Knowledge of the main computational algorithms for the solution of standard numerical problems like integration, differentiation, root finding, Monte Carlo simulation, differential equations. As a first step, the algorithms will be described from a theoretical point of view. As a second step, the main features and issues related to the numerical implementation of these algorithms will be explored through examples taken mainly from Finance.
The students will develop the ability to solve common problems numerically by using the algorithms described in the lectures (possibly modified and optimized for the specific problem at hand).
The students will be able to critically analyze computational algorithms and their numerical outcomes (efficiency, accuracy, stability, rate of convergence, rounding and machine-precision effects).
Prerequisiti
Basic calculus (functions of real variables, limits, derivatives, integrals, differential equations). Basic notions of linear algebra (linear systems). Basics notions of probability and statistics (probability distributions, mean and standard deviation calculation, expectation values).
Metodi didattici
Frontal teaching. Classes will cover both the theoretical foundations of the main algorithms, and their numerical implementation with Matlab in lab-like sessions.
Verifica Apprendimento
Computer-based test focused on the solution of simple numerical problems via Matlab programming.
Testi
[1] P. Brandimarte: Numerical Methods in Finance and Economics, Wiley. [2] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery: Numerical Recipes, Cambridge University Press. [3] Slides provided by the teacher. [4] Free online resources.
Contenuti
Introduction to Matlab and its main commands. Numbers and their computer representation. Deterministic integration. Numerical derivatives. Root finding. Optimization problems. Fitting of functions and related problems (overfitting). Fitting of probability distributions and maximum likelihood methods. Monte Carlo simulation and integration: pseudo random-number generation, path simulation, statistical foundations of the Monte Carlo method (main applications: option pricing). Differential equations (ODE/PDE).
