Numerical Methods for Finance and Risk Management

A.Y. 2019/2020
Overall hours
Learning objectives
Module Numerical Methods for Finance
The first part of the course aims to provide a good knowledge of stochastic calculus and no arbitrage principles that constitute the foundations in the pricing of financial derivatives. We first discuss the Wiener process, then we move to the construction of stochastic integrals. We also introduce the concept of a martingale measure and its connection with the Fundamental Theorem of Asset Pricing.
The second part of the course aims to introduce students to the main numerical methods for the estimation of stochastic processes and to the numerical evaluation of contingent claims. The main topics presented are: Monte Carlo simulation, parameter estimation of stochastic processes, model selection and calibration.

Module Risk Management
At the end of the course, the student will possess an adequate mathematical terminology, learned the main quantitative and computational tools to be able to work in the risk management unit of a bank or insurance company.
Expected learning outcomes
Module Numerical Methods for Finance
At the end of the course, students should have acquired the fundamentals of stochastic calculus and the main numerical methods for the evaluation of contingent claims. Students should be able to produce scripts in the R programming language for the estimation of a stochastic process that describes the asset price dynamics and evaluate numerically contingent claims based on no arbitrage principles.

Module Risk Management
At the end of the course, the student will know the basic elements of the Basel and Solvency regulatory frameworks for banks and insurance companies; will possess an adequate mathematical terminology and learned the main quantitative tools related to the study of risk variables and measures in quantitative risk management; will be able to recognize statistically the presence of an elliptical or heavy-tailed distribution and determine its influence on a risk portfolio; will be able to code a software for the computation of the capital reserve needed by a financial institution to comply with the above regulatory frameworks; will be aware of the basic quantitative tools to perform the stochastic aggregation of various typologies of risks.
Course syllabus and organization

Single session

Lesson period
Second trimester
Prerequisites for admission
The students must have some preliminary knowledge of calculus, standard financial mathematics, linear algebra and optimization.
Elementary Probability and Integration
Assessment methods and Criteria
I module: Written exam composed of practical exercises and theoretical questions.
Through the theoretical questions, the students have to show that they
understood correctly the theory behind the construction of an optimal portfolio.
As practical exercises, students have to choose the best method

II module: Written test + oral interview + extra assignments
Module Numerical Methods for Finance
Course syllabus
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
Martingale measures and their relations to pricing
Fundamental theorem of asset pricing
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
Introduction to Lévy processes: properties, simulation and estimation
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
Module Risk Management
Course syllabus
Overview of Basel 2, Basel 3 and Solvency 2. Basic Concept in Risk Management: Risk Measures (VaR and ES).
Light tailed versus Heavy tailed distributions. Regularly varying distributions, EVT: the POT method.
Modeling dependence with copulas.
Multivariate Modelling: ''if Only the World Were Elliptical'' - Coherent Measures of Risk .
Standard methods for Market Risk .
Risk Aggregation and Model Uncertainty.
Operational Risk: some case studies.
Teaching methods
Lectures + R programming
Teaching Resources
AJ McNeil, R Frey and P Embrechts,
Quantitative Risk Management: Concepts, Techniques, Tools. Revised Edition.
Princeton University Press, Princeton, 2015;
EXTRA MATERIAL will be provided by the instructor
Module Numerical Methods for Finance
SECS-S/01 - STATISTICS - University credits: 6
Lessons: 40 hours
Professor: Saredi Viola Luisa
Module Risk Management
Lessons: 40 hours
Professor: Puccetti Giovanni
Educational website(s)