Uncertain Volatility Pricing¶

Background¶

uncertain_volatility_pricing.py

This module implements Uncertain Volatility Pricing.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" uncertain_volatility_pricing.py

This module implements Uncertain Volatility Pricing.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

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

def uncertain_volatility_pricing(): """ Uncertain Volatility Pricing.

This function demonstrates the key concepts and computational techniques
for uncertain volatility pricing.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Uncertain Volatility Pricing
print(f"Computing Uncertain Volatility Pricing...")

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

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Uncertain Volatility Pricing"
}

return results

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

# 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("Uncertain Volatility Pricing")
ax.grid(True, alpha=0.3)

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

return results

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

Exercises¶

Exercise 1. In the uncertain volatility model (UVM), volatility is unknown but bounded: $\sigma(t) \in [\sigma_{\min}, \sigma_{\max}]$. Write the Black-Scholes-Barenblatt (BSB) PDE for the upper price bound.

Solution to Exercise 1

The BSB equation for the upper price $\bar{V}(t, S)$ is:

\[ \frac{\partial \bar{V}}{\partial t} + rS\frac{\partial \bar{V}}{\partial S} + \frac{1}{2}\hat{\sigma}^2 S^2 \frac{\partial^2 \bar{V}}{\partial S^2} - r\bar{V} = 0, \]

where the volatility is chosen adversarially:

\[ \hat{\sigma}^2 = \begin{cases} \sigma_{\max}^2 & \text{if } \frac{\partial^2 \bar{V}}{\partial S^2} > 0, \\ \sigma_{\min}^2 & \text{if } \frac{\partial^2 \bar{V}}{\partial S^2} < 0. \end{cases} \]

This nonlinear PDE selects the worst-case volatility at each point in the $(t, S)$ grid.

Exercise 2. For a European call option (which has $\Gamma > 0$ everywhere), what is the BSB upper price?

Solution to Exercise 2

Since $\Gamma = \partial^2 C/\partial S^2 > 0$ for a European call (convex payoff), the BSB equation selects $\hat{\sigma} = \sigma_{\max}$ everywhere. The upper bound is simply the Black-Scholes price at $\sigma_{\max}$:

\[ \bar{V} = C_{\text{BS}}(S, K, T, r, \sigma_{\max}). \]

Similarly, the lower bound uses $\sigma_{\min}$: $\underline{V} = C_{\text{BS}}(S, K, T, r, \sigma_{\min})$.

Exercise 3. For a portfolio that is long a call and short a put (a risk reversal), the Gamma changes sign across the strike range. Explain why the BSB equation becomes a genuinely nonlinear PDE.

Solution to Exercise 3

The risk reversal has $\Gamma > 0$ near the call strike (call's positive Gamma dominates) and $\Gamma < 0$ near the put strike (short put's negative Gamma dominates). The BSB equation switches between $\sigma_{\max}$ and $\sigma_{\min}$ depending on the sign of $\Gamma$ at each $(t, S)$ point:

\[ \hat{\sigma}^2(t, S) = \sigma_{\max}^2 \cdot \mathbf{1}_{\Gamma > 0} + \sigma_{\min}^2 \cdot \mathbf{1}_{\Gamma < 0}. \]

This makes the PDE nonlinear because the diffusion coefficient depends on the solution itself (through $\Gamma$). Standard analytical solutions do not exist, and numerical methods (finite differences) must solve the equation iteratively.

Exercise 4. If $\sigma_{\min} = 15\%$ and $\sigma_{\max} = 25\%$, the Black-Scholes call price at $20\%$ is $5.00, at $15\%$ is $4.00, and at $25\%$ is $6.20. What is the UVM price range, and what does it imply about model risk?

Solution to Exercise 4

The UVM price range is $[\$4.00, \$6.20]$. The width is $\$2.20$, representing the maximum model risk from volatility uncertainty. The midpoint model price ($5.00) could be off by up to $\$1.20$ ($24\%$ of the price). This is substantial model risk.

Implications: (1) the bid-ask spread should be at least $2.20 to cover model uncertainty; (2) hedging should use the robust approach (delta at the appropriate boundary volatility); (3) if the uncertainty range can be narrowed (e.g., by trading more vanilla options to pin down implied vol), the price range tightens and model risk decreases.