Statistical methods for finance

A.A. 2021/2022
6
Crediti massimi
40
Ore totali
SSD
SECS-S/01
Lingua
Inglese
Obiettivi formativi
The main goal of this course is to give students the necessary statistical instruments required to deal with multivariate data in modern quantitative finance, focusing in particular on multivariate probability distributions and dependence measures.
The course will first introduce and review the general basic concepts related to multivariate random variables and then will analyze some multivariate models that have found wide application in quantitative finance (multivariate normal model and generalizations thereof). Then, the course will present copulas as a statistical tool for building flexible multivariate models and defining new dependence measures that can better fit and explain specific features present in financial data.
Risultati apprendimento attesi
At the end of the course, the student should know the basic theory of multivariate random variables and the genesis and properties of some noteworthy families of probability distributions, such as the multivariate normal distribution, the multivariate normal variance mixtures, the spherical and elliptical distributions. The student should be also acquainted with the concept of copula and its use in the construction of multivariate distribution, and with copula-based dependence measures which overtake the shortcomings of Pearson's correlation coefficient.
The students is expected to be able to apply this theoretical knowledge by evaluating the applicability of different models from a scientific perspective and choosing the most appropriate distribution for modeling multivariate data in the financial field.
Programma e organizzazione didattica

Edizione unica

Responsabile
Periodo
Terzo trimestre
In relazione alle modalità di erogazione delle attività formative per l'a.a. 2021/22, verranno date indicazioni più specifiche nei prossimi mesi, in base all'evoluzione della situazione sanitaria.
Programma
1.Review of basic concepts for univariate and bivariate random variables
Basic notions of univariate random variables. Bivariate distributions, discrete case: bivariate and marginal probability mass functions. Bivariate distributions, continuous case: bivariate density function, bivariate cumulative distribution function, marginal density functions and cumulative distribution functions. Continuous uniform r.v. Distributions of the minimum and maximum of two independent uniform random variables in (0,1). Skewness and Kurtosis (leptokurtic, mesokurtic and platykurtic distributions). Generalized inverse function and quantile function. Transformation of random variables: methods for recovering the pdf/cdf of a function of a random variable (univariate case). The case of monotone functions; the case of Y=X^2. Characteristic function: definition and main properties. Characteristic function for the normal rv and for the sum of independent normal rvs. Characteristic function and central limit theorem. Inversion theorems. Independence and cf. Property of pdf and cf for symmetrical distributions. Properties and asymptotic distribution of the ecdf. Kolmogorov-Smirnov test. Stable distributions: definition, parametrization, generalization of central limit theorem, link with infinite divisibility.
2.Standard multivariate models
Introduction to multivariate models: joint, marginal and conditional distributions; independence; moments (mean vector and covariance and correlation matrices); linear transformations. Standard estimators of the mean vector and of covariance and correlation matrices. Multivariate transformation method. The multivariate Normal distribution. Definition/construction. Joint density function. Stochastic simulation. Properties of the multivariate Normal distribution: linear transformations, marginal distributions, conditional distributions, quadratic form, convolution. The bivariate case: joint density function, conditional distributions, joint cumulative distribution function (quadrant probability). Exact and approximate distribution of the correlation coefficient for a bivariate normal distribution. Testing normality: 1) univariate case: QQplot; theoretical and sample skewness and kurtosis; Jarque-Bera test 2) multivariate case: Mahalanobis distance and its asymptotic distribution. Multivariate skewness and kurtosis; Mardia test.
Weaknesses of the multivariate normal model. Mixture models: generalities (finite mixtures and compound distributions). Multivariate normal variance mixture models: genesis and first main properties. Characteristic function, linear transformation, density, uncorrelation/independence for multivariate variance mixture models. Examples of mixtures. The univariate and multivariate Student's t distribution. Multivariate normal mean-variance mixture models. Spherical distributions: definitions and chartacterizations, also in terms of rvs R and S. Joint density of a spherical rv. Elliptical distributions: definition. Properties: stochastic representation, characteristic function, linear operations, marginal distributions, conditional distributions, convolutions, quadratic form. Estimating the location vector and dispersion matrix. Testing for elliptical symmetry: QQplots and numerical tests.
3.Copulas
Copulas: introduction and basic properties. Quantile transformation and probability transformation. Sklar's theorem. Copula for a random vector of continuous distributions; copulas and discrete distributions. Invariance of copulas for strictly increasing transformations. Frechet lower and upper bounds. Example of copulas: fundamental copulas (independence copula, comonotonicity copula, countermonotonicity copula); implicit copulas (Gaussian and t copulas). Examples of explicit copulas (Gumbel and Clayton). Meta-distributions: joining arbitrary margins together through a copula; simulation of meta distributions. Survival copulas. Radial Symmetry. Conditional distributions of copulas. Copula density. Exchangeability. Perfect dependence: comonotonicity and countermonotonicity. Dependence Measures. Pearson's correlation: definition. First fallacy of Pearson's rho: The marginal distributions and pairwise correlations of a random vector do not determine its joint distribution. Second fallacy of Pearson's rho: For two given univariate margins and a correlation coefficient in [-1,+1] it is not always possible to construct a joint distribution with those margins and that rho. Attainable correlations for rho. Examples. Correlation and extremal properties of bivariate normal distribution. Kendall's tau; Spearman's rho: definitions and main properties; relationship with the copula C of a bivariate random vector. Relationship between Pearson's rho, Kendall's tau and Spearman's rho for the Gaussian copula. Coefficients of Upper and Lower Tail Dependence: definition and their relation to the copula C of a bivariate random vector. Archimedean copulas. Fitting copulas to data. The method-of-moments approach. The maximum likelihood method and the two-step approach. Step 1: estimating the margins (parametrically or non-parametrically), step 2: estimating the copula parameter via pseudo-sample from the copula. Examples: estimating the Gaussian and t copulas. The R package "copula".
Prerequisiti
Lo studente dovrebbe conoscere le basi di algebra lineare e calcolo differenziale e integrale, di calcolo delle probabilità e statistica inferenziale, assieme a delle minime capacità di programmazione.
Metodi didattici
Lezioni ed esercitazioni. Lezioni teoriche sono sempre combinate con un momento di esperienza pratica, che consiste nella risoluzione di esercizi numerici o nell'implementazione di modelli teorici e metodologie nell'ambiente di programmazione R.

A lezione, il docente utilizza la lavagna e proietta al PC le slide che ha precedentemente caricato nella pagina Ariel del corso; utilizza il PC anche per illustrare l'implementazione in R dei modelli e metodi statistici presentati.
Gli studenti sono incoraggiati a verificare e migliorare le loro abilità attraverso il materiale supplementare fornito loro (principalmente, esercizi), testi di esami precedenti e bibliografia addizionale.
Materiale di riferimento
A.J. McNeil, R. Frey and P. Embrechts: Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton University Press, 2005
A.J. McNeil, R. Frey and P. Embrechts: Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton University Press, 2nd Edition, 2015
J.-F. Mai, M. Scherer: Financial Engineering with Copulas Explained, Palgrave Macmillan, New York, 2014
M. Hofert, I. Kojadinovic, M. Machler, J. Yan: Elements of Copula Modeling with R, Springer, New York, 2018
R.G. Gallager, Stochastic Processes for Applications, Cambridge University Press, 2013
T. Mazzoni, A First Course in Quantitative Finance, Cambridge University Press, 2018
Modalità di verifica dell’apprendimento e criteri di valutazione
L'esame finale consiste di un test scritto che può essere sostenuto in una qualsiasi sessione di esame. Esso consiste di
- un numero (solitamente 15) di domande a scelta multipla; per ogni domanda sono fornite 4 possibili risposte di cui una sola è corretta
- una o più domande teoriche con risposta aperta (diciamo, 150 parole)
- uno o più esercizi numerici
Le domande coprono l'intero programma del corso e bilanciando la teoria con la pratica dovrebbero sperabilmente consentire di valutare le competenze complessive dello studente.
SECS-S/01 - STATISTICA - CFU: 6
Lezioni: 40 ore
Docente/i
Ricevimento:
Per il primo trimestre, lunedì 10-13.
stanza 33, 3° piano DEMM