Robust Hedging Simulation¶

Background¶

robust_hedging_simulation.py

This module implements Robust Hedging Simulation.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" robust_hedging_simulation.py

This module implements Robust Hedging Simulation.

Author: Financial Math Library """

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

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

def robust_hedging_simulation(): """ Robust Hedging Simulation.

This function demonstrates the key concepts and computational techniques
for robust hedging simulation.

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

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Robust Hedging Simulation"
}

return results

def main(): """Main execution function.""" results = robust_hedging_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("Robust Hedging Simulation")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. Robust hedging seeks a hedging strategy that performs well under model uncertainty. Explain the difference between delta hedging under a specific model and robust hedging.

Solution to Exercise 1

Delta hedging under a specific model (e.g., Black-Scholes) computes $\Delta = \partial C/\partial S$ using the model's assumptions (constant volatility). If the true dynamics differ (stochastic volatility, jumps), the hedge is imperfect and P&L residuals can be large. Robust hedging considers a set of possible models $\mathcal{M}$ and finds the strategy that minimizes the worst-case hedging error:

\[ \min_{\Delta} \max_{m \in \mathcal{M}} \mathbb{E}^m[(\text{hedge error})^2]. \]

This min-max approach ensures the hedge performs reasonably regardless of which model is correct, at the cost of potentially sub-optimal performance under any single model.

Exercise 2. In the uncertain volatility model with $\sigma \in [\sigma_{\min}, \sigma_{\max}]$, describe the super-replication strategy for a European call.

Solution to Exercise 2

The seller of a call needs to super-replicate (hedge against the worst case). For a convex payoff like a call, higher volatility increases the option value. The worst case for the seller is $\sigma = \sigma_{\max}$. The super-replication strategy uses:

\[ \sigma^*(t, S) = \begin{cases} \sigma_{\max} & \text{if } \Gamma > 0, \\ \sigma_{\min} & \text{if } \Gamma < 0, \end{cases} \]

where $\Gamma = \partial^2 V/\partial S^2$. For a call ($\Gamma > 0$ everywhere), the hedge uses $\sigma_{\max}$ throughout. The super-replication price is the Black-Scholes price at $\sigma_{\max}$.

Exercise 3. Explain why robust hedging is more conservative (and hence more expensive) than standard hedging.

Solution to Exercise 3

Robust hedging prepares for the worst-case scenario within the uncertainty set $\mathcal{M}$. This means:

The initial cost (super-replication price) is higher because it must cover all possible model realizations.
The hedge ratios are adjusted to be safe under adversarial conditions, which may involve over-hedging relative to any single model.
The P&L variance is reduced but the expected P&L is more negative (higher cost for the hedger).

The conservatism is the price of model robustness -- a form of insurance against model misspecification.

Exercise 4. If the Black-Scholes price of a call is $5.00 at $\sigma = 20\%$ and the robust super-replication price using $\sigma_{\max} = 30\%$ is $6.50, what is the "model uncertainty premium"?

Solution to Exercise 4

The model uncertainty premium is $\$6.50 - \$5.00 = \$1.50$, or $30\%$ of the base price. This premium represents the cost of hedging against the possibility that volatility could be as high as $30\%$ rather than the assumed $20\%$. If the realized volatility turns out to be $20\%$, the robust hedger has overpaid by $1.50. If volatility reaches $30\%$, the robust hedger breaks even while the standard hedger would face a loss. The premium is justified if the hedger cannot precisely determine volatility and wants protection against the upside risk.