Mathematical methods for finance
A.A. 2025/2026
Obiettivi formativi
The aim of the course is to teach students the main techniques to approach multivariable optimization problems, both constrained and unconstrained, and to make students able to solve systems of differential equations and optimal control problems. The theoretical part of each module of the course will be enriched by a numerical part, where the goal is to get students acquainted with the main ideas and methodologies of numerical solutions with Matlab and Julia.
Risultati apprendimento attesi
At the end of the course, students will be expected to possess and be able to use the main techniques for solving multivariable optimization problems, both constrained and unconstrained. Moreover, they will have the necessary backgrounds to categorize and, whenever possible, to solve analytically systems of differential equations and optimal control problems. Everywhere the analytical solution is out of reach, students will be equipped with the necessary numerical tools available in Matlab and Julia.
Periodo: Primo trimestre
Modalità di valutazione: Esame
Giudizio di valutazione: voto verbalizzato in trentesimi
Corso singolo
Questo insegnamento non può essere seguito come corso singolo. Puoi trovare gli insegnamenti disponibili consultando il catalogo corsi singoli.
Programma e organizzazione didattica
Edizione unica
Responsabile
Periodo
Primo trimestre
Programma
Synthetic Contents
1. Taylor Polynomials and Multivariable Calculus
2. Linear Algebra and Vector Spaces
3. Elements of Topology
4. Optimization
5. Linear Programming
6. Dynamical Systems and Differential Equations
7. Optimal Control Theory
8. Network Theory and Financial Applications
Syllabus Outline
1. Taylor Polynomials and Multivariable Calculus
o Taylor polynomials on the real line
o Functions of several variables: domains, graphs, level curves
o Partial derivatives, directional derivatives, gradients, Hessian matrix
o Convexity and concavity; Taylor's formula for functions of two variables
o Implicit and inverse function theorems
o Visual interpretation and geometric representation
2. Linear Algebra and Vector Spaces
o Systems of linear equations
o Matrix operations (addition, multiplication, inverse, transpose)
o Vector spaces, subspaces, linear independence, basis, rank
o Determinant, trace, eigenvalues, eigenvectors, diagonalization
o Quadratic forms and their applications
3. Elements of Topology
o Convex sets, open and closed sets in n-dimensional space
o Continuity and compactness (brief review, as relevant for optimization)
4. Optimization
o Unconstrained optimization: first and second order conditions, global extrema
o Constrained optimization:
Equality constraints: Lagrange multipliers, bordered Hessian, Weierstrass theorem
Inequality constraints: Karush-Kuhn-Tucker (KKT) conditions, economic examples
Concave programming and sufficient conditions
5. Linear Programming
o Motivation and applications in economics and finance
o Formulation: objective function, constraints, feasible region
o Graphical solution (two variables)
o The simplex method: concepts and intuition
o Duality: introduction and economic interpretation
o Applications: portfolio optimization, resource allocation
6. Dynamical Systems and Differential Equations
o Introduction to ordinary and partial differential equations, difference equations
o First-order ODEs: separable, linear, exact; existence and uniqueness
o Second-order ODEs; planar systems; equilibrium points and stability
o Linear stability analysis, phase portraits, trajectories, fixed points, classification of steady states
o Nonlinear systems: linearization, stability
o Examples: Logistic equation, contagion models, and population growth
o Matlab for ODEs: ODE45, symbolic toolbox
7. Optimal Control Theory
o Introduction to optimal control problems
o Pontryagin's Maximum Principle (statement and intuition)
o Regularity and sufficient conditions
o Applications in economics and finance
8. Network Theory and Financial Applications
o Graphs and networks: nodes, edges, adjacency and incidence matrices
o Network measures: degree, centrality, clustering coefficients, metrics
o Types of networks: random, small-world, scale-free
o Information diffusion processes on networks
o Applications: interbank networks, contagion, systemic risk, market structure
o Case studies and computational exercises (Matlab, Python, R)
9. Optional: Stochastic Processes and Introduction to Stochastic Calculus
o Probability spaces, random variables, Brownian motion
o Basic stochastic processes in finance (e.g., geometric Brownian motion)
o Itô's Lemma (statement and basic application)
o Applications: Black-Scholes model (intuition, no proofs), risk-neutral pricing
1. Taylor Polynomials and Multivariable Calculus
2. Linear Algebra and Vector Spaces
3. Elements of Topology
4. Optimization
5. Linear Programming
6. Dynamical Systems and Differential Equations
7. Optimal Control Theory
8. Network Theory and Financial Applications
Syllabus Outline
1. Taylor Polynomials and Multivariable Calculus
o Taylor polynomials on the real line
o Functions of several variables: domains, graphs, level curves
o Partial derivatives, directional derivatives, gradients, Hessian matrix
o Convexity and concavity; Taylor's formula for functions of two variables
o Implicit and inverse function theorems
o Visual interpretation and geometric representation
2. Linear Algebra and Vector Spaces
o Systems of linear equations
o Matrix operations (addition, multiplication, inverse, transpose)
o Vector spaces, subspaces, linear independence, basis, rank
o Determinant, trace, eigenvalues, eigenvectors, diagonalization
o Quadratic forms and their applications
3. Elements of Topology
o Convex sets, open and closed sets in n-dimensional space
o Continuity and compactness (brief review, as relevant for optimization)
4. Optimization
o Unconstrained optimization: first and second order conditions, global extrema
o Constrained optimization:
Equality constraints: Lagrange multipliers, bordered Hessian, Weierstrass theorem
Inequality constraints: Karush-Kuhn-Tucker (KKT) conditions, economic examples
Concave programming and sufficient conditions
5. Linear Programming
o Motivation and applications in economics and finance
o Formulation: objective function, constraints, feasible region
o Graphical solution (two variables)
o The simplex method: concepts and intuition
o Duality: introduction and economic interpretation
o Applications: portfolio optimization, resource allocation
6. Dynamical Systems and Differential Equations
o Introduction to ordinary and partial differential equations, difference equations
o First-order ODEs: separable, linear, exact; existence and uniqueness
o Second-order ODEs; planar systems; equilibrium points and stability
o Linear stability analysis, phase portraits, trajectories, fixed points, classification of steady states
o Nonlinear systems: linearization, stability
o Examples: Logistic equation, contagion models, and population growth
o Matlab for ODEs: ODE45, symbolic toolbox
7. Optimal Control Theory
o Introduction to optimal control problems
o Pontryagin's Maximum Principle (statement and intuition)
o Regularity and sufficient conditions
o Applications in economics and finance
8. Network Theory and Financial Applications
o Graphs and networks: nodes, edges, adjacency and incidence matrices
o Network measures: degree, centrality, clustering coefficients, metrics
o Types of networks: random, small-world, scale-free
o Information diffusion processes on networks
o Applications: interbank networks, contagion, systemic risk, market structure
o Case studies and computational exercises (Matlab, Python, R)
9. Optional: Stochastic Processes and Introduction to Stochastic Calculus
o Probability spaces, random variables, Brownian motion
o Basic stochastic processes in finance (e.g., geometric Brownian motion)
o Itô's Lemma (statement and basic application)
o Applications: Black-Scholes model (intuition, no proofs), risk-neutral pricing
Prerequisiti
Prerequisites
The following prerequisites are required to ensure students are well-prepared for the advanced material presented in this course:
· Calculus: Solid understanding of single-variable and multivariable calculus, including differentiation, integration, Taylor expansions, and an introduction to ordinary differential equations. Students should be familiar with concepts such as partial derivatives, gradients, and optimization in two variables.
· Basic Linear Algebra: Familiarity with matrices, matrix operations, determinants, vector spaces, and the solution of systems of linear equations. Knowledge of eigenvalues and eigenvectors is beneficial.
· Probability and Statistics: Basic knowledge of probability theory, random variables, distributions, and elementary statistical concepts relevant for interpreting financial data.
· Mathematical Reasoning: Ability to follow and construct rigorous proofs, work with abstract mathematical concepts, and apply logical reasoning to problem-solving.
· Computational Skills: While prior experience with a programming environment (such as Matlab, Julia, R, or Python) is advantageous, it is not strictly required. Students should be willing to engage with computational tools, as numerical methods and hands-on exercises are an integral part of the course.
· Economics and Finance Fundamentals: Basics of economics and classical financial mathematics are useful for contextualizing mathematical models, but are not mandatory.
This preparation will enable students to effectively engage with both the theoretical and practical components of the course.
The following prerequisites are required to ensure students are well-prepared for the advanced material presented in this course:
· Calculus: Solid understanding of single-variable and multivariable calculus, including differentiation, integration, Taylor expansions, and an introduction to ordinary differential equations. Students should be familiar with concepts such as partial derivatives, gradients, and optimization in two variables.
· Basic Linear Algebra: Familiarity with matrices, matrix operations, determinants, vector spaces, and the solution of systems of linear equations. Knowledge of eigenvalues and eigenvectors is beneficial.
· Probability and Statistics: Basic knowledge of probability theory, random variables, distributions, and elementary statistical concepts relevant for interpreting financial data.
· Mathematical Reasoning: Ability to follow and construct rigorous proofs, work with abstract mathematical concepts, and apply logical reasoning to problem-solving.
· Computational Skills: While prior experience with a programming environment (such as Matlab, Julia, R, or Python) is advantageous, it is not strictly required. Students should be willing to engage with computational tools, as numerical methods and hands-on exercises are an integral part of the course.
· Economics and Finance Fundamentals: Basics of economics and classical financial mathematics are useful for contextualizing mathematical models, but are not mandatory.
This preparation will enable students to effectively engage with both the theoretical and practical components of the course.
Metodi didattici
Teaching Methods
The course employs an interactive teaching methodology designed to foster both theoretical understanding and practical skills:
· Lectures: Traditional lectures introduce and develop theoretical concepts, with a focus on rigorous mathematical reasoning and financial applications.
· Interactive Problem-Solving: In-class exercises, guided problem sets, and worked examples encourage active student participation and immediate feedback.
· Computational Labs: Dedicated sessions using Matlab, Julia, and R for numerical experimentation, simulation, and data analysis, bridging theory and real-world practice.
· Online Resources: Supplementary materials and code notebooks provided via the course platform.
· Tutoring and Office Hours: Weekly faculty reception hours and scheduled tutoring sessions (individual or small-group) support personalized learning and address specific student questions.
· Continuous Feedback: Frequent tests, exercises, prompts, and opportunities for interactive discussions enable students to monitor their progress and reinforce their learning throughout the course.
The course employs an interactive teaching methodology designed to foster both theoretical understanding and practical skills:
· Lectures: Traditional lectures introduce and develop theoretical concepts, with a focus on rigorous mathematical reasoning and financial applications.
· Interactive Problem-Solving: In-class exercises, guided problem sets, and worked examples encourage active student participation and immediate feedback.
· Computational Labs: Dedicated sessions using Matlab, Julia, and R for numerical experimentation, simulation, and data analysis, bridging theory and real-world practice.
· Online Resources: Supplementary materials and code notebooks provided via the course platform.
· Tutoring and Office Hours: Weekly faculty reception hours and scheduled tutoring sessions (individual or small-group) support personalized learning and address specific student questions.
· Continuous Feedback: Frequent tests, exercises, prompts, and opportunities for interactive discussions enable students to monitor their progress and reinforce their learning throughout the course.
Materiale di riferimento
Main reference
· Course notes provided by the instructor.
· Learning materials available through Ariel.
Core Texts:
· Hammond et al., Essential Mathematics for Economic Analysis, 5th Edition.
· Hammond et al., Further Mathematics for Economic Analysis, 2nd Edition.
· Salsa & Squellati, Dynamical Systems and Optimal Control, A Friendly Introduction.
· Deisenroth et al., Mathematics for Machine Learning (available online).
· Strogatz, Nonlinear Dynamics and Chaos, 2nd Edition.
· Attaway, Matlab: A Practical Introduction to Programming and Problem Solving, 6th Edition.
Additional Suggested Readings:
· Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research.
· Jackson, M. O. (2010). Social and Economic Networks.
· Estrada, E. (2012). The structure of complex networks: theory and applications.
· Course notes provided by the instructor.
· Learning materials available through Ariel.
Core Texts:
· Hammond et al., Essential Mathematics for Economic Analysis, 5th Edition.
· Hammond et al., Further Mathematics for Economic Analysis, 2nd Edition.
· Salsa & Squellati, Dynamical Systems and Optimal Control, A Friendly Introduction.
· Deisenroth et al., Mathematics for Machine Learning (available online).
· Strogatz, Nonlinear Dynamics and Chaos, 2nd Edition.
· Attaway, Matlab: A Practical Introduction to Programming and Problem Solving, 6th Edition.
Additional Suggested Readings:
· Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research.
· Jackson, M. O. (2010). Social and Economic Networks.
· Estrada, E. (2012). The structure of complex networks: theory and applications.
Modalità di verifica dell’apprendimento e criteri di valutazione
Assessment Methods
1. Written Exam:
The written exam is mandatory for all students and is structured as follows:
o Five multiple-choice questions (maximum 3 points each)
o Three open-ended exercises/problems (maximum 5 points each)
o One honors question (maximum 3 points)
Two midterm examinations are scheduled for students taking the course. The overall evaluation of the two midterms will be the arithmetic mean of their respective evaluations. Both the full and midterm exams will be one and a half hours long.
2. Oral Exam:
The oral examination is optional. Only students who have obtained a grade of 15 or higher in the written exam may register for the oral exam. Students may individually submit a project in one of the following forms (but not limited to):
o Case Studies: Students analyze a real-world financial scenario or problem, applying mathematical methods to draw conclusions or make recommendations.
o Data Analysis: Students use software (e.g., Matlab, R, Python) to analyze a real-world economic/financial dataset, reflecting professional practice.
The oral exam will consist of a presentation and discussion of the project. The project must demonstrate application of, and comprehensive understanding of, the concepts learned in the course. The exam will be more of a discussion, allowing students to defend their reasoning or approach and simulate real-world financial decision-making, focusing on analytical reasoning, clarity, and depth.
All students who take the oral exam will receive a second, independent evaluation. The final grade will be the arithmetic mean of the written and oral grades. If the average of the written and oral grades is below 18, the student is considered to have failed the exam and must retake the written exam in a subsequent session. In no case will the same written exam be valid for more than one oral exam session.
3. Continuous Assessment:
Short quizzes, exercises, and assignments will be given throughout the quarter to encourage interactive discussion. These will not be evaluated numerically. Students will be encouraged to present and share their solutions or results to promote continued engagement and develop communication skills that are important in finance.
4. Self-Assessment:
Students will be encouraged to submit brief reflections on their learning processes or exam performances to help them understand their strengths and areas for improvement.
1. Written Exam:
The written exam is mandatory for all students and is structured as follows:
o Five multiple-choice questions (maximum 3 points each)
o Three open-ended exercises/problems (maximum 5 points each)
o One honors question (maximum 3 points)
Two midterm examinations are scheduled for students taking the course. The overall evaluation of the two midterms will be the arithmetic mean of their respective evaluations. Both the full and midterm exams will be one and a half hours long.
2. Oral Exam:
The oral examination is optional. Only students who have obtained a grade of 15 or higher in the written exam may register for the oral exam. Students may individually submit a project in one of the following forms (but not limited to):
o Case Studies: Students analyze a real-world financial scenario or problem, applying mathematical methods to draw conclusions or make recommendations.
o Data Analysis: Students use software (e.g., Matlab, R, Python) to analyze a real-world economic/financial dataset, reflecting professional practice.
The oral exam will consist of a presentation and discussion of the project. The project must demonstrate application of, and comprehensive understanding of, the concepts learned in the course. The exam will be more of a discussion, allowing students to defend their reasoning or approach and simulate real-world financial decision-making, focusing on analytical reasoning, clarity, and depth.
All students who take the oral exam will receive a second, independent evaluation. The final grade will be the arithmetic mean of the written and oral grades. If the average of the written and oral grades is below 18, the student is considered to have failed the exam and must retake the written exam in a subsequent session. In no case will the same written exam be valid for more than one oral exam session.
3. Continuous Assessment:
Short quizzes, exercises, and assignments will be given throughout the quarter to encourage interactive discussion. These will not be evaluated numerically. Students will be encouraged to present and share their solutions or results to promote continued engagement and develop communication skills that are important in finance.
4. Self-Assessment:
Students will be encouraged to submit brief reflections on their learning processes or exam performances to help them understand their strengths and areas for improvement.
SECS-S/06 - METODI MATEMATICI DELL'ECONOMIA E DELLE SCIENZE ATTUARIALI E FINANZIARIE - CFU: 9
Lezioni: 59.4 ore
Docente:
Bartesaghi Paolo
Docente/i