Learning outcomes: Explain the purpose of copula functions and the translation of the copula equation. Describe the Gaussian copula and explain how to use it to derive the joint probability of default of two assets. Summarize the process of finding the default time of an asset correlated to all other assets in a portfolio using the Gaussian copula
Questions:
505.1. What is a copula function?
a. A copula correlates a stock price (S) and its volatility (σ) which allows it to model equity implied volatility skew
b. A copula function joins (n) univariate distributions to one multivariate (n-dimensional) distribution
c. A copula function solve for the conditional P[X|Y] as function of the product of P[Y|X] and the ratio of marginal probabilities P[X]/P[X]
d. A copula function is a limiting (special) case of the the Pearson model, such that it only analyzes linear relationship between variables
505.2. About the Gaussian copula utilized for financial applications, Meissner says each is true EXCEPT which is not?
a. The Gaussian copula has low tail dependence which is a weakness because dependencies (including correlations) increase in a crisis
b. The Gaussian copula is difficult to calibrate to market prices; for example, it is difficult to calibrate CDO tranches with a single correlation model
c. The Gaussian copula is principally static and consequently allows only limited risk management; i.e., there is no stochastic process for the critical underlying variables’ default intensity and default correlation
c. The Gaussian copula is limited to market risk applications because it requires (n*n) pairwise correlation parameters, which is natural to a covariance matrix, but in credit risk there is no theoretical way to assume these values when they are pairwise default correlations
(Source: Gunter Meissner, Correlation Risk Modeling and Management, (New York: Wiley, 2014))
505.3. Assume the marginal one-year default probabilities for asset (A) and asset (B), respectively, are 5.0% and 7.0%. If the Gaussian correlation coefficient is 0.30, which is nearest to the joint probability of default in year one, assuming the two assets are jointly bivariately distributed? (please note this requires a spreadsheet or software; e.g., )
a. 0.240%
b. 0.480%
c. 1.353
d. 2.141%
Answers:
Questions:
505.1. What is a copula function?
a. A copula correlates a stock price (S) and its volatility (σ) which allows it to model equity implied volatility skew
b. A copula function joins (n) univariate distributions to one multivariate (n-dimensional) distribution
c. A copula function solve for the conditional P[X|Y] as function of the product of P[Y|X] and the ratio of marginal probabilities P[X]/P[X]
d. A copula function is a limiting (special) case of the the Pearson model, such that it only analyzes linear relationship between variables
505.2. About the Gaussian copula utilized for financial applications, Meissner says each is true EXCEPT which is not?
a. The Gaussian copula has low tail dependence which is a weakness because dependencies (including correlations) increase in a crisis
b. The Gaussian copula is difficult to calibrate to market prices; for example, it is difficult to calibrate CDO tranches with a single correlation model
c. The Gaussian copula is principally static and consequently allows only limited risk management; i.e., there is no stochastic process for the critical underlying variables’ default intensity and default correlation
c. The Gaussian copula is limited to market risk applications because it requires (n*n) pairwise correlation parameters, which is natural to a covariance matrix, but in credit risk there is no theoretical way to assume these values when they are pairwise default correlations
(Source: Gunter Meissner, Correlation Risk Modeling and Management, (New York: Wiley, 2014))
505.3. Assume the marginal one-year default probabilities for asset (A) and asset (B), respectively, are 5.0% and 7.0%. If the Gaussian correlation coefficient is 0.30, which is nearest to the joint probability of default in year one, assuming the two assets are jointly bivariately distributed? (please note this requires a spreadsheet or software; e.g., )
a. 0.240%
b. 0.480%
c. 1.353
d. 2.141%
Answers:
