Intensity Model Simulation¶

Background¶

intensity_model_simulation.py

This module implements Intensity Model Simulation.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" intensity_model_simulation.py

This module implements Intensity Model Simulation.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm

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

def intensity_model_simulation(): """ Intensity Model Simulation.

This function demonstrates the key concepts and computational techniques
for intensity model simulation.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Intensity Model Simulation
print(f"Computing Intensity Model Simulation...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Intensity Model Simulation"
}

return results

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

# 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("Intensity Model Simulation")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. In the intensity-based (reduced-form) model, default is the first jump of a Poisson process with stochastic intensity \(h(t)\). Write the survival probability in terms of the integrated intensity.

Exercise 2. Describe the simulation algorithm for generating a default time from a stochastic intensity \(h(t)\).

Exercise 3. If the intensity follows a CIR process \(dh = \kappa(\bar{h} - h)\,dt + \sigma_h\sqrt{h}\,dW\), explain why this is popular for credit modeling.

Exercise 4. Simulate 5 default times using the following procedure: \(E_i \sim \text{Exp}(1)\) values are \(\{0.5, 2.1, 0.8, 3.5, 1.2\}\) and the constant intensity is \(h = 0.03\). Which entities default before year 10?