This course is divided into 3 parts. In Module I (Zezza) we will analyze the symplex algorithm and we will study the programming language GAMS, with financial applications in view. In Module II (Livieri) we will study standard portfolio theories (Markowitz, CAPM, C-CAPM APT). In Module III (Scandolo) we will employ binomial trees and Monte Carlo simulation for option pricing.
(Module II - Livieri) Notes provided by the teacher
(Module III - Scandolo) "Numerical methods in finance and economics: a MATLAB-based introduction", P. Brandimarte, Wiley Ed., 2006 (2nd edition)
(Module I - Zezza) The general purpose is to make the student familiar with some numerical solution methods and to use a specific modelling language by analysing a case study and preparing the required code.
(Module II - Livieri) The general purpose is to get acquainted the student with the different
portfolio theories and discuss a research article relative to the topic.
(Module III - Scandolo) The goal of this part is to make the student able to numerically solve, in Matlab, some basic pricing/hedging problems, by employing binomial trees or Monte Carlo simulation.
(Module I - Zezza) The math prerequisite for the course is a basic knowledge of Linear Algebra.
(Module II - Livieri) The are basically no prerequisite for this part of the course
(Module III - Scandolo) It is required to have a basic knowledge of Probability and of derivatives pricing (basically, the contents of the course "Quantitative Finance and Derivatives")
(Module I - Zezza) Academic lessons and practical coding sessions
(Module II - Livieri) Academic lessons
(Module III - Scandolo) Academic lessons and coding sessions
Module I is 6 CFU; Modules II and III are 3 CFU each.
The final grade will be the weighted average of the grades of the 3 Modules.
Type of Assessment
(Module I - Zezza) Linear Programming: Two Homework (5%) - A short presentation of 15 minutes about a given topic (5%) - A Final written exam followed by a short discussion on it (40%)
GAMS: One Homework (10%) - A case study to examine autonomously, including data retrieval and coding (40%).
(Module II - Livieri) Study and (a short) presentation of a research paper
on some of the topics presented in the course.
(Module III - Scandolo) A short essay on some follow-up topic with presentation/discussion.
(Module I - Zezza) This module will be divided into two parts, each weighting 3 CFU each. The general purpose is to make the student familiar with some numerical solution methods. To reach this goal, we will study a specific kind of model and a specific algorithm in the first part while in the second part we will describe a modelling language, available on line, which will be used to solve different kind of models with different kinds of algorithms.
PART 1. Linear Programming. By a case study in finance we will analyse Linear Models which are characterised by a linear objective function and by a set of linear equality and/or inequality constraints. We will introduce the Simplex algorithm which can be used to solve numerically these problems and we will analyse in depth how an algorithm works, the problems which can arise while using it and its efficiency. The course requires some previous knowledge of Linear Algebra.
PART 2. GAMS. The General Algebraic Modelling System, is a modelling language which can be used to describe many different types of models (not only linear ones) and we will use it to describe financial models and to solve them by calling some other computer programs implementing the algorithms.
(Module II - Livieri) This part of the course is mainly theoretical and it is devoted to a review and presentation of the standard portfolios theories. More precisely, We will start by describing the mechanism of the choices under risk and the criterion of expected utility theory. Then, We will move to the so-called mean-variance criterion introduced by Markowitz (1952) and to a comparison of the latter with the criterion of expected utility. In the third and fourth lecture We will give the definition of efficient portfolios and We will study (in-depth) the construction of the efficient frontier. Finally, We will move to analyse the Capital Asset Pricing Model (CAPM), the consumption CAPM (C-CAPM) and the Arbitrage Pricing Theory (APT).
(Module III - Scandolo) (References are to Brandimarte textbook) Summary of the main concepts in derivatives pricing/hedging by no-arbitrage, Black-Scholes model and formula (Sections 2.1,2.5,2.6,2.7). Binomial trees method for pricing european options, calibration issues, pricing of american options and exchange options, trinomial trees (Chapetr 7). Principles of Monte Carlo simulation, pseudo-random numbers, generation of random variates, variance reduction techniques (Sections 4.2,4.3,4.4,4.5). Simulation of a Brownian Motion, Monte Carlo method for pricing/hedging of european, path-dependent, exchange options within Black-Scholes model (Chapter 8). Monte Carlo for option pricing under alternative models (local/stochastic volatility, Heston, SABR).