Default Correlation (Gaussian Copula)¶

Background¶

default_correlation_copula.py

This module implements Default Correlation Gaussian Copula.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" default_correlation_copula.py

This module implements Default Correlation Gaussian Copula.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

======================================================================¶

def default_correlation_copula(): """ Default Correlation Gaussian Copula.

This function demonstrates the key concepts and computational techniques
for default correlation gaussian copula.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Default Correlation Gaussian Copula
print(f"Computing Default Correlation Gaussian Copula...")

# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Default Correlation Gaussian Copula"
}

return results

def main(): """Main execution function.""" results = default_correlation_copula()

# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("Default Correlation Gaussian Copula")
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("/tmp/default_correlation_copula.png", dpi=150)
print(f"Figure saved to /tmp/default_correlation_copula.png")
plt.close()

return results

if name == "main": main() ```

Exercises¶

Exercise 1. The Gaussian copula model generates correlated default times using the formula \(\tau_i = F_i^{-1}(\Phi(X_i))\) where \(X_i = \sqrt{\rho}\,M + \sqrt{1-\rho}\,Z_i\). Explain each component.

Exercise 2. If \(\rho = 0.3\), \(M = -2\) (economic downturn), and \(Z_i = 0\) (average idiosyncratic risk), compute \(X_i\) and the implied default probability percentile.

Exercise 3. Explain the "correlation smile" problem with the Gaussian copula: why does a single \(\rho\) fail to match the market prices of CDO tranches across all detachment points?

Exercise 4. How would you modify the model to address the correlation smile? Name two alternative copula specifications.