Numerical Methods for Finance

A.Y. 2020/2021
Overall hours
Learning objectives
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.
Expected learning outcomes
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.
Course syllabus and organization

Single session

Lesson period
Second trimester
The two lessons will be held exclusively on Microsoft Teams and can be followed either synchronously or asynchronously because it will be recorded and left available to students on the same platform.


Program and material
The program and the material will not change.

The exam rules will be communicated at the beginning of the course.
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
Prerequisites for admission
The students must have some preliminary knowledge of calculus, standard financial mathematics, linear algebra and optimization.
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
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
SECS-S/01 - STATISTICS - University credits: 6
Lessons: 40 hours
Professor: Mercuri Lorenzo
Tuesday from 2.30 p.m. to 5.30 p.m.
Microsoft office Teams (