Learning objectives
The interaction between statistics, probability and finance is an ongoing process: the solution of statistical problems is the necessary condition for evaluating the elements of uncertainty in the markets as well as probability theory is fundamental to model uncertainty in the market. The course aims to provide the basic tools most suitable for the analysis and modelization of some key aspects of monetary and financial markets. In the first part of the course particular attention will be paid to the historical series of financial issues: exchange rates, interest rates, prices and stock returns, prices and yields of derivatives.
In the second part, the student will learn the basic principles of arbitrage pricing and completeness in the market, and the mathematical modeling of an elementary market model. In addition, the student will be familar with the representation of preferences for a rational decision maker and the optimal selection of a portfolio, given the returns and covariances of the traded assets.
Prerequisites
Basic elements of calculus, financial mathematic and statistics.
Course unit content
MODULO 1; Elementary theory of stochastic processes for stationary series
1. Recalls elements of probability 'for random vectors.
2. Transformation of univariate and multivariate random variables.
3. Gaussian and White Noise processes.
4. Brief introduction to non-stationary processes of type Random Walk
/ Empirical evidence of the observed time series
1. Empirical characteristics of the time series of financial returns. Formulas combinations of multi-period returns.
2. The shape of the distribution of returns. Test of symmetry, kurtosis, and normality .
3. The time dependence (linear and nonlinear) of returns. Autocorrelation function and tests of significance 'associates.
4. Autoregressive processes for stationary series of returns and transforms associated with them.
/ Overview of analysis of the trend of stock market prices and moving averages
MODULO 2
Introduction to probability theory: the various approaches. The axiomatic approach. conditional probability and Bayes'theorem. Random numbers: the discrete case 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's model.
Full programme
MODULO 1
Elementary theory of stochastic processes for stationary series
1. Recalls elements of probability 'for random vectors.
2. Transformation of univariate and multivariate random variables.
3. Gaussian and White Noise processes.
4. Brief introduction to non-stationary processes of type Random Walk
/ Empirical evidence of the observed time series
1. Empirical characteristics of the time series of financial returns. Formulas combinations of multi-period returns.
2. The shape of the distribution of returns. Test of symmetry, kurtosis, and normality .
3. The time dependence (linear and nonlinear) of returns. Autocorrelation function and tests of significance 'associates.
4. Autoregressive processes for stationary series of returns and transforms associated with them.
Overview of analysis of the trend of stock market prices and moving averages
MODULO 2
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, with zero e non-zero interest rate.
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's model.
Bibliography
MODULO 1:Slides on all topics of the course
MODULO 2
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
During the classes, a theoretical exposition of the contents of the course will be given.
In addition a great number of examples and exercises will be discussed,
with a particular focus on the financial examples. The students will be asked to discuss and propose possible solutions to the exercises.
Assessment methods and criteria
Written exam. The student will be asked to solve problems and to answer to theoretical questions on the topics covered during the course.
Other information
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