Market Comovements and Copula Families

1. Market Comovements and Copula Families

2. Measures of Associations • Definition 1:(concordant, discordant) Observations (x1, y1) and (x2, y2) are concordant if (x1 – x2)(y1 – y2) > 0. They are discordant if (x1 – x2)(y1 – y2) < 0. • Definition 2: MX,Y = MC is a measure of concordance between random variables X and Y, with copula C iff • It is defined for every pair of r.v.’s (completeness) • MX,Y  [-1, 1]

3. Measures of Associations • Definition 2 (cont.) • MX,Y = MY,X • If X and Y are independent, then MX,Y = 0 • M-X,Y = MX,-Y = -MX,Y • If {(Xn, Yn)} is a sequence of continuous rnadom variables with copulas Cn, and lim Cn = C, then lim MXn,Yn = MX,Y • If C1 C2 then MC1 < MC2

4. Measures of Associations • Theorem 1. Measure of concordance is invariant under monotonic transformation of the random variables. • Theorem 2. If X and Y are comonotone, then MX,Y = 1; if they are countermonotone, then MX,Y = -1.

5. Measures of Associations • Kendall’s :Note: for Frechet’s lower bound copula  = -1, and for Frechet upper bound copula  = 1.

6. Measures of Associations • Theorem 3.3. Let C1 and C2 be copulas. ThenAn unbiased estimator of  is the Kendall’s sample :

7. Measures of Associations • Theorem 3.4. The Kendall’s  of a copula and its associated survival copula coincide. • Definition: Spearman’s .Let X and Y be random variables with distributions F1 and F2 and joint distribution F. The population Spearman’s rank correlation is given by s = (F1(X), F2(Y)), where  is the usual linear correlation.

8. Measures of Associations • The sample estimator of the Spearman’s rank correlation is given byOther alternative formulae for s: • Theorem 3.5. Spearman’s  of a copula and ists associated survival copula coincide.

9. Measures of Associations • Linear Correlation: • Theorem 3.6. Linear correlation satisfies all the axioms of the definition2 except the 6th one. • Property 1: XY is invariant under linear transformation, and not under non-linear transformations. • Example: Let (X, Y) have a bivariate standard normal distribution, with correlation coefficient XY. Let Z1= (X), and Z2 = (Y), where (.) is the standard normal cdf. Then (X)(Y) = (6/)arcsin(XY/2).

10. Measures of Associations • Hoeffding expression for covariance:where D = Dom F1 Dom F2. • Property 2. XY is bounded: l≤ XY≤ u, where

11. Tail Dependence • Definition: Let X and Y be random variables with cdf’s F1 and F2. The coefficient of upper (lower) tail dependence of X and Y is: Alternatively:

12. Positive Quadrant Dependency • Definition: The r.v.’s X and Y are positive quadrant iff C(u,v)  uv for all (u, v) in I2.Note that PQD implies the non-negativity og Kendall’s , Spearman’s , and of the linear correlation coefiicient.

13. The Gaussian Copula • Definition: The Gaussian copula is defined as follows: • Theorem 3.6. The Gaussian copula generates the joint normal standard distribution function, via Skar’ theorem, iff the marginals are standard normal. • Since the Gaussian copula is parameterized by the linear coefficient correlation , and that  respects concordance, the Gaussian copula is positively ordered in the following sense:

14. The Gaussian Copula • For  = -1, the Frechet lower bound is obtained, and for  = 1, the Frechet upper bound copula is obtained. Therefore the Gaussian Copula is comprehensive. • For any other marginal choice, the Gaussian copula does not give a standard jointly normal vector. For example the effect of marginal Student distribution is to increase the tail probabilities.

15. The bivariate Student’s t copula • Definition: The bivariate Student’s copula T,v is given by:this copula converges to the Gaussian copula as the degree of freedom increases.The Student’s t copula presents more observations in the tails than the Gaussian copula.

16. The Frechet family • Definition: The Frechet family of copula is the following two parameter copula: CF(u,v) = pmax(u + v -1, 0) + (1 – p – q)uv +qmin(u,v)The Frechet density is CF(u,v) = 1 – p – q