Implied Volatility (Figures)¶

Background¶

Black Scholes Iv Figures

Educational script demonstrating black scholes iv figures concepts.

Code¶

```python """ Black Scholes Iv Figures

Educational script demonstrating black scholes iv figures concepts. """

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

black_scholes_IMPLIED_VOLATILITY_6_FIGURES.py¶

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

import black_scholes as bs

Initialize model¶

if name == "main": model = bs.BlackScholesImpliedVol( S0=17.6639, # VSTOXX spot value K=18.0, # Placeholder T=0.25, # Placeholder r=0.01, # Risk-free rate sigma=0.2, # Initial volatility tol=0.5 # Moneyness tolerance )

# Load real VSTOXX data (with auto-download if needed)
model.load_data(auto_download=True)  # This will download if file doesn't exist

# Compute implied volatilities on real data
model.compute_implied_volatility_batch(sigma_0=2.0)

# Get summary statistics
stats = model.get_summary_statistics()
print(f"📊 Results Summary:")
print(f"   ✅ Valid computations: {stats['count']:,}")
print(f"   📈 Mean volatility: {stats['mean']:.4f} ({stats['mean']*100:.2f}%)")
print(f"   📏 Range: {stats['min']:.4f} - {stats['max']:.4f}")
print(f"   📊 Std deviation: {stats['std']:.4f}")

# Plot 1: Volatility Smiles (creates its own figure)
model.plot_volatility_smiles("Real VSTOXX Implied Volatility Smiles")

# Plot 2: 3D Surface (creates its own figure)
model.plot_3d_surface(
    title="Real VSTOXX - 3D Volatility Surface",
    save_path="./data/real_3d_surface.png"
)

# Plot 3: 3D Smiles (creates its own figure)
model.plot_3d_smiles(
    title="Real VSTOXX - 3D Volatility Smiles",
    save_path="./data/real_3d_smiles.png"
)

```

Exercises¶

Exercise 1. The volatility smile plots IV versus strike for fixed maturities. Explain why different maturities show different smile shapes.

Exercise 2. The 3D volatility surface plots IV as a function of both strike and maturity. Describe the typical shape for equity indices.

Exercise 3. Explain why the 3D surface may have artifacts or missing data points, and how interpolation handles these gaps.

Exercise 4. If you observe IV of 110% for a 1-month VSTOXX option and 90% for a 1-year option, compute the forward implied volatility from month 1 to month 12 using the variance additive rule.