#
Statistical Methods for Finance

A.Y. 2020/2021

Learning objectives

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.

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.

Expected learning outcomes

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.

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.

**Lesson period:** Third trimester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Third trimester

The lessons will be held on the Microsoft Teams platform and can be attended both synchronously, according to the third trimester timetable, and asynchronously, because they will be registered and left available to students on the same platform.

If, for some reason, synchronous class is not possible, the lesson will be recorded and uploaded asynchronously.

The program and the reference material will not change.

The exam will always be written, but taken remotely, via the Exam.net and Microsoft Teams platforms. The structure of the exam will not change.

If, for some reason, synchronous class is not possible, the lesson will be recorded and uploaded asynchronously.

The program and the reference material will not change.

The exam will always be written, but taken remotely, via the Exam.net and Microsoft Teams platforms. The structure of the exam will not change.

**Course syllabus**

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".

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".

**Prerequisites for admission**

Students are required to be familiar with linear algebra and differential and integral calculus, with the basics of probability theory and inferential statistics, and to have elemental programming skills.

**Teaching methods**

lectures and practical classes. Theoretical classes are always combined with practical experience, consisting of numerical exercises (to be solved by hand) or implementation of theoretical models and methods in the R programming environment.

During the class, the istructor uses the whiteboard and presents the slides he has provided to the students on the Ariel webpage; the instructor uses the PC to illustrate software implementation of statistical models and methods in the R programming environment.

Students are encouraged to test and enhance their skills through the provided supplementary material (mainly solved exercises), past exams and suggested bibliographical references.

During the class, the istructor uses the whiteboard and presents the slides he has provided to the students on the Ariel webpage; the instructor uses the PC to illustrate software implementation of statistical models and methods in the R programming environment.

Students are encouraged to test and enhance their skills through the provided supplementary material (mainly solved exercises), past exams and suggested bibliographical references.

**Teaching Resources**

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

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

**Assessment methods and Criteria**

The final exam is a unique written test that can be taken during any of the official exam dates. It consists of

- a number (usually 15) of multiple choice questions; for each question, four possible answers are provided of which only one is correct

- one or more theoretical questions with an open answer (say, about 150 words)

- one or more numerical exercises

The questions cover the whole course programme and by balancing theory with practice are hopefully able to check the overall student's competences.

- a number (usually 15) of multiple choice questions; for each question, four possible answers are provided of which only one is correct

- one or more theoretical questions with an open answer (say, about 150 words)

- one or more numerical exercises

The questions cover the whole course programme and by balancing theory with practice are hopefully able to check the overall student's competences.

Professor(s)

Reception:

Office hours are on Monday 10.30-12.30 and 2.30-3.30. Office hours in presence are suspended but they are carried out via Teams, by sending a chat message

Room 33, 3rd floor DEMM