Finance and insurance

The "Methods and Models for Finance" course aims to provide the mathematical tools to describe and analyze some of the most important mathematical models of continuous-time finance used for the valuation of derivative securities in major financial markets, while also providing appropriate numerical applications. Upon completion of the course, students will have the skills to move from theory to the implementation of continuous-time financial models to obtain the fair value of a large number of derivative securities. After passing the examination, students will be able to recognize the most suitable model to describe a given financial context and determine the most efficient methodologies for dealing with problems in that context. After analyzing the main methods and models, students will be able to independently analyze the financial context, evaluate possible resolution methodologies, and interpret the results obtained. Furthermore, students will be able to adequately describe the topics learned during the course, both orally and in written documents. Both lectures and individual activities ensure that students develop a method for autonomously acquiring new knowledge and skills in economics and finance, both theoretical and practical. PART 1: Stochastic calculus Continous-time martingales Girsanov theorem PART 2: Investment strategies and self-financing portfolios First and second fundamental theorems of the Asset Pricing Theory and non-arbitrage principle The Black-Scholes-Merton model Stochastic volatility models PART 3: Term structure of interest rates Interest rates derivatives Heath-Jarrow-Morton theorem Exponential affine models PART 4: Credit risk Credit derivatives Risky bonds evaluation

The course has no formal prerequisites. However, to successfully attend the lectures and pass the exam, students are expected to have a basic knowledge of probability theory (random variables, probability distributions, moments, moment-generating functions, and characteristic functions), real analysis in several variables (Taylor expansion and solution of ordinary differential equations), linear algebra, as well as the fundamentals of financial mathematics and quantitative finance (force of interest, introductory notions on derivative instruments). These prerequisites are normally acquired through the first-year courses, such as Advanced Mathematics, Probability and Stochastic Processes for Insurance and Finance, and Quantitative Finance.

I. Oliva e R. Renò (2021) Principi di Finanza Quantitativa, Maggioli Editore F.D. Rouah (2013) The Heston model and its extensions in Matlab and C#, Wiley. A.J. McNeil, R. Frey e P. Embrechts (2005) Quantitative Risk Management: Concepts, Techniques, Tools, Princeton University Press. The supplementary teaching material (lecture slides, any recordings, recommended readings) is available at the following page: https://classroom.google.com/c/ODA2MzYxMTIzOTcy

The course is based on standard classroom lectures

The course is delivered in person: the instructor conducts lectures in the Faculty classrooms, making use of the available IT equipment. Attendance is not mandatory but is strongly recommended.

The examination consists of a written test lasting 90 minutes, structured into three questions. Each question may in turn include several sub-questions. The questions may involve exercises or open-ended questions. In the case of open-ended questions, students are expected to provide comprehensive answers, contextualizing the techniques used and the applied models. For exercises, students are required to show all mathematical steps leading to the final result, accompanied by appropriate comments. The grading scale ranges from 0 to 30 with honors (“lode”). A score of 18 or higher is considered a passing grade. The examination procedures are the same for both attending and non-attending students.

A. Pascucci (2011) PDE and martingale methods in option pricing, Springer. M. Avellaneda e P. Laurence (2000) Quantitative modeling of derivative securities, Chapman&Hall/CRC. J. Hull (2000) Opzioni, futures e altri derivati, Pearson Ed. S. J. Shreeve (2000) Stochastic calculus for finance II -- continuous-time models, Springer.

The course is delivered through in-person lectures, complemented by guided discussions, practical exercises, and occasional seminars led by industry experts, with the aim of deepening specialized knowledge and fostering engagement with real-world applications. Lectures may be recorded and made available to students.