Finance and insurance

The course Methods and Models for Finance aims to provide the mathematical tools needed to describe and analyse some of the most important continuous-time quantitative finance models used for pricing derivative securities in modern financial markets, including appropriate numerical illustrations. The course covers the following topics: Part 1 – Stochastic Calculus. Overview of measure theory: measurable spaces and measure spaces, outer measures and positive measures, measurable functions, Lp spaces, product measures and product spaces, Fubini’s and Tonelli’s theorems (without proof). The probability space as a particular example of a measure space; equivalent measures. Review of probability theory: definition and properties of conditional expectation with respect to σ-algebras, filtrations, and filtered probability spaces. Stochastic processes and their continuity and adaptiveness properties. Stochastic differential equations. The Wiener process and its quadratic variation; stochastic integration with respect to Brownian motion and Itô’s isometry; Itô processes and Itô’s Lemma. Martingales and the exponential martingale in financial applications. Girsanov’s Theorem and its financial implications. Part 2 – Arbitrage Theory. State variables, investment portfolios and self-financing trading strategies. Statement and proof of the First Fundamental Theorem of Asset Pricing and its financial implications. Statement and applications of the Second Fundamental Theorem of Asset Pricing. Change of numéraire and forward measure. Part 3 – Continuous-Time Market Models. The Black–Scholes–Merton model for equity markets: physical and risk-neutral dynamics, replicating portfolio, closed-form pricing of European options, and Greeks. Affine short-rate models and the fundamental equation for the term structure of interest rates. The Vasicek and CIR short-rate models: physical and risk-neutral dynamics, economic interpretation of parameters, conditional and unconditional distributions and moments, and closed-form zero-coupon bond pricing. The Hull–White model: condition ensuring deterministic long-term mean. HJM term-structure models: forward-rate dynamics, drift condition, and application to the Ho–Lee model. Derivatives on stochastic interest rates: fair valuation of IRS, Cap, and Floor. The Heston model for equity markets: physical and risk-neutral dynamics, replicating portfolio, and closed-form valuation of European options. Part 4 – Introduction to Credit Risk. Structural models: default probability in the Merton model and valuation of credit default swaps (CDS). Reduced-form (intensity-based) models: hazard rate, survival probability, and valuation of defaultable zero-coupon bonds both with and without recovery.

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.