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Department of Economics, Social Studies, Applied Mathematics and Statistics

# Quantitative Finance and Insurance

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## NUMERICAL AND STATISTICAL METHODS FOR FINANCE

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Course ID
ECO0152
Teachers
Stefano Favaro (Lecturer)
Degree course
Finance
Insurance and Statistics
Year
1st year
Teaching period
Annual
Type
Distinctive
Credits/Recognition
12
Course disciplinary sector (SSD)
SECS-S/01 - statistics
Delivery
Formal authority
Language
English
Attendance
Obligatory
Type of examination
Written
Type of learning unit
corso
Prerequisites
It is very important for the students to be familiar with the basic topics in mathematics, probability and statistics acquired in the three-year undergraduate program. These topics are presented in the short course "Essentials of Mathematics and Probability" usually given in September: see www.masters-finins.unito.it/ for more details.
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## Course objectives

Ability to solve, through the use of simulation tools, some standard problems in probability and statistical inference. Ability to apply statistical concepts and statistical techniques with respect to the point estimation, hyphotesis testing and confidence sets. Ability to the code with the language R and to use some of its main packages.

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## Results of learning outcomes

Knowledge and understanding
Advances knowledge of statistical modeling via point estimation, hypothesis testing and confidence intervals; basic knowlegde of Monte Carlo simulation techniques for statistical models; basic knowlegde of the language R.

Applying knowledge and understanding
Ability to convert various problems of practical interest into statistical models and make inference on it; ability to implement a Monte Carlo simulation of a statistical model using the language R.

Making judgements
Students will be able to discern the different aspects of statistical modeling and of  Monte Carlo simulation with the language R.

Communication skills
Students will properly use statistical and probabilistic language arising from the classical statistics and Monte Carlo simulation; students will properly use the language R.

Learning skills
The skills acquired will give students the opportunity of improving and deepening their knowledge of the different aspects of statistical modeling and Monte Carlo simulation using the language R.

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## Program

1. Statistics: The module deals with some key themes of the theory of statistical inference, with emphasis on the role of the likelihood function. Topics include

• Random samples and their distributions, the statistical model, the likelihood function, exponential family.
• Sufficient statistics and minimal sufficient statistics, finite properties for estimators, asymptotic properties for estimators, methods for evaluating estimators.
• Methods for constructing point estimators: method of moments and generalizations, method of the least square errors, method of maximum likelihood, methods of minimum distance.
• Hypothesis testing: probabilistic structure of hypothesis testing, Neyman-Pearson lemma, likelihood ration tests, asymptotic tests, confidence sets; nonparametric tests

2. Simulation: this module introduces various computational statistical methods. In particular, the program includes some computationally intensive methods in statistics, such as Monte Carlo methods. An important part of the module will be devoted to practicals. All the methods discussed during the course will be implemented in the R software.

Topics include:

• Preliminaries:
• Random variables/vectors and probability distributions;
• Theorems for sequencies of random variables.
• Transformations of random variables/vectors.
• Introduction to R software.
• Pseudo-random number generators.
• Generating discrete and continuous random variables:
• The Inverse-transform method;
• The Transformation method;
• The Acceptance-Rejection method;
• The Polar Method for generating Normal random variables;
• The Composition method.
• Generating continuous random vectors:
• The Multivariate Normal;
• Copulas.
• Monte Carlo integration methods.
• Variance reduction techniques:
• Antithetic Variables;
• Control Variates;
• Importance Sampling.
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## Course delivery

Until further notice, for the AY 2022/2023 the teaching modality is foreseen to be in presence.

With regards to statistics, the course is composed of 48 hours of lectures, including lectures dedicated to excercices. Non-mandatory 30 hours of TA sessions adjoint to the lectures are also provided.

With regards to simulation, lectures are mainly devoted to the probability theory and theory of Monte Carlo simulation. The course is composed of 48 hours of lectures and laboratories, where students can practice on R coding with the supervision of the instructor. Non-mandatory 30 hours of TA sessions adjoint to the lectures are also provided.

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## Learning assessment methods

With regards to statistics, the exam has the durantion of 1 hour and 15 minutes and it consists of three parts

1) an exercise on the topics (probability) presented during the preliminary course taught by Cecilia Scarinzi; the maximum score for the excercise is 3/33

2) a question requiring a formal discussion of one of the main topics of statistical infence based on the likelihood function; the maximum score for this question is 18/33

3) an exercise on the topics (statistics) presented during the course; the maximum score for the excercise is 12/33

Until further notice, for the AY 2021/2022 the exam is foreseen to be in presence.

With regards to simulation, the exam has the duration of 1:30 hour and it consists of exercises and theory questions. Points attributed to each question/exercise depend on the complexity of the exercises/questions they refer to. Exercises can require to solve theoreticl problems, to draft an R-script, to complete a given code or to comment an output. More specific instructions will be uploaded on Moodle.

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## Support activities

No extra activities.

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Book
Title:
Introduction to scientific programming and simulation using R
Year of publication:
2014
Publisher:
Chapman & Hall/CRC
Author:
Jones, Owen Dafydd - Maillardet, Robert - Robinson, Andrew P.
ISBN
Notes:
Disponibile online su piattaforma Ebsco
Required:
No

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Book
Title:
Statistical computing with R
Year of publication:
2019
Publisher:
Chapman & Hall/CRC
Author:
Rizzo, Maria L.
ISBN
Notes:
Disponibile online su piattaforma Ebsco
Required:
No

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Book
Title:
Simulation
Year of publication:
2013
Publisher:
Author:
Ross, Sheldon M.
ISBN
Notes:
Disponibile online su piattaforma Sciencedirect
Required:
No

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## Class schedule

Enroll
• Open
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Last update: 07/05/2024 11:47
Location: https://www.finance-insurance.unito.it/robots.html