SABR Calibration to Market Smiles¶

Background¶

sabr_calibration_market_smiles.py

This module implements SABR Calibration to Market Smiles.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" sabr_calibration_market_smiles.py

This module implements SABR Calibration to Market Smiles.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt from scipy.optimize import minimize

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

def sabr_calibration_market_smiles(): """ SABR Calibration to Market Smiles.

This function demonstrates the key concepts and computational techniques
for sabr calibration to market smiles.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of SABR Calibration to Market Smiles
print(f"Computing SABR Calibration to Market Smiles...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "SABR Calibration to Market Smiles"
}

return results

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

# 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("SABR Calibration to Market Smiles")
ax.grid(True, alpha=0.3)

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