NUMERICAL METHODS AND MACHINE LEARNING FOR FINANCE
Syllabus
EN
IT
Learning Objectives
LEARNING OUTCOMES:
Knowledge and understanding: Gain deep understanding of numerical methods and machine learning algorithms used in finance.
Applying knowledge and understanding: Apply computational techniques to real-world problems in pricing, risk management, calibration of parametrical models and financial forecasting.
Making judgments: Develop critical assessment of quantitative methods used in financial models.
Communication skills: Effectively present results from numerical and predictive models.
Learning skills: Acquire independent learning capabilities to keep up with evolving computational and AI tools in finance.
Knowledge and understanding: Gain deep understanding of numerical methods and machine learning algorithms used in finance.
Applying knowledge and understanding: Apply computational techniques to real-world problems in pricing, risk management, calibration of parametrical models and financial forecasting.
Making judgments: Develop critical assessment of quantitative methods used in financial models.
Communication skills: Effectively present results from numerical and predictive models.
Learning skills: Acquire independent learning capabilities to keep up with evolving computational and AI tools in finance.
DAVIDE ERMINIO PIRINO
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 2: Machine Learning
Week 1: 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 2: Neural Network Theory and Numerical Optimization Methods
- Universal approximation theorem
- Estimation and approximation limits
- Gradient descent class algorithms
Week 3: 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 1: 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 2: Neural Network Theory and Numerical Optimization Methods
- Universal approximation theorem
- Estimation and approximation limits
- Gradient descent class algorithms
Week 3: 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
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.
Blanka Horvath, Aitor Muguruza & Mehdi Tomas (2021) Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models, Quantitative Finance, 21:1, 11-27, DOI: 10.1080/14697688.2020.1817974
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.
Blanka Horvath, Aitor Muguruza & Mehdi Tomas (2021) Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models, Quantitative Finance, 21:1, 11-27, DOI: 10.1080/14697688.2020.1817974
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 and the evaluation of an individual project. 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.
The project involves analyzing the theoretical results of a paper assigned during the course and in the inumerical of theoretical results, and is discussed during the oral exam.
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.
The project involves analyzing the theoretical results of a paper assigned during the course and in the inumerical of theoretical results, and is discussed during the oral exam.
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.
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