Learning objectives
The aim is to provide the necessary tools to evaluate financial instruments. During the course, students will learn the basic concepts of probability theory employed to construct and analyze financial market models under uncertainty, and the basic principles of arbitrage-free valuation and market completeness, notions which will be analyzed in detail in an elementary model, but which can be easily extended to more complicated models. Finally, we will illustrate how to represent preferences of a rational decision maker and how to optimally select of a portfolio when asset returns and covariances are given. The methodologies described will also be implemented in Matlab environment. At the end of the course, the student will be able to construct an elementary financial market model in presence of uncertainty, to analyze its properties and compute under this framework the prices of the derivatives and portfolio strategies.
Prerequisites
Basic elements of calculus and financial mathematics.
Course unit content
Introduction to probability theory: the various approaches. The axiomatic approach, conditional probability and Bayes' theorem. Random numbers: the discrete and the continuous case. Random vectors. Basic notions on financial markets. One-period financial market. Fundamental theorems of asset pricing. Pricing of derivatives. Introduction to expected utility theory. Portfolio selection: Markowitz model.
Full programme
Introduction to probability theory. Classical, empirical and subjective approaches. Axiomatic approach: sample space, sigma-algebra and probability measure. Axioms of probability. Conditional probability, Bayes' theorem. Random numbers, measurability. Distribution function. Discrete random numbers: probability mass function. Continuous random numbers: density function.
Expectation, variance and standard deviation. Moments of a random number.
Random vectors. Independent random numbers. Covariance and correlation.
Introduction to financial market. A 1-period financial market: case without and with interest rate.
The law of one price. Arbitrage and completeness. State price densities and risk-neutral probabilities. Fundamental theorems of asset pricing. Derivatives: call and put options. Put-call parity. Forward contracts and forward prices.
Introduction to expected utility theory. Von-Neumann-Morgenstern axioms. Expected Utility theorem. Portfolio selection: Mean-variance principle. Markowitz model.
Bibliography
E. CASTAGNOLI, Brevissimo Abbecedario di Matematica Finanziaria, scaricabile dalla sezione "materiali didattici" o disponibile presso il Centro fotocopie della Facoltà.
E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Probability. A Brief Introduction, 2° edizione, Egea, 2009
Teaching methods
The lectures will be recorded and the relevant link to the video will be available on Elly. The theoretical contents of the course will be presented rigorously. They will be accompanied by a wide discussion of examples and exercises to be solved with just pen and paper and Matlab software, with particular attention to those of a more financial nature. To solicit the participation of the students, they will be asked to solve these exercises.
Assessment methods and criteria
The exam consists of various assignments administered during the lectures and a final Matlab test. The final grade derives 70% from the grade obtained in the final test (out of thirty) and 30% from the grade of the assignments (out of thirty).
The solution of the final test will be made available to Elly within a week of the exam. The result of the exam will be published on Elly within 10 days of the exam. For the rules on the attribution of the final grade and praise, please refer to the syllabus of the entire course.
Other information
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