Volatility of Volatility Sensitivity Analysis¶

Background¶

vol_of_vol_sensitivity.py

This module implements Vol-of-Vol Sensitivity Analysis.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" vol_of_vol_sensitivity.py

This module implements Vol-of-Vol Sensitivity Analysis.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

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

def vol_of_vol_sensitivity(): """ Vol-of-Vol Sensitivity Analysis.

This function demonstrates the key concepts and computational techniques
for vol-of-vol sensitivity analysis.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Vol-of-Vol Sensitivity Analysis
print(f"Computing Vol-of-Vol Sensitivity Analysis...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Vol-of-Vol Sensitivity Analysis"
}

return results

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

# 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("Vol-of-Vol Sensitivity Analysis")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. In the Heston model, the vol-of-vol parameter \(\sigma_v\) controls the curvature of the implied volatility smile. Explain this connection qualitatively.

Exercise 2. The SABR model uses \(dF = \alpha F^\beta\,dW_1\), \(d\alpha = \nu\alpha\,dW_2\). Explain the role of \(\beta\) in controlling the smile shape.

Exercise 3. Explain the leverage effect: why do stock returns and volatility tend to be negatively correlated (\(\rho < 0\))?

Exercise 4. Compare the Heston and SABR models for calibrating to equity index smiles. Which is better for short maturities? For long maturities?