Laurea Magistrale (M.Sc.) in Stochastics and Data Science

Computational methods for statistics

Computational methods for statistics

Academic year 2022/2023

• NUMERICAL AND STATISTICAL METHODS FOR FINANCE (ECO0152)
  Quantitative Finance and Insurance

Sommario del corso

• Course objectives
• Results of learning outcomes
• Course delivery
• Learning assessment methods
• Program
• Suggested textbooks and readings
• Notes
• Teaching tools

Course objectives

Simulation is the imitation of a real-world process and it represent a tool for analyzing a stochastic system. In finance, a basic model for the evolution of stock prices, interest rates, exchange rates etc. would be necessary to determine a fair price of a derivative security. In insurance, detailed analyses of data are needed to quantify risks. This course furnishes the building blocks of the basic techniques used to model the behavior of random outcomes (e.g., lifetime duration, severity of risk, correlation between assets, portfolio losses) and to conduct quantitative and statistical analyses to evaluate (market, credit, liquidity) risk by using the statistical software R (open source).

Results of learning outcomes

• Knowledge and understanding
  Knowledge of Monte Carlo simulation techniques for statistical models; basic knowledge of the language R.
• Applying knowledge and understanding
  Ability to convert various problems of practical interest into statistical models; 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. Students will gain ability to solve, through the use of simulation tools, some standard problems in probability and statistical inference and ability to co­­de with the language R by some of its main packages.

Course delivery

This course is borrowed from the simulation module (6 creditis) of the course   (NUMERICAL AND STATISTICAL METHODS FOR FINANCE (ECO0152)  of the Master in Quantitative Finance and Insurance. The two modules, Statistics and Simulations, of the course Numerical and Statistical Methods for Finance are delivered in parallel. Students have to follow only the Lectures on Simulations. 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.

A module of the course, included in the overall courseload, will be taught by Visiting Professor Marie Kratz (ESSEC Business School) on Extreme Values and Copulas.

All classes are foreseen to be taught in presence. All course material will be posted on Moodle.

Learning assessment methods

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 theoretical problems, to draft an R-script, to complete a given code or to comment an output. More specific instructions will be uploaded on Moodle.

Program

• 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:
    
    • Control Variates;
    • Importance Sampling.

1. Simulation 4th edition, Ross. S.M. (2006) -- Academic Press.
2. Statistical Computing with R  (Second Edtion), Rizzo, M.L. (2015) -- Chapman & Hall/CRC The R Series.
3. Introduction to scientific programming and simulation usig R, Jones, O., Maillardet, R. and Robinson A. (2009) -- Chapman and Hall/CRC.

Class schedule

Note

Class schedule available here.

• Teaching materials
• Exams and finals
• Go to Moodle
• Visit the forums