Delta Hedging Profit and Loss Simulation¶

Background¶

delta_hedging_pnl_simulation.py

This module implements Delta Hedging P&L Simulation.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" delta_hedging_pnl_simulation.py

This module implements Delta Hedging P&L Simulation.

Author: Financial Math Library """

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

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

def delta_hedging_pnl_simulation(): """ Delta Hedging P&L Simulation.

This function demonstrates the key concepts and computational techniques
for delta hedging p&l simulation.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Delta Hedging P&L Simulation
print(f"Computing Delta Hedging P&L Simulation...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Delta Hedging P&L Simulation"
}

return results

def main(): """Main execution function.""" results = delta_hedging_pnl_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("Delta Hedging P&L Simulation")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. In a delta hedging simulation, the hedger holds \(\Delta_t\) shares at each rebalancing time. Explain the P&L evolution equation and the role of the risk-free rate.

Exercise 2. Why does the hedging P&L distribution have mean near zero but nonzero variance? What determines the variance?

Exercise 3. If the hedging is performed with 1000 time steps over 1 year, estimate the standard deviation of the hedging error for an ATM call with \(S_0 = 1\), \(\sigma = 0.2\).

Exercise 4. The histogram of final P&L is centered near zero and symmetric. Under what conditions would it become asymmetric?