structure and purpose of derivative markets [Chaps 1, 2] • Linear derivatives: Forward contracts, futures, vanilla swaps; pricing and use in hedging strategies [Chaps 3, 4, 5, 6, 7] • Introduction to options: Basic properties, trading strategies and pricing by binomial lattices [Chaps 10, 11, 12, 13 and Björk, T, 2009] • 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, and Björk, T, 2009] • The Black-Scholes-Merton model: BSM pricing formula; extension to indexes and futures options; Greeks [Chaps 15, 17, 18, 19 and Björk, T, 2009] •Completeness of the BS model [Björk, T, 2009] •Incomplete markets [Björk, T, 2009] • Loss and profit of a financial portfolio and risk measures: Value at risk and Expected shortfall [Chap 22] • Credit risk [Chap 24]

The course consists of lectures and exercises integrated by the solution of sample exam problems.

Main references: - John C. Hull. Options, futures, and other derivatives (10th edition). Pearson, 2018. The book contains several problems, and a student solution manual is also available. Additional text: -Björk, Tomas. Arbitrage theory in continuous time. Oxford university press, 2009. - Marek Capinski, and Tomasz Zastawniak. "Mathematics for finance." An Introduction (2003): 118-124. When the book needs to be integrated by additional material, slides will be provided.

Exam: Written test;

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. 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, 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. If the final points are 30, the final mark is 30 or 30 with honors (30L) if all the answers are clearly and completely motivated. 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.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.