Developments in modelling dependence

1. Developments in modelling dependence Bas J.M. Werker Tilburg University 31 March 2008

2. Dependence in finance • Dependence of stochastic/random quantities is a key concept in finance • Between different asset returns:Diversification • Between asset return and risk factor:Hedging • This tutorial: two random variables • more random variables: curse of dimensionality

3. Applications • Joint risk measures • see Paul’s presentation earlier • Joint modelling of rare events • Credit risk • CDO valuation • Note: if you are a truly mean-variance agent, you don’t care • effect on asset allocation will depend on utility function

4. Application: Joint risk measure • Daily Nasdaq/S&P500 returns 1972-2007 • Correlation 79%

5. Probability of joint extremes • What’s the probability of both Nasdaq and S&P500 daily return below their respective quantiles?

6. Important questions • How to measure dependence? • correlation is about the worst idea one can have • Is dependence a one-dimensional concept? • No, full description by copula • Extremal dependence • diversification works least when you need it most • Time-variation in dependence

7. Measuring dependence Zero dependence according to most measures

8. Bond price versus yield Negative though perfect nonlinear dependence (corr. -93.8%)

9. Black-Scholes call price vs moneyness Positive perfect nonlinear dependence (corr. 98.1%)

10. Significance of 90% correlation • Consider two individual assets with identical risk/return properties • say mean return 8%, volatility 40% • Mean-variance agent will always choose equal weights • in case of perfect dependence vol = 40% • in case of 90% dependence • Certainty equivalent (γ=2): 80bp

11. Correlation: deficiencies • Correlation is an intrinsically linear concept • non-linear transformation of variables changes correlation • Correlation needs ‘existence of variances’ • excludes seriously fat tails • Generally: Correlation is fine if and only if data is Gaussian/normally distributed • … and not focusing on extremes (later)

12. Alternative dependence measures • Kendall’s tau • (normalized) difference between number of concordant and discordant data pairs • Spearman’s rho • correlation between ranks of data • Key idea: ranks contain all information about dependence • and order statistics about the marginals

13. Alternative dependence measures 2 • With respect to correlation • these measures are invariant to marginal transformations (bond price vs yield) • but still one-dimensional and thus only summarize dependence information • cannot be used to simulate data • Full information on dependence is provided in copula

14. Sklar’s theorem and copulas • Copulas provide a way to disentangle information about • marginal distribution of X • marginal distribution of Y • dependence of X and Y • Definition: A copula is the joint distribution of X and Y after they have (marginally) been transformed to uniform random variables • Probability transform

15. Mathematics • Let X be a random variable with a given probability distribution function F

16. Mathematics 2 • F(X) then has a standard uniform distribution • In case of X and Y with distribution functions FX and FY we can construct two uniform random variables • U = FX(X) • V = FY(Y) • The copula is by definition the joint distribution of U and V

17. Sklar’s theorem • ‘Any’ multivariate distribution can uniquely be described by • each of the marginal distributions • the copula

18. Copulas • Independence copula • Perfect positive dependence copula • X = H(Y) with H increasing • Perfect negative dependence copula • X = H(Y) with H decreasing • Gaussian copula • t-copula • has nothing to do with marginal fait tails! • Archimedian copulas

19. Gaussian copula: rho = 0

20. Gaussian copula: rho = 0.5

21. Gaussian copula: rho = -0.5

22. Gaussian copula: rho = 0.9

23. Which copula to use? • Empirical question • no convincing answer yet • fortunately in some cases (exotic derivative valuation) the answers seem not too sensitive to the copula in some situations • Popular choice: t copula • Which copula not to use? • the Gaussian since extremal indepence

24. Empirical copula • Empirical copula daily Nasdaq/S&P500 returns

25. Empirical copula 2 • Extremal dependence • Recall VaR calculations before • Asymmetry lower-left versus upper-right • Not too visible graphically due to large number of observations

26. Extremal dependence • Riskmanagement is often about extreme events • what happens ‘in the left tail’ • extreme value theory is traditionally univariate • Multivariate extremes have been introduced by Laurens de Haan (Tilburg) and Sid Resnick (Cornell) • This concept easily illustrates failure of Gaussian copula for financial modelling • extremal independence

27. Extremal dependence 50 bivariate random variables and cut-off

28. Extremal dependence 2 Correlation of 1,000,000 normal variables below cut-off

29. Mathematics • In the left-tail (joint crashes) the multivariate normal distribution/copula induces independence • This can, clearly, not be solved by using fat-tailed marginal distributions • Use copula with ‘extremal dependence’ • t-copula • some Archimedian copulas • …

30. Time-varying dependence • Copulas can be used with models for stochastic volatility etc. • copula should then be interpreted as copula for some innovations • Ample evidence of time-varying dependence • periods of high/low ‘correlation’ • dynamic correlation models • re-do this with ‘descent’ dependence measure

31. Summary and conclusions • Traditional dependence concept “correlation” is intrinsically linear • Modern concepts will use classical (1950s) probabilistic ideas • Full dependence model is provided by copula • Which one to use: empirical issue • calibration, but certainly not Gaussian one

32. Further reading • Nelsen (1999), “An Introduction to Copulas “, Springer • “Credit Lyonnaise” papers series • Cherubini, Luciano, Vecchiato (2004), “Copula methods in finance”, Wiley • Many, many other papers …