Statistical Methods for Finance

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.
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. 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.
4.Extreme value theory
Extreme value theory: distribution of maximum and minimum of n iid random variables. Convergence of sums and convergence of normalized maxima. The Generalized Extreme Value (GEV) distribution and its three types: Frechet, Gumbel, Weibull. Maximum domain of attraction and Gnedenko theorem as a counterpart of the Central Limit Theorem. Examples of determination of the limiting distribution. Characterization of the three limits (Frechet, Gumbel Weibull). Fitting the GEV distribution: maxima on n-blocks; maximum likelihood estimation.