Numerical Methods for Finance and Portfolio Optimization
A.Y. 2023/2024
Learning objectives
The first part of the course (Portfolio Optimization) aims to introduce students to optimization methods for the construction of optimal portfolios. The identification of the optimal strategies will be presented under discrete time setup. In this context, specific methodologies will be discussed based on the nature of the assets in the portfolio.
The second part of the course (Numerical Methods for Finance) aims to provide a good knowledge of stochastic calculus and no arbitrage principles that constitute the foundations in the pricing of financial derivatives. The main numerical methods for pricing contingent claims will be presented during the course.
The second part of the course (Numerical Methods for Finance) aims to provide a good knowledge of stochastic calculus and no arbitrage principles that constitute the foundations in the pricing of financial derivatives. The main numerical methods for pricing contingent claims will be presented during the course.
Expected learning outcomes
At the end of the course students will be able to use the main tools for pricing contingent claims and for constructing optimal portfolio strategies. They will possess a proper terminology and will acquire mathematical tools that allow to cope with numerical/financial problems that arise in financial institutions or in insurance companies. Finally they should be also able to produce scripts in the R programming language for financial analysis.
Lesson period: Second trimester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second trimester
Course syllabus
Numerical Methods for Finance
Binomial tree model for option pricing
Introduction to continuous time stochastic processes
Wiener process
Stochastic differential equations
Ito stochastic integral and Ito formula
Black and Scholes Model
Simulation of stochastic differential equations
Monte Carlo approach to option pricing
Parameter estimation from discretely observed stochastic differential equations
Historical and implied volatility
Calibration Methods.
Portfolio Optimization
1 Asset-Liability Management
1.a Review of Bond Evaluation, Duration, Convexity
1.b Immunization Theory
- Fisher and Weil Theorem
- Redington Theorem
- Advances
2 Optimal Portfolio Selection: preliminaries
2.a Preferences Representation and Risk Aversion
2.b Stochastic Dominance
2.c Mathematics of Portfolio Frontier
3. Optimal portfolio in one period models.
3.a Finite State Market model
3.b No-arbitrage Condition and Complete Market: Equivalent Martingale Measure
3.c Standard Portfolio Optimization methods.
3.d Portfolio Optimization using the Equivalent Martingale Measure
4 Introduction of Optimal Dynamic Portfolio Selection in a Discrete-Time Framework
Binomial tree model for option pricing
Introduction to continuous time stochastic processes
Wiener process
Stochastic differential equations
Ito stochastic integral and Ito formula
Black and Scholes Model
Simulation of stochastic differential equations
Monte Carlo approach to option pricing
Parameter estimation from discretely observed stochastic differential equations
Historical and implied volatility
Calibration Methods.
Portfolio Optimization
1 Asset-Liability Management
1.a Review of Bond Evaluation, Duration, Convexity
1.b Immunization Theory
- Fisher and Weil Theorem
- Redington Theorem
- Advances
2 Optimal Portfolio Selection: preliminaries
2.a Preferences Representation and Risk Aversion
2.b Stochastic Dominance
2.c Mathematics of Portfolio Frontier
3. Optimal portfolio in one period models.
3.a Finite State Market model
3.b No-arbitrage Condition and Complete Market: Equivalent Martingale Measure
3.c Standard Portfolio Optimization methods.
3.d Portfolio Optimization using the Equivalent Martingale Measure
4 Introduction of Optimal Dynamic Portfolio Selection in a Discrete-Time Framework
Prerequisites for admission
The students must have some preliminary knowledge of calculus, standard financial mathematics, linear algebra and optimization.
Elementary Probability and Integration
Elementary Probability and Integration
Teaching methods
Classroom and laboratories
Teaching Resources
Bjork, T. (2009) Arbitrage Theory in Continuous Time. Oxford University press, 2009
Iacus, S.M. (2011) Option Pricing and Estimation of Financial Models with R, John Wiley & Sons, Ltd., Chichester, 472 page, ISBN: 978-0-470-74584-7
Iacus, S.M. (2008) Simulation and Inference for Stochastic Differential Equations: with R examples, Springer Series in Statistics, Springer NY, 300 pages, ISBN: 978-0-387-75838-1
Barucci E., Fontana C. "Financial Markets Theory: Equilibrium
Efficiency and Information" Second Edition Springer (Chapters 2,3,6)
Cornuéjols G., Pena J., Tutuncu R. "Optimization Methods in Finance" Second Edition
Cambrige University Press (Chapters 3,5,6,7,11,12,14)
Iacus, S.M. (2011) Option Pricing and Estimation of Financial Models with R, John Wiley & Sons, Ltd., Chichester, 472 page, ISBN: 978-0-470-74584-7
Iacus, S.M. (2008) Simulation and Inference for Stochastic Differential Equations: with R examples, Springer Series in Statistics, Springer NY, 300 pages, ISBN: 978-0-387-75838-1
Barucci E., Fontana C. "Financial Markets Theory: Equilibrium
Efficiency and Information" Second Edition Springer (Chapters 2,3,6)
Cornuéjols G., Pena J., Tutuncu R. "Optimization Methods in Finance" Second Edition
Cambrige University Press (Chapters 3,5,6,7,11,12,14)
Assessment methods and Criteria
Written exam composed of practical exercises and theoretical questions + assignments (oral exam for students who do not submit the assignments)
SECS-S/01 - STATISTICS - University credits: 6
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES - University credits: 6
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES - University credits: 6
Lessons: 80 hours
Professor:
Mercuri Lorenzo
Educational website(s)
Professor(s)
Reception:
Thursday from 1.15 pm to 4.15 pm
3rd floor room 33 or Teams (send me an email to schedule a meeting)