NUMERICAL METHODS AND MACHINE LEARNING FOR FINANCE
Syllabus
EN
IT
Learning Objectives
LEARNING OUTCOMES:
The course provides knowledge of the fundamentals of programming, with an emphasis on financial applications. During the course, the linear regression model will also be analyzed from both a theoretical and an applied perspective.
Students will apply their knowledge to write code aimed at solving classical problems in the field of quantitative finance.
1) KNOWLEDGE AND UNDERSTANDING
Students are required to understand the basics of programming and the theory of linear regression. They are also required to understand elementary MATLAB code with statistical applications and to be able to formulate linear regression models to be applied to the study of finance-related problems, such as, for example, the Capital Asset Pricing Model.
2) APPLYING KNOWLEDGE AND UNDERSTANDING
By the end of the course, students must be able to formulate the analysis of a simple financial problem in terms of linear regression, derive the theoretical properties of the model formulated — in particular, of the estimators used to estimate the model — and produce efficient code capable of carrying out such analyses on real datasets.
3) MAKING JUDGEMENTS
By the end of the course, students are expected to have the tools needed to assess the economic contexts in which it is appropriate to apply linear regression models and, more specifically, the ability to judge which formulation is best suited to the context under consideration, such as time series of asset prices, GDP, and similar data.
4) COMMUNICATION SKILLS
By the end of the course, students are expected to be able to express themselves using rigorous, precise, and concise language, particularly with regard to the theory of OLS estimators and their asymptotic properties.
5) LEARNING SKILLS
The course provides the tools needed to work independently on basic econometric models and to translate such problems into code useful for estimating these models on time series of various kinds. By the end of the course, students will have developed a good level of independent learning ability in relation to scientific articles in the field of financial econometrics.
The course provides knowledge of the fundamentals of programming, with an emphasis on financial applications. During the course, the linear regression model will also be analyzed from both a theoretical and an applied perspective.
Students will apply their knowledge to write code aimed at solving classical problems in the field of quantitative finance.
1) KNOWLEDGE AND UNDERSTANDING
Students are required to understand the basics of programming and the theory of linear regression. They are also required to understand elementary MATLAB code with statistical applications and to be able to formulate linear regression models to be applied to the study of finance-related problems, such as, for example, the Capital Asset Pricing Model.
2) APPLYING KNOWLEDGE AND UNDERSTANDING
By the end of the course, students must be able to formulate the analysis of a simple financial problem in terms of linear regression, derive the theoretical properties of the model formulated — in particular, of the estimators used to estimate the model — and produce efficient code capable of carrying out such analyses on real datasets.
3) MAKING JUDGEMENTS
By the end of the course, students are expected to have the tools needed to assess the economic contexts in which it is appropriate to apply linear regression models and, more specifically, the ability to judge which formulation is best suited to the context under consideration, such as time series of asset prices, GDP, and similar data.
4) COMMUNICATION SKILLS
By the end of the course, students are expected to be able to express themselves using rigorous, precise, and concise language, particularly with regard to the theory of OLS estimators and their asymptotic properties.
5) LEARNING SKILLS
The course provides the tools needed to work independently on basic econometric models and to translate such problems into code useful for estimating these models on time series of various kinds. By the end of the course, students will have developed a good level of independent learning ability in relation to scientific articles in the field of financial econometrics.
KATIA COLANERI
Prerequisites
Probability: random variables, joint laws and conditional laws.
Basics of Financial Mathematics.
Characteristics of financial derivatives.
Stochastic Processes: Brownian motions and its characteristics, Ito Formula.
Mathematical models for finance: Binomial model, Black and Scholes Model, risk neutral probability, arbitrage and market completeness, hedging strategies.
Knowledge of the Matlab language.
Basics of Financial Mathematics.
Characteristics of financial derivatives.
Stochastic Processes: Brownian motions and its characteristics, Ito Formula.
Mathematical models for finance: Binomial model, Black and Scholes Model, risk neutral probability, arbitrage and market completeness, hedging strategies.
Knowledge of the Matlab language.
Program
Part 1: Numerical Methods for Finance
Week 1: Lattice (tree) methods
- Binomial trees
-Trinomial trees
Week 2: Monte Carlo methods
- Generating random variables
- Correlated Gaussian random variables
- Random paths simulation and option pricing
- Simulation and estimation error
- Variance reduction methods
- Option pricing: application to European down and out barrier option under Black-Scholes, and Bond pricing with the CIR model
Week 3: Finite difference methods for PDEs
- Explicit scheme, Implicit scheme, Crank-Nicolson scheme
- Stability and convergence analysis
- Numerical solution of systems of linear equations.
Week 1: Lattice (tree) methods
- Binomial trees
-Trinomial trees
Week 2: Monte Carlo methods
- Generating random variables
- Correlated Gaussian random variables
- Random paths simulation and option pricing
- Simulation and estimation error
- Variance reduction methods
- Option pricing: application to European down and out barrier option under Black-Scholes, and Bond pricing with the CIR model
Week 3: Finite difference methods for PDEs
- Explicit scheme, Implicit scheme, Crank-Nicolson scheme
- Stability and convergence analysis
- Numerical solution of systems of linear equations.
Books
Glasserman, Paul. Monte Carlo methods in financial engineering, Springer-Verlag, 2003.
Goodfellow, Ian; Bengio, Yoshua; Courville, Aaron (2016). Deep Learning. Cambridge, Massachusetts: MIT Press.
Goodfellow, Ian; Bengio, Yoshua; Courville, Aaron (2016). Deep Learning. Cambridge, Massachusetts: MIT Press.
Bibliography
K.E. Atkinson. An introduction to numerical analysis, Second Edition. Wiley, 1989.
P. Billingsley. Convergence of probability measures (2nd ed.). John Wiley & Sons, 1999.
R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and
exponential L´evy models. SIAM Journal On Numerical Analysis, 43(4): 1596-1626, 2005.
Barron, A.R., Approximation and estimation bounds for artificial neural networks. Mach. Learn., 1994, 14(1), 115–133.
Horvath, B., Muguruza A. and Tomas M. (2021) Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models, Quantitative Finance, 21:1, 11-27.
Hornik, K., Stinchcombe, M. and White, H., Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw., 1990, 3(5), 551.560
P. Billingsley. Convergence of probability measures (2nd ed.). John Wiley & Sons, 1999.
R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and
exponential L´evy models. SIAM Journal On Numerical Analysis, 43(4): 1596-1626, 2005.
Barron, A.R., Approximation and estimation bounds for artificial neural networks. Mach. Learn., 1994, 14(1), 115–133.
Horvath, B., Muguruza A. and Tomas M. (2021) Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models, Quantitative Finance, 21:1, 11-27.
Hornik, K., Stinchcombe, M. and White, H., Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw., 1990, 3(5), 551.560
Teaching methods
Traditional lectures, with the option to carry out projects at home and discuss them in class.
Exam Rules
Student learning outcomes are assessed through an oral exam.
This ensures the effective acquisition of the expected learning outcomes, particularly regarding:
understanding of numerical methods and machine learning techniques (knowledge and understanding);
their application to real financial problems (applying knowledge);
critical thinking and autonomy (making judgments);
clarity and accuracy in presentation (communication skills).
The oral exam covers theoretical aspects of the course and evaluates comprehension, reasoning ability, and independent judgment.
Students are allowed to reject their final ONLY ONCE.
Criteria for grade (out of 30 points):
Fail: Major gaps and/or inaccuracies in understanding; limited analysis and synthesis skills; generalizations prevail.
18-20: Barely sufficient understanding; adequate analysis, synthesis, and judgment skills.
21-23: Routine understanding; correct analytical and synthesis skills; coherent logical reasoning.
24-26: Fair knowledge; good analytical and synthesis skills; rigorous reasoning.
27-29: Comprehensive understanding; strong analytical, synthesis, and judgment skills.
30-30L: Excellent understanding; outstanding analytical, synthesis, and independent thinking skills; original arguments.
This ensures the effective acquisition of the expected learning outcomes, particularly regarding:
understanding of numerical methods and machine learning techniques (knowledge and understanding);
their application to real financial problems (applying knowledge);
critical thinking and autonomy (making judgments);
clarity and accuracy in presentation (communication skills).
The oral exam covers theoretical aspects of the course and evaluates comprehension, reasoning ability, and independent judgment.
Students are allowed to reject their final ONLY ONCE.
Criteria for grade (out of 30 points):
Fail: Major gaps and/or inaccuracies in understanding; limited analysis and synthesis skills; generalizations prevail.
18-20: Barely sufficient understanding; adequate analysis, synthesis, and judgment skills.
21-23: Routine understanding; correct analytical and synthesis skills; coherent logical reasoning.
24-26: Fair knowledge; good analytical and synthesis skills; rigorous reasoning.
27-29: Comprehensive understanding; strong analytical, synthesis, and judgment skills.
30-30L: Excellent understanding; outstanding analytical, synthesis, and independent thinking skills; original arguments.
Attendance Rules
Presence in the classroom.
JODI DIANETTI
Program
Part 2: Machine Learning
Week 4: Introduction to Machine Learning Methods
- Basic concepts of AI and machine learning
- Feedforward neural networks: definition and properties
- Simple examples of neural network training
Week 5: Neural Network Theory and Numerical Optimization Methods
- Universal approximation theorem
- Estimation and approximation limits
- Gradient descent class algorithms
Week 6: Network Calibration and Model Estimation
- Hyperparameter tuning of a neural network
- Parametric stochastic processes in continuous time and discretization
- Estimation via Simulated Method of Moments
- Estimation via Deep Neural Networks
Week 4: Introduction to Machine Learning Methods
- Basic concepts of AI and machine learning
- Feedforward neural networks: definition and properties
- Simple examples of neural network training
Week 5: Neural Network Theory and Numerical Optimization Methods
- Universal approximation theorem
- Estimation and approximation limits
- Gradient descent class algorithms
Week 6: Network Calibration and Model Estimation
- Hyperparameter tuning of a neural network
- Parametric stochastic processes in continuous time and discretization
- Estimation via Simulated Method of Moments
- Estimation via Deep Neural Networks
Exam Rules
Student learning outcomes are assessed through an oral exam. This ensures the effective acquisition of the expected learning outcomes, particularly regarding:
understanding of numerical methods and machine learning techniques (knowledge and understanding);
their application to real financial problems (applying knowledge);
critical thinking and autonomy (making judgments);
clarity and accuracy in presentation (communication skills).
The oral exam covers theoretical aspects of the course and evaluates comprehension, reasoning ability, and independent judgment.
Students are allowed to reject their final ONLY ONCE.
Criteria for grade (out of 30 points):
Fail: Major gaps and/or inaccuracies in understanding; limited analysis and synthesis skills; generalizations prevail.
18-20: Barely sufficient understanding; adequate analysis, synthesis, and judgment skills.
21-23: Routine understanding; correct analytical and synthesis skills; coherent logical reasoning.
24-26: Fair knowledge; good analytical and synthesis skills; rigorous reasoning.
27-29: Comprehensive understanding; strong analytical, synthesis, and judgment skills.
30-30L: Excellent understanding; outstanding analytical, synthesis, and independent thinking skills; original arguments.
understanding of numerical methods and machine learning techniques (knowledge and understanding);
their application to real financial problems (applying knowledge);
critical thinking and autonomy (making judgments);
clarity and accuracy in presentation (communication skills).
The oral exam covers theoretical aspects of the course and evaluates comprehension, reasoning ability, and independent judgment.
Students are allowed to reject their final ONLY ONCE.
Criteria for grade (out of 30 points):
Fail: Major gaps and/or inaccuracies in understanding; limited analysis and synthesis skills; generalizations prevail.
18-20: Barely sufficient understanding; adequate analysis, synthesis, and judgment skills.
21-23: Routine understanding; correct analytical and synthesis skills; coherent logical reasoning.
24-26: Fair knowledge; good analytical and synthesis skills; rigorous reasoning.
27-29: Comprehensive understanding; strong analytical, synthesis, and judgment skills.
30-30L: Excellent understanding; outstanding analytical, synthesis, and independent thinking skills; original arguments.