Gamma Hedging with Options¶

Background¶

gamma_hedging_with_options.py

This module implements Gamma Hedging with Options.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" gamma_hedging_with_options.py

This module implements Gamma Hedging with Options.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

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

def gamma_hedging_with_options(): """ Gamma Hedging with Options.

This function demonstrates the key concepts and computational techniques
for gamma hedging with options.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Gamma Hedging with Options
print(f"Computing Gamma Hedging with Options...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Gamma Hedging with Options"
}

return results

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

# 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("Gamma Hedging with Options")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. A delta-neutral portfolio has \(\Gamma = -0.05\) (short gamma). To gamma-hedge, you buy calls with individual gamma \(\gamma_c = 0.025\). How many calls are needed?

Exercise 2. Explain why gamma hedging requires options (not just the underlying asset) and what tradeoff the hedger faces.

Exercise 3. For a gamma-neutral, delta-neutral portfolio, what is the dominant remaining risk? How would you hedge it?

Exercise 4. A portfolio is short 100 ATM calls (\(\Gamma = -0.02\) each, total \(\Gamma = -2.0\)). ATM puts have \(\Gamma = 0.02\). Compute the put quantity needed for gamma neutrality and the resulting theta cost if each put has \(\Theta = -5\)/year.