Mathematical Engineering

The use of quantitative tools in the financial sector has become increasingly significant in recent years. Mathematical and statistical models, together with computational methods, are nowadays of fundamental importance and widely used in finance in a vast array of applications, such as the valuation of derivative securities, portfolio management, risk management, the construction of financial products, fintech innovations, and the use of big data.

Within this framework, the Department of Mathematics of Politecnico di Milano offers a curriculum in Quantitative Finance through a specific Quantitative Finance track in the Bachelor’s and Master’s degree programs in Mathematical Engineering. There are also double degree opportunities, see below.

This track features courses (see the list below) designed to provide students with advanced skills in quantitative and mathematical finance, covering derivative pricing and structuring, risk management, portfolio management, life and non-life insurance, and econometrics.

Particular emphasis is placed on the applications of probability and continuous-time stochastic calculus to mathematical finance, as well as on the use of advanced statistical and computational methods in quantitative finance.

This program also addresses the transformative impact of the fintech revolution, equipping students with the necessary skills to innovate and excel in this rapidly evolving sector. By integrating cutting-edge technologies such as blockchain, artificial intelligence, and data analytics into the curriculum, the courses prepare students to be at the forefront of the transformation of financial services.

The result is a program that prepares students for successful careers both in the job market (including roles in investment banks and commercial banks, investment funds, central banks and supranational institutions, insurance companies, consulting firms, private equity firms, and pension funds) and in research. It encompasses all professional profiles requiring expertise in quantitative finance, especially in the following areas: valuation and structuring of derivative products (equity, interest rate, and credit), risk management, portfolio management, structuring of insurance contracts, trading and sales of financial products, financial analysis, compliance, and business analytics.

Moreover, thanks to continuously updated courses and activities (such as seminars, workshops, and conferences) organized by the Quantitative Finance Laboratory (QFinLab), graduates are well positioned to contribute to advancements in digital banking, smart contracts, automated investment services, regulatory technology, and green finance, which are reshaping the global financial landscape.

Upon completing the program, graduates are financial engineers who not only understand the modern financial system but are also capable of driving the technological innovations that will shape the future of finance.

Courses

Finanza Matematica I (Prof. Emilio Barucci)

FINANZA MATEMATICA I

10 crediti

Mix didattico: 64 ore di lezione, 20 di esercitazioni e 20 ore di didattica innovativa.

Modalità di valutazione:

L’esame si compone di tre parti: la valutazione dei compiti assegnati durante la didattica innovativa, una prova scritta ed una prova orale. Ciascuna prova d’esame scritta si compone di sei esercizi e/o domande.

La parte di didattica innovativa pesa per 9/30, l’esame scritto per 21/30. Il voto che ne risulta può essere accettato dallo studente oppure costui può presentarsi all’orale. La valutazione della didattica innovativa può essere mantenuta soltanto per l’anno accademico in cui è stato seguito il corso.

Programma:

L’insegnamento copre la teoria classica dei mercati finanziari e si compone di sei moduli:

1) Struttura dei tassi di interesse e valutazione di obbligazioni (5 lezioni)

- leggi di capitalizzazione

- tassi di interesse a pronti e a termine

- valutazione di obbligazioni-scelta di investimenti

- rischio di tasso di interesse, duration, indicatori di variabilità

- teoria dell’immunizzazione

2) Scelte di portafoglio in condizioni di rischio (6 lezioni)

- teoria dell’utilità attesa

- avversione al rischio

- analisi media-varianza

- scelte di portafoglio tra un titolo rischioso e uno privo di rischio

- scelte di portafoglio tra un titolo privo di rischio e N titoli rischiosi

- scelta di assicurazione

3) Frontiera dei portafogli (4 lezioni)

- frontiera dei portafogli con titoli rischiosi

- frontiera dei portafogli con un titolo privo di rischio

- scelte di portafoglio e frontiera dei portafogli

4) Asset pricing, modelli di equilibrio e non arbitraggio (6 lezioni)

- capital asset pricing model

- assenza opportunità di arbitraggio

- teorema fondamentale dell’asset pricing

- arbitrage pricing theory

- albero binomiale

5) Gestione del rischio (3 lezioni)

- mapping del rischio: portafoglio azionario, obbligazionario, derivati

- Value at Risk, Expected shortfall

- principali metodologie per la stima del VaR (full evaluation e metodo delta-normal), backtesting del VaR

6) Introduzione ai mercati finanziari (8 lezioni)

- funzionamento sistema finanziario

- intermediazione finanziaria e creditizia

- mercato monetario, obbligazionario, azionario, dei cambi

- ruolo della Banca Centrale

- politica monetaria

- gestione di un intermediario finanziario/creditizio

- Seminario su Rapporto Stabilità Finanziaria (BDI)

Moduli didattica innovativa:

1) Struttura dei tassi di interesse e valutazione di obbligazioni (5 ore)

- bootstrap della curva, tassi equivalenti, tassi a termine (relazione con quelli a pronti),

- valutazione di obbligazioni

- TIR, TAEG

- Rendimento investimento, tasso cedolare, tasso di rendimento (HP, scadenza e aspettative di tasso)

- rischio di tasso di interesse, duration, immunizzazione

2) Scelte di portafoglio in condizioni di rischio e Frontiera dei Portafogli (5 ore)

- individuazione del grado di avversione al rischio

- costruzione frontiera

- analisi al variare della correlazione, dei titoli, lunghezza delle serie storiche

- determinazione del portafoglio tangente

- scelte di portafoglio e frontiera dei portafogli con grado di avversione al rischio

3) Asset pricing, modelli di equilibrio e non arbitraggio (5 ore)

- albero binomiale: opzioni americane, payoff complessi

- bound di non arbitraggio su opzioni

- copertura tramite strategie con derivati

4) Gestione del rischio (5 ore)

- stima del VaR, ES: metodi parametrici, simulazione stroica

- metodo delta-normal

- Backtesting del VaR

Bibliografia:

Emilio Barucci, Claudio Marsala, Matteo Nencini, Carlo Sgarra, Ingegneria Finanziaria, Editore: EGEA, Anno edizione: 2009, ISBN: 978-88-238-2095-1

Frederic Mishkin, Stanley Eakins, Giancarlo Forestieri, Istituzioni e mercati finanziari, Editore: Pearson

MOOC educazione finanziaria http://www.imparalafinanza.it/mooc/

Mathematical Finance II (Prof. Alessandro Calvia)

MATHEMATICAL FINANCE II (Master Level, Mathematical Engineering)

Prof. Alessandro Calvia

AIM of the Course:

The aim of the course is to make the students familiar with the mathematical and, more in general, with the quantitative methods adopted in describing the dynamics of financial markets and in valuating and hedging financial derivatives.

SUBJECTS of the Course:

The No Arbitrage Principle in a continuous time setting. The fundamental Theorems of Asset Pricing. The Geomeric Brownian motion as a limit of a binomial random walk. The Black-Scholes-Merton analysis and the Black-Scholes formula for European Options. The Black-Scholes equation solved via PDE methods and via the Feynman-Kac representation formula. Complete and incomplete market models. Completeness of the Black-Scholes model and its relation with the martingale representation theorem. Hedging in the Black-Scholes setting: the Greeks.

Portfolio dynamics and self-financing strategies. The optimal control problems in the Black-Scholes setting: the optimal-consumption-investment problem and the related Hamilton-Jacobi-Bellman equation. The two-funds theorem: the case with no risk-free asset and the case with a risk-free asset. The Merton problem and its generalizations.

Early exercise features and American option pricing. The free boundary problem for the Black-Scholes equation and its formulation as an optimal stopping problem.

Exotic and Path-Dependent options: Barrier, Lookback, Asian options valuation for the lognormal model. The multidimensional BS model and the valuation of basket options. Forward and Futures: Black-76 formula for options on futures. Currency derivatives.

Fixed-Income derivatives. Short rate models and bond valuation: affine models and affine term structures. The Vasicek, Ho-lee, Cox-Ingersoll-Ross, Hull and White Models. Forward rate models: the Heath-Jarrow-Morton approach. The LIBOR market model. Valuation of Caps, Floors and Swaptions in the LIBOR market model.

Valuation of forward and futures in stochastic interest rate models.

Limitations of the Black-Scholes model: empirical evidences on lorgeturns distributions: fat tails, aggregational Gaussianity, volatility clustering and the leverage effect. Qualitative discussion on the volatility smile and the volatility term structure of option prices.

The course will include exercise sessions in which examples and applications of the general theory to specific concrete cases will be provided and discussed.

EXAMINATION PROCEDURE. The exam will be perormed in two steps: a written part, mainly related to exercises and applications, and an oral part mainly related to the general theory.

REFERENCES::

1) T. Bjork, ARBITRAGE THEORY IN CONTINUOUS TIME, Oxford University Press, 4th Ed., 2009.

2) E. Rosazza Gianin, C. Sgarra, MATHEMATICAL FINANCE: THEORY REVIEW AND EXERCISES, Springer, 2013.

PREVIOUS KNOWLEDGE: Basic notions of probability, mathematical statistics, differential and integral calculus; basic notions on stochastic calculus: Ito integral, Ito lemma, stochastic differential equations. Basic notions on PDE: the diffusion equation and its fundamental solution. Basic definitions of the most popular financial products available on the market.

LANGUAGE: ENGLISH.

Financial Engineering (Prof. Roberto Baviera)

FINANCIAL ENGINEERING (Master Level, Mathematical Engineering)

Prof. Roberto Baviera in collaboration with Laura Locatelli (Unicredit), Alessandro Montinari (Mediobanca) & Giona Soldati (Banor) and also Michele Domenico Massaria (PoliMi).

Main obiectives and contents

From theory to practice in finance with a case-study approach:

1. Credit Risk: single-name and multi-name products;

2. Quantitative Risk Management (RM): from RM Measures to RM Techniques;

3. Structured products: calibration, valuation and some hedging issues.

Suggestions by previous year students (before starting the course) can be found below. Pls check also for course prerequisites. For a glimpse into the course, check out one of the last editions of the Best Pitch Award.

Description of main arguments

0. Basic derivatives' concepts

Forward & option: Exchange-traded Markets vs OTC markets, Forward vs Futures. Forward Price: deduction via a no-arbitrage argument. European Option (Call/Put): decomposition in Intrinsic Value & Time Value; Put Call Parity. CRR & Black Model and examples. Monte-Carlo technique.

Main Greeks: Delta, Gamma, Vega e Theta. Volatility Smile.

Basic Interest Rate (IR) instruments: Fundamental Year-fractions in IR Derivatives. Depos, FRA, STIR Futures, Interest Rate Swaps & Fwd Swap, Cap/Floor, Swaptions.

IR bootstrap. Sensitivities: BPV, DV01 and duration. Sensitivity analysis and hedging of IR risk with IRS.

1. Credit Risk

Introduction to credit risk.

Basic Fixed Income instruments in presence of Credit Risk: Fixed Coupon Bond, Floater Coupon Bond, Asset Swap, CDS. SPOL, CDS, ASW relations. Bootstrap Credit Curve.

Firm-value (Merton, KMV calibration, Black-Cox) & Intensity models (Jarrow-Turnbull, inhomogeneous Poisson).

Multiname products (ABS, MBS, CDO) and models for HP and LHP.

Copula approach and Li model with examples (Archimedean and Gaussian Copulas), CDO Implied Correlation.

2. Quantitative Risk Management

Basel Accords, Risk Management Policy.

VaR/Expected Shortfall (ES): Variance-Covariance method, Historical Simulation, Weighted Historical Simulation, Bootstrap, Full valuation Monte-Carlo, Delta-normal & Delta Gamma method, plausibility check and Scaling rule. Coherent measures: assioms, VaR subadditivity (counterexample, elliptic case), ES coherence.

Backtest VaR: Basel approach, unconditional & conditional backtest.

Capital Allocation: Euler Principle & Contribution to VaR & ES.

3. Structured products

Certificates, Equity and IR Structured bond: the general Monte-Carlo approach for pricing not-callable structured products. Callable & Autocallable products.

Deal structuring and Issuer hedging.

Digital Risk: Slope impact & Black Correction in Autocallable products, FFT technique. Lewis formula for option pricing and analytic strip via an example: Exponential Levy model and characteristic function (NIG & VG). Global calibration and pricing via a Monte-Carlo (NIG). Sticky Strike & Sticky Delta. Parsimony and smile symmetry.

IR products and models: plain vanilla and exotics. HJM models: Main equation under Risk Neutral measure; equivalence with a ZC bond approach. Fundamental Lemmas and examples:

o Market models: BMM, LMM and SMM. Calibration: Flat Vol vs Spot Vol in Cap/Floor markets.

o Hull White model (Extended Vasicek): Cap/Floor & Swaption solution. Calibration. Pricing: Trinomial Tree.

Course Prerequisites

Mathematical Finance II, Stochastic Differential Equations, proficiency in Matlab/basic knowledge in Python

More in detail:

- Arbitrage Pricing Theory

- Forwards, Futures. Physical delivery & Cash settlement contracts. Call/Put European, American and Barrier Options.

- CRR and Tree pricing approach

- Fixed Coupon Bonds, Floaters and their sensitivities (e.g. duration), caplet/floorlet and swaption

- Filtration, Stochastic Ito Calculus, change of measure and Girsanov Theorem, Stochastic Fubini-Tonelli theorem

- Integration rules in the complex plain

- Proficiency in Matlab (it's also suggested a basic knowledge in Python)

Course organization

- Written & oral examination

- The 10 CFU version uses some "Innovative teaching methods" designed in collaboration with the financial industry.

This version could end also with a final project; starting from the AY 2019-2020 the best pitch has been awarded. Also a second, a third, a forth edition and a fifth edition of the Best Pitch Award has been conferred for the AY 2020-2021, AY 2021-2022, AY 2022-2023 and AY 2023-2024. Last but not least, it has been followed by a sixth edition for the AY 2024-2025.

Suggestions (and answers to some FAQs) by previous year students: "Should I choose FE?"

... and do not hesitate to contact previous years' students for any further question and/or clarification!

Textbooks

1. J. Hull (2009), Options, futures and other derivatives, Pearson Prentice Hall, 7th Ed.

2. A.J. McNeil, R. Frey, P. Embrects (2005), Quantitative RM: concepts, techniques and tools, Princeton Univ. Press

3. P.J. Schonbucher (2003), Credit Derivatives Pricing Models, Wiley

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Computational Finance (Prof. Michele Azzone and Daniele Marazzina)

Insurance and Econometrics (Prof. Michele Azzone and Alessandro Calvia)

INSURANCE AND ECONOMETRICS (Master Degree, Mathematical Engineering)

Prof. Michele Azzone (Econometrics) and Prof. Alessandro Calvia (Insurance)

AIM of the Course:

The aim of this course is to cover two main topics: insurance markets and econometrics.

Lectures and coding sessions will allow students to acquire the following competences:

- Knowledge and understanding

basic and advanced concepts of actuarial mathematics;

basic and advanced concepts of econometrics.

- Ability in applying knowledge and understanding

know how to compute the fair premium of an insurance contract;

know how to analyze in a proper way a financial time series.

- Making judgements

be able to judge the main financial risks in insurance contracts;

find proper modeling assumptions to analyze financial time series.

- Communication skills

to be able to express mathematical and financial concepts in a clear and rigorous way.

SUBJECTS of the Course:

On the insurance markets part, we will address some modelization issues that make the market different from the banking market and we will deal with the Solvency II regulation. We will address the issue of pricing life products and the definition of reserves. We then consider the main features of damage insurances, and the differents w.r.t. life insurances.

The econometrics part is intended to supply the students with basic econometric tools for financial time series modeling. Stationary processes and the Wold theorem will be described to pursue ARMA models selection, estimation and forecast. We will introduce unit root processes, vector models and cointegration. We address heteroskedastic (ARCH-GARCH) models for market volatility to be applied to portfolio hedging problems. We will consider factor models for large portfolios and the principal component approach.

EXAMINATION PROCEDURE:

a) a project on the insurance part;

b) a project on the econometrics part;

c) an oral exam on both the insurance and the econometrics part.

For both projects, the maximum mark is 30/30. In order to be admitted to the oral exam the average mark of the two projects must be at least 12/30. The objective of projects is to let students work in groups, applying the approaches and principles taught in class. Projects will be assigned during the semester. Project outputs are expected to be released at fixed deadlines that will be defined by the time the project will be assigned. The evaluation of projects will be based on the produced outputs (documentation, code, …).

The oral exam is mandatory, and it could result in a maximum increase of the final mark of 3/30 points.

The exam has the goal of checking whether the student has acquired the following skills:

- knowledge of basic and advanced concepts of actuarial mathematics;

- knowledge of the EU regulamentation for Insurance markets;

- knowledge of basic and advanced concepts of econometrics;

- ability to compute the fair premium of an insurance contract, as well as its profit for the insurance company;

- ability to analyze in a proper way financial time series;

- ability to judge the main sources of financial risk in insurance contracts;

- ability to express mathematical and financial concepts in a clear and rigorous way.

Fintech (Prof. Daniele Marazzina)

Fintech (Master Level, Mathematical Engineering)

AIM of the Course:

The aim of the course is to make students familiar with the new frontiers of Fintech. Financial technology (Fintech) is the introduction of new technologies and innovations in the traditional financial methods and services. Within the financial services industry, some of the used technologies include artificial intelligence (AI), big data, and blockchain. The aim of the course is to show use cases were these techniques are applied to specific financial problems, overcoming traditional methods.

The course is divided in two parts: 50% on blockchain and big data, and (50%) on machine learning applications on financial problems.

SUBJECTS of the Course:

Bitcoin and Blockchain Technologies

Hash functions and elliptic curve cryptography; Blockchain and Merkle tree; Distributed consensus and mining; Transactions; Bitcoin Core; Wallets; Token; Stable coins.

BigData

How to use data to prevent financial problems: lapse risk in the insurance sector.

How to use data to obtain credit rate for small and medium enterprise.

Machine Learning

How to face problems in capital markets and the energy sector via a Machine Learning approach.

EXAMINATION PROCEDURE:

a) projects on the fintech topics;

b) projects presentation and oral exam.

The objective of projects is to let students work in groups, applying the approaches and principles taught in class. Projects will be assigned through the semester. Project artifacts are expected to be released at fixed deadlines that will be defined by the time the project will be assigned. The evaluation of projects will be based on the produced artifacts (documentation, code, …).

The oral exam is mandatory.

The exam has the goal of checking whether the student has acquired the following skills:

- knowledge of advanced concepts of stochastic calculus, which are essential in quantitative finance;

- knowledge of how to choose modeling assumptions to solve a financial problem properly;

- ability to write a code/algorithm to solve a financial problem, exploiting the developed theoretical knowledgements;

- ability to find proper modeling assumption to describe a financial asset (like interest rate, volatility, commodities, etc.);

- ability to express mathematical and financial concepts in a clear and rigorous way.

LANGUAGE: ENGLISH.

Advanced mathematical models in finance (Prof. Roberto Baviera)

Double degree

For students enrolled in the Quantitative Finance track of the Master of Science in Mathematical Engineering, there are excellent opportunities to pursue a double degree. Notably, we offer a direct agreement with the École Nationale Supérieure d’Informatique pour l’Industrie et l’Entreprise (ENSIIE) in France. This specific program allows highly motivated first-year students to split their studies, spending one year at Politecnico di Milano and one year at ENSIIE. Upon completion, graduates are awarded both the “Master of Science in Mathematical Engineering” from PoliMi and the “Diplôme d’Ingénieur ENSIIE.” Additionally, students in this exchange can apply for a concurrent Ingénieur/Master program at ENSIIE, allowing them to obtain a second Master’s degree from Paris-Saclay University in specialized tracks such as Financial Mathematics.

Beyond this direct partnership, students can also take advantage of PoliMi’s general international mobility agreements, which are open to the wider student body (view the map here). Within these general agreements, several prestigious partner institutions offer strong quantitative finance-oriented programs. Prominent examples include CentraleSupélec, ENSTA Paris, École des Ponts ParisTech, Sorbonne University, and KTH Royal Institute of Technology.